Gabriel Téllez

Fabio Deelan Cunden

Ada Altieri

David Dean

Lara Koehler

Saptarshi Majumdar

Saverio Bocini

Li Gan

Vincenzo Schimmenti

Federico Lanza

Lorenzo Rosso

First, I would like to express my gratitude to my

Titre: Physique Statistique de systèmes de particules en interactions à longue portée Mots clés: systèmes en interaction à longue portée, gaz de Coulomb, variables fortement corrélées, principe de grandes déviations, plasma unidimensionnel à une composante Résumé: Dans cette thèse, nous étudions l'équilibre thermodynamique et les propriétés dynamiques de particules chargées en une dimension interagissant via des interactions à longue portée. Tout d'abord, nous nous concentrons sur le plasma unidimensionnel à une composante (1dOCP) à l'équilibre, également connu sous le nom de modèle du jellium. Il consiste en N charges sur une ligne en présence d'un potential harmonique et interagissant via des interactions de Coulomb répulsives par paire. Tout d'abord, nous introduisons quelques propriétés d'équilibre bien connues de ce modèle. Nous présentons ensuite la méthode du gaz de Coulomb, que nous utilisons dans la suite de cette thèse pour analyser le comportement des grandes déviations de différentes observables dans la limite d'un grand nombre de particules. Nous trouvons la forme de ces grandes déviations pour une statistique linéaire générale des positions des particules, et nous dérivons la formule exacte de sa variance. Nous poursuivons en calculant la distribution des fluctuations typiques de l'écart entre deux particules près de l'origine (dans le "bulk") ainsi que les grandes déviations associées. Nous analysons également les fluctuations typiques et atypiques du nombre de particules à l'intérieur d'un intervalle symétrique, toujours dans le "bulk". La dernière observable d'équilibre que nous étudions est la statistique linéaire tronquée, qui décrit la somme des positions des N ′ particules les plus à droite. Dans le dernier chapitre, nous étudions la dynamique de Langevin du modèle du jellium en l'absence d'un potentiel de confinement. Nous trouvons que, aux temps longs, il existe une correspondance entre le système dynamique et le 1dOCP à l'équilibre. Nous utilisons cette connexion pour étudier les fluctuations des positions des particules dans le système dynamique.

Title: Statistics of particles with long-ranged pairwise interactions

Keywords: Long-range interacting systems, Coulomb gases, strongly correlated variables, large deviation theory, one-dimensional one-component plasma Abstract: In this thesis, we investigate the equilibrium and dynamical properties of charged particles in one dimension interacting via long-range interactions. First, we focus on the equilibrium one-dimensional one-component plasma (1dOCP), also known as the jellium model. It consists of N harmonically confined charges on a line interacting via pairwise repulsive Coulomb interactions. First, we introduce some well-known equilibrium properties of the model. Then we present the Coulomb gas method, which we later use to analyze the behavior of large deviations of different observables in the limit of a large number of particles. We find the large deviation form for the general full linear statistics, and we derive the exact formula for its variance. We continue by computing typical as well as large deviations of the gap between two middle particles. We also analyze typical and atypical fluctuations of the number of particles inside of a symmetric interval that is in the bulk of the system. The last equilibrium observable that we study is the truncated linear statistics, which describes the sum of the positions of the N ′ rightmost particles. In the last chapter, we investigate the Langevin dynamics of the jellium model without the confining potential. We find that, for large times, there exists a correspondence between the dynamical system and the equilibrium 1dOCP. We use this connection to study fluctuations of the particle positions in the dynamical system.

Introduction and summary of main results

Introduction

An abundance of natural phenomena are governed by interactions that decay slowly with distance. Obvious examples of such interactions are the gravitational and the electrostatic forces. Due to the ubiquitous presence of long-range interacting systems, they have been studied by various scientific communities, such as astrophysics [START_REF] Padmanabhan | Statistical mechanics of gravitating systems[END_REF][START_REF] Chavanis | Statistical Mechanics of Twodimensional Vortices and Collisionless Stellar Systems[END_REF][START_REF] Chavanis | Phase transitions in self-gravitating systems and bacterial populations with a screened attractive potential[END_REF][START_REF] Hertel | A soluble model for a system with negative specific heat[END_REF][START_REF] Lynden-Bell | The gravo-thermal catastrophe in isothermal spheres and the onset of red-giant structure for stellar systems[END_REF][START_REF] Marcos | Collisional relaxation of two-dimensional self-gravitating systems[END_REF], plasma physics [START_REF] Nicholson | Introduction to Plasma Theory[END_REF][START_REF] Elskens | Microscopic Dynamics of Plasmas and Chaos[END_REF], free electron lasers [START_REF] Antoniazzi | Free electron laser as a paradigmatic example of systems with long-range interactions[END_REF][START_REF] Barré | Statistical theory of high-gain free-electron laser saturation[END_REF], hydrodynamics [START_REF] Miller | Statistical mechanics of Euler equations in two dimensions[END_REF][START_REF] Ellis | Nonequivalent statistical equilibrium ensembles and refined stability theorems for most probable flows[END_REF][START_REF] Robert | Statistical equilibrium states for two-dimensional flows[END_REF][START_REF] Kulkarni | Hydrodynamics of cold atomic gases in the limit of weak nonlinearity, dispersion, and dissipation[END_REF], and cold atoms physics [START_REF] Chalony | Long-range onedimensional gravitational-like interaction in a neutral atomic cold gas[END_REF][START_REF] Zhang | Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator[END_REF]. The defining characteristic of a long-range interacting system is an interaction potential that changes as a power of the inverse interparticle distance with an exponent smaller than the dimension of the embedding space. As it turns out, systems with such interactions behave significantly differently than systems with interactions that decay fast with the interparticle distance. Possibly the most notable consequence of long-range interactions is that the energy becomes non-additive. This means that the energy of two systems that are in contact can not be calculated simply by adding both energies. Lack of additivity leads to many unusual properties of systems with long-range interactions, including the non-extensive behavior of energy as well as the possibility of negative specific heat [START_REF] Loladze | Effective negative specific heat by destabilization of metastable states in dipolar systems[END_REF][START_REF] Staniscia | Negative specific heat in the canonical statistical ensemble[END_REF][START_REF] Thirring | Systems with negative specific heat[END_REF], ensemble inequivalence [START_REF] Lederhendler | Long-Range correlations and ensemble inequivalence in a generalized ABC model[END_REF][START_REF] Barré | Inequivalence of ensembles in a system with long-range interactions[END_REF][START_REF] Bouchet | Classification of phase transitions and ensemble inequivalence, in systems with long range interactions[END_REF][START_REF] Leyvraz | Ensemble inequivalence in systems with long-range interactions[END_REF][START_REF] Baldovin | Physical interpretation of the canonical ensemble for long-range interacting systems in the absence of ensemble equivalence[END_REF], and unusually slow relaxation to the equilibrium [START_REF] Gupta | Quasistationarity in a model of classical spins with long-range interactions[END_REF][START_REF] Van Den | Relaxation timescales and decay of correlations in a long-range interacting quantum simulator[END_REF][START_REF] Gupta | Slow relaxation in long-range interacting systems with stochastic dynamics[END_REF][START_REF] Patelli | Linear response theory for longrange interacting systems in quasistationary states[END_REF]. While some of the peculiar features present in some specific systems have been understood long ago, a more general theoretical understanding of the statistical physics of systems with long-range interactions has been developed more recently [START_REF] Campa | Statistical mechanics and dynamics of solvable models with long-range interactions[END_REF][START_REF] Campa | Physics of Long-Range Interacting Systems[END_REF][START_REF] Dauxois | Dynamics and Thermodynamics of Systems with Long-Range Interactions: An Introduction[END_REF][START_REF] Defenu | Longrange interacting quantum systems[END_REF][START_REF] Mukamel | Statistical mechanics of systems with long range interactions[END_REF].

A useful platform for investigating the behavior of long-ranged systems is the family of confined Riesz gases [START_REF] Riesz | Intégrales de Riemann-Liouville et potentiels[END_REF]. It is one of the most general models describing particles interacting via repulsive interactions that vary with the distance as a power law ∼ |x ix j | -k . By varying the parameter k, we can change the nature of interactions from shortranged to long-ranged, which allows us to observe different effects that emerge when interactions start to decay slower and slower with the distance. However, it is rather complicated to obtain exact results that are valid for general k. For example, the solution for the density profile of the Riesz gas in the limit of a large number of particles was computed only a few years ago [START_REF] Agarwal | Harmonically Confined Particles with Long-Range Repulsive Interactions[END_REF][START_REF] Hardin | Large Deviation Principles for Hypersingular Riesz Gases[END_REF]. This result was later extended by computing the average density profile of the Riesz gas in the presence of a wall [START_REF] Kethepalli | Harmonically confined long-ranged interacting gas in the presence of a hard wall[END_REF]. In both cases, they find significantly different behavior of the density for different values of k. Other important results include complete characterization of the fluctuations of the rightmost particle [START_REF] Kethepalli | Edge fluctuations and third-order phase transition in harmonically confined longrange systems[END_REF] and the expression for the pair correlation function [START_REF] Beenakker | Pair correlation function of the one-dimensional Riesz gas[END_REF]. Very recently, the dynamical fluctuations in Riesz gas have also been investigated [START_REF] Dandekar | Dynamical fluctuations in the Riesz gas[END_REF][START_REF] Huse | Spatiotemporal spread of perturbations in power-law models at low temperatures: Exact results for classical out-of-time-order correlators[END_REF]. Interestingly, the Riesz gases are not investigated only in the physics community. In fact, the study of the Riesz gases is also an active research domain in mathematics [START_REF] Hardin | Large Deviation Principles for Hypersingular Riesz Gases[END_REF][START_REF] Leblé | Large deviation principle for empirical fields of Log and Riesz gases[END_REF][START_REF] Lewin | Coulomb and Riesz gases: The known and the unknown[END_REF][START_REF] Chafaï | Aspects of Coulomb gases[END_REF][START_REF] Dereudre | Number-Rigidity and beta-Circular Riesz gas[END_REF][START_REF] Boursier | Decay of correlations and thermodynamic limit for the circular Riesz gas[END_REF][START_REF] Boursier | Optimal local laws and CLT for the circular Riesz gas[END_REF].

Despite being very insightful, analytical results for general k are, in many cases, still unreachable. Often it is more realistic to focus on specific values of k. In fact, for some special values of k, the Riesz gas corresponds to some renowned physical models, which have been extensively investigated in their own regard. Two prominent examples of longranged models belonging to Riesz gases are the one-dimensional one-component plasma (1dOCP) (k = -1) [START_REF] Lenard | Exact statistical mechanics of a one-dimensional system with Coulomb forces[END_REF][START_REF] Prager | The One-Dimensional Plasma[END_REF][START_REF] Baxter | Statistical mechanics of a one-dimensional Coulomb system with a uniform charge background[END_REF][START_REF] Choquard | One-Dimensional Coulomb Systems BT -Physics in One Dimension[END_REF] and the Dysons's log-gas (k → 0) [START_REF] Mehta | Random Matrices and the Statistical Theory of Spectra[END_REF][START_REF] Forrester | Log-Gases and Random Matrices[END_REF][START_REF] Dyson | Statistical theory of the energy levels of complex systems. I[END_REF][START_REF] Dyson | Statistical theory of the energy levels of complex systems. II[END_REF]. A famous shortranged model that belongs to the Riesz gas family is the integrable Calogero-Moser system [START_REF] Calogero | Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials[END_REF][START_REF] Calogero | Exactly solvable one-dimensional many-body problems[END_REF][START_REF] Moser | Three integrable Hamiltonian systems connected with isospectral deformations[END_REF][START_REF] Diejen | Calogero-Moser-Sutherland Models[END_REF][START_REF] Sutherland | Exact results for a quantum many-body problem in one dimension[END_REF], which is recovered for k = 2.

The study of Dyson's log-gas has produced many insights into the behavior of longranged systems. The popularity of this model can be largely attributed to its connection to the random matrix theory. As established by Dyson and Mehta [54,[START_REF] Dyson | Statistical theory of the energy levels of complex systems. II[END_REF][START_REF] Dyson | Statistical theory of the energy levels of complex systems. III[END_REF][START_REF] Mehta | Statistical theory of the energy levels of complex systems. IV[END_REF][START_REF] Mehta | Statistical Theory of the Energy Levels of Complex Systems. V[END_REF], there is a mapping between the joint probability distribution of the eigenvalues of invariant random matrices and the positions of confined particles in one dimension interacting via repulsive logarithmic potential. By reformulating the questions that arise in random matrix theory in terms of the log-gas, one can use a variety of tools from statistical mechanics to investigate the properties of large random matrices. A method that exploits this connection is known as the Coulomb gas method. A particular advantage of this method is that it allows one to investigate atypically large fluctuations of different observables in the limit of a large number of particles. Over the past two decades, numerous new results regarding the large deviations of different quantities in log-gas or, equivalently, random matrix ensembles have been derived by using the Coulomb gas technique. Some examples are the large deviation tails of the probability distribution of the largest eigenvalue, which corresponds to the rightmost particle in the log-gas picture [START_REF] Majumdar | Top eigenvalue of a random matrix: large deviations and third order phase transition[END_REF][START_REF] Vivo | Large deviations of the maximum eigenvalue in Wishart random matrices[END_REF][START_REF] Majumdar | Large deviations of the maximum eigenvalue for Wishart and Gaussian random matrices[END_REF][START_REF] Majumdar | Large deviations of the top eigenvalue of large Cauchy random matrices[END_REF][START_REF] Nadal | A simple derivation of the Tracy-Widom distribution of the maximal eigenvalue of a Gaussian unitary random matrix[END_REF], the results connected to the large deviations of the linear statistics [START_REF] Krajenbrink | Linear statistics and pushed Coulomb gas at the edge of β-random matrices: Four paths to large deviations[END_REF][START_REF] Grabsch | Distribution of spectral linear statistics on random matrices beyond the large deviation function -Wigner time delay in multichannel disordered wires[END_REF][START_REF] Grabsch | Capacitance and charge relaxation resistance of chaotic cavities -Joint distribution of two linear statistics in the Laguerre ensemble of random matrices[END_REF][START_REF] Chen | Distribution of linear statistics in random matrix models (metalic conductance fluctuations)[END_REF][START_REF] Cunden | Universality of the weak pushedto-pulled transition in systems with repulsive interactions[END_REF][START_REF] Cunden | A shortcut through the Coulomb gas method for spectral linear statistics on random matrices[END_REF], the truncated linear statistics [START_REF] Grabsch | General truncated linear statistics for the top eigenvalues of random matrices[END_REF][START_REF] Grabsch | Truncated Linear Statistics Associated with the Eigenvalues of Random Matrices II. Partial Sums over Proper Time Delays for Chaotic Quantum Dots[END_REF][START_REF] Grabsch | Truncated linear statistics associated with the top eigenvalues of random matrices[END_REF], and the counting statistics [START_REF] Dean | Extreme value statistics of eigenvalues of Gaussian random matrices[END_REF][START_REF] Dean | Large deviations of extreme eigenvalues of random matrices[END_REF][START_REF] Majumdar | How many eigenvalues of a Gaussian random matrix are positive?[END_REF][START_REF] Fogler | Probability of an eigenvalue number fluctuation in an interval of a random matrix spectrum[END_REF][START_REF] Marino | Number statistics for β -ensembles of random matrices: Applications to trapped fermions at zero temperature[END_REF][START_REF] Marino | Phase transitions and edge scaling of number variance in gaussian random matrices[END_REF]. The Coulomb gas approach allows us to obtain the large deviation form of a given observable by solving the optimization problem determined by the constrained Coulomb gas. In many cases, the constrained system exhibits different phases which correspond to different regimes of the probability distribution [START_REF] Majumdar | Top eigenvalue of a random matrix: large deviations and third order phase transition[END_REF][START_REF] Nadal | A simple derivation of the Tracy-Widom distribution of the maximal eigenvalue of a Gaussian unitary random matrix[END_REF][START_REF] Cunden | A shortcut through the Coulomb gas method for spectral linear statistics on random matrices[END_REF][START_REF] Allez | Index distribution of the Ginibre ensemble[END_REF][START_REF] Cunden | Universal covariance formula for linear statistics on random matrices[END_REF][START_REF] Cunden | Universality of the third-order phase transition in the constrained Coulomb gas[END_REF][START_REF] Cunden | Joint statistics of quantum transport in chaotic cavities[END_REF][START_REF] Nadal | Phase transitions in the distribution of bipartite entanglement of a random pure state[END_REF][START_REF] Nadal | Statistical Distribution of Quantum Entanglement for a Random Bipartite State[END_REF][START_REF] Colomo | Third-order phase transition in random tilings[END_REF].

The second long-ranged model that belongs to the family of Riesz gases is the onedimensional one-component plasma. In this case, the interaction potential grows linearly with the interparticle distance, which means that the long-range effects are even stronger than in the log-gas case. The 1dOCP is an especially interesting representative of onedimensional Riesz gases since the linear interaction potential corresponds to the Coulomb electrostatic interaction in one dimension. The statistical properties of the 1dOCP have been first studied in the sixties by Lenard, Prager, Baxter, and Choquard [START_REF] Lenard | Exact statistical mechanics of a one-dimensional system with Coulomb forces[END_REF][START_REF] Prager | The One-Dimensional Plasma[END_REF][START_REF] Baxter | Statistical mechanics of a one-dimensional Coulomb system with a uniform charge background[END_REF][START_REF] Choquard | One-Dimensional Coulomb Systems BT -Physics in One Dimension[END_REF]. Since then, different aspects of it have been investigated [START_REF] Dean | Effects of dielectric disorder on van der Waals interactions in slab geometries[END_REF][START_REF] Téllez | Screening like charges in one-dimensional Coulomb systems: Exact results[END_REF][START_REF] Téllez | Like-charge attraction in one-and two-dimensional Coulomb systems[END_REF][START_REF] Varela | Like-charge attraction at short distances in a chargeasymmetric two-dimensional two-component plasma: Exact results[END_REF][START_REF] Varela | One-dimensional colloidal model with dielectric inhomogeneity[END_REF][START_REF] Varela | Configurational and energy landscape in onedimensional Coulomb systems[END_REF][START_REF] Varela | Relaxation dynamics of two interacting electrical double-layers in a 1D Coulomb system[END_REF][START_REF] Chafaï | At the edge of a one-dimensional jellium[END_REF][START_REF] Rojas | Universal behavior of the full particle statistics of one-dimensional Coulomb gases with an arbitrary external potential[END_REF]. Very recently, the authors of Ref. [START_REF] Dhar | Exact extremal statistics in the classical 1d Coulomb gas[END_REF][START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF] successfully adapted the Coulomb gas technique to study the large deviations of the rightmost particle, the number of particles on the positive axis, and the gap between two particles near the edge of the system. However, compared to the abundance of results describing the large deviations in the log-gas, there are many observables in the one-dimensional one-component plasma that have not yet been explored from this perspective. The aim of this thesis is to present new results that complement the existing knowledge about the properties of the one-dimensional one-component plasma. We use the Coulomb gas technique to study large deviations of the full linear statistics, the bulk gap, the full counting statistics, and the truncated linear statistics. In some cases, we complement this approach with other techniques that allow us to probe typical fluctuations. The research that led to the findings presented in this thesis has been conducted during the three years of my Ph.D. together with my advisors Satya N. Majumdar and Grégory Schehr and collaborators. Some of the results obtained under the supervision of my advisor Sergei Nechaev and other collaborators are not presented in this manuscript. We made this decision because the investigated topics diverge significantly from the central theme of this thesis. For the sake of completeness, we include the preprint based on this work [START_REF] Flack | Generalized Devil ' s staircase and RG flows[END_REF] in Appendix B of the thesis.

Overview of the thesis

This thesis is structured as follows. In Ch. 1 we define the one-dimensional onecomponent plasma or the jellium model. We also present the Coulomb and Riesz gases. In the last part of the chapter, we summarize some well-known equilibrium properties of the jellium model.

In Ch. 2 we introduce the Coulomb gas technique, which we use throughout the following chapters to compute the large deviations of different observables. We present the discrete as well as continuous formulations of the method.

Ch. 3, Ch. 4, Ch. 5, and Ch. 6 are based on the results published in four different articles provided at the end of each chapter. In Ch. 3, we focus on the full linear statistics. Using the Coulomb gas technique, we derive the expression for the rate function of general linear statistics. By analyzing the behavior of the obtained rate function around its minimum, we find an exact formula for the variance. We continue by checking the validity of this result by explicitly calculating the large deviation forms and rate functions for different choices of the test functions (f (x) = x, f (x) = x 2 , f (x) = x 3 , and f (x) = |x|). By expanding the rate functions around their minima, we obtain the variance for the specific choices of f (x). We find that the results match the general formula that we derive in the first part of the chapter.

In Ch. [START_REF] Hertel | A soluble model for a system with negative specific heat[END_REF] we focus on the probability distributions of two observables: the gap between two neighboring particles that are deep in the bulk of the system and the full counting statistics (FCS), which measures the number of particles inside of a given interval. Using the Coulomb gas technique, we first derive the large deviation form describing the atypically large fluctuations of the bulk gap. We continue by finding the typical fluctuations of the bulk gap that are not captured by the large deviation form. In the second part of the chapter, we focus on the FCS. Again, we first derive the large deviation form describing atypically large fluctuations. Next, we derive the scaling form that captures the behavior of typical fluctuations. At the end of the chapter, we discuss the connection between both observables, the bulk gap, and the FCS.

In Ch. 5, we study the truncated linear statistics, which is given by the sum of N ′ rightmost particles. Given that the system is composed of N particles, we denote the fraction of particles included in the sum by κ = N ′ /N . We again use the Coulomb gas method to derive the rate function for all possible values of κ ∈ [0, 1]. The main result of this chapter is the explicit expression for the rate function, which has different forms for 0 ≤ κ ≤ 1/2 and 1/2 < κ ≤ 1. We show that for a fixed κ, the rate function displays four different regimes. We also analyze asymptotic behaviors of the rate function and compare them to previously known results for the position of the rightmost particle (κ → 0) and the full linear statistics (κ → 1). At the end of the chapter, we describe the algorithm that we use for numerically computing the full probability distribution of the truncated linear statistics.

Finally, in Ch. 6, we move away from the equilibrium and explore the Langevin dynamics of particles in one dimension that interact via repulsive one-dimensional Coulomb interaction potential. This process is also known as the repulsive ranked diffusion. First, we introduce the model and discuss the properties of the three regimes that emerge at different timescales. We continue by analyzing the probability density function of the positions at late times. We show that at late times there is an analogy between the dynamical model and the equilibrium jellium system. At the end of the chapter, we use this connection to predict the typical fluctuations of the middle and edge gaps in the ranked diffusion problem.

Summary of the main results

In this section, we provide the main results of the thesis. The model that we study is called the one-dimensional one-component plasma (1dOCP) or the jellium model. The one-dimensional one-component plasma consists of N equally charged particles confined to a line in one dimension. They interact via long-range repulsive Coulomb interaction potential and are confined by an outside harmonic potential. If we denote the positions of the particles by x 1 , • • • , x N , we can write the energy of such a gas as

βE[{x i }] = N 2 2 N i=1 x 2 i -N α i̸ =j |x i -x j | , (1)
where we rescaled the positions so that x i = O(1) and both terms scale in the same way with N . This is important because we are interested in the limit of large N . For convenience, we expressed the energy in units of the inverse temperature β = 1/(k B T ), and α is a constant that determines the strength of the interaction. At equilibrium, the probability of a given configuration of particles {x i } is given by the Boltzmann-Gibbs weight

P({x i }) = 1 Z N e -βE[{x i }] , (2)
with the normalization given by the partition function Z N defined as

Z N = ∞ -∞ dx 1 ∞ -∞ dx 2 • • • ∞ -∞ dx N e -βE[{x i }] . ( 3
)
When the system is in equilibrium, the charges settle down in a finite region between x = -2α and x = 2α. The most probable configuration is the one where particles are equidistantly spaced. Their equilibrium positions are [START_REF] Lenard | Exact statistical mechanics of a one-dimensional system with Coulomb forces[END_REF][START_REF] Prager | The One-Dimensional Plasma[END_REF][START_REF] Baxter | Statistical mechanics of a one-dimensional Coulomb system with a uniform charge background[END_REF] x i = x * i = 2α N (2i -N -1), for i = 1, 2, .., N .

(4)

In the large N limit, we can describe the configuration of particles by a continuous function, namely the average macroscopic density profile ρ(x). The continuous analog of Eq. ( 4) thus reads

ρ * (x) = 1 4α , -2α ≤ x ≤ 2α . ( 5
)
By considering the average density profile, we can compute the mean values of different observables. However, the values of the observables fluctuate around their average values.

We investigate these fluctuations by computing their full probability distributions.

An exact formula for the variance of linear statistics

In Ch. 3 we derive an exact formula for the variance Var(s) = ⟨(s-s) 2 ⟩, of the rescaled full linear statistics defined as

s = 1 N N i=1 f (x i ) . ( 6
)
Considering the equilibrium density profile from Eq. ( 5) we can easily find the average value of s given by

s = 1 4α 2α -2α f (x)dx . ( 7
)
By using the Coulomb gas method, we derive the formula for the variance in the limit of large N that reads

Var(s) ≈ 1 4αN 3 2α -2α f ′ (x)
2 dx , [START_REF] Elskens | Microscopic Dynamics of Plasmas and Chaos[END_REF] where [f ′ (x)] 2 should exists and be integrable. We also obtain the rate function ϕ(s), which captures the large deviation behavior of the full distribution P(s, N ) ∼ e -N 3 ϕ(s) . The formula for the variance from Eq. ( 8) is extracted from the rate function ϕ(s) by expanding the rate function around its minimum at s = s. For a few choices of f (x), we explicitly compute the large deviation form P(s, N ) ∼ e -N 3 ϕ(s) . First, we test two simple examples f (x) = x, and f (x) = x 2 . We find that both the rate functions ϕ(s) and the variances agree with the prediction from our general formula given in Eq. [START_REF] Elskens | Microscopic Dynamics of Plasmas and Chaos[END_REF]. During the derivation of the formula for the variance, we make two assumptions. First, we assume that the saddle-point density of the constrained Coulomb gas is supported on a single interval. Second, we assume that the effective potential given by V eff (x) = x 2 /2 + µf (x) is always confining. However, we show that in some cases, our general formula for the variance holds even if these two assumptions are not true. We discuss two such cases, f (x) = x 3 , and f (x) = |x|. Especially interesting is the choice f (x) = |x|, where we discover a third-order phase transition of the underlying Coulomb gas at s = s. These results are published in Ref. [START_REF] Flack | An exact formula for the variance of linear statistics in the one-dimensional jellium model[END_REF].

Gap probability and full counting statistics

In Ch. [START_REF] Hertel | A soluble model for a system with negative specific heat[END_REF] we focus on two observables: the bulk gap and the full counting statistics. Given the ordered configuration of particles where x 1 ≤ • • • ≤ x N , we define the bulk gap as the distance between the two middle particles g = x ⌊N/2⌋+1 -x ⌊N/2⌋ . [START_REF] Antoniazzi | Free electron laser as a paradigmatic example of systems with long-range interactions[END_REF] We expect that the results for the middle gap are valid for all gaps that are deep in the bulk of the system where 1 ≪ i ≪ N . From the uniform equilibrium configuration in Eq. ( 5) we can easily determine the average value of the bulk gap for large N that is given by ⟨g⟩ ≈ 4α N . [START_REF] Barré | Statistical theory of high-gain free-electron laser saturation[END_REF] We show that the typical fluctuations of the bulk gap scale as O(1/N ) and the large fluctuations as O [START_REF] Padmanabhan | Statistical mechanics of gravitating systems[END_REF]. We can summarize our results in the following way

P gap,bulk (g, N ) ∼        N H α (g N ), g ∼ O(1/N ),
e -N 3 ψ bulk (g) , g ∼ O( 1) , [START_REF] Miller | Statistical mechanics of Euler equations in two dimensions[END_REF] where the scaling function H α (z) is given by

H α (z) = θ(z)B A 2 (α) ∞ -∞ dyF α (y + 4α)F α (8α -y -z)e -y 2 /2-(y+z-4α) 2 /2 . ( 12
)
The constant B is the normalization and the function F α (x) is the unique solution of the non-local differential equation [START_REF] Baxter | Statistical mechanics of a one-dimensional Coulomb system with a uniform charge background[END_REF][START_REF] Dhar | Exact extremal statistics in the classical 1d Coulomb gas[END_REF][START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF] dF α (x) dx = A(α)F α (x + 4α)e -x 2 2 , [START_REF] Robert | Statistical equilibrium states for two-dimensional flows[END_REF] with the boundary conditions F α (x → ∞) → 1 and F α (x → -∞) → 0. This equation can be thought of as an eigenvalue equation, with A(α) as the unique eigenvalue for which there exists a solution that satisfies both boundary conditions. The second line of Eq. [START_REF] Miller | Statistical mechanics of Euler equations in two dimensions[END_REF] describes the atypical fluctuations of order O [START_REF] Padmanabhan | Statistical mechanics of gravitating systems[END_REF] where the rate function ψ bulk (g) is given explicitly by

ψ bulk (g) = g 3 96α , g ≥ 0 . ( 14
)
It is interesting to compare the fluctuations of the bulk gap to the previously known results for the gap near the edge published in Ref. [START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF]. Interestingly, both the typical and large fluctuations change when we move from the edge toward the bulk of the system. The large deviations of the gap near the edge are Gaussian and are independent of the interaction strength α. However, in the bulk, the tails of the probability distribution decay faster than Gaussian and depend on α (see the second line of Eq. ( 11) and Eq. ( 14)).

The second observable we study in Ch. 4 is the full counting statistics N I . It measures the number of particles inside an interval I = [-L, L]. We only consider intervals that are inside of the bulk of the system. We define the FCS as

N I = N l=1 I I (x l ), (15)
where I I (x l ) is an indicator function which takes the value 1 if the i-th particle is inside the interval I and zero otherwise. Again, we study the probability distribution of the typical and large fluctuations of N I around its mean NI = L 2α N . We show that the typical size of the fluctuations N I -NI is of order O(1) for large N , while the large fluctuations are of order O(N ). These two behaviors are summarized in the following expression

P F CS (N I , N ) ∼        2αU α [2α (N I -NI )], |N I -NI | = O(1) , (typical) e -N 3 ψ(N I /N ) , |N I -NI | = O(N ) , (atypical) . ( 16
)
The scaling function U α (z) is given by

U α (z) = F 2 α (-z + 2α)F 2 α (z + 2α) ∞ -∞ F 2 α (-z ′ + 2α)F 2 α (z ′ + 2α) dz ′ , ( 17
)
where the function F α (x) is defined in Eq. ( 13). The rate function ψ(z) in the second line of Eq. ( 16) is given explicitly by

ψ(z) = 2 3 α 2 z - L 2α 3 . ( 18
)
An interesting feature of the probability distribution of the FCS described by the Eq. ( 16) is that the third derivative of the rate function ψ(z) has a jump at N I = NI , which indicates a third order phase transition in the underlying Coulomb gas. The described results are published in Ref. [START_REF] Flack | Gap probability and full counting statistics in the one dimensional one-component plasma[END_REF].

Truncated linear statistics in the one-dimensional one-component plasma In Ch. 5 we explore the behavior of the fluctuations of the truncated linear statistics (TLS). It is given by the sum of N ′ rightmost particles where we set N ′ = κN with 0 ≤ κ ≤ 1. The expression for the rescaled truncated linear statistics reads

s = 1 N N i=N -N ′ +1 x i = 1 N N i=N (1-κ)+1 x i , ( 19
)
where the x i 's denote the positions of particles in the ordered configuration. By using the equilibrium positions of the particles (see Eq. ( 4)), we can find the average value of the TLS that reads for large N

s ≃ 2ακ(1 -κ) . ( 20
)
The truncated linear statistics is particularly interesting observable. By changing κ form 0 to 1 it interpolates between the full linear statistics describing the center of mass, and the position of the rightmost particle. The probability distribution of the full linear statistics is purely Gaussian for all values of s. On the other hand, the probability distribution of the rightmost particle is much more complex and was computed in Refs. [START_REF] Dhar | Exact extremal statistics in the classical 1d Coulomb gas[END_REF][START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF]. By describing the fluctuations of the truncated linear statistics for all κ ∈ [0, 1] we explore the transition between the two behaviors.

Once again, we employ the Coulomb gas method to find that the atypically large fluctuations of the TLS are given by the large deviation form P N,κ (s) ∼ e -N 3 ϕκ(s) . As depicted in Fig. 1, we find a very rich phase diagram with five different phases in κ-s phase space. The typical densities of the constrained Coulomb gas in each of the phases are plotted in the upper right corner of the right panel of Fig. 1. We find that if we fix κ and change s from ∞ to -∞ we start in Phase I, where the particles are organized in two blocks. The density in each of them is uniform. When we reach Phase II, the two blocks merge. The density is thus supported on a single interval. It consists of uniformly distributed charges and an additional δ-peak. By further decreasing s, we come from Phase II to Phase IIIa (κ > 1/2) or Phase IIIb (κ ≤ 1/2). In these two phases, the location of the δ-peak shifts from the bulk to the edge of the system. In Phase IIIa to the left edge and in Phase IIIb to the right one. Finally, if we keep decreasing s, we reach Phase IV. In this phase, all the charges are gathered in one single point.

In each of the phases, the rate function is described by different expression. If 0 ≤ κ ≤ 1/2 the rate function is given by

ϕ κ (s) =                                            (s -2ακ(1 -κ)) 2 2κ , (Phase I) 4 -64 3 α 2 κ 3 -8α 2 (κ -1)κ 2 -4ακ s + 2 3 √ 2α (2ακ(5κ -1) + s) 3/2 , (Phase II) (2ακ -s) 3 24ακ 3 , (Phase IIIb) 2α 2 3 + s 2 2κ 2 , (Phase IV) (21)
Phases I, II, IIIb, and IV correspond to phases depicted in Fig. 1. On the other hand if 1/2 < κ ≤ 1 the rate function reads In the center, we see a depiction of the probability distribution of the truncated linear statistics P N,κ (s). Four different regimes that appear for 0 < κ ≤ 1/2 are separated by dotted vertical lines. In the upper right corner, the associated saddle-point densities are plotted. In the upper left corner, the large deviation function ϕ κ (s) (see Eq. ( 21)) is plotted as a function of s, together with the results from numerical simulation (red dots).

ϕ κ (s) =                                                            (s -2ακ(1 -κ)) 2 2κ , (Phase I) 4 -64 3 α 2 κ 3 -8α 2 (κ -1)κ 2 -4ακ s + 2 3 √ 2α (2ακ(5κ -1) + s) 3/2 , (Phase II) 4α 2 (κ -1) 2 8κ 4 -8κ 3 + 4κ -1 + 12α(κ -1)(2κ -1)(2(κ -1)κ + 1)s 6(2κ -1) 3 + s 2 2(2κ -1) + 4 √ 2 α(κ -1) 3 2α(κ -1)κ 2 + (2κ -1)s 3/2 3α(κ -1) 3 (2κ -1) 3 , (Phase IIIa) 2α 2 3 + s 2 2κ 2 , ( Phase
The phases I, II, IIIa and IV again correspond to those shown in the left panel of Fig. 1. The right panel of Fig. 1 depicts a schematic plot of the probability distribution of s for κ < 1/2. We denoted the values of s where the regimes change by vertical dotted lines. At each of these boundaries, the rate function ϕ κ (s) displays a non-analytical behavior. While the first and the second derivatives are continuous, the third derivative ϕ ′′′ κ (s) is discontinuous. This indicates a third-order phase transition in the underlying Coulomb gas. The inset in the upper left corner of the right panel of Fig. 1 shows the corresponding rate function ϕ κ (s) for κ = 0.3 as given in Eq. [START_REF] Bouchet | Classification of phase transitions and ensemble inequivalence, in systems with long range interactions[END_REF]. The red points are obtained by numerical simulations based on the importance sampling algorithm.

Repulsive ranked diffusion

In Ch. 6 we investigate out-of-equilibrium properties of particles in one dimension that interact via linear repulsive Coulomb interaction potential. In contrast to the jellium model studied in the previous chapters, in this case, the particles are not confined by any external potential. The Langevin equation describing the time evolution of the i-th particle is given by

dx i dt = -c N j=1 sgn(x j -x i ) + √ 2T ξ i (t) . ( 23
)
Here ξ i (t)'s denote the N unit independent white noises with zero mean ⟨ζ i (t)⟩ = 0 and delta-correlator

⟨ζ i (t)ζ j (t ′ )⟩ = δ i,j δ(t -t ′ )
. By convention we use sgn(0) = 0. The strength of the interaction is determined by c. We focus on the case where particles repel each other, therefore c > 0. The strength of the Coulomb force acting on the i-th particle in 0.0 0.5 1.0 1.5 2.0 2.5 the ordered configuration is proportional to the number of particles ahead of it minus the number of particles behind. This kind of diffusion is also known as ranked diffusion. We focus on the initial condition where all particles start from the same point at t = 0 and analyze the dynamics governed by Eq. ( 23). In the upper panel of Fig. 2 we show examples of trajectories obtained by numerically simulating the time evolution of particles given by Eq. ( 23). We find that there are two characteristic time scales separating three distinct regimes

t * 1 ∼ 1 N 2 T c 2 , and t * 2 ∼ T c 2 . ( 24
)
For t < t * 1 , the gas is very dense, and the particles cross very often. This regime, which we call the dense regime or Regime I, corresponds to plots a) in Fig. 2. When t > t * 1 , the system enters into Regime II. The long-range interactions start to play an important role, and the size of the system starts to grow linearly with time as 2cN t. In plot b) in the lower panel of Fig. 2, we see a typical density profile measured in Regime II. It starts to approach a square shape, however we observe the presence of boundary layers that are wider than the typical interparticle distance. The second characteristic timescale t * 2 (see Eq. ( 24)) marks the crossover between Regime II and Regime III. For t > t * 2 , the particles are well separated, and they form an equidistantly spaced lattice. As depicted in plots c) of Fig. 2, the typical density profile in this regime is square-shaped. The boundary layers that are present in Regime II are not noticeable anymore.

In the main part of the chapter, we analyze the late time limit of the expression for the joint probability density function P (⃗ x, t) of the particles evolving according to Eq. [START_REF] Leyvraz | Ensemble inequivalence in systems with long-range interactions[END_REF]. By using the saddle-point approximation valid for t → ∞ we find

P (⃗ x, t) ≃ 1 (4πt) N/2 e -t 4 N j=1 (z j -c(2j-N -1)) 2 1≤a<b≤N z b -z a z b -z a + 2c , for z 1 ≤ z 2 • • • ≤ z N ,( 25
)
where T = 1, z j = x j t and O(z j ) = 1. This expression is valid for late times c 2 t ≫ 1 and any N . From the exponential in Eq. ( 25), we can determine the most probable positions of particles that are given by

x j t = z j = c(2j -N -1) . ( 26
)
This result indicates that in the late time limit, the system forms an expanding crystal with equidistantly positioned particles. We continue by approximating the expression in Eq. ( 25) by neglecting the double product prefactor, which is subleading. This leads us to the following probability density function

P (⃗ x, t) ∝ e -t 4 N j=1 (z j -c(2j-N -1)) 2 , for z 1 < z 2 < • • • < z N . ( 27
)
We notice that this expression closely resembles the joint probability distribution of the ordered particles in the equilibrium jellium model

P J [⃗ y] ∝ e -βE J [⃗ y] = e -N 2 2 N i=1 [yi-2α N (2i-N -1)] 2 , for y 1 < y 2 < • • • < y N , ( 28
)
where ⃗ y denotes the configuration of particles and α is parameter that controls the strength of the interactions. By comparing Eqs. ( 27) and ( 28), we see that they are formally equivalent if we use the following relations

z i = 2 t N y i , x i = z i t = √ 2tN y i , α = c √ t 2 √ 2 . ( 29
)
By inspecting the third relation in Eq. ( 29) we conclude that the large time limit in the dynamical problem corresponds to the strong coupling limit in the equilibrium jellium model. By using this analogy between the ranked diffusion and the equilibrium jellium model, we predict the fluctuations of the gaps between particles in the dynamical problem. We use the expressions for the typical fluctuations of the bulk gap from Ch. 4 and the edge gap from Ref. [START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF]. We find that for late times and large N the leading order fluctuations of the gaps in the ranked diffusion are Gaussian with the width ∼ √ t. Therefore, we can describe the gaps in the ranked diffusion by the following Gaussian random variable

G RD mid-gap = x N/2+1 (t) -x N/2 (t) = tg RD mid-gap ≈ 2c t + 2 √ t N (0, 1) . ( 30
)
We confirm this prediction by numerically simulating the dynamics governed by Eq. ( 23) and finding the numerical distributions of the bulk and edge gaps. Comparing the numerical data to Eq. ( 30) we find a very good match between both.

11

Chapter 1

One-dimensional one-component plasma or the jellium model

The model known as the one-dimensional one-component plasma (1dOCP) or the jellium model [START_REF] Lenard | Exact statistical mechanics of a one-dimensional system with Coulomb forces[END_REF][START_REF] Baxter | Statistical mechanics of a one-dimensional Coulomb system with a uniform charge background[END_REF][START_REF] Choquard | One-Dimensional Coulomb Systems BT -Physics in One Dimension[END_REF] is the central and reoccurring theme in this thesis. This chapter aims to introduce the fundamental properties of the model while also contextualizing it within a wider framework. After defining the model, we present two different possible generalizations of our model: the Coulomb gases and the Riesz gases [START_REF] Riesz | Intégrales de Riemann-Liouville et potentiels[END_REF][START_REF] Lewin | Coulomb and Riesz gases: The known and the unknown[END_REF]. At the end of the chapter, we derive two important quantities used throughout this work, namely, the configuration with the lowest energy and the partition function.

The one-dimensional one-component plasma consists of N equally charged particles confined to a line in one dimension. The charged particles, sometimes referred to as repulsive electrons, interact via long-range pairwise interaction that behaves linearly with the distance between the particles. Additionally, all the charges are subjected to an outside confining harmonic potential or harmonic trap. Without it, due to the pairwise repulsion, the particles would disperse to infinity. However, the interplay between the pairwise repulsion and the confinement leads to charges settling in a finite interval. In Fig. 1.1 we present a schematic drawing of such a system. The red dots represent the particles confined to a line, and the dashed parabola depicts the harmonic trap. The figure shows the ordered configuration of particles where the position of the leftmost particle is y 1 and the position of the last particle is y N .

If we denote the position of the i-th particle by y i and the configuration of the whole system by {y 1 , ..., y N } = {y i }, we can write the energy of the system as [START_REF] Lenard | Exact statistical mechanics of a one-dimensional system with Coulomb forces[END_REF][START_REF] Prager | The One-Dimensional Plasma[END_REF][START_REF] Baxter | Statistical mechanics of a one-dimensional Coulomb system with a uniform charge background[END_REF][START_REF] Choquard | One-Dimensional Coulomb Systems BT -Physics in One Dimension[END_REF]

E[{y

i }] = 1 2 N i=1 y 2 i -α i̸ =j |y i -y j | . (1.1)
The first term in the above equation represents the potential energy due to the harmonic trap, and the second term is the repulsive linear interaction between particles. The parameter α > 0 is of order one, and it controls the strength of the interaction. Because we wish to study the phenomena arising from the competition between the two effects, we must rescale the positions y i so that both terms scale in the same way with N . This is essential since we are interested in the system's behavior in the limit of many particles N → ∞. We rescale the positions as y i → ax i , where x i are of order one. To determine the value of a, we insert this into the energy equation (1.1)

E[{x i }] = a 2 2 N i=1 x 2 i -αa i̸ =j |x i -x j | . (1.2)
Since there are N terms of order one in the first sum, it is of order a 2 N . The second term includes a double sum with N (N -1) ≈ N 2 of O(1) terms. Therefore, it is of order aN 2 . By equating the two terms, we can conclude that the constant a needs to be of order N . Henceforth, we can write the final version of the energy

βE[{x i }] = N 2 2 N i=1 x 2 i -N α i̸ =j |x i -x j | . (1.3)
For convenience, we expressed the energy in the units of inverse temperature β = 1/(k B T ).

As we explain later on, this model can be interpreted as interacting electrons moving in the uniform neutralizing background of opposite charges. A similar model, known as Wigner's jellium, was introduced by Wigner to describe delocalized electrons in metals [START_REF] Wigner | Effects of the electron interaction on the energy levels of electrons in metals[END_REF]. Maybe even more widely known is the earlier introduced Thomson's model of the atom [START_REF] Thomson | XXIV. On the structure of the atom: an investigation of the stability and periods of oscillation of a number of corpuscles arranged at equal intervals around the circumference of a circle; with application of the results to the theory of atomic structure[END_REF] that is based on the same idea. Wigner's idea was to model the structure of the atom by a positively charged sphere with negatively charged particles embedded inside. This model is often compared to a plum pudding with small pieces of dried fruit, usually raisins, trapped in the jelly that symbolizes the uniformly spread background charge leading to the name jellium model.

The first part of this thesis mainly explores the equilibrium properties of the model with the energy given in Eq. (1.3), which belongs to the class of long-range interacting models. Despite the complexity arising from the strong correlations, we will see that many of its properties can be computed analytically, especially in the limit of a large number of particles. This makes the 1dOCP an attractive model for studying phenomena connected to long-range interactions from the perspective of statistical physics as well as mathematics.

The first appearances of the jellium model in physics literature can be traced back to A. Lenard, S. Prager and R. J. Baxter who were studying it in the sixties [START_REF] Lenard | Exact statistical mechanics of a one-dimensional system with Coulomb forces[END_REF][START_REF] Prager | The One-Dimensional Plasma[END_REF][START_REF] Baxter | Statistical mechanics of a one-dimensional Coulomb system with a uniform charge background[END_REF][START_REF] Edwards | Exact statistical mechanics of a one-dimensional system with coulomb forces. II. The method of functional integration[END_REF].

In these papers, the authors provided a complete thermodynamical treatment of a onedimensional system of electrons, including the derivation of the partition function. Another historically important contribution from P. Choquard titled One-Dimensional Coulomb Systems was published in 1981 [START_REF] Choquard | One-Dimensional Coulomb Systems BT -Physics in One Dimension[END_REF], where the differences between one-component and two-component one-dimensional plasmas are discussed in detail. Different perspectives of the 1dOCP have been sparking interest ever since [START_REF] Dean | Effects of dielectric disorder on van der Waals interactions in slab geometries[END_REF][START_REF] Téllez | Screening like charges in one-dimensional Coulomb systems: Exact results[END_REF][START_REF] Téllez | Like-charge attraction in one-and two-dimensional Coulomb systems[END_REF][START_REF] Varela | One-dimensional colloidal model with dielectric inhomogeneity[END_REF][START_REF] Varela | Configurational and energy landscape in onedimensional Coulomb systems[END_REF][START_REF] Varela | Relaxation dynamics of two interacting electrical double-layers in a 1D Coulomb system[END_REF][START_REF] Chafaï | At the edge of a one-dimensional jellium[END_REF][START_REF] Rojas | Universal behavior of the full particle statistics of one-dimensional Coulomb gases with an arbitrary external potential[END_REF][START_REF] Dhar | Exact extremal statistics in the classical 1d Coulomb gas[END_REF][START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF]. One of the prominent research directions recently is finding the full probability distributions together with the large deviations of different observables in the jellium model, where the effects of strongly long-ranged interactions can be analyzed analytically. This includes, for example, the distribution of the rightmost particle, the number of particles in a given interval, and the gap between two particles near the edge [START_REF] Dhar | Exact extremal statistics in the classical 1d Coulomb gas[END_REF][START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF].

. Jellium, Coulomb and Riesz gases

. Coulomb gases

The jellium model introduced in the beginning of the chapter contains charged particles that interact via linear repulsive interaction, which corresponds to one-dimensional electrostatic Coulomb potential. Perhaps more familiar is the electrostatic Coulomb potential in three dimensions. In this case the interaction potential of a point charge is proportional to 1/r where r is the distance from the charge. We can extend the Coulomb interaction potential g(y) to d dimensions by defining it as the fundamental solution of Poission's equation

-∇ 2 g(y) = C d δ(y) , (1.4)
where

∇ 2 = ∂ 2 1 + ... + ∂ 2 d is the d-dimensional Laplacian, C d = 2π d/2 /Γ(d/2
) is a surface area of a d-dimensional unit ball and y is a d-dimensional vector with the norm |y|. The solution of this equation is

g(y) =    -log |y|, d = 2, 1 (d-2)|y| d-2 , otherwise. (1.5)
This means that we can write the energy of the Coulomb gas in any dimension as

E d [{y i }] = α N i̸ =j g(y i -y j ) + N i=1 V (y i ) , (1.6)
with V (y i ) being the external potential which in our case reads V (y) = y 2 /2. For the review of the mathematical perspective on Coulomb gases, see [START_REF] Lewin | Coulomb and Riesz gases: The known and the unknown[END_REF] as well as [START_REF] Leblé | Large deviation principle for empirical fields of Log and Riesz gases[END_REF][START_REF] Armstrong | Local laws and rigidity for Coulomb gases at any temperature[END_REF][START_REF] Chafaï | First-order global asymptotics for confined particles with singular pair repulsion[END_REF]. Inserting d = 1, 2, 3 for the first three dimensions, the interaction simplifies to

g(y) =        -|y|, d = 1, -log |y|, d = 2, 1 |y| , d = 3 , (1.7)
which matches the energy of the 1dOCP for d = 1 (see Eq. (1.1)). The case d = 2 is also especially interesting since it can be mapped to the Ginibre ensemble of random matrices [START_REF] Mehta | Random Matrices and the Statistical Theory of Spectra[END_REF][START_REF] Forrester | Log-Gases and Random Matrices[END_REF], which consists of matrices with independent complex entries drawn from a normal distribution. There have been many recent developments in understanding this type of systems [START_REF] Allez | Index distribution of the Ginibre ensemble[END_REF][START_REF] Cunden | Universality of the third-order phase transition in the constrained Coulomb gas[END_REF][START_REF] Varela | Like-charge attraction at short distances in a chargeasymmetric two-dimensional two-component plasma: Exact results[END_REF][START_REF] Armstrong | Remarks on a Constrained Optimization Problem for the Ginibre Ensemble[END_REF][START_REF] Butez | Extremal particles of two-dimensional Coulomb gases and random polynomials on a positive background[END_REF][START_REF] Lacroix-A-Chez-Toine | Intermediate deviation regime for the full eigenvalue statistics in the complex Ginibre ensemble[END_REF][START_REF] Lacroix-A-Chez-Toine | Extremes of 2d Coulomb gas: Universal intermediate deviation regime[END_REF][START_REF] Lacroix-A-Chez-Toine | Rotating trapped fermions in two dimensions and the complex Ginibre ensemble: Exact results for the entanglement entropy and number variance[END_REF][START_REF] Téllez | A two-dimensional one component plasma and a test charge : polarization effects and effective potential[END_REF][START_REF] Téllez | Expanded Vandermonde Powers and Sum Rules for the Two-Dimensional One-Component Plasma[END_REF][START_REF] Grimaldo | Relations Among Two Methods for Computing the Partition Function of the Two-Dimensional One-Component Plasma[END_REF][START_REF] Salazar | Exact mean-energy expansion of Ginibre's gas for coupling constants Γ = 2×(odd integer)[END_REF][START_REF] Can | Exact and asymptotic features of the edge density profile for the one component plasma in two dimensions[END_REF][START_REF] Ferrero | Screening of an electrically charged particle in a twodimensional two-component plasma at Γ = 2[END_REF][START_REF] Fantoni | Two-dimensional one-component plasma on a Flamm's paraboloid[END_REF].

Charge neutrality and harmonic trap

Regardless of the dimension of the Coulomb system, there is one question that needs to be addressed when dealing with systems containing electric charge, and that is charge neutrality. When we say that a system is charge neutral, we mean that its net charge amounts to zero. This is a common assumption that is made when studying Coulomb gases and also, specifically, in the 1dOCP. It is also an assumption that holds true for all of the results presented in the later chapters of this thesis.

Looking back at the energy of the 1dOCP in Eq. (1.1) one might not think that the net charge is zero in this case. Yet, digging deeper reveals that the quadratic term in energy comes exactly from assuming that there is an oppositely charged, uniform background that neutralizes the charge of N particles.

But let us first examine a more intuitive charge-neutral version of one-dimensional plasma, the one-dimensional two-component plasma. Instead of being composed of just one species, it includes both positively and negatively charged particles. By including the same number of charges from each type, we can ensure that the total charge of the system vanishes. The energy of this model reads

E[{ỹ i }, {y i }] = -B i̸ =j |y i -y j | -B i̸ =j |ỹ i -ỹj | + B |y i -ỹj | . ( 1.8)
The particles with negative charge have positions {y i }, and those with a positive charge are at positions {ỹ i }. The parameter B controls the interaction strength. Opposite charges attract each other, and particles of the same charge have repulsive interactions. This kind of model was already studied in the early papers by Lenard, Baxter, Prager, and Choquard [START_REF] Lenard | Exact statistical mechanics of a one-dimensional system with Coulomb forces[END_REF][START_REF] Prager | The One-Dimensional Plasma[END_REF][START_REF] Baxter | Statistical mechanics of a one-dimensional Coulomb system with a uniform charge background[END_REF][START_REF] Choquard | One-Dimensional Coulomb Systems BT -Physics in One Dimension[END_REF]. However, in the large N limit, we can use an approximation where one species of particles -in our case, positive ones -is replaced by a uniform background density ρ b (y) on the interval [-l, l], so that the neutrality is preserved. With this in mind, we can simplify the interaction term between positive and negative charges, namely,

B i̸ =j |y i -ỹj | ∼ BN ρ b N i=1 l -l |y i -ỹ|dỹ.
(1.9)

Evaluating the last integral gives

B i̸ =j |y i -ỹj | ∼ BN ρ b N i=1 (l 2 + y 2 i ) ∝ N i=1 y 2 i + const. (1.10)
The expression for the interaction term between positive and negative charges is now expressed only in terms of the positions of the negative particles y i , and the full energy of negative charges is .11) Comparing this equation to the energy of the 1dOCP in Eq. (1.1), we see that the contribution to the energy from the uniform background density is precisely the term quadratic in y i 's that was introduced in the beginning as the harmonic trap. In contrast to twocomponent plasma, models with only one type of charged particle are usually called onecomponent plasma.

E[{y i }] = C 1 N i=1 y 2 i -B i̸ =j |y i -y j |, -l < y i < l . ( 1
Although it is common to assume charge neutrality, one can also inquire about systems with nonzero total charge. Recently in Ref. [START_REF] Chafaï | At the edge of a one-dimensional jellium[END_REF], the authors presented results for the jellium model where negative particles can populate the region beyond the interval where the background charge is spread. In addition, they do not assume charge neutrality but allow the background charge to be greater than the charge of electrons. Here we focused on the one-dimensional case, but the same logic can also be applied to higher dimensional Coulomb gases (see, for example, Ref. [START_REF] Chafaï | Aspects of Coulomb gases[END_REF]).

. Riesz gases

Before focusing on the properties of the one-dimensional jellium model, we want to introduce another family of models, known as the Riesz gases. They were introduced by Hungarian-born mathematician Marcel Riesz in 1938 [START_REF] Riesz | Intégrales de Riemann-Liouville et potentiels[END_REF] and are considered as one of the most general and widely applicable models, perfectly suited for the study of systems with many interacting particles. Consequently, they are often explored in physical [35, 37-40, 122, 123] as well as mathematical literature [START_REF] Hardin | Large Deviation Principles for Hypersingular Riesz Gases[END_REF][START_REF] Leblé | Large deviation principle for empirical fields of Log and Riesz gases[END_REF][START_REF] Lewin | Coulomb and Riesz gases: The known and the unknown[END_REF][START_REF] Chafaï | Aspects of Coulomb gases[END_REF].

All Riesz gases are composed of N interacting particles confined by an external potential. The only requirement is that the strength of interaction varies with distance as a power law. Therefore the energy of the Riesz gas is given by [START_REF] Riesz | Intégrales de Riemann-Liouville et potentiels[END_REF]

E({y

i }) = A N i=1 y 2 i + sgn(k)B i̸ =j 1 |y i -y j | k , A, B > 0.
(1.12)

where {y i } with i ∈ {1, . . . , N } denote positions of particles. The parameter k determines the strength of the interaction and should be greater than -2. Note that the factor sgn(k) ensures that the pairwise interaction is always repulsive for k > -2. The lower bound k > -2 ensures that the quadratic potential can confine the particles, and they do not fly off to ±∞. We can immediately see from Eq. (1.12) that the 1dOCP is recovered for k = -1.

In this subsection, we focus on one-dimensional Riesz gases, however, the definition of Riesz gas can be extended to higher dimensions, where we can see the connection between Riesz gases and Coulomb gases. In fact, d-dimensional Coulomb gas corresponds to d-dimensional Riesz gas with k = d -2.

Returning to the one-dimensional case, it is natural to ask what changes when k is varied. From Eq. (1.12) we can deduce that changing the value of k in the allowed range from -2 to ∞, changes the interactions in the system from long-ranged for k < 1 to short-ranged for k > 1. Consequently, Riesz gases display various phenomena depending on the parameter k. Moreover, techniques and methods used for investigating properties of Riesz gases with different k differ depending on the value of the parameter. Despite this, recent analytical results have been obtained for general Riesz gases. For example, in Ref. [START_REF] Agarwal | Harmonically Confined Particles with Long-Range Repulsive Interactions[END_REF], the authors compute the average density profile for systems with a large number r is the distance between two particles. The force is given by the first derivative of the potential. We can clearly see the difference between k = 2, which falls into the shortranged regime, k = -1.8, which is strongly long-ranged, and k = -0.5 which belongs to the weakly long-ranged regime. The Orange colored line represents two-particle interaction for the 1dOCP. In this case, the force is just a constant.

of particles and any k > -2. They find that the typical scale of the positions of particles N α k depends on k as

α k = 1 k+2 , -2 < k < 1 k k+2 , k > 1 .
(

By setting k = -1 we obtain the typical scale N , which agrees with our conclusion for the 1dOCP (see Eq. (1.3) and Eq. (1.2)). For a large number of particles, the average density of the Riesz gas converges to a scaling form

⟨ρ N (x)⟩ = 1 N α k ρk (x/N α k ) , (1.14)
where the support of the scaling function ρk (y) is [-l k /2, l k /2] and

ρk (y) = 1 l k F k (y/l k ) , (1.15) with F k (z) = 1 B(γ k + 1, γ k + 1) ( 1 4 -z 2 ) γ k , (1.16)
where -1/2 < z < 1/2 and B(a, b) is a standard beta function. The parameter l k that controls the length of the support depends explicitly on the interaction strength and k.

For the explicit expression, see Ref. [START_REF] Agarwal | Harmonically Confined Particles with Long-Range Repulsive Interactions[END_REF]. The parameter γ k describes the vanishing of the density at two edges and is given by

γ k = k+1 2 , -2 < k < 1 1 k , k > 1 .
(1.17)

Looking at the equations (1.13) and (1.17), we can see a change of behavior at k = 1.

In fact, the authors found three distinct regimes, -2 < k < 1, k = 1, and k > 1.

Figure 1.3: The horizontal axis represents the value of the parameter k determining the interaction in a Riesz gas. The range k ≤ -2 is not allowed (red dashed line). For -2 < k < 1, the long-range interactions are most important, and for k > 1, the gas behaves as a model with short-range interacting particles. For three specific values of k -1, 0 and 2 that correspond to well-known models, we schematically show the shape of the equilibrium particle density.

In the regime, k > 1, the power-law interactions can be well approximated by shortranged interactions. The average density in this regime displays a dome-like shape. When -2 < k < 1, the interactions are truly long-ranged. We can further separate this regime into weakly long-ranged -1 < k < 1 and strongly long-ranged for -2 < k < -1. In the first case, the density stays dome-shaped like in the short-range regime. On the other hand, for -2 < k < -1, the average density obtains two peaks at the edges of the support and has a minimum in the middle. Different regimes are schematically presented in Fig. 1.3. Understanding of different regimes can provide an interesting context for studying the 1dOCP (k = -1), since we see that it is on the border between strongly and weakly long-ranged models.

Other observables that have been computed for a general Riesz gas include the average density in the presence of a hard wall [START_REF] Kethepalli | Harmonically confined long-ranged interacting gas in the presence of a hard wall[END_REF], fluctuations of the rightmost particle [START_REF] Kethepalli | Edge fluctuations and third-order phase transition in harmonically confined longrange systems[END_REF], pair correlation function [START_REF] Beenakker | Pair correlation function of the one-dimensional Riesz gas[END_REF] and gap statistics [START_REF] Santra | Gap statistics for confined particles with power-law interactions[END_REF].

Another interesting aspect of the Riesz gas is that it includes three very well-known systems for special values of k. In addition to the 1dOCP for k = -1, we can recover the classical Calogero-Moser model if we set k = 2 [START_REF] Calogero | Exactly solvable one-dimensional many-body problems[END_REF][START_REF] Moser | Three integrable Hamiltonian systems connected with isospectral deformations[END_REF]. Last but not least, in the limit k → 0, we obtain very famous Dyson's log-gas [START_REF] Mehta | Random Matrices and the Statistical Theory of Spectra[END_REF][START_REF] Forrester | Log-Gases and Random Matrices[END_REF][START_REF] Dyson | Statistical theory of the energy levels of complex systems. I[END_REF]. We will briefly introduce both models since we believe that knowing their properties can help with the interpretation of the results that we later present for the 1dOCP.

Dysons's log-gas

Perhaps the most well-known system belonging to Riesz gases is Dyson's log-gas. We obtain the energy for it by looking at the limit k → 0 + in Eq. (1.12). We can approximate the interaction term as sgn(k)B i̸ =j

|y i -y j | -k ≈ B i̸ =j (1 -k log |y i -y j |) .
(1.18)

By setting B = B/k, we obtain, up to an additive constant, the energy for the log-gas

E[{y i }] = A N i=1 y 2 i -B i̸ =j log |y i -y j | , (1.19)
with the Boltzmann weight

P({y i } = 1 Z N e -βE[{y i }] . (1.20)
The most striking feature of this system is that the positions of the particles can be mapped to the eigenvalues of the Gaussian random matrices. The joint distribution of the eigenvalues of Gaussian ensembles reads [START_REF] Mehta | Random Matrices and the Statistical Theory of Spectra[END_REF]

P(λ 1 , ..., λ N ) ∝ i<j |λ i -λ j | β e -1 2 N i=1 λ 2 i , β ∈ {1, 2, 4} , (1.21)
where λ i are the eigenvalues of random matrices and β ∈ {1, 2, 4} corresponds to three Gaussian ensembles: Gaussian orthogonal ensemble, Gaussian unitary ensemble, and Gaussian symplectic ensemble. By appropriately rescaling the eigenvalues, we can rewrite the equation (1.21) as a Gibbs weight of Dyson's log-gas

P(λ 1 , ..., λ N ) ∝ e -βN 2 2 E[{λ i }] , (1.22)
with E given in Eq. (1.19) with A = 1/N and B = 1/N 2 . The connection between log-gases and random matrices is very useful since it allows one to use tools available in statistical mechanics to derive results for eigenvalues of random matrices. Since Dyson introduced this relationship in his famous papers [START_REF] Dyson | Statistical theory of the energy levels of complex systems. II[END_REF][START_REF] Dyson | Statistical theory of the energy levels of complex systems. III[END_REF][START_REF] Mehta | Statistical theory of the energy levels of complex systems. IV[END_REF][START_REF] Mehta | Statistical Theory of the Energy Levels of Complex Systems. V[END_REF], it has been heavily exploited [53, 64, 69-71, 74-78, 80, 83, 85, 124-126]. One of the most known properties of log-gases is the semi-circular average density profile, which was derived by Wigner [START_REF] Wigner | Effects of the electron interaction on the energy levels of electrons in metals[END_REF] and is also known as Wigner's semicircular law. The semicircular shape is schematically shown in Fig. 1.3. Dyson's log-gas has an important place in the analysis of the results presented in this thesis due to multiple reasons. Like 1dOCP, it is also part of the long-range regime of Riesz gases, and therefore it is reasonable to compare the results for both systems. On the other hand, thanks to the link with random matrices, log-gases have been studied thoroughly in the past. Consequently, the observables that we study (variance of linear statistics, truncated linear statistics, full counting statistics, gaps) in the context of the 1dOCP have already been explored for Dyson's log-gases. This creates a great opportunity for comparison and contextualization of our findings.

Calogero-Moser model

The third well-known member of the Riesz gas family is the Calogero-Moser model [START_REF] Calogero | Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials[END_REF][START_REF] Calogero | Exactly solvable one-dimensional many-body problems[END_REF][START_REF] Moser | Three integrable Hamiltonian systems connected with isospectral deformations[END_REF][START_REF] Diejen | Calogero-Moser-Sutherland Models[END_REF][START_REF] Sutherland | Exact results for a quantum many-body problem in one dimension[END_REF]. We recover it by setting k = 2 in Eq. (1.12). As we can see in Fig. 1.3, it can be effectively described as a model with particles that interact only at short distances. It is an important example of an exactly solvable many-body interacting system. Furthermore, it is an example of a classically integrable system. However, many techniques that can be applied to the Calogero-Moser model exploit the integrable nature of the system and are thus rather different than the ones used for the 1dOCP or log-gases. Surprisingly, there are still some similarities between log-gas and the Calogero-Moser model. For example, the average density converges to Wigner's semicircle in both cases. This and other unexpected similarities between the two models are explored in Ref. [START_REF] Agarwal | Harmonically Confined Particles with Long-Range Repulsive Interactions[END_REF][START_REF] Agarwal | Some Connections Between the Classical Calogero-Moser Model and the Log-Gas[END_REF].

. Some equilibrium properties of the 1dOCP

This last section of the first chapter is dedicated to two main properties of the jellium model: the average density and the partition function. Both of them are fundamental properties of the 1dOCP and are needed for later computations. First, we derive them in a discrete setting, and then we provide a derivation using a continuous description of the model, valid in the large N limit.

In equilibrium, the probability of a specific configuration of particles {x i } is given by the Boltzmann-Gibbs weight

P({x i }) = 1 Z N e -βE[{x i }] , (1.23)
where the energy βE is given by Eq. (1.3). The normalization is determined by the partition function of the system

Z N Z N = ∞ -∞ dx 1 ∞ -∞ dx 2 • • • ∞ -∞ dx N e -βE[{x i }] . ( 1.24)
Since we are dealing with the limit of large N , we expect that the maximum of the integrand dominates the multiple integrals over x i . Consequently, finding the configuration {x * i } that minimizes the energy E[{x i }] determines the main contribution to Z N in the limit N → ∞ as

Z N ∼ e -βE[{x * i }] . (1.25)
Therefore our first task is to find the configuration {x * i }.

Minimum of E[{x i }]

Our first goal is to find the configuration of charges resulting in the lowest energy of the system from Eq. (1.3). The expression for the energy simplifies significantly if we order the particles by their positions: x 1 < x 2 < ... < x N . The immediate benefit of ordering is that we can drop the absolute value in the energy equation Eq. (1.3), i.e.,

βE[{x

i }] = N 2 2 N i=1 x 2 i -2N α N i=1 j<i (x i -x j ) . (1.26)
We replaced the double sum over i, j with the sum over i and j < i multiplied by a factor of two. We can evaluate the double sum in Eq. (1.26), by looking at both terms separately

T 1 = i x i j<i 1 = N i=1 (i -1)x i , (1.27) T 2 = j x j i>j 1 = N j=1
(N -j)x j .

(1.28)

0 -2 α 2 α ρ(x) 1 4α
Figure 1.4: A schematic drawing of 1dOCP at equilibrium. The particles represented as red circles are in the energy minimizing equidistantly spaced configuration {x * i } given in Eq. (1.32). The box-shaped macroscopic density of this configuration ρ * (x) given in Eq. (1.48) is shown with a solid line.

We can exchange the indices i and j in T 2 and combine everything to get

βE[{x i }] = N 2 2 N i=1 x 2 i -2N α(T 1 -T 2 ) = N 2 2 N i=1 x 2 i -2N α N i=1 (2i -N -1)x i . (1.29)
The simplification of the interaction term by ordering the particles and carrying out one summation was proposed by Baxter and is sometimes known as Baxter's combinatorial identity [START_REF] Baxter | Statistical mechanics of a one-dimensional Coulomb system with a uniform charge background[END_REF]. We can further simplify the equation (1.29) by rewriting it in the following way

βE[{x i }] = N 3 1 2N N i=1 x i - 2α N (2i -N -1) 2 -C N (α) , (1.30)
with .31) This formulation allows us to determine the configuration that minimizes the energy easily. The term C N does not depend on any position x i therefore, we need only to minimize the sum of squares in the first term of Eq. (1.30). It reaches its minimum when all of the terms in the sum are zero, which occurs at [START_REF] Lenard | Exact statistical mechanics of a one-dimensional system with Coulomb forces[END_REF][START_REF] Prager | The One-Dimensional Plasma[END_REF][START_REF] Baxter | Statistical mechanics of a one-dimensional Coulomb system with a uniform charge background[END_REF]

C N (α) = 2α 2 N i=1 (2i -N -1) 2 = 2α 2 3 N 3 - 2 3 α 2 N . ( 1
x i = x * i = 2α N (2i -N -1), for i = 1, 2, .., N . (1.32)
This means that the lowest energy configuration is the one where all of the particles are equidistantly spaced in the interval [L 1 , L 2 ]

L 1 = x * 1 = -2α + 2α N ∼ -2α, (1.33
) .34) This crystal-like solution is depicted in Fig. 1.4.

L 2 = x * N = 2α -2α N ∼ 2α. ( 1
The partition function Z N After obtaining the energy-minimizing configuration {x * i }, it is straightforward to compute the partition function of the system in the limit of a large number of particles. Following Eq. (1.25) we use the saddle-point method to compute the most significant contribution to the partition function, defined in Eq. (1.24). We estimate the dominant contribution to the N -fold integral as

Z N ∼ e -βE[{x * i }] , (1.35)
where E[{x * i } denotes the lowest energy of the jellium model. We calculate it by inserting the equilibrium positions x * i from Eq. (1.32) into the equation for the energy (see Eq. (1.30)). The first term given by the sum of squares vanishes, and we are left only with the term C N (α). Thus the lowest energy of the jellium model reads

βE[{x * i }] = - 2 3 α 2 N 3 + 2 3 α 2 N ∼ - 2 3 α 2 N 3 + o(N 3 ). ( 1
Z N ∼ e -βE[{x * i }] = e 2 3 α 2 N 3 +o(N 3 ) . (1.37)
Continuous description in large N limit In this short section, we re-derive the partition function by using the continuous description of the 1dOCP valid in the large N limit. When the number of particles is very large, it is natural and convenient to describe the state of the system by a continuous function, called the particle density, instead of specifying the position of each particle. The continuum approximation can often be more useful for calculations and is used in some of the further chapters of this thesis.

To construct the macroscopic description, we first introduce the normalized macroscopic density of particles .38) This function counts the fraction of particles in the interval [x, x + dx] and is a continuous field in the large N limit. Another closely related observable that we need to define is the averaged macroscopic density profile

ρN (x) = 1 N N i=1 δ(x -x i ) . ( 1
ρN (x) = 1 N N i=1 δ(x -x i ) , (1.39)
where the averaging is done over the Boltzmann-Gibbs measure given in Eq. (1.23). In fact, one can show that the macroscopic density profile is a self-averaging quantity, and therefore when N is large, we can substitute ρN (x) by ρN (x). Moreover, it has been shown that for large N , the average macroscopic density becomes independent of the number of particles [48-51, 100, 101]. Therefore we drop the index N and denote the N -independent form of the averaged macroscopic density as ρ(x).

Utilizing the averaged macroscopic density allows us to exchange the sums with the integrals as

N i=1 f (x i ) = N ρN (x)f (x)dx ----→ large N N ρ(x)f (x)dx , ( 1.40)
where f (x) represents any function of the positions. In this way, we can express the energy in Eq. (1.3) by the following energy functional

βE[ρ(x)] = N 3 E[ρ(x)] = N 3 x 2 2 ρ(x) dx -α ρ(x)ρ(y)|x -y| dx dy . (1.41)
To find the density profile corresponding to the lowest energy ρ * (x) or, in this case equivalently, the average density profile, we set the functional derivative of E with respect to ρ(x) to zero. Additionally, we need to impose the normalization constraint, which we achieve by introducing a Lagrange multiplier µ 0

Ẽ[ρ(x)] = E[ρ(x)] + µ 0 ρ(x)dx -1 . (1.42)
The two equations determining ρ * (x) are given by the derivatives over ρ(x) and µ 0

δ Ẽ[ρ(x)] δρ(x) ρ * (x) = 0 and ∂ Ẽ[ρ(x)] ∂µ 0 ρ * (x) = 0 . (1.43)
The functional derivative over ρ(x) gives

• : x 2 2 -2α ρ * (y)|x -y|dy = 0 . (1.44)
To solve the equation, we use a standard trick of taking two derivatives over x to get

d dx • :
x -2α ρ * (y)sgn(x -y)dy = 0, (1.45)

d 2 dx 2 • : 1 -4α ρ * (y)δ(x -y)dy = 0 , ( 1.46)
where we used d|x|/dx = sgn(x) and dsgn(x)/dx = 2δ(x). Resolving the integral over y brings us to .47) This result does not fully characterize the density profile. In fact, we are missing the edges of the support where the density has a non-zero value. To obtain the edge points, we consider the normalization given by the second equation in Eq. (1.43). We easily see that the full solution to the equations is a flat density profile that is non-zero in the interval [-2α, 2α]

ρ * (x) = 1 4α . ( 1
ρ * (x) = 1 4α , -2α ≤ x ≤ 2α . (1.48)
This box-shaped density profile is plotted in Fig. 1.4. Comparing this result to Eq. (1.32) we see that ρ * (x) is a continuum version of the previous result obtained in the discrete picture. Interestingly, it can be shown that the flat average density of particles is a general feature of any dimensional Coulomb gas and does not appear exclusively in one dimension [START_REF] Cunden | Universality of the third-order phase transition in the constrained Coulomb gas[END_REF][START_REF] Chafaï | First-order global asymptotics for confined particles with singular pair repulsion[END_REF]. The next step is to compute the partition function. In order to do this, we need to exchange the variables {x i } present in the discrete formulation in Eq. (1.25) by the macroscopic density function ρ(x). Consequently, the integrals over x i 's are replaced by the functional integral over all normalized density profiles. This process is done in two steps. First, we fix the macroscopic density profile ρ(x) and sum over all possible microscopic configurations. As explained in Appendix A, this gives us the entropic contribution to the integral e -N dxρ(x) log ρ(x) . In the second step, we integrate over all possible normalized density profiles. Thus, the partition function is given by the following functional integral

Z N ≈ Dρ(x)e -N 3 E[ρ(x)]-N dxρ(x) log ρ(x) δ dxρ(x) -1 . (1.49)
Comparing the two terms in the exponential of Eq. (1.49), we notice that the entropic term that scales as O(N ) is subleading with respect to the energy that scales as O(N 3 ).

Since we are interested in the large N limit, we neglect the subleading entropic term. Note that in the case of short-range interacting systems where the energy is extensive, both terms would scale as O(N ), and one should not neglect the entropic contribution.

After neglecting the entropic term, we can find the leading contribution to the partition function in Eq. (1.49) by using the saddle-point approximation. We insert ρ * (x) into the energy functional from Eq. (1.41) and find

Z N ≈ e -N 3 E[ρ * (x)] = e 2 3 α 2 N 3 . (1.50)
This is the same as the result obtained in the discrete picture in Eq. (1.37).

With this result, we conclude the introduction of one-dimensional one-component plasma. In the next chapter, we focus on the Coulomb gas method, which can be a powerful tool for obtaining analytical results for probability distributions of different observables in systems with long-range interactions.

25

Chapter 2

Coulomb gas technique for linear statistics in the 1dOCP

In this chapter, we further delve into the details of the Coulomb gas technique, which allows us to probe atypically large fluctuations of linear statistics which is defined as

S N = N i=1 f (x i ) , ( 2.1)
where the function f (x i ) can be chosen freely and is not necessarily a linear function. Specifically, we discuss the Coulomb gas method applied to the 1dOCP, which was recently done in Refs. [START_REF] Dhar | Exact extremal statistics in the classical 1d Coulomb gas[END_REF][START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF]. In this case, the x i are the positions of charges, and the probability of a configuration {x i } is given by

P({x i }) = 1 Z N e -βE[{x i }] , ( 2.2)
where

βE[{x i }] = N 2 2 N i=1 x 2 i -N α i̸ =j |x i -x j | . (2.
3)

The Coulomb gas method can also be applied to other systems, such as the two-dimensional Coulomb gas [START_REF] Allez | Index distribution of the Ginibre ensemble[END_REF][START_REF] Lacroix-A-Chez-Toine | Intermediate deviation regime for the full eigenvalue statistics in the complex Ginibre ensemble[END_REF][START_REF] Lacroix-A-Chez-Toine | Extremes of 2d Coulomb gas: Universal intermediate deviation regime[END_REF][START_REF] Cunden | Fluctuations in the two-dimensional one-component plasma and associated fourth-order phase transition[END_REF] or Dyson's log-gas. In fact, the idea was introduced by Dyson [START_REF] Dyson | Statistical theory of the energy levels of complex systems. I[END_REF][START_REF] Dyson | Statistical theory of the energy levels of complex systems. II[END_REF][START_REF] Dyson | Statistical theory of the energy levels of complex systems. III[END_REF] to study spectral linear statistics of random matrices by exploiting their connection to Dyson's log-gas. More recently, the method was successfully used to extract large deviation behavior of many different observables in the context of random matrix theory [START_REF] Majumdar | Top eigenvalue of a random matrix: large deviations and third order phase transition[END_REF][START_REF] Vivo | Large deviations of the maximum eigenvalue in Wishart random matrices[END_REF][START_REF] Majumdar | Large deviations of the maximum eigenvalue for Wishart and Gaussian random matrices[END_REF][START_REF] Majumdar | Large deviations of the top eigenvalue of large Cauchy random matrices[END_REF][START_REF] Krajenbrink | Linear statistics and pushed Coulomb gas at the edge of β-random matrices: Four paths to large deviations[END_REF][START_REF] Chen | Distribution of linear statistics in random matrix models (metalic conductance fluctuations)[END_REF][START_REF] Grabsch | General truncated linear statistics for the top eigenvalues of random matrices[END_REF][START_REF] Grabsch | Truncated linear statistics associated with the top eigenvalues of random matrices[END_REF][START_REF] Dean | Extreme value statistics of eigenvalues of Gaussian random matrices[END_REF][START_REF] Majumdar | How many eigenvalues of a Gaussian random matrix are positive?[END_REF][START_REF] Fogler | Probability of an eigenvalue number fluctuation in an interval of a random matrix spectrum[END_REF][START_REF] Majumdar | Index Distribution of Gaussian Random Matrices[END_REF][START_REF] Chen | Asymptotic level spacing of the Laguerre ensemble: A Coulomb fluid approach[END_REF][START_REF] Cunden | Large deviations of spread measures for Gaussian matrices[END_REF]. The Coulomb gas technique is also useful for obtaining analytical results for systems that can be described by random matrix theory. Prominent examples of this are: chaotic transport through a cavity [START_REF] Grabsch | Distribution of spectral linear statistics on random matrices beyond the large deviation function -Wigner time delay in multichannel disordered wires[END_REF][START_REF] Grabsch | Capacitance and charge relaxation resistance of chaotic cavities -Joint distribution of two linear statistics in the Laguerre ensemble of random matrices[END_REF][START_REF] Grabsch | Truncated Linear Statistics Associated with the Eigenvalues of Random Matrices II. Partial Sums over Proper Time Delays for Chaotic Quantum Dots[END_REF][START_REF] Cunden | Joint statistics of quantum transport in chaotic cavities[END_REF][START_REF] Texier | Wigner time-delay distribution in chaotic cavities and freezing transition[END_REF][START_REF] Damle | Phase transitions in the distribution of the Andreev conductance of superconductor-metal junctions with multiple transverse modes[END_REF][START_REF] Vivo | Distributions of conductance and shot noise and associated phase transitions[END_REF][START_REF] Vivo | Probability distributions of linear statistics in chaotic cavities and associated phase transitions[END_REF], the Rényi entropy in a random pure state of a bipartite system [START_REF] Nadal | Phase transitions in the distribution of bipartite entanglement of a random pure state[END_REF][START_REF] Nadal | Statistical Distribution of Quantum Entanglement for a Random Bipartite State[END_REF][START_REF] Pasquale | Phase transitions and metastability in the distribution of the bipartite entanglement of a large quantum system[END_REF][START_REF] Facchi | Entropy-driven phase transitions of entanglement[END_REF][START_REF] Facchi | Phase transitions of bipartite entanglement[END_REF] or fermions in a trap [START_REF] Marino | Number statistics for β -ensembles of random matrices: Applications to trapped fermions at zero temperature[END_REF][START_REF] Marino | Phase transitions and edge scaling of number variance in gaussian random matrices[END_REF]. This chapter starts with a short explanation of some concepts from the large deviation theory relevant for understanding the Coulomb gas method. The main body of the chapter is divided into two parts. First, we describe the Coulomb gas method in terms of discrete configurations, and later we explain the method again, however, we frame it in the continuous setting. We already discussed the transition from discrete to continuous in Ch. 1 at the end of Sec. 1.2. In this chapter, we extend these ideas and apply them to the computation of linear statistics.

. Some concepts from the large deviation theory

At its core, the large deviation theory deals with the correct description of the probabilities for exponentially unlikely events. Often we are satisfied by using the central limit theorem and describing the fluctuations around the most probable value by the Gaussian approximation. However, this approximation is only valid for typical fluctuations not too far away from the mean value. If one wants to describe the probabilities of very rare events, it is necessary to employ some new techniques. The large deviation theory offers us powerful tools to study such rare events.

It emerged as a mathematical theory; however, later, it was shown that it could be an extremely useful framework for studying statistical physics. For a review dealing with the application of the large deviation approach to statistical physics, see Ref. [START_REF] Majumdar | Large deviations[END_REF][START_REF] Touchette | The large deviation approach to statistical mechanics[END_REF] and references therein. In this section, we aim to introduce some concepts from the large deviation theory that are necessary to understand the Coulomb gas method. We start with a small example borrowed from [START_REF] Majumdar | Large deviations[END_REF][START_REF] Touchette | The large deviation approach to statistical mechanics[END_REF] that is meant to show how we can describe the rare fluctuations by using the large deviation form that consists of the speed and the rate function.

. Example: tossing coins

Imagine tossing N independent coins. Each coin has a probability 1/2 to have value one, and probability 1/2 to have value zero. We are interested in the probability distribution for the fraction of coins with the value of one. This observable is a simple linear statistics given by the following sum

S N = 1 N N i=1 s i , (2.4)
where s i is either 1 or 0 depending on the i-th coin. We know that the probability of obtaining a certain fraction of coins with the value 1 is given by the binomial distribution

P(S N = s) = 1 2 N N sN = N ! 2 N (sN )!(N -sN )! . (2.5)
In the denominator, we have 2 N , which is the total number of possible configurations {s i }.

In the numerator is the binomial symbol which counts the number of configurations {s i } with the fixed number of coins with the value one.

In the next step, we make an important assumption that the number of coins is large. When this is true, we can use Stirling's approximation N ! ≈ N N e -N √ 2πN . By inserting this in the expression for the probability distribution, we get to leading order in N

P(S N = s) ∼ 2 -N s -sN (1 -s) -N (1-s) .
(2.6) (2.8) (magenta line), and the large deviation approximation in Eq. (2.7) (green line). On the vertical axis, we plot the logarithm of probability, and the horizontal axis shows the number of coins with value 1, which equals to sN , where N = 1000 in this case. The main plot shows the full range of sN , from 0 to 1000. We can see that for rare events, the Gaussian approximation starts to fail. The inset shows just a narrow region around N/2, where both approximations agree well with the exact solution.

We can formulate this expression in terms of the exponential function as

P(S N = s) ∼ e -N ϕ(s) , ϕ(s) = ln 2 + s ln s + (1 -s) ln(1 -s) . ( 2.7)
Using Stirling's approximation valid for large N , we managed to write the probability distribution in an exponential form, referred to as the large deviation form, that plays an important role in the large deviation theory. The function ϕ(s), defined in Eq. (2.7) is called the rate function and N is the speed.

The exponential function indicates that the probabilities of rare events, where one type of coin is much more common than the other, are decaying exponentially with speed N . We can see in Fig. 2.1 that the large deviation form given in Eq. (2.7) and shown with green, correctly captures the large deviation tails of the probability distribution of S N .

Further examination of the rate function reveals that ϕ(s) is a convex function with a minimum at 1/2. This means that the most probable configuration is the one where the number of coins 1 is equal to the number of coins 0, which is expected. Another useful property of the rate function is that it also contains information about typical fluctuations around the mean value. We know from the Central Limit Theorem for the sum of random independent variables that the typical fluctuations around the mean have a Gaussian shape. In the case of coins, we have

P Gauss (S N = s) ≈ 2N π e -2N (s-1 2 ) 2 , s - 1 2 ∼ O 1 √ N . (2.8)
This approximation is valid when s-1/2 is of order 1/ √ N . For values of s that are further away from the mean value, it fails. We can see this in Fig. 2.1, where we plot the Gaussian approximation with a solid magenta line. We can recover the Gaussian behavior from the large deviation form by expanding the rate function around its minimum at s = 1/2. The leading contribution to the rate function when s → 1/2 gives us the quadratic form 2(s -1/2) 2 . The inset of Fig. 2.1 shows more closely the region around the value s = 1/2. We can see that the large deviation form successfully captures the quadratic behavior of the typical fluctuations.

It should be mentioned that not all probability distributions can be approximated by the exponential large deviation form in the limit of large N . In some cases, the tails of the probability distribution do not decay exponentially, and therefore large deviation principle does not hold. In fact, one of the main goals of the large deviation theory from a mathematical perspective is to determine conditions under which a probability distribution admits the large deviation form.

. Summary of the Coulomb gas method

This section is dedicated to the presentation of the Coulomb gas technique applied to the one-dimensional one-component plasma. Since we use this method to obtain the majority of the results featured in this manuscript, we present it in a detailed manner. We give two perspectives on the method. First, we discuss the discrete and then the continuous version of the same method.

. Discrete formulation

Let us start in the same way as we did in the first chapter and first present the method retaining the discrete nature of the model. Our goal is to obtain the full probability distribution of the linear statistics described by the large deviation form. Since we want to handle quantities that are of order one, we introduce the rescaled linear statistic

s = 1 N N i=1 f (x i ) .
(2.9)

In the case of the jellium model, x i are the rescaled positions of the particles and f (x) is any function. We start by writing down the formal definition of the full probability distribution of the linear statistics s

P(s, N ) = dx 1 dx 2 • • • dx N P({x i })δ s - 1 N N i=1 f (x i ) . (2.10)
The joint probability distribution of the positions of the particles in the jellium model, P({x i }), was introduced in Eq. (1.23). We can insert it into the above equation to get

P(s, N ) = dx 1 dx 2 • • • dx N e -βE[{x i }] δ s -1 N N i=1 f (x i ) dx 1 dx 2 • • • dx N e -βE[{x i }] , ( 2.11)
where the energy reads

βE[{x i }] = N 2 2 N i=1 x 2 i -N α i̸ =j |x i -x j | . (2.
12)

The denominator is the partition function Z N that we already encountered in Ch.1 where we computed it in the limit of large N . We recall the result given in Eq. (1.37) here

Z N ∼ e 2 3 α 2 N 3 +o(N 3 ) . (2.13)
Focusing on the numerator of Eq. (2.11) we can express it more conveniently by using the integral representation of the delta function

δ(x) = N 3 Γ dµ 2πi e N 3 µx , (2.14)
where Γ is a Bromwich contour going along the imaginary axis in the complex µ-plane.

Plugging in the integral representation of δ(x) in Eq. (2.11) yields up to an overall constant factor

P(s, N ) ∼ 1 Z N dµ dx 1 • • • dx N e N 3 µs e -βEµ[{x i }] = Γ dµe N 3 µs Z N (µ) Z N . (2.15)
Here we introduced a new partition function ] that belongs to a system with the energy

Z N (µ) = dx 1 • • • dx N e -βEµ[{x i }
E µ [{x i }] given by βE µ [{x i }] = N 2 2 N i=1 x 2 i -N α i̸ =j |x i -x j | + N 3 µ 1 N N i=1 f (x i ) = N 2 N i=1 x 2 i 2 + µ f (x i ) -N α i̸ =j |x i -x j | = N 3 Ẽµ [{x i }] , (2.16)
where we denoted Ẽµ [{x i }] = βE µ [{x i }]/N 3 . The energy βE µ [{x i }] from Eq. (2.16) describes the system of particles where the interaction term is the same as in the 1dOCP, but the effective confining potential has an additional µ dependent term

V eff = x 2 2 + µf (x) .
(2.17)

The parameter µ can be thus understood as an amplitude of the perturbation of the original harmonic potential. Indeed, for µ = 0 the energy in Eq. (2.16) reduces to the original 1dOCP energy E[{x i }] (see Eq. (2.12)).

To solve our problem, we need to find the partition function Z N (µ) of the perturbed system with the energy Eq. (2.16) and then integrate over µ. The general idea is the same as for obtaining the partition function Z N in Ch. 1. We want to find the leading contribution to the multiple integrals over x i 's in the limit of large N by using the saddlepoint approximation. We expect that the integrals are dominated by the configuration

{x * i } that has the lowest energy Ẽµ [{x * i }]. The saddle-point solution is fixed by N saddle- point equations ∂ Ẽµ [{x i }] ∂x i {x * i } = 0 . (2.18)
After solving the system of equations we can approximate the ratio of the two partition functions as

Z N (µ) Z N ≈ e -N 3 ( Ẽµ[{x * i }]+ 2 3 α 2 ) = e -N 3 Φ(µ) . (2.19)
Here we inserted the result for the partition function Z N from Eq. (2.13). The function Φ(µ) is also known as the cumulant generating function. Inspecting Eq. (2.15) we see that we still need to resolve the integral over µ

P(s, N ) ∼ Γ dµe N 3 µs Z N (µ) Z N ∼ Γ dµe N 3 µs e -N 3 Φ(µ) . (2.20)
We again use the saddle-point approximation. The dominant contribution to the integral comes from the minimum along the imaginary axis. This corresponds to the maximum along the real µ axis. Thus

Γ dµe N 3 µs e -N 3 Φ(µ) ∼ e -N 3 max µ∈R [Φ(µ)-µs] . (2.21)
We find the maximum by setting the derivative of Φ(µ) -µs over µ to zero, which yields

∂ (Φ(µ) -µs) ∂µ µ * = ∂ Ẽµ [{x i }] + 2 3 α 2 -µs ∂µ µ * = 0 (2.22)
By inserting the energy from Eq. (2.16) for Ẽµ [{x i }] we get the following constraint

1 N N i=1 f (x * i ) -s = 0 , (2.23)
which ensures that the linear statistics has a given value s. Substituting the positions {x * i } that depend on µ in the above relation determines µ * as a function of s at the saddle-point. We can thus obtain the final solution as

P(s, N ) ∼ e -N 3 ϕ(s) , ϕ(s) = Φ(µ * (s)) -µ * (s)s .
(2.24)

Note that this method involves evaluating the energy at the saddle-point (Eq. (2.19)), which can be technically complicated. However, we can derive another relation that gives us the rate function. It reads [START_REF] Grabsch | Distribution of spectral linear statistics on random matrices beyond the large deviation function -Wigner time delay in multichannel disordered wires[END_REF][START_REF] Grabsch | Capacitance and charge relaxation resistance of chaotic cavities -Joint distribution of two linear statistics in the Laguerre ensemble of random matrices[END_REF][START_REF] Cunden | A shortcut through the Coulomb gas method for spectral linear statistics on random matrices[END_REF][START_REF] Cunden | Joint statistics of quantum transport in chaotic cavities[END_REF]

ϕ(s) = - s s µ * (s)ds , (2.25)
where s denotes the mean value of s. We can derive this relation by taking the total derivative of ϕ(s)

dϕ(s) ds = ∂[ Ẽµ [{x * i }] -µ * (s)s + 2 3 α 2 ] ∂s + N i=1 ∂[ Ẽµ [{x * i }] -µ * (s)s + 2 3 α 2 ] ∂x * i ∂x * i ∂s + + ∂[ Ẽµ [{x * i }] -µ * (s)s + 2 3 α 2 ] ∂µ * ∂µ * ∂s .
(2.26)

Let us examine each term separately. The first term is simply

∂[ Ẽµ [{x * i }] -µ * (s)s + 2 3 α 2 ] ∂s = -µ * (s) . (2.27)
The second term equals to zero, which can be seen by using the saddle-point equation given in Eq. (2.18). We have

N i=1 ∂[ Ẽµ [{x * i }] -µ * (s)s + 2 3 α 2 ] ∂x * i ∂x * i ∂s = N i=1 ∂ Ẽµ [{x * i }] ∂x * i ∂x * i ∂s = 0 . (2.28)
The third term also equals to zero, which can be seen from

∂[ Ẽµ [{x * i }] -µ * (s)s + 2 3 α 2 ] ∂µ * ∂µ * ∂s = 1 N N i=1 f (x * i ) -s ∂µ * ∂s = 0 , (2.29)
where we used the constraint for the linear statistics from Eq. (2.23). Summing all three terms, we have dϕ(s) ds = -µ * (s) .

(2.30)

By integrating this relation and fixing the integration constant so that ϕ(s) = 0, we arrive at the main result announced in Eq. (2.25). Note that when using the shortcut in Eq. (2.30), we obtain the relation µ * (s) by evaluating the linear statistics at the saddlepoint (see Eq. (2.23)), which gives us s(µ * ). By inverting this expression, we obtain µ * (s), which we can use in Eq. (2.30).

. Continuous formulation

In all of the examples presented in this thesis, we use the Coulomb gas technique for systems with a large number of particles, where making a continuum approximation can be justified. Sometimes such approximation is useful since it can be easier to minimize the energy functional rather than deal with the discrete configurations. As we explained, we need to solve N saddle-point equations (see Eqs. (2.18)) to find the large deviation form. Often this is very complicated or impossible to do. Instead, in the large N limit, we can describe the system with the macroscopic density profile ρ(x) already introduced in Ch. 1. This reduces the system of N equations to one integral equation that can sometimes be analytically solved. We already discuss the transition from discrete to continuous in the first chapter. In this chapter, we build on the previously introduced ideas, expanding them to establish the Coulomb gas method in a continuous framework.

In Ch. 1 we introduced the continuous energy functional of the jellium model in Eq. (1.41), and it reads

βE[ρ(x)] = N 3 x 2 2 ρ(x) dx -α ρ(x)ρ(y)|x -y| dx dy = N 3 E[ρ(x)] , (2.31)
where

E[ρ(x)] = x 2 2 ρ(x) dx -α ρ(x)ρ(y)|x -y| dx dy . (2.32)
To write the continuous version of the expression for the probability distribution of s given in Eq. (2.10), we need to change N integrals over x i 's to a functional integral over the macroscopic density profile ρ(x). As argued in Ch. 1 and explained in detail in Appendix A we do this in two steps. First, we fix ρ(x) and sum over all microscopic configurations {x i } that correspond to a given ρ(x). In the second step, we integrate over all possible normalized macroscopic density profiles. In this way, we express the probability distribution of the linear statistics as [START_REF] Dean | Extreme value statistics of eigenvalues of Gaussian random matrices[END_REF][START_REF] Allez | Index distribution of the Ginibre ensemble[END_REF]

P(s, N ) ∼ ∼ 1 Z N Dρ e -N 3 E[ρ(x)]-N dxρ(x) log ρ(x) δ ρ(x)f (x)dx -s δ ρ(x)dx -1 .
(2.33)

The entropy term in the exponent N dxρ(x) log ρ(x) originates from the summation of all possible microscopic configurations corresponding to a given macroscopic density profile. We neglect the entropy term linear in N , since in the large N limit, it is negligible compared to the energy term that scales as O(N 3 ). The second delta function in Eq. (2.33) ensures that we are integrating only over the normalized density profiles.

Our goal is to find a density profile ρ * µ (x) that minimizes the energy but still fulfills the two constraints, namely the normalization and the constraint imposed by the linear statistics itself. In other words, we are looking for a macroscopic configuration that is most likely to happen out of all configurations where ρ * µ (x)f (x)dx = s. Finding this state allows us to calculate the dominant contribution of the integral over ρ(x) and therefore obtain the full distribution of s in the large N limit.

We again replace the delta function constraints in Eq. (2.33) by their integral representations given in Eq. (2.14) yielding

P(s, N ) ∼ 1 Z N Dρ dµ dµ 0 e -N 3 S[ρ(x),µ,µ 0 ] , ( 2.34)
with the effective action that reads

S[ρ(x), µ, µ 0 ] = x 2 2 ρ(x) dx -α ρ(x)ρ(y)|x -y| dx dy+ + µ ρ(x)f (x)dx -s + µ 0 ρ(x)dx -1 . (2.35)
The density profile ρ * µ (x) that minimizes the effective action in Eq. (2.35) is fixed by the following saddle-point equations

δS[ρ(x), µ, µ 0 ] δρ(x) ρ * µ ,µ * ,µ * 0 = 0 , ∂S[ρ(x), µ, µ 0 ] ∂µ ρ * µ ,µ * ,µ * 0 = 0, ∂S[ρ(x), µ, µ 0 ] ∂µ 0 ρ * µ ,µ * ,µ * 0 = 0 . (2.36)
We can insert the solution ρ * µ (x) back into the functional from Eq. (2.35) and evaluate S[ρ * (x)]. Note that when the constraints are satisfied, the two terms with µ and µ 0 are equal to zero and, therefore

S[ρ * µ (x), µ * , µ * 0 ] = E[ρ * µ (x)] , (2.37)
where E[ρ(x)] is given in Eq. (2.32). Following the saddle-point approximation, we calculate the dominant contribution from the integrals over ρ(x), µ, and µ 0 by inserting E[ρ * µ (x)] into Eq. (2.34). Hence, we obtain

P(s, N ) ≈ e -N 3 E[ρ * µ (x)] e N 3 2 3 α 2 , ( 2.38)
where we used the expression for the partition function of the unconstrained 1dOCP Z N from Eq. (2.13). The final large deviation form is thus given by

P(s, N ) ≈ e -N 3 ϕ(s) , ϕ(s) = E[ρ * µ (x)] + 2 3 α 2 .
(2.39)

Note that we can again use the shortcut presented in the previous section in Eq. (2.25) to circumvent the evaluation of the action at the saddle-point. We can derive the relation in a similar way as in the discrete case. The derivative of ϕ(s) with respect to s is given by

dϕ(s) ds = ∂[S + 2 3 α 2 ] ∂s + dx δ[S + 2 3 α 2 ] δρ * µ (x) ∂ρ * µ (x) ∂s + ∂[S + 2 3 α 2 ] ∂µ * ∂µ * ∂s , ( 2.40)
where we have inserted

ϕ(s) = S[ρ * µ (x), µ * , µ * 0 ] + 2 3 α 2 .
Looking at the action in Eq. (2.35) we find that the first term equals to -µ * (s). The second term is zero, which can be seen by using the first saddle-point equation in Eq. (2.36). Therefore we get

dϕ(s) ds = -µ * (s) + ρ * µ (x)dx -s ∂µ * ∂s = -µ * (s) . ( 2.41)
The second term in Eq. (2.41) vanishes since the constraint for the linear statistics is satisfied in the saddle-point (see the second saddle-point equation in Eq. (2.36)).

With this remark, we conclude this chapter, where we presented two different formulations of the Coulomb gas technique applied to the calculation of linear statistics in the 1dOCP. In Sec. 2.2.1, we explained the method by using a discrete description, and in Sec. 2.2.2, we presented it in the continuous framework. We showed that the main step of the method consists of using the saddle-point approximation to find a dominant contribution to the N + 1 fold integral that appears in the numerator of the probability distribution of the linear statistics P(s, N ) (see Eq. (

Chapter 3

An exact formula for the variance of linear statistics in the 1dOCP

This chapter aims to expand the previously introduced Coulomb gas technique to derive a formula for the variance of the rescaled linear statistics in the 1dOCP

Var(s) = ⟨(s -s) 2 ⟩, with s = 1 N N i=1 f (x i ) , (3.1)
where s is the average value of s given by

s = ρ * (x)f (x)dx = 1 4α 2α -2α f (x)dx , ( 3.2)
and ρ * (x) = 1/(4α) stands for the average density of the 1dOCP. We show that in the large N limit, the variance of the linear statistics s reads

Var(s) ≈ 1 4αN 3 2α -2α f ′ (x) 2 dx , (3.3) if [f ′ (x)
] 2 exists and is integrable. In addition to deriving this result, we discuss its validity for different choices of the test functions f (x). For a few examples of f (x), we calculate the full large deviation form given by ∼ e -N 3 ϕ(s) . By examining the behavior of the rate function ϕ(s) around its minimum at s = s, we find the variance and compare the result to the general formula from Eq. (3.3). Moreover, we confirm the result by Monte Carlo simulations. The presented results have been published in [START_REF] Flack | An exact formula for the variance of linear statistics in the one-dimensional jellium model[END_REF], which is included at the end of the chapter. This chapter is organized as follows. First, we summarize already-known results for the variance of linear statistics in Dyson's log-gas and Coulomb gases in dimensions higher than one. The main part of Ch. 3 consists of the derivation of the expression (3.3) together with analyzing two simple cases: f (x) = x and f (x) = x 2 . In the last part of the chapter, we discuss the validity of our expression for the variance. We argue that it can predict correct results even in some cases where we would not expect it. We show two such examples: f (x) = x 3 and f (x) = |x|. We end the chapter by giving a few concluding remarks.

. Formula for the variance in other systems

Unlike in the jellium model, the fluctuations of linear statistics have been studied extensively in the case of Dyson's log-gas. As explained in the first chapter, the positions of particles in Dyson's log-gas can be mapped to the eigenvalues of Gaussian random matrices. Thus, the results derived for Dyson's log-gas directly apply to the linear statistics of eigenvalues of random matrices. In fact, one of the first results about the variance of the linear statistics of Dyson's log-gas was given by Dyson and Mehta in [START_REF] Mehta | Statistical theory of the energy levels of complex systems. IV[END_REF], where they computed the variance for spectral linear statistics of Gaussian random matrices

Var(s) ≈ 1 βπ 2 N 2 ∞ 0 k | f (k)| 2 dk , ( 3.4)
where f (k) = ∞ -∞ e ikx f (x) dx is the Fourier transform of f (x) and the parameter β = 1, 2, 4 depends on Gaussian ensemble (Orthogonal, Unitary, Symplectic). Later Dyson's result was extended, and new formulas relevant for different ensembles of random matrices were derived. In the context of quantum transport, Beenaker derived an expression valid for Jacobi N × N random matrices [START_REF] Beenakker | Universality in the Random-Matrix Theory of Quantum Transport[END_REF][START_REF] Beenakker | Random-matrix theory of mesoscopic fluctuations in conductors and superconductors[END_REF]. Later, he generalized his results and derived an expression valid for all classical invariant ensembles of random matrices [START_REF] Beenakker | Universality of Brézin and Zee's spectral correlator[END_REF]. In the language of log-gases, this translates to a formula for the variance that holds for systems with general confining potential V (x). Furthermore, the variance converges to a N -independent form for large N and the typical fluctuations around the mean are Gaussian [START_REF] Chen | Distribution of linear statistics in random matrix models (metalic conductance fluctuations)[END_REF][START_REF] Vivo | Probability distributions of linear statistics in chaotic cavities and associated phase transitions[END_REF][START_REF] Ameur | Fluctuations of eigenvalues of random normal matrices[END_REF][START_REF] Lytova | Central limit theorem for linear eigenvalue statistics of random matrices with independent entries[END_REF][START_REF] Pastur | Limiting laws of linear eigenvalue statistics for Hermitian matrix models[END_REF][START_REF] Politzer | Random-matrix description of the distribution of mesoscopic conductance[END_REF][START_REF] Soshnikov | The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities[END_REF][START_REF] Costin | Gaussian fluctuation in random matrices[END_REF][START_REF] Baker | Finite N Fluctuation Formulas for Random Matrices[END_REF]. A new formula for the covariance of two linear statistics, also valid for general V (x), was derived in Ref. [START_REF] Cunden | Universal covariance formula for linear statistics on random matrices[END_REF]. Other results describing the fluctuations of the linear statistics have been derived in the context of random matrix theory, see for example [START_REF] Grabsch | General truncated linear statistics for the top eigenvalues of random matrices[END_REF][START_REF] Vivo | Probability distributions of linear statistics in chaotic cavities and associated phase transitions[END_REF][START_REF] Baker | Finite N Fluctuation Formulas for Random Matrices[END_REF][START_REF] Basor | Variance calculations and the Bessel kernel[END_REF][START_REF] Johansson | On fluctuations of eigenvalues of random hermitian matrices[END_REF][START_REF] Sommers | Statistics of conductance and shot-noise power for chaotic cavities[END_REF].

The formula for the variance of linear statistics has also been derived for the Ginibre ensemble of random matrices [START_REF] Forrester | Fluctuation formula for complex random matrices[END_REF]. In this case the matrix elements are independent complex numbers drawn from Gaussian distribution. As discussed in Ch. 1, eigenvalues of such matrices can be mapped into positions of electrons of a two-dimensional Coulomb gas in a harmonic potential. Interestingly, up to an overall constants and N dependent factors the formula for the variance in this case matches with our expression in Eq. (3.3) and is given by

V ar(s) ∼ [∇f (⃗ r)] 2 d⃗ r, with s = 1 N N i=1 f (⃗ r i ) . (3.5)
This expression was rigorously proved in [START_REF] Armstrong | Remarks on a Constrained Optimization Problem for the Ginibre Ensemble[END_REF][START_REF] Ameur | Fluctuations of eigenvalues of random normal matrices[END_REF][START_REF] Forrester | Fluctuation formula for complex random matrices[END_REF][START_REF] Rider | The noise in the circular law and the Gaussian free field[END_REF][START_REF] Leblé | Fluctuations of Two Dimensional Coulomb Gases[END_REF] and successfully generalized for Coulomb gases of dimensions higher than two d ≥ 2 [START_REF] Armstrong | Local laws and rigidity for Coulomb gases at any temperature[END_REF]. However, the techniques that were used do not extend to one dimension.

. Computing the rate function

In this section, we compute the rate function of the linear statistics by using the continuous formulation of the Coulomb gas technique presented in Ch. 2. The first step of the derivation is determining the saddle-point density ρ * µ (x) that minimizes the energy functional

βE[ρ(x)] = N 3 x 2 2 ρ(x) dx -α ρ(x)ρ(y)|x -y| dx dy = N 3 E[ρ(x)] , (3.6)
and at the same time satisfies two constraints: the normalization ρ * µ (x)dx = 1 and s = ρ * µ (x)f (x)dx. As we explained, we achieve this by introducing two Lagrange multipliers µ 0 and µ and minimizing the action functional given in Eq. (2.35)

S[ρ(x)] = x 2 2 ρ(x) dx -α ρ(x)ρ(y)|x -y| dx dy + (3.7) +µ ρ(x)f (x)dx -s + µ 0 ρ(x)dx -1 = = E[ρ(x)] + µ ρ(x)f (x)dx -s + µ 0 ρ(x)dx -1
The density profile that minimizes the action is determined by three saddle-point equations from Eq.(2.36)

δS[ρ(x), µ, µ 0 ] δρ(x) ρ * µ ,µ * ,µ * 0 = 0 , ∂S[ρ(x), µ, µ 0 ] ∂µ ρ * µ ,µ * ,µ * 0 = 0, ∂S[ρ(x), µ, µ 0 ] ∂µ 0 ρ * µ ,µ * ,µ * 0 = 0 . (3.8)
After obtaining ρ * µ (x) we can substitute it back into the energy functional E[ρ(x)] and obtain the rate function as a

ϕ(s) = E[ρ * µ (x)] -E[ρ * (x)] = E[ρ * µ (x)] + 2 3 α 2 , ( 3.9)
where ρ * (x) = 1/(4α) represents the saddle-point density of the unconstrained 1dOCP calculated in Eq. (1.48).

Returning back to finding ρ * µ (x), we start by using the fist equation in (3.8) and doing the functional derivative over ρ(x), which yields

• : x 2 2 -2α ρ * µ (y)|x -y|dy + µf (x) + µ 0 = 0 . (3.10)
This equation holds for all x that are inside of the support of the density profile ρ(x).

Taking two derivatives over x in Eq. (3.10) we find

d dx • : x -2α ρ * µ (y)sgn(x -y)dy + µf ′ (x) = 0 , (3.11) d 2 dx 2 • : 1 + µf ′′ (x) -4 αρ * µ (x) = 0 . (3.12)
The second equality implies that

ρ * µ (x) = 1 4α 1 + µf ′′ (x) , L 1 ≤ x ≤ L 2 , ( 3.13)
Here we assumed that ρ * µ (x) is supported on a single interval [L 1 , L 2 ] as is shown in Fig. 3.1. To fully characterize the saddle-point density we still need to fix the four parameters L 1 , L 2 , µ 0 and µ. We can determine one relation by setting x = L 2 in Eq. (3.11). Since L 2 is the right edge of the density, we can set sgn(L 2 -y) = 1 for all y ≤ L 2 . Using the normalisation

L 2 L 1 ρ * µ (x) dx = 1, one gets L 2 -2α + µf ′ (L 2 ) = 0 . (3.14)
Similarly, by setting x = L 1 in Eq. (3.11) and using sgn(L 1 -y) = -1 for all y ≥ L 1 gives

L 1 + 2α + µf ′ (L 1 ) = 0 . ( 3.15)
Having four unknowns (L 1 , L 2 , µ and µ 0 ) and two equations, we are missing another two independent relations. We can obtain the third one by setting x = L 2 in Eq. (3.10) and using

|L 2 -y| = L 2 -y for all y ≤ L 2 µ 0 = - L 2 2 2 + 2αL 2 - 1 4 (L 2 2 -L 2 1 ) - µ 2 L 2 L 1 yf ′′ (y)dy -µf (L 2 ) . (3.16)
We use the constraint

L 2 L 1 f (x)ρ * µ (x) dx = s , ( 3.17)
to find the fourth relation. Inserting the saddle-point density from Eq. (3.13) and performing the integral gives the desired fourth relation

µ 4α L 2 L 1 f (x)f ′′ (x)dx = s - 1 4α L 2 L 1 f (x)dx . ( 3
S[ρ * µ (x)] = L 3 2 -L 3 1 48α + L 2 2 -L 2 1 8 + L 2 2 4 -αL 2 + + µ 4 1 4α L 2 L 1 x 2 f ′′ (x)dx -2s + 2f (L 2 ) + L 2 L 1 xf ′′ (x)dx . (3.19)

40

Note that because the evaluation of action is not complicated in this specific case, we did not use the shortcut introduced in Eq. (2.25). However, we will use the shortcut in other derivations later on. Thus, the rate function is given by the difference between the energy functional evaluated at saddle-point density ρ * µ (x) and the one evaluated at the average density for the unconstrained 1dOCP

ϕ(s) = E[ρ * µ (x)] -E[ρ * (x)] = = L 3 2 -L 3 1 48α + L 2 2 -L 2 1 8 + L 2 2 4 -αL 2 + 2 3 α 2 + + µ 4 1 4α L 2 L 1 x 2 f ′′ (x)dx -2s + 2f (L 2 ) + L 2 L 1 xf ′′ (x)dx , ( 3.20)
where L 1 , L 2 and µ are functions of s and are determined from the relations (3.14), (3.15), (3.16) and (3.18). This expression gives the exact rate function for the linear statistics s = (1/N ) N i=1 f (x i ) with arbitrary f (x) as long as ρ * µ (x) is supported on a single interval. Expression for the rate function in Eq. (3.20) is one of the main results presented in this chapter. However, due to the relations between L 1 , L 2 , and µ, it is not very easy to interpret. As we explained in Ch. 2 when describing the simple example of random coins, the minimum of the rate function corresponds to the typical value for an observable. We expect that in our case the typical value equals the equilibrium value for the 1dOCP given in Eq. (3.2). By inspecting Eq. (3.18) together with Eq. (3.14) and (3.15), we see that when s goes towards the equilibrium value s, µ approaches zero. Furthermore, the edges of the density support L 1 and L 2 approach L 1 → -2α, L 2 → 2α, which are the values for the equilibrium 1dOCP. Returning to the rate function ϕ(s) in Eq. (3.20), we can see that it approaches zero when µ → 0. Since the rate function is positive and convex s = s must be its minimum.

. Extracting the variance

In the previous section, we derived the expression for the rate function of the linear statistics s, given in Eq. (3.20). In this section, we want to investigate typical fluctuations around the minimum at s = s = 1 4α 2α -2α f (x)dx. Given that the rate function is twice differentiable at s = s we can expand the rate function around the minimum and obtain Gaussian approximation for typical fluctuations as

ϕ(s) ≈ 1 2 ϕ ′′ (s)(s -s) 2 -→ P typ (s, N ) ≈ e -N 3 2 ϕ ′′ (s)(s-s) 2 . (3.21)
The variance of s can be extracted from the Gaussian form and is given by

Var(s) ≈ ϕ ′′ (s) N 3 . (3.22)
When s is close to the mean value, it is useful to write it as

s = s + ϵ , (3.23)
where ϵ is very small. We expect that a small change in s results in a small change in the position of the edges

L 1 = -2α + δ 1 (ϵ) , L 2 = 2α + δ 2 (ϵ) , (3.24)
where δ 1 (ϵ) and δ 2 (ϵ) are small as well. To find the expansion of the rate function, we need to express L 1 , L 2 , and µ as functions of ϵ. We can do this by inserting Eq. (3.24) into the relations (3.18), (3.14), and (3.15). Keeping terms up to O(ϵ 2 ) we have

µ = - 4α ϵ +2α -2α [f ′ (x)] 2 dx + O(ϵ 2 ) (3.25) δ 1 (ϵ) = 4αf ′ (-2α) ϵ +2α -2α [f ′ (x)] 2 dx + O(ϵ 2 ) (3.26) δ 2 (ϵ) = 4αf ′ (2α) ϵ +2α -2α [f ′ (x)] 2 dx + O(ϵ 2 ) . (3.27)
Substituting them back into Eq. (3.20) and keeping only terms with ϵ 2 would give us a second-order expansion of the rate function. Although possible, this approach involves cumbersome computations and cancelation of terms O(ϵ). We can avoid these complications by using the shortcut introduced in Eq. (2.25)

ϕ(s) = -µ * (s)ds + const, (3.28)
Since the rate function is an integral of µ * (s), we can reach the second order in ϵ, by evaluating µ only up to O(ϵ). The constant is fixed by the condition ϕ(s = s) = 0. Straightforward integration of Eq. (3.25) leads us to the final result

ϕ(s) = 1 2b (s -s) 2 + O((s -s) 3 ) , with b = 1 4α 2α -2α [f ′ (x)] 2 dx . (3.29)
Thus the variance of the linear statistics s is

Var(s) = b N 3 = 1 N 3 4α 2α -2α [f ′ (x)] 2 dx . (3.30)
For a better understanding of the result (3.30), we explicitly calculate the rate function and the variance for two simple examples: f (x) = x and f (x) = x 2 . Furthermore, we check our prediction by comparing it with numerical simulations.

. Example:

f (x) = x First, let us choose f (x) = x,
which results in linear statistics given by

s = 1 N N i=1 x i . (3.31)
The effective potential felt by the particles of the constrained gas is given by

V eff (x) = 1 2 x 2 + µx . (3.32)
This potential is confining regardless of the value of µ. We can see the shape of the effective potential for µ = 0 and µ = -1 plotted in Fig. 3

ρ * µ (x) = 1 4α I [L 1 ,L 2 ] (x), L 1 = s -2α , L 2 = s + 2α , µ = -s , ( 3.33)
where

I [L 1 ,L 2 ] (x) is an indicator function, which is 1 if x ∈ [L 1 , L 2 ]
and 0 otherwise. This solution is depicted in Fig. 3.2 for two values of s: 0 and 1. When s = 0 also µ = 0 and the density profile is the same as for the unperturbed 1dOCP. For s = 1 or µ = -1 the shape stays the same, but all the charges are translated, which is expected since the minimum of the effective potential shown in the top left panel of Fig. 3.2 is also shifted by µ.

The rate function for this case can be computed directly by using the shortcut with the relation between µ and s given in Eq. (3.33). This yields

ϕ(s) = sds = s 2 2 , ( 3.34)
where we fixed the constant by imposing ϕ(s = s) = 0. Thus the large deviation form reads

P(s, N ) ∼ e -N 3 s 2 2 , ( 3.35)
which implies that the variance equals to Var(s) = 1/N 3 . This finding agrees with the prediction from our formula for the variance in Eq. (3.30)

Var(s) = 1 N 3 4α 2α -2α [f ′ (x)] 2 dx = 1 N 3 .
(3.36)

. Example:

f (x) = x 2
In this case, the effective potential is given by

V eff (x) = ( 1 2 + µ)x 2 . ( 3.37)
For µ = 0.006 and µ = -0.442, we plotted V eff (x) in Fig. 3.2 in top right panel. Unlike the case f (x) = x, the minimum stays in the position x = 0 for both values of µ. Note that if µ ≤ 1/2, the potential is not confining anymore, and the particles would fly off to ±∞ due to repulsive interactions. However, as we shall see, µ is always above -1/2, and consequently, the potential is confining for all allowed values of µ. The saddle-point density from Eq. (3.13) reads

ρ * µ (x) = 1 2 √ 3 s I [L 1 ,L 2 ] (x) , (3.38)
with the parameters L 1 , L 2 and µ again determined respectively from Eqs. (3.14), (3.15) and (3.18). We get

L 1 = - √ 3 s , L 2 = √ 3 s , µ = - 1 2 + α √ 3
ϕ(s) = s 2 - 2α √ 3 √ s + 2α 2 3 . (3.40)
The variance obtained by expanding it to the second order agrees with our prediction in Eq. (3.36) and reads The height of the maximum is 1/(54µ). The inset shows the saddle-point density ρ * µ (x) calculated from Eq. (3.44) (solid black line) compared with Monte-Carlo simulation (red dots). For simulation, we used N = 1000 particles and set µ = 0.1 and α = 1/10. We averaged over 4 × 10 7 realizations.

Var(s) = 1 N 3 16α 2 3 . (3.41)

. The validity of the formula for the variance

During the derivation of the formula for the variance given in Eq. (3.30), we made a few assumptions that allowed us to reach the final result. For the formula in Eq. (3.30) to hold, the integral 2α -2α dx[f ′ (x)] 2 should exists and be finite. The next assumption is that the effective potential V eff (x) = x 2 /2 + µf (x) felt by the constrained system with energy given in Eq. (2.16), is confining for all possible values of µ. If this is not the case, the global minimum of energy is reached when particles fly off to ∞ or -∞ depending on V eff (x). The third assumption was made after Eq. (3.13), when we assumed that the density is supported in a single interval between L 1 and L 2 .

In this section, we show that our formula for the variance predicts the correct result even in some cases where one of the above assumptions does not hold. We focus on two interesting examples: f (x) = x 3 , for which the assumption about confining V eff (x) does not hold, and f (x) = |x|, where the saddle-point density has double support for µ < 0.

. f (x) = x 3

Choosing f (x) = x 3 results in an effective potential given by

V eff (x) = x 2 2 + µx 3 , (3.42)
which is non-confining for any µ. In Fig. 3.3, we show an example of the effective potential for µ = 0.1. For all µ > 0 the potential diverges negatively for x → -∞. Conversely, for µ < 0 it diverges negatively for x → ∞. From now on, we focus on the case µ > 0, which has a local minimum at x = 0 and a local maximum at x = -1/(3µ). Its height changes with µ as

V eff x = - 1 3µ = 1 54 µ 2 . ( 3
ρ * µ (x) = 1 4α (1 + µf ′′ (x)) = 1 4α (1 + 6µx), (3.44
)

L 1 = -1 + √ 1 -24αµ 6µ , L 2 = -1 + √ 1 + 24αµ 6µ . (3.45)
In the inset of Fig. 3.3 we compare this result with data gathered from Monte-Carlo simulations. We see that our analytical prediction successfully describes the numerical data. Furthermore, we can use Eq. (3.30) for the variance to find x 3 dx = 0 , (3.47) which corresponds to µ = 0. For small enough µ, the potential barrier is high enough and all charges stay at the minimum around x = 0. Consequently, our formula for the variance correctly captures the small fluctuations around s. However, the expression for the full rate function from Eq (3.20), which is defined for all s does not hold in this case.

Var(s) = b N 3 = 1 N 3 1 4α 2α -2α (3x 2 ) 2 dx = 1 N 3 144 5 α 4 . ( 3
V ef f (x) µ = -1 V ef f (x) V ef f (x) µ = 0 µ = 1 x x x
The same arguments apply to other odd functions f (x) diverging faster than x 2 , such as x 5 , x 7 ... We expect that in these cases, our formula predicts the correct variance since all of these functions have a local minimum at x = 0.

3.4.2 . f (x) = |x| Choosing f (x) =
|x| results in the effective potential given by

V eff (x) = x 2 2 + µ|x| , (3.48)
This potential displays one minimum at x = 0 for µ > 0 and two minima at ±|µ| for µ < 0 as we can see in Fig. 3.5. We expect the saddle-point density ρ * µ (x) to have a single support when there is a single minimum for µ > 0. However, for µ < 0 we expect that ρ * µ (x) is supported on two intervals centered around the two minima. We confirm these two claims by explicitly obtaining ρ * µ (x) for µ > 0 and µ < 0. In order to proceed, we need to return to Eq. (3.11), which still holds for any type of support. By inserting f (x) = |x| we get

x -2α ρ * µ (y)sgn(x -y)dy + µ sgn(x) = 0 , ( 3.49)
which holds for all x inside of the support of the saddle-point density. From now on, we separate the two cases µ > 0 and µ < 0.

The case µ > 0: In this case we assume single support of ρ * µ (x) over [-ℓ, +ℓ]. We expect that the support is symmetric around x = 0 due to the symmetric effective potential. The steps of the calculation are similar as in the two examples f (x) = x and f (x) = x 2 . Taking one more derivative of Eq. (3.49) with respect to x, and using d dx sgn(x) = 2δ(x), we get Note, that our formula for the density for general f (x) form Eq. (3.13) correctly predicts the density profile given in Eq. (3.50) and Eq. (3.51). As we will see in the next section this is not the case for µ < 0 where the density has two disjoint supports. For s = 0.025 we compared our result for the saddle-point density with Monte-Carlo simulations and we observed a good agreement as is shown in the right panel of Fig. 3.6.

ρ * µ (x) = 1 4α + µ 2α δ(x) . ( 3
The case µ < 0: In this case, we assume that the saddle-point density is supported on two intervals [-L, -a] ∪ [a, L] with 0 ≤ a ≤ L. We expect that the two intervals are symmetric with respect to x = 0, because the two minima of V eff (x) are symmetric around x = 0 as well. Taking a derivative with respect to x of Eq. (3.49) we get

ρ * µ (x) = 1 4α for -L < x < -a or a < x < L . (3.52)
Once again we need to determine the edges of the support, this time given by L and a. Together with µ we have three unknown parameters that can be fixed by using the normalization condition, Eq. for f (x) = |x|. We compare this solution with numerical data in the left panel of Fig. 3.6.

To summarize we found the saddle-point density ρ * µ (x) for both cases µ > 0 (or equivalently s < s) and µ < 0 (or equivalently s > s). When µ < 0 the density is supported on two intervals positioned symmetrically with respect to x = 0. When µ is approaching zero (or when s is approaching s) the gap between two supports is diminishing until it vanishes at µ = 0 or s = s. When we reach the single support regime for µ > 0 the delta spike at x = 0 appears. Further increasing µ results in more charges moving to x = 0 and thus increasing the height of the spike.

Knowing the saddle-point density we can quickly determine the rate function by using the shortcut ϕ(s) = -µ * (s)ds + const. We already computed the relations between µ and s in both regimes in Eqs. (3.51) and (3.53). To summarize, we have

µ * (s) =        2α - √ 4αs , s < s = α α -s , s > s = α . (3.54)
Integrating with respect to s yields the rate function

ϕ(s) =          -2α s + 4 3 √ αs 3/2 + 2 3 α 2 , s < α = s , 1 2 (s -α) 2 , s > α = s . (3.55)
It is interesting to check how the transition from a double-to a single-support regime affects the rate function. The first and second derivatives are continuous at the transition point s = s. However, the third derivative displays a discontinuity

ϕ ′′′ (s) =          - 1 2α , s → s-, 0 , s → s+ . (3.56)
From this observation, we can conclude that there is a third-order phase transition at s = s. This type of phase transition, also known as pulled to pushed phase transition, has been observed in many other Coulomb and log-gases [START_REF] Majumdar | Top eigenvalue of a random matrix: large deviations and third order phase transition[END_REF][START_REF] Cunden | Universality of the weak pushedto-pulled transition in systems with repulsive interactions[END_REF][START_REF] Cunden | Universality of the third-order phase transition in the constrained Coulomb gas[END_REF]. We will again encounter third-order phase transitions of this type in the following chapters of this thesis. We end this section by comparing the variance calculated by expanding Eq. MC as a function of increasing N . We find that it decreases to zero as N → ∞, thus verifying the theoretical prediction. The correct prediction of our formula for the variance (Eq. (3.30)) can be explained by the following reasoning. We showed that the phase transition between single-and double-supported phase shows up as a discontinuity in the third derivative of the rate function ϕ(s) (see Eq. (3.56)). Since the variance is completely described by the second derivative of the rate function, we expect that it is not affected by the third-order phase transition.

. Conclusion of Chapter 3

In this chapter, we derived the expression for the variance of the rescaled linear statistics s = (1/N ) N i=1 f (x i ) in one-dimensional one-component plasma. The two main results are the expression for the variance given in Eq. (3.30) and the full rate function in Eq. (3.20). We derived these two results by assuming a single support of the saddle-point density and confining effective potential. However, we argue that the expression for the variance predicts correct results even for some cases where these assumptions do not hold. First, we discuss the case f (x) = x 3 , where the effective potential diverges negatively at -∞ or ∞. Next, we examine the case f (x) = |x|, which displays a phase transition from single to the double supported density at s = s.

At the end of this chapter, we mention some extensions of the presented results. Similarly, as in the case of invariant random matrix ensembles, we can derive an expression for the variance that holds for general confining potential V (x). In Ref. [START_REF] Flack | An exact formula for the variance of linear statistics in the one-dimensional jellium model[END_REF], we show how we can use an alternative derivation based on the generating function of s to find the variance of linear statistics with general V (x). Interestingly, up to an overall N -dependent factor, the formula for the variance derived in this chapter for V (x) = x 2 also holds for general V (x). The result is not explicitly dependent on the potential, which enters in the equation through the edges of the support of the saddle-point density, L 1 , and L 2 . The same is true for the variance of the linear statistics in log-gases.

Another interesting question is, how does the variance of s in the 1dOCP compare with the results of other Riesz gases? The answer to this can be found in a recent paper by Beenakker [START_REF] Beenakker | Pair correlation function of the one-dimensional Riesz gas[END_REF], where he computes the variance and covariance of linear statistics for

.

Abstract:

We consider the jellium model of N particles on a line confined in an external harmonic potential and with a pairwise one-dimensional Coulomb repulsion of strength α > 0. Using a Coulomb gas method, we study the statistics of s = (1/N ) N i=1 f (x i ) where f (x), in principle, is an arbitrary smooth function. While the mean of s is easy to compute, the variance is nontrivial due to the long-range Coulomb interactions. In this paper we demonstrate that the fluctuations around this mean are Gaussian with a variance Var(s) ≈ b/N 3 for large N . In this paper, we provide an exact compact formula for the constant b = 1/(4α) 2α

-2α [f ′ (x)] 2 dx. In addition, we also calculate the full large deviation function characterising the tails of the full distribution P(s, N ) for several different examples of f (x). Our analytical predictions are confirmed by numerical simulations.

Chapter 4 The probability distributions of the gap and the full counting statistics

The purpose of this chapter is to focus on two observables describing the bulk of the jellium model: the gap between the positions of the two particles that are deep in the bulk of the system and the full counting statistics (FCS), that measures the number of particles in an interval [-L, L] that is also in the bulk of the system. For both observables, we study typical fluctuations around the mean as well as large atypical fluctuations that can be analyzed using the saddle-point method. In order to analyze the gap and the full counting statistics, we need to look at the ordered configuration of particles where x 1 < • • • < x N and x i is the position of the i-th particle. The energy of such a configuration is given in Ch. 1 in Eq. (1.30).

The first observable that we focus on is the gap between i-th and (i + 1)-th particles, which is given by

g i = x i+1 -x i . (4.1)
The average value of g i does not depend on the index i and is easily calculated from the uniform equilibrium configuration of the jellium model (see Eq. (1.32) and Eq. (1.48))

⟨g⟩ ≃ 4α N . (4.2)
The value of the gap g i fluctuates around this average value, and our goal is to describe the distribution of these fluctuations. We expect that the distribution is the same for all gaps that are in the bulk of the system. We consider that a gap g i is in the bulk if 1 ≪ i ≪ N . For the sake of simplicity, we choose to study the gap between two particles in the middle of the system given by

g = x ⌊N/2⌋+1 -x ⌋N/2⌋ . ( 4.3)
A schematic depiction of the bulk gap between the two middle particles is shown in Fig. 4.1.

Studying the bulk gap is a natural extension of the work presented in Ref. [START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF], where

x 1

x N x N/2 x N/2+1 0 g Figure 4.1: A symbolic example of the ordered configuration of the 1dOCP. The particles are labeled from left to right so that

x 1 ≤ x 2 ≤ • • • ≤ x N .
The bulk gap g is given by the difference of positions of two middle particles g = x ⌊N/2⌋+1 -x ⌊N/2⌋ . the authors found large and typical fluctuations of the gap between the rightmost particle and its neighbor

g edge = x N -x N -1 . (4.4)
They showed that the typical fluctuations of the edge gap are of order O(1/N ) for large N , while the atypically large fluctuations are of order O(1). These two behaviors are summarised as follows [START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF] P gap,edge (g, N )

≈        N h α (g N ) g = O(1/N ) (typical) , e -N 2 ψ edge (g) g = O(1) (atypical) , (4.5)
where the scaling function h α (z) is given by

h α (z) = θ(z)A(α) ∞ -∞ dy(y + z -4α)e -(y+z-4α) 2 /2 F α (y) . ( 4

.6)

Here A(α) is a constant that depends on the interaction strength α and F α (x) is a function that appears rather regularly in the computation of several observables in the 1dOCP [START_REF] Baxter | Statistical mechanics of a one-dimensional Coulomb system with a uniform charge background[END_REF][START_REF] Dhar | Exact extremal statistics in the classical 1d Coulomb gas[END_REF][START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF]. The function F α (x) is the unique solution of the non-local differential equation [START_REF] Baxter | Statistical mechanics of a one-dimensional Coulomb system with a uniform charge background[END_REF][START_REF] Dhar | Exact extremal statistics in the classical 1d Coulomb gas[END_REF][START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF]]

dF α (x) dx = A(α)F α (x + 4α)e -x 2 2 , ( 4.7)
with the boundary conditions F α (x → ∞) → 1 and F α (x → -∞) → 0. This equation can be thought of as an eigenvalue equation, with A(α) as the unique eigenvalue for which there exists a solution that satisfies both boundary conditions. The asymptotic behaviors of the function F α (x) are given by

1 -F α (x) ∼ e -x 2 /2+o(x 2 ) , x → +∞ , F α (x) ∼ e -|x| 3 /(24α)+o(x 3 ) , x → -∞ . (4.8)
The full explicit expression for F α (x) is not known, however, it is possible to determine the solutions of Eq. (4.7) numerically. More specifically, one can use the shooting method [START_REF] Press | Numerical Recipes 3rd Edition: The Art of Scientific Computing[END_REF], which is a numerical method for solving boundary value problems. To find the solution of Eq. (4.7) we first guess the eigenvalue A(α) and numerically integrate Eq. (4.7). In this way we find the value of F α (x) at large x for the current guess A(α). We proceed with different guesses for the eigenvalue until we find the optimal solution for which the function F α (x) satisfies the boundary conditions given in Eq. (4.8). In (4.9)

The probability of the atypical fluctuations of the gap, for g = O(1), is described by the second line of Eq. (4.5) where the rate function ψ edge (g) is given by

ψ edge (g) = g 2 2 , g ≥ 0 , ( 4.10)
and is independent of the interaction strength α. By inserting Eq. (4.9) into the first line of Eq. (4.5), we find that the the tail of the distribution that describes the typical fluctuations of the edge gap matches with the large deviation form given in the second line of Eq. (4.5) and the rate function from Eq. (4.10). As we show in the following sections, both typical and atypical fluctuations of the gap get modified when we move from the edge of the system to the bulk. Most notably, the tails of the distribution describing large fluctuations are no longer Gaussian. Furthermore, typical and atypical fluctuations become dependent on the interaction strength α. Gaps between particles have also been analyzed in other strongly correlated systems. For example, there has been a lot of interest in the gaps between different eigenvalues of random matrices or equivalently between particles in log-gases. There are known results for the gap statistics, both in the bulk and near the edge [START_REF] Mehta | Random Matrices and the Statistical Theory of Spectra[END_REF][START_REF] Majumdar | How many eigenvalues of a Gaussian random matrix are positive?[END_REF][START_REF] Forrester | Fluctuation formula for complex random matrices[END_REF][START_REF] Witte | Joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles[END_REF][START_REF] Perret | Near-Extreme Eigenvalues and the First Gap of Hermitian Random Matrices[END_REF]. In this case it has been established long ago that the fluctuations of the appropriately scaled bulk gap are well approximated by the Wigners surmise [START_REF] Mehta | Random Matrices and the Statistical Theory of Spectra[END_REF], which describes the distribution of the gap in the simple case of two particles. Going beyond Dyson's log-gas, there are some recent results for the gap in the general Riesz gas, given in [START_REF] Santra | Gap statistics for confined particles with power-law interactions[END_REF], which indicate that the Wigner's surmise does not hold for other members of Riesz gas family.

N L -L N I = N κ I L N R x N L x N L+1 x N L +N I x N L +N I +1 x N L +N I x N L +N I +1 0 x 1 x N
The second observable that we study in this chapter, namely the full counting statistics, is defined as

N I = N l=1 I I (x l ), (4.11)
where I I (x l ) is an indicator function which takes the value 1 if the i-th particle is inside the interval I and zero otherwise. In Our goal in this chapter is to describe the fluctuations of N I around this value.

The FCS is a quantity with many possible applications and has thus been previously studied in many different contexts. For example:

• Quantum transport and quantum dots, where the FCS is connected to the transferred charge and electric current [START_REF] Levitov | Charge distribution in quantum shot noise[END_REF][START_REF] Levitov | Electron counting statistics and coherent states of electric current[END_REF][START_REF] Groth | Counting statistics of coherent population trapping in quantum dots[END_REF][START_REF] Gustavsson | Counting statistics of single electron transport in a quantum dot[END_REF].

• Spin chains, where FCS can be used to characterize quantum fluctuations see, for example, [START_REF] Ivanov | Characterizing correlations with full counting statistics: Classical Ising and quantum XY spin chains[END_REF][START_REF] Stéphan | Full counting statistics in the Haldane-Shastry chain[END_REF][START_REF] Groha | Full counting statistics in the transverse field Ising chain[END_REF][START_REF] Gamayun | Fredholm determinants, full counting statistics and loschmidt echo for domain wall profiles in one-dimensional free fermionic chains[END_REF].

• Ensembles of random matrices [START_REF] Forrester | Log-Gases and Random Matrices[END_REF][START_REF] Fogler | Probability of an eigenvalue number fluctuation in an interval of a random matrix spectrum[END_REF][START_REF] Costin | Gaussian fluctuation in random matrices[END_REF]. The FCS is especially interesting in this context since the eigenvalues of some random matrices can be mapped to noninteracting fermions in a harmonic trap [START_REF] Marino | Number statistics for β -ensembles of random matrices: Applications to trapped fermions at zero temperature[END_REF][START_REF] Marino | Phase transitions and edge scaling of number variance in gaussian random matrices[END_REF][START_REF] Dean | Noninteracting fermions in a trap and random matrix theory[END_REF]. Moreover, the variance of the full counting statistics in these systems is closely related to the entanglement entropy of the subsystem [-L, L] with the rest of the system at zero temperature [START_REF] Vicari | Entanglement and particle correlations of Fermi gases in harmonic traps[END_REF][START_REF] Eisler | Universality in the full counting statistics of trapped fermions[END_REF][START_REF] Calabrese | Random matrices and entanglement entropy of trapped Fermi gases[END_REF]. Since the FCS was proved to be a useful observable in fermionic systems, it has been computed for various systems of interacting and non-interacting fermions [START_REF] Lacroix-A-Chez-Toine | Rotating trapped fermions in two dimensions and the complex Ginibre ensemble: Exact results for the entanglement entropy and number variance[END_REF][START_REF] Smith | Noninteracting trapped fermions in double-well potentials: Inverted-parabola kernel[END_REF][START_REF] Smith | Counting statistics for noninteracting fermions in a d -dimensional potential[END_REF][START_REF] Smith | Full counting statistics for interacting trapped fermions[END_REF][START_REF] Smith | Counting statistics for non-interacting fermions in a rotating trap[END_REF][START_REF] Gouraud | Hole probability for non-interacting fermions in a d-dimensional trap[END_REF].

A quantity related to the full counting statistics, called the index, was already computed for the 1dOCP [START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF]. The index is defined as the number of particles located to the left of the origin, denoted by N -. It can be interpreted as a full counting statistics measuring the number of particles inside of an asymmetric interval [-∞, 0]. The authors of Ref. [START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF] found the complete description of the fluctuations around the average N-= N/2 summarized by [START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF]

P(N -, N ) ≈    4αf α (4α(N --N-)) |N --N-| = O(1) (typical), e -N 3 ψ index ( N - N ) |N --N-| = O(N ) (atypical) , ( 4.13)
where the rate function describing the large deviation is given by

ψ index (c) = 8α 2 3 |c -1/2| 3 , 0 < c < 1/2 . (4.14)
Instead, the typical fluctuations of the index are described by the following scaling function

f α (z) = F α (z + 2α) F α (-z + 2α) ∞ -∞ F α (z ′ + 2α) F α (-z ′ + 2α) dz ′ , (4.15)
where F α (x) is defined in Eq. (4.7). As we show, the distribution of the typical fluctuations of the FCS is given in terms of the same scaling function f α (z). Furthermore, when computing the typical fluctuations of the FCS, we closely follow the derivation given in Ref. [START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF]. The rest of this chapter is organized as follows. First, we investigate the probability distribution of the bulk gap. After obtaining the large deviations of order O(1), we continue by analyzing the typical fluctuations that are of order O(1/N ). In the second part of the chapter, we focus on the full counting statistics. We again derive the atypical fluctuations that are of order O(N ) as well as typical fluctuations of order O(1). We end the chapter by discussing the connection between the bulk gap and the FCS.

. Distribution of the bulk gap

. Large fluctuations

In this section, we use the saddle-point approximation, which allows us to obtain the expression describing the large fluctuations of the bulk gap g, defined in Eq. (4.3). We start with the probability distribution of the middle gap that is given by a N -fold integral The Heaviside theta functions θ(z) ensure that the positions of particles x i are ordered. The energy of the jellium model βE to the leading order in N reads

P gap,bulk (g, N ) = = ∞ -∞ dx 1 ... ∞ -∞ dx N e -βE[{x i }] δ(g -(x N/2+1 -x N/2 )) N i=2 θ(x j -x j-1 ) ∞ -∞ dx 1 ... ∞ -∞ dx N e -β E[{x i }] N i=2 θ(x j -x j-1 ) = = N N D N . ( 4
βE[{x i }] = N 2 2 N i=1 x 2 i -N α i̸ =j |x i -x j | = = N 3 1 2N N i=1 x i - 2α N (2i -N -1) 2 - 2α 2 3 N 3 = N 3 Ẽ[{x i }] , (4.17)
where we introduced Ẽ[{x i }] that is of order one. The expression in the denominator of Eq. (4.17) is proportional to the partition function of the jellium model given in Ch. 1 in Eq. (1.37). To the leading order in N it reads

D N ∼ e 2 3 α 2 N 3 . (4.18)
The numerator N N in Eq. (4.16) can be understood as a partition function of a jellium model under a constraint ensuring that the gap between two middle particles equals to g. Our strategy for evaluating N N is to apply the saddle-point approximation for large N . Thus, we need to find the configuration of the charges that minimizes the energy in Eq. (4.17) and at the same time fulfills the gap constraint. We approach this problem by first looking at the results from Monte-Carlo simulations shown in the right panel of Fig. 4.4. From the numerical data, we learn that the configuration that we are looking for has the following properties:

• The density has two disjoint supports that are symmetrical with respect to x = 0. This means that half of the particles are contained in the left and the other half in the right support.

• Each block of particles is composed of two parts. The "bulk" that consists of equidistantly spaced charges supported on [-a, -g/2] and [g/2, a], shown with blue and green color in the left panel of Fig. 4.4. At both edges of the gap, we see a δ-peak of intensity Y . It contains particles that are displaced due to the creation of the gap.

• The height of the density in each block is the same and has the value h.

By combining these observations we can make the following ansatz for the saddle-point density

ρ * (x) =        h + Y δ x + g 2 , -a ≤ x ≤ -g 2 , h + Y δ x -g 2 , g 2 ≤ x ≤ a . (4.19)
Additionally, we have an equation that fixes the number of particles in each block to N/2. Looking at the left block, shown in blue color in Fig. (4.4), we notice that we can write the number of particles in this block as a sum of the particles located in the δ-peak and those in the "bulk". Since we assume that the particles in the bulk are uniformly distributed between -a and -g/2 we can write

Y + h a - g 2 = 1 2 . ( 4.20)
Our goal is first to evaluate the energy Ẽ[{x i }] in Eq. (4.17) for the density given in Eq. (4. [START_REF] Thirring | Systems with negative specific heat[END_REF]). Then we find the optimal values of the parameters Y , a and h by minimizing the energy Ẽ[{x i }]. We find that it is the most convenient to evaluate the energy separately for the δ-peak and the uniformly distributed blocks. Due to the symmetry between the left and right parts of the density, we can evaluate the energy only for the charges in the left and multiply the result by two. In this way, the final result is calculated as

Ẽ * = 2( Ẽleft,bulk + Ẽleft,delta ) - 2α 2 3 . (4.21)
Here we compute Ẽleft,bulk and Ẽleft,delta by evaluating the first term in the second line of Eq. (4.17). The constant 2α 2 /3 is subtracted at the end. Let us start with the contribution from the left bulk. We know that the spacing between two particles is 1/(hN ). Consequently, the position of the i-th charge in the left block is

x i = -a + 1 hN i . (4.22)
Inserting this in the energy gives us

Ẽleft,bulk = 1 2N N/2-Y N i=1 i hN -a - 2α N (2i -N -1) 2 , ( 4.23)
which can be expressed by an integral over z = i/N in the large N limit Ẽleft,bulk ≃ 1 2

1/2-Y 0 1 h -4α z + 2α -a 2 dz . (4.24)
In the sum of Eq. (4.23), the index i runs from 1 to N/2 -Y N . Out of N/2 particles in the left support Y N are in the δ-peak, and the rest is in the uniform bulk. Next, we evaluate the contribution from Y N charges concentrated in the δ-peak located at x = -g/2. Their energy reads

Ẽleft,delta = 1 2N N/2 i=N/2-Y N - g 2 - 2α N (2i -N -1) 2 ≃ 1 2 1/2 1/2-Y - g 2 -4αz + 2α 2 dz . (4.25)
The final expression for the energy is given by adding both contributions according to Eq. (4.21)

Ẽ * = = 1/2-Y 0 1 h -4α z + 2α -a 2 dz + 1/2 1/2-Y - g 2 -4αz + 2α 2 dz - 2α 2 3 . (4.26)
We can reduce the number of parameters from three to two by using the relation (4. Solving these equations leads us to a simple solution

a * = 2α , h * = 1 4α , Y * = g 8α , ( 4.28)
where we used the relation (4.20) to find the value of Y * . In the next step, we evaluate Ẽ * at the saddle-point by inserting the values a * , h * , and Y * . In this case, this step is simple since the only non-vanishing contributions come from the Ẽ * left, delta and the constant term. Finally, the energy Ẽ * reads

Ẽ * = g 3 96α - 2α 2 3 . (4.29)
Knowing this result, we can return to Eq. (4.16) and evaluate P gap,bulk (g, N ) for large N . The large deviation form reads

P gap,bulk (g, N ) ∼ e -N 3 ψgap(g) , ψ gap (g) = g 3 96α . ( 4

.30)

Comparing this expression with the equivalent for the edge gap given in Eqs. (4.5) and (4.10), we see that, unlike the rate function for the edge gap, the rate function of the bulk gap depends on the interaction strength α. Furthermore, we find that the probability for atypical fluctuations decays faster than Gaussian in the bulk gap case.

. Typical fluctuations

After obtaining the large deviations form for the bulk gap, we proceed with analyzing the typical fluctuations of order O(1/N ) around the mean bulk gap ⟨g⟩ = 4α/N . Fluctuations of order O(1/N ) are not accessible by the Coulomb gas technique that relies on the saddle-point approximation of the integrals in Eq. (4.16). Indeed, when the gap is of order O(1), the saddle-point of the numerator changes with respect to the denominator, which results in obtaining the rate function. However, when the gap is of order O(1/N ), the saddle-point of the numerator stays the same compared to the unperturbed jellium. Hence, in order to describe the typical fluctuations, we need to evaluate more precisely the N -fold integral in the numerator.

We again start with the formal expression for the probability distribution of the gap

P gap,bulk (g, N ) = = N ! Z N ∞ -∞ dx 1 . . . ∞ -∞ dx N e -βE[{x i }] δ(g -(x N/2+1 -x N/2 )) N i=2 θ(x j -x j-1 ) , (4.31)
where

βE[{x i }] = N 3 1 2N N i=1 x i - 2α N (2i -N -1) 2 -C N (α) . (4.32)
The term C N (α) is given in Eq. (1.31) and we recall it here

C N (α) = 2α 2 N i=1 (2i -N -1) 2 = 2α 2 3 N 3 - 2 3 α 2 N . ( 4.33)
We slightly reformulated the expression from Eq. (4.16) by writing Z N = N !D N , where Z N is the partition function of the jellium model from Eq. (1.37). The factor N ! appears because the energy of the 1dOCP is symmetric under permutations of x i 's. Therefore we can only consider the ordered configuration and multiply the result by N !. Following the steps presented in Ref. [START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF] we introduce new variables

ϵ i = N x i -2α(2i -N -1) . (4.34)
Looking at the energy of the jellium in Eq. (4.17), we notice that in terms of these new variables, the energy simplifies to βE[{x i }] = 1 2 i ϵ 2 i . The expression for the gap distribution can therefore be written as

P gap,bulk (g, N ) = = N 1-N N ! e C N (α) Z N ∞ -∞ dϵ 1 . . . ∞ -∞ dϵ N e -1 2 N k=1 ϵ 2 k δ(ϵ N/2+1 -ϵ N/2 -gN + 4α) × N j=2 θ(ϵ j -ϵ j-1 + 4α) . (4.35)
Thus, in terms of the variables ϵ i 's, the gas becomes effectively short-ranged. The next step consists of separating the integrals over the two middle particles ϵ N/2 and ϵ N/2+1 , all particles to the left of ϵ N/2 (the left bulk), and all particles to the right of The inset shows the same data, however, we plot it with a logarithmic scale on the H α (z) axis.

ϵ N/2+1 (the right bulk). The integrals over the left and right bulks can be expressed in terms of two functions already introduced in Ref. [START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF]

D α (x, n) = x -∞ dϵ n ϵn+4α -∞ dϵ n-1 ... ϵ 2 +4α -∞ dϵ 1 e -1 2 n k=1 ϵ 2 k , (4.36) E α (x, n) = ∞ x dϵ 1 ∞ ϵ 1 -4α dϵ 2 ... ∞ ϵ n-1 -4α dϵ n e -1 2 n k=1 ϵ 2 k . (4.37)
They can be interpreted as partition functions of a gas confined in the semi-infinite domain. The function D α (x, n) describes the gas confined to (-∞, x], symmetrically E α (x, n) corresponds to the system confined to [x, +∞). The two functions are related by

E α (-x, n) = D α (x, n) . (4.38)
Returning to the distribution of the gap, we can now formulate it as

P gap,bulk (g, N ) = N 1-N N ! e C N (α) Z N ∞ -∞ dϵ N/2+1 ∞ -∞ dϵ N/2 e -1 2 (ϵ 2 N/2+1 +ϵ 2 N/2 ) ×δ(ϵ N/2+1 -ϵ N/2 -gN + 4α)θ(ϵ N/2+1 -ϵ N/2 + 4α) (4.39) ×D α (ϵ N/2 + 4α, N/2 -1) × E α (ϵ N/2+1 -4α, N/2 -1) .
We separated the integrals over ϵ N/2 and ϵ N/2+1 and wrote the rest of the integrals in terms of D α (x, n) and E α (x, n). We can further simplify the expression by using the symmetry relation in Eq. (4.38) and changing the variable ϵ N/2+1 → -ϵ N/2+1 . This yields

P gap,bulk (g, N ) = N 1-N N ! e C N (α) Z N ∞ -∞ dϵ N/2+1 ∞ -∞ dϵ N/2 e -1 2 (ϵ 2 N/2+1 +ϵ 2 N/2 ) ×δ(-ϵ N/2+1 -ϵ N/2 -gN + 4α)θ(-ϵ N/2+1 -ϵ N/2 + 4α) (4.40) ×D α (ϵ N/2 + 4α, N/2 -1)D α (ϵ N/2+1 + 4α, N/2 -1) .
In order to continue with the computation, we now consider the scaling limit N → ∞, g → 0 where we keep the product g N fixed. Following the idea from Ref. [START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF] we introduce the ratio

F α (x, M ) = D α (x, M ) D α (∞, M ) . ( 4

.41)

The denominator D α (∞, M ) is by definition in Eq. (4.36) proportional to the partition function Z M of the 1dOCP gas with M particles. More precisely, it is easy to see that

Z M = M !e C M (α) M M D α (∞, M ) . ( 4

.42)

The ratio F α (x, M ) in Eq. (4.41) then represents the cumulative probability that the position of the rightmost particle is less than x. In particular, F α (x → +∞, M ) = 1 and

F α (x → -∞, M ) = 0.
In terms of this function F α (x, M ), the gap distribution in (4.40) becomes

P gap,bulk (g, N ) = N D 2 α (∞, N/2 -1) D α (∞, N ) ∞ -∞ dϵ N/2+1 ∞ -∞ dϵ N/2 e -1 2 (ϵ 2 N/2+1 +ϵ 2 N/2 ) ×δ(-ϵ N/2+1 -ϵ N/2 -gN + 4α)θ(-ϵ N/2+1 -ϵ N/2 + 4α) ×F α (ϵ N/2 + 4α, N/2 -1)F α (ϵ N/2+1 + 4α, N/2 -1) , (4.43)
where we used the relation (4.42). Our goal is to evaluate the expression in Eq. (4.43) in the large N limit. First, we want to estimate the large N behavior of the prefactor in Eq. (4.43). In order to achieve this, we need to estimate the ratio in Eq. (4.41) for large N . Based on the observation that D α (∞, M ) is proportional to the partition function of a short-ranged gas, we expect that its free energyln D α (∞, M ) is extensive in M for large M . To the leading order in M we thus expect

D α (∞, M ) ∼ B[A(α)] -M , ( 4.44)
where B is a constant independent of N . We note thatln A(α) is a free energy per particle of the short-ranged gas. Based on Eq. (4.44) the large N behavior of the prefactor from Eq. (4.43) is given by lim

N →∞ N D 2 α (∞, N/2 -1) D α (∞, N ) ≈ N [A(α)] 2 B . (4.45)
To make further progress, we want to find the large M form of the function F α (x, M ). By taking a derivative with respect to x of Eq. (4.41), we find

dF α (x, M ) dx = D α (∞, M -1) D α (∞, M ) e -x 2 /2 F α (x + 4α, M -1) ≈ ≈ A(α)e -x 2 /2 F α (x + 4α, M -1) . (4.46)
We expect that the function F α (x, M ) converges to a M -independent limiting form F α (x) for large M . In this case, Eq. (4.46) becomes equivalent to the non-local eigenvalue equation in Eq. (4.7). The asymptotic behavior of F α (x) is therefore given in Eq. (4.8).

Setting further x = ϵ N/2+1 and y = ϵ N/2 , we then get for large N

P gap,bulk (g, N ) ≈ N B[A(α)] 2 ∞ -∞ dx ∞ -∞ dye -1 2 (x 2 +y 2 ) × δ(-x -y -g N + 4α)θ(-x -y + 4α)F α (y + 4α)F α (x + 4α) . (4.47)
This clearly has the scaling form

P bulk,gap (g, N ) ≈ N H α (gN ) , ( 4.48)
with the scaling function

H α (z) = B[A(α)] 2 ∞ -∞ dx ∞ -∞ dye -1 2 (x 2 +y 2 ) ×δ(-x -y -z + 4α)θ(-x -y + 4α)F α (y + 4α)F α (x + 4α) . (4.49)
We can further simplify the expression by removing the theta function since it is automatically satisfied when the delta function δ(-x -y -z + 4α) is non-zero. Integrating over x yields

H α (z) = θ(z)B A 2 (α) ∞ -∞ dyF α (y + 4α)F α (8α -y -z)e -y 2 /2-(y+z-4α) 2 /2 , ( 4.50)
which is our final result describing the typical fluctuations of the middle gap. Let us remark that the scaling form H α (z) is non-zero only for z > 0, which is expected since we are dealing with the ordered particles, and therefore the gap can be only positive. We tested this prediction by comparing it to the Monte-Carlo simulation, and we found very good agreement, as shown in Fig. 4.5.

We can get a better intuition about the scaling function if we look at the asymptotic behavior when z → ∞ and z → 0. When z → 0, the function approaches a constant, and when z → ∞, the scaling function decays. As explained at the end of Sec. 3.2 of Ref. [START_REF] Flack | Gap probability and full counting statistics in the one dimensional one-component plasma[END_REF], we can derive the tail of the function for z → ∞. It is given by

H α (z) ∼ e -z 3 96α +o(z 3 ) . (4.51)
By comparing this result with the large deviation form given in Eq. (4.30) we notice that they perfectly match. With this result, we conclude the section about the middle-gap statistics in the 1dOCP.

. Full counting statistics

As we announced in the beginning of this chapter, the second observable that we investigate is the full counting statistics defined by the following equation

N I = N l=1 I I (x l ) . (4.52)
where I I (x l ) is an indicator function which takes the value 1 if the i-th particle is inside the interval I and zero otherwise. We will consider only intervals symmetric around x = 0 that are deep in the bulk of the system, |L| < 2α. Like in the previous section, we first compute the large deviation form that describes the atypically large fluctuations. In the second part of this section, we derive the scaling form that describes the typical fluctuations of the full counting statistics.

. Large fluctuations

The starting point of the derivation is the formal expression for the probability distribution of the full counting statistics given by

P FCS (N I , N ) = = ∞ -∞ dx 1 . . . ∞ -∞ dx N e -βE[{x i }] N i=2 θ(x j -x j+1 )δ N I -N l=1 I I (x l ) ∞ -∞ dx 1 . . . ∞ -∞ dx N e -βE[{x i }] N i=2 θ(x j -x j-1 ) = ÑN D N , ( 4.53)
with the energy βE[{x i }] given in Eq. (4.17). The denominator D N in Eq. (4.53) is again proportional to the partition function of the jellium model and reads in the limit of large N

D N ∼ e 2 3 α 2 N 3 . ( 4.54)
The numerator, however, differs from the bulk gap case. Our goal is to evaluate the numerator ÑN in the limit of large N , which we do by using the saddle-point method. In order to find the correct saddle-point density, we need to handle separately two cases:

• The overpopulated regime where N I > NI or κ I > κI ,

• The underpopulated regime where N I < NI or κ I < κI , where we used N I = κ I N .

Overpopulated regime

Let

ρ * I (x) =        h, -a ≤ x ≤ -b, h + Bδ(x + L) + Bδ(x -L), -L ≤ x ≤ L , h, b ≤ x ≤ a , (4.55)
with four unknown parameters, B, b, a, and h. See the left panel of Fig. 4.7 for a schematic depiction of the ansatz. We determine the unknown parameters by inserting the density ρ * I (x) in the 1dOCP energy in Eq. (4.17) and solving the saddle-point equations

∂ Ẽ * ∂a = 0, ∂ Ẽ * ∂b = 0, ∂ Ẽ * ∂B = 0, ∂ Ẽ * ∂h = 0 . (4.56)
By solving the equations, we obtain the optimal values of the parameters given by

a * = 2α, b * = 2ακ I , B * = 2ακ I -L 4α , h * = 1 4α . (4.57)
After obtaining ρ * I (x), we need to evaluate the saddle-point energy Ẽ * . Luckily, this is not a hard task since the only non-zero contributions come from the two δ-peaks. Due to the symmetry, we can conclude that the two contributions are the same and only compute the contribution from one peak. We chose the left one and insert x i = -L for all x i in the first term of the second line in Eq. (4.17). We know that in the left block (blue color in Fig. 4.7), there are N 2 (1 -κ i ) charges. Consequently, the sum over the particles in the left peak starts with i = N 2 (1 -κ I ). Since the charges accumulated in the peak originate from the depleted area [-b, -L], we can determine the number of particles in the peak by calculating the number of particles in the interval [-b, -L] given the uniform density. Finally, we can conclude that the indices of the particles in the peak run from i = (1 -κ I )N/2 to i = (1 -L/(2α))N/2. Taking this into account, we obtain the following saddle-point energy

Ẽ * = 1 N (1-L/(2α))N/2 i=(1-κ I )N/2 -L - 2α N (2i -N -1) 2 - 2α 2 3 . (4.58)
This sum can be replaced by an integral in the large N limit which leads to

Ẽ * = (1-L/(2α))/2 (1-κ I )/2 (-L -4α z + 2α) 2 dz - 2α 2 3 = (2ακ I -L) 3 12α - 2α 2 3 . (4.59)
Now we can insert the saddle-point energy Ẽ * into the numerator ÑN to find the dominant contribution to the integrals in Eq. (4.53). Together with the denominator D N given in Eq. (4.54) we get

P FCS (N I = κ I N, N ) ∼ e -N 3 Ẽ * e 2 3 α 2 N 3 = e -N 3 ψ(κ I ) , κ I > κ = L 2α , (4.60)
where the rate function ψ(z) is given by

ψ(z) = 2 3 α 2 z - L 2α 3 , for z > L/(2α
(x) x x κ I > κ κ I < κ -2α 2α -2α 2α -L L -b b -L L -c c

Underpopulated regime

The system is in the underpopulated regime when κ I < κI . This means that the number of particles in the interval [-L, L] is smaller than the average value. The Monte-Carlo simulations presented in the right panel of Fig. 4.6 indicate that the excess charge from the interval gathers just on the outside of the edges of the interval, creating two δ-peaks located at -L and L. Consequently, the support of the charges in the middle interval shrinks to [-c, c]. The remaining charges seem to stay unperturbed. These observations lead us to the following ansatz for the saddle-point density

ρ * I (x) =        h + Cδ(x + L) -a ≤ x ≤ -L, h, -c ≤ x ≤ +c , h + Cδ(x -L), L ≤ x ≤ a . (4.62)
A schematic drawing of the ansatz is presented in the right panel of Fig. 4.7. We determine all four unknown parameters h, c, a and C by evaluating the energy in Eq. (4.17) for the density ρ * I (x) and solving the saddle-point equations

∂ Ẽ * ∂a = 0, ∂ Ẽ * ∂c = 0, ∂ Ẽ * ∂C = 0, ∂ Ẽ * ∂h = 0 . (4.63)
Solving these equations gives us the saddle-point values of the parameters given by

a * = 2α, c * = 2ακ I , h * = 1 4α , C * = L -2ακ I 4α . (4.64)
Again, the only non-vanishing contributions to the saddle-point energy come from the two δ-peaks. Since they are symmetric, we focus on the left peak, which contains particles

with indices from i = N 4α (2α -L) to N 4α (2α -L) + CN . This results in Ẽ * = 1 N N 4α (2α-L)+C * N i= N 4α (2α-L) -L - 2α N (2i -N -1) 2 - 2 3 α 2 , (4.65)
which can be written in terms of an integral for large

N Ẽ * = 1 4α (2α-L)+C * 1 4α (2α-L) (-L -4αz + 2α) 2 dz - 2 3 α 2 = 1 12α (L -2α κ I ) 3 - 2 3 α 2 . (4.66)
This saddle-point energy allows us to obtain the large deviation form for the FCS in the underpopulated regime as

P FCS (N I = κ I N, N ) ∼ e -N 3 Ẽ * e 2 3 α 2 N 3 = e -N 3 ψ(κ I ) , κ I < κI = L 2α , (4.67)
where the rate function ψ(z) is given by

ψ(z) = 2 3 α 2 L 2α -z 3 , for z < L/(2α) . (4.68)
Comparing this expression with the result for the overpopulated regime in Eqs. (4.60) and (4.61) we see that they can be combined in a single expression given by

P FCS (N I = κ I N, N ) ∼ e -N 3 ψ(κ I ) where ψ(z) = 2 3 α 2 z - L 2α 3 . (4.69)
As expected, the minimum of the rate function occurs at z = L 2α , which coincides with the mean value of the FCS. An interesting feature of the rate function is its non-Gaussian behavior around the minimum. In fact, the third derivative of ψ(z) is discontinuous at z = L/(2α) indicating a third-order phase transition. The jump of the third derivative is given by

ψ ′′′ O (z = l 2α ) -ψ ′′′ U (z = l 2α ) = 8α 2 , (4.70)
where we denoted by ψ O (z) the rate function of the overpopulated regime given in Eq. (4.61) and by ψ U (z) the rate function for the underpopulated regime from Eq. (4.68). A similar third-order phase transition has been observed in the case of the index in the twodimensional Coulomb gas [START_REF] Allez | Index distribution of the Ginibre ensemble[END_REF].

. Typical fluctuations

We expect the typical fluctuations around the mean to be of order O(1), meaning that N κ I -N κi ∼ O(1). We can not access these fluctuations by approximating the multiple integrals in Eq. (4.53) by the saddle-point method as discussed in the previous section about the bulk gap. To find the typical fluctuations, we resolve the integrals more precisely.

To begin with, let us reevaluate the situation. By looking at the figure Fig. 4.3, we realize that we can divide the charges in the following way. We have N L particles positioned left of the interval [-L, L] (in blue color), N R particles on the right side of the middle interval (in green color), and there are N I particles inside of the interval (in red color). All three observables N L , N R , and N I are random variables, however, they are not independent. Indeed, we know that

N = N R + N I + N L . (4.71)
We can first find the joint distribution of N R and N L given that the total number of particles is N , denoted by P joint (N L , N R |N ). We can then obtain the marginal distribution P F CS (N I , N ) by summing over

N L P F CS (N I , N ) = N N L =0 P joint (N L , N R = N -N L -N I |N ) . (4.72)
We start by expressing the joint distribution P F CS (N I , N ) in terms of the multiple integrals over x i as

P joint (N L , N R |N ) = N ! Z N ∞ -∞ N k=1 dx k e -β E[{x i }] N k=2 θ(x k -x k-1 ) × θ(-L -x N L )θ(L + x N L +1 )θ(L -x N L +N I )θ(x N L +N I +1 -L) , (4.73)
where the product of theta functions enforces the ordering of the particles. The overall factor N ! counts all possible permutations, and Z N is the partition function of the 1dOCP given in Eq. (1.37). We denote by βE[{x i }] the energy of the jellium model, given, for example, in Eq. (4.17). Importantly, the last four theta functions in Eq. (4.73) make sure that all the N L particles are positioned on the left of -L, all N I particles are inside of the interval [-L, L] and the remaining N R particles are on the right of L. We order the integrals in Eq. (4.73) by splitting them into three parts in the following way These integrals, valid for any N , are rather difficult to solve. However, we show that for large N the two variables N L and N R become effectively uncorrelated. This simplification allows us to find the solution for the integrals and give an expression for the typical fluctuations of the FCS. We support our claim that N L and N R are uncorrelated by numerical simulations. We show that the covariance of the two variables, given by

P joint (N L , N R |N ) = N ! Z N I 1 I 2 I 3 = N ! Z N   ∞ -∞ N L k=1 dx k e -βE[{x i }] N L k=2 θ(x k -x k-1 )θ(-L -x N L )   ×   ∞ -∞ N L +N I k=N L +1 dx k e -βE[{x i }] N L +N I k=N L +2 θ(x k -x k-1 )θ(x N L +1 + L)θ(L -x N L +N I )   ×   ∞ -∞ N k=N L +N I +1 dx k e -βE[{x i }] N k=N L +N I +2 θ(x k -x k-1 )θ(x N L +N I +1 -L)   . ( 4
C N L ,N R (N ) = N L N R -NL NR , (4.75)
decreases exponentially when we increase N . These results are shown in Fig. 4.8. If N L and N R are independent, we can approximate their joint distribution as

P joint (N L , N R |N ) ≈ P(N L |N ) P(N R |N ) , ( 4.76)
where P(N L |N ) and P(N R |N ) are the distributions of the number of particles to the left of -L and to the right of +L respectively. Substituting this approximation, valid for large N , in Eq. ( 4.72), we can express the marginal distribution of N I as

P F CS (N I , N ) ≈ N N L =0 P(N L |N ) P(N R = N -N L -N I |N ) . (4.77)
By symmetry, one would expect that the sum will be dominated by configurations where

N L = N R = (N -N I )/2
. Hence, we can approximate the marginal distribution of N I by where C is a constant entropy factor, independent of N I . Thus, we can obtain the expression for the typical fluctuations of N I , computing only the distribution of the number of particles left of -L. This problem is very similar to the recently solved problem of the index in Ref. [START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF], where they compute typical and atypical fluctuations of the number of particles left of the origin (x = 0). See also Ref. [START_REF] Rojas | Universal behavior of the full particle statistics of one-dimensional Coulomb gases with an arbitrary external potential[END_REF] for similar derivation. To answer our question about the probability of N L particles being left of -L, we need to follow the same steps with minor adjustments. The final result reads

P F CS (N I , N ) ≈ C P N L = N -N I 2 N P N R = N -N I 2 N = C P 2 N L = N -N I 2 N , ( 4
P(N L , N ) ≈ 4αf α (4α(N L -NL )) where NL = 2α -L 4α N , (4.79)
where f α (z) is the same as the scaling function describing the typical fluctuation of the index introduced in Eq. (4.15). After plugging in this result into Eq. ( 4.78) we arrive to the final expression for the typical fluctuations of the FCS that have the following scaling form

P F CS (N I , N ) ∼ 2αU α (2α(N I -NI )) where NI = L 2α N , (4.80)
with the normalized scaling function

U α (z) = F 2 α (2α + z)F 2 α (2α -z) ∞ -∞ F 2 α (2α + z ′ )F 2 α (2α -z ′ ) dz ′ . (4.81)
We check this result for the typical fluctuations by comparing it to numerical data obtained from Monte-Carlo simulations. We can see very good agreement between both, as shown in Fig. 4.9. Let us stress again that our result is valid only in the bulk of the system, meaning that the edges of the interval [-L, L] are far away from -2α and 2α, which means that N L = O(N ) and N R = O(N ). We can check that the tails of the scaling function given in Eq. (4.81) match the large deviation tails computed before and given in Eq. (4.69). Using the asymptotic behavior of the function F α (z) given in Eq. (4.8) we can easily see that for large z the scaling function U α (z) displays non-Gaussian behavior given by

U α (z) ≈ e -|z| 3 12α . (4.82)
A striking feature of our result for the fluctuations of the FCS given in Eq. (4.80) is that they are of order O(1) independent of L and N , as long as L is in the bulk and N is large. This differs from the result describing the variance of the FCS in the log-gas, where the variance of the typical fluctuations scales as log NI [START_REF] Mehta | Statistical theory of the energy levels of complex systems. IV[END_REF][START_REF] Fogler | Probability of an eigenvalue number fluctuation in an interval of a random matrix spectrum[END_REF][START_REF] Marino | Number statistics for β -ensembles of random matrices: Applications to trapped fermions at zero temperature[END_REF][START_REF] Marino | Phase transitions and edge scaling of number variance in gaussian random matrices[END_REF][START_REF] Costin | Gaussian fluctuation in random matrices[END_REF]. Moreover, in the non-interacting case, the FCS has a Poisson statistics, and its variance grows linearly with NI . Comparing all three cases, the 1dOCP, the log-gas, and the non-interacting case, we notice that compared to the non-interacting case, the fluctuations of the number statistics in both the 1dOCP and the log-gas are suppressed. Such systems where the fluctuations of the number of particles in a given interval are suppressed are known as the hyperuniform gases [START_REF] Torquato | Hyperuniform states of matter[END_REF]. Furthermore, we can conclude that the fluctuations of the FCS in the jellium model are more strongly suppressed than in the log-gas. This is, in a way, not surprising since the strength of the interactions in the jellium model decays slower with the interparticle distance.

. Connection between the bulk gap and the FCS

Before concluding the chapter, let us show that the gap and the full counting statistics are connected in the limit of large N . The relation is not obvious at first glance since the full counting statistics is defined as a number of particles in a fixed interval [-L, L] and the bulk gap is defined with respect to two specific particles, x N/2+1 and x N/2 . We can express the probability that no particle is inside of the interval as

P F CS (N I = 0, N ) = N N L =0 Prob. [x N L ≤ -L, g N L ≥ -L -x N L + 2L] . (4.83)
If there are N L particles left of -L, the interval [-L, L] is empty when the x N L -th particle is positioned before -L and the gap to the next particle is bigger than the length of the interval 2L plus the distance from the particle to -L. Summing over all possible N L gives us the final probability of the empty interval. For large N and big gaps, the relation between the gap and the FCS given in Eq. (4.83) is simplified. In this case, the sum is dominated by the saddle-point configuration, with N/2 particles left of -L and N/2 particles right of L. The excess charge from the interval [-L, L] is gathered at the edges at -L and L. Such a configuration is shown in the right panel of Fig. 4.7, however, if κ I = 0, the red block is not present. In this case, the probability of the middle gap being 2L is actually equivalent to the probability of the FCS with N I = 0 which implies the following relation

P F CS (N I = 0, N ) = P gap,bulk (g = 2L, N ) . (4.84)
We can check this relation by examining the equation Eq. (4.69) describing the large deviations of the FCS and inserting κ I = 0. Comparing the result to Eq. (4.30) for g = 2L, we indeed obtain the same expression.

. Conclusion of Chapter 4

In this chapter, we derived the expressions for the typical and atypical fluctuations of two variables: the bulk gap and the full counting statistics. For the bulk gap, the atypically large fluctuations of order O( 1 Especially interesting is our finding that the gap between particles in the bulk does not display the same behavior as the gap near the edge computed in Ref. [START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF]. For the gap between the edge particles, the authors found that the distribution has Gaussian tails given by ∼ e -N 2 g 2 /2 , while we found that the probability for large gaps in the bulk decays faster as ∼ e -N 3 g 3 /(96α) . Furthermore, the distribution becomes dependent on the interaction strength α. We understand these differences as a consequence of strong long-range interactions that have more effect in the bulk than at the edge of the system. Moreover, our result for the distribution of the bulk gap, given in Eqs. (4.48) and Eq. (4.51), confirms that the Wigner's surmise [START_REF] Mehta | Random Matrices and the Statistical Theory of Spectra[END_REF] does not hold for general Riesz gas. The Wigner's surmise would predict a gap distribution with a Gaussian tail, which is not in agreement with our exact formula.

Our results for the distribution of the FCS show an interesting third-order phase transition when we approach the critical value NI = L/(2α). This type of transition is commonly observed in Coulomb gases [START_REF] Majumdar | Top eigenvalue of a random matrix: large deviations and third order phase transition[END_REF][START_REF] Cunden | Universality of the weak pushedto-pulled transition in systems with repulsive interactions[END_REF][START_REF] Cunden | Universality of the third-order phase transition in the constrained Coulomb gas[END_REF]. Another interesting observation can be made upon examining the result for the typical fluctuations of the full counting statistics given in Eq. (4.80). In fact, the fluctuations are of order O(1) and do not depend on the size of the interval given by L or the number of particles in the system. This means that the one-dimensional jellium model belongs to the group of hyperuniform systems. In such systems, the variance of the number of particles inside of a spherical observational window with radius R grows slower than the volume of the sphere for large R [START_REF] Torquato | Hyperuniform states of matter[END_REF]. This is in agreement with recent results published in Ref. [START_REF] Boursier | Optimal local laws and CLT for the circular Riesz gas[END_REF], where the authors show that all one-dimensional Riesz gases with long-range interactions (s < 1) are hyperuniform.

Chapter 5

Truncated linear statistics

In this chapter, we explore the probability distribution of the truncated linear statistics (TLS). It is given by the sum of N ′ rightmost particles where we set N ′ = κN with 0 ≤ κ ≤ 1. The expression for the rescaled truncated linear statistics reads

s = 1 N N i=N -N ′ +1 x i = 1 N N i=N (1-κ)+1
x i , (5.1)

where the x i 's denote the positions of particles in the ordered configuration where

x 1 < • • • < x N .
The truncated linear statistics is an interesting observable since it interpolates between two extremes. If we set N ′ = N , the sum includes all particles and the TLS coincides with the full linear statistics. We explored the general full linear statistics in Ch. 1, and we focused on the choice f (x) = x in the example in section 3.3.1. We found that the large deviation form is Gaussian and is given by (see Eq. (3.35))

P(s, N ) ∼ e -N 3 s 2 2 , (5.2)
with the variance Var[s] = 1/N 3 . On the other hand, when N ′ = 1, we are dealing with the position of the rightmost particle. The full probability distribution of the position of the edge particle, denoted by x N = x max , was recently computed in Ref. [START_REF] Dhar | Exact extremal statistics in the classical 1d Coulomb gas[END_REF][START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF]. The authors found that the typical fluctuations of x max around its average position at x = 2α are of order O(1/N ). They are described by the following scaling form Prob.(x max = w, N ) ≃ N f α (N (w -2α)) , (5.3) where the scaling function f α (x) has asymmetric tails given by

f α (x) ≃ exp [-|x| 3 /(24α) + O(x 2 )] , as x → -∞ exp [-x 2 /2 + O(x)]
, as x → +∞ .

(5.4)

The atypically large fluctuations are of order x max -2α = O(1). The left and right large deviation forms are also not symmetric. The full probability distribution is given by

Prob.(x max = w, N ) ≃        e -N 3 Φ -(w)+O(N 2 ) , 0 < 2α -w = O(1) N f α (N (w -2α)) , |w -2α| = O(1/N ) e -N 2 Φ + (w)+O(N ) , 0 < w -2α = O(1) .
(

The rate function describing the left large deviation tail is given by [START_REF] Dhar | Exact extremal statistics in the classical 1d Coulomb gas[END_REF][START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF] Φ

-(w) =            (2α -w) 3 24α , -2α ≤ w ≤ 2α w 2 2 + 2 3 α 2 , w ≤ -2α , (5.6)
while the right one is [START_REF] Dhar | Exact extremal statistics in the classical 1d Coulomb gas[END_REF][START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF] Φ

+ (w) = (w -2α) 2 2 , w ≥ 2α . ( 5.7)
The three regimes are separated by discontinuities in the third derivative of the large deviation function at w = ±2α. Comparing the distributions of the full linear statistics for f (x) = x (see Eq. (5.2)) and the distribution of the rightmost particle (see Eq. (5.5)), we notice that the distribution of the full linear statistics admits a simple Gaussian form, however, the probability distribution of x max is rather complex. In this chapter, we aim to explore the transition from one to the other by considering the full range of the parameter κ, from κ = 0 to κ = 1.

In the existing literature, the truncated linear statistics was already investigated in the context of the Wishart-Laguerre ensemble of random matrices. The probability distribution of the truncated linear statistics was calculated for the arbitrary monotonic test function f (x) [START_REF] Grabsch | Truncated linear statistics associated with the top eigenvalues of random matrices[END_REF]. Very recently, the results were extended to non-monotonic functions as well [START_REF] Grabsch | General truncated linear statistics for the top eigenvalues of random matrices[END_REF]. In both cases, the authors found interesting phases and phase transitions in the κ-s plane. The truncated linear statistics can also be viewed as a special case of the linear statistics calculated for the eigenvalues of the "thinned random matrix ensembles" [START_REF] Bohigas | Randomly incomplete spectra and intermediate statistics[END_REF][START_REF] Charlier | Thinning and conditioning of the circular unitary ensemble[END_REF][START_REF] Berggren | Mesoscopic Fluctuations for the Thinned Circular Unitary Ensemble[END_REF]. In these articles, the authors study linear statistics of the eigenvalues of random matrices, but with a certain probability, they remove elements from the sum. A similar problem was also studied in the context of chaotic quantum dots in Ref. [START_REF] Grabsch | Truncated Linear Statistics Associated with the Eigenvalues of Random Matrices II. Partial Sums over Proper Time Delays for Chaotic Quantum Dots[END_REF]. Interestingly, the TLS also appears in the computation of the ground-state energy of non-interacting trapped fermions in a random potential [START_REF] Krajenbrink | Linear statistics and pushed Coulomb gas at the edge of β-random matrices: Four paths to large deviations[END_REF][START_REF] Schawe | Ground-state energy of noninteracting fermions with a random energy spectrum[END_REF].

In this chapter, we use the Coulomb gas method to investigate atypically large fluctuations of the TLS, as defined in Eq. (5.1), around the average value s, which is given by s

= 1 N N i=N -N ′ +1 x * i ≃ 2α N 2 N i=N -N ′ +1 (2i -N -1) = 2ακ(1 -κ) . ( 5.8)
Here the x * i 's denote the average positions of the particles in the equilibrium configuration of the jellium model (see Eq. (1.32)). We show that the probability distribution of the TLS admits a large deviation from e -N 3 ϕκ(s) and we provide the analytical expression for the rate function ϕ κ (s). Note that the typical fluctuation of the TLS are instead given by the following scaling from

P N,κ (s) ≃ N 3/2 f G (s -s)N 3/2 , ( 5.9)
with a Gaussian scaling function f G (z) given by For the derivation of the typical fluctuations, see Appendix B of Ref. [START_REF] Flack | Truncated linear statistics in the one dimensional one-component plasma[END_REF]. We organize the remainder of this chapter in the following way. First, we identify configurations that minimize the energy of the constrained gas. This leads us to a very rich phase diagram with five different phases, presented in Fig. 5.1. Next, we find the expression for the rate function, which is given by a different functional form in each of the phases. We also analyze the limiting behaviors at κ → 1 and κ → 0 and compare them to the known results for the full linear statistics and the distribution of the rightmost particle. In the last section, we describe the importance sampling algorithm that we use to numerically verify the expression for the rate function of the TLS.

f G (z) = 1 √ 2πκ e -z 2 2κ . ( 5

. Minimum energy configurations

Once again, we start the derivation by writing the formal expression for the probability distribution of the truncated linear statistics

P N,κ (s) = N ! Z N dx 1 • • • dx N e -βE[{x i }] δ   s - 1 N N i=N (1-κ)+1 x i   , ( 5.11)
with

βE[{x i }] = N 2 2 N i=1 x 2 i -N α i̸ =j |x i -x j |, and x 1 < • • • < x N .
(5.12)

The factor N ! accounts for all possible permutations of the labels x 1 , ..., x N . The partition function Z N was computed in Ch. 1 and for large N reads

Z N ≈ e 2 3 α 2 N 3 . (5.13)
We approach this calculation in the same way as before, by using the integral representation of the delta function given in Eq. (2.14). This yields

P N,κ (s) = N 3 N ! Z N Γ dµ 2πi dx 1 • • • dx N e -βEµ[{x i }] , and x 1 < • • • < x N , (5.14)
where

βE µ [{x i }] = N 2 2 N i=1 x 2 i -α N i̸ =j |x i -x j | + µN 2   N i=N (1-κ)+1 x i -s N   = βE[{x i }] + µN 2   N i=N (1-κ)+1 x i -s N   = N 3 Ẽµ [{x i }] . ( 5.15)
As we will see later, it is useful to rewrite the energy by separating the sums over x 1 , ...x N -N ′ and over x N -N ′ +1 , ..., x N . Using the quadratic form derived in Eq. (1.30) we get

βE µ [{x i }] = F µ [{x i }] + C 1 , (5.16)
with

F µ [{x i }] = N 2 2 N -N ′ i=1 x i - 2α N (2i -N -1) 2 + N i=N -N ′ +1 x i - 2α N (2i -N -1) -µ 2 , ( 5.17)
where N ′ = κ N and C 1 is a constant with respect to x i 's given by

C 1 = -2α 2 N i=1 (2i -N -1) 2 - N 2 2 N i=N (1-κ)+1 µ 2 - 4α N (2i -N -1)µ = N 3 - 2α 2 3 - µ 2 κ 2 + 2ακ(1 -κ)µ + 2α 2 3 N . ( 5.18)
Using the saddle-point method, we compute the dominant contribution to the N + 1 integrals in Eq. (5.14) as

P N,κ (s) ∼ e -(βEµ[{x * i }]+ 2 3 α 2 N 3 ) , ( 5.19)
where we used Eq. (5.13) for Z N and {x * i } is the configuration of the ordered charges that minimizes the energy E µ . This configuration is fixed by the system of N + 1 equations

∂ Ẽµ ∂x i = 0 =⇒ ∂E ∂x i x * i + µN 2 = 0 (5.20) ∂ Ẽµ ∂µ = 0 =⇒ N i=N (1-κ)+1
x * i = s N .

(5.21) The second equation ensures that the TLS of the saddle-point configuration has the given value s. In order to simplify the notation, in this chapter we denote the saddle-point value µ * (s), determined by Eq. (5.21), simply as µ.

In this section, we find the saddle-point configurations {x * i } for all allowed values of the parameters κ and µ. We proceed by fixing κ and varying µ in the κ-µ space, which allows us to complete the phase diagram depicted in Fig. 4.1. It turns out that we need to separate two cases. First, we analyze the simpler regime, µ < 0, where we find one phase (Phase I). Then we solve the more complex case µ > 0, where we observe four different phases (Phase II, Phase IIIa, Phase IIIb, and Phase IV).

. µ < 0

In this regime, we can find the configuration that minimizes the energy by closely looking at the expression for the energy given in Eqs. (5.16), and (5.17). Since C 1 has no dependence on the positions of the charges, we can focus only on minimizing F µ [{x i }]. We proceed by minimizing both terms of Eq. (5.17) separately. Since the sums are expressed by quadratic terms, we reach the minimum by setting each term to zero. We can check that the resulting configuration satisfies the ordering of the particles (which is not true when µ > 0). Following this reasoning, we obtain a solution composed of two blocks: the left block L (depicted in orange color in Fig. 5.2) containing particles x 1 , ..., x N -N ′ , and the right block R (depicted in purple color in Fig. 5.2) containing particles x N -N ′ +1 , ..., x N , where N ′ denotes the number of particles included in the TLS (see Eq. (5.1)). More precisely, the positions of the particles that minimize the energy Eq. (5.17) are

x * i L = 2α N (2i -N -1), i = 1, 2, • • • , (1 -κ) N (5.22) and x * i R = 2α N (2i -N -1) -µ, i = (1 -κ)N + 1, • • • , N .
(5.23)

In the large N limit we can express this solution in terms of the macroscopic density profile given by

ρ(x) =      1 4α , -2α < x < 2α(1 -2κ), 1 4α , 2α(1 -2κ) -µ < x < 2α -µ, 0 , elsewhere.
(

As we also see in Fig. 5.2, the saddle-point density for µ < 0 is composed of two blocks, each containing uniformly distributed charges. They are separated by a gap of length -µ = |µ| > 0. This solution is valid for all allowed κ. For a fixed κ when µ approaches zero, the gap between both blocks is shrinking. At µ = 0, we recover the original equilibrium density profile of the 1dOCP. When µ becomes positive, the solution from Eq. (5.24) becomes invalid since the two blocks start to merge and the ordering condition is violated.

. µ > 0

When µ > 0 we expect a uniform density profile with a single support and an additional δ-peak consisting of the excess charge from both left and right blocks. This kind of configuration is schematically depicted in Fig. 5.3 and can be parametrised by the following seven parameters:

• the position of the leftmost particle: l,

• the position where the delta-function occurs, i.e., where fractions of charges from the left and right pile up: d,

• the position of the rightmost particle: r,

• the fraction of charges that are on the left of d: n l ,

• the fraction of charges that are on the right side of d: n r ,

• the fraction of particles at the point d that came from the left: nl ,

• the fraction of particles at the point d that came from the right: nr .

• Phase IIIa

Here, we set 1/2 ≤ κ ≤ 1 and the solution is given by

l = -2α -µ + 2 2α µ(1 -κ) nl = 1 -κ nr = -2α(1 -κ) + 2αµ(1 -κ) 2α , (5.32)
with the condition that 2α(1 -κ) < µ < 2α/(1 -κ), which ensures that nl and nr satisfy the inequalities 0 ≤ nl ≤ 1 -κ and 0 ≤ nr ≤ κ. The corresponding saddle-point configuration is shown in Fig. 5.4 in the inset with a green frame. It is characterized by the δ-peak located at the left edge of the density support. This solution is valid in

8α(1 -κ) < µ < 2α 1 -κ with 1 2 ≤ κ ≤ 1 . ( 5.33)
In Fig. 5.4, we show this area in green color. The lower boundary is shared with Phase II.

• Phase IIIb Here, we set 0 ≤ κ ≤ 1/2 and the solution is given by

l = -2α , nl = -2 √ α κ + √ 2µκ 2 √ α , nr = κ , ( 5.34)
with the condition that 2ακ < µ < 2α/κ, so that nl and nr satisfy the inequalities 0 ≤ nl ≤ 1 -κ and 0 ≤ nr ≤ κ. In this configuration, the δ-peak occurs at the right edge of the support (see the inset of Fig. 5.4 with the yellow frame). In the κ-µ plane, this phase exists in the yellow-colored region in Fig. 5.4. The equations determining the boundaries are

8α κ < µ < 2α κ with 0 ≤ κ ≤ 1 2 . ( 5.35)
• Phase IV Here the solution is given by l = -κ µ , nl = 1 -κ , nr = κ . (5.36) This solution corresponds to the case where all the particles are in the δ-peak, as indicated in the inset with the red border of Fig. 5.4. This solution is valid in the region (red color in Fig. 5.4)

µ ≥ 2α min(κ, 1 -κ) . ( 5.37)
Thus, we completely determined the phase diagram in the κ-µ space. In the next section, we translate the diagram into the κ-s plane.

. Phase diagram

In order to transform the phase diagram depicted in Fig. 5.4 into the phase diagram shown in Fig. 5.1, we need to use the second line in Eq. (5.21). By evaluating the truncated linear statistics s at the saddle-point we find the relation s(µ). Inverting it gives us the relation µ(s), which can be inserted in the boundaries given in the previous section. Furthermore, we will also need the relation µ(s) later to compute the full rate function in the next section.

. Case µ < 0

When µ < 0, we have only Phase I. We determined the saddle-point configuration {x * i } in Eqs. (5.23), (5.22). Evaluating the TLS for this configuration in the large N limit yields

s = 1 N N i=N (1-κ)+1 x * i = 2α N 2 N i=N (1-κ)+1 2i -1 -1 + µ 2α N ≃ ≃ 2α 1 - µ 2α κ -κ 2 . (5.38)
Inverting this relation, we find

µ(s) = 2α(1 -κ) - s κ .
(5.39)

Since in this phase µ ≤ 0, this translates to the region

s ≥ 2ακ(1 -κ) , (5.40)
which is depicted by the blue colour in the κ-s plane in Fig. 5.1. Hence, the boundary of Phase I is s = 2ακ(1 -κ).

. Case µ > 0

For positive values of µ we found four different phases. The saddle-point configuration in each phase can be described by three independent parameters l, nl , and nr . However, for the purpose of evaluating the TLS at the saddle-point it is more intuitive to use the parameters d, n r , and nr (see Fig. 5.3) that describe the "right" particles, which are the ones included in the TLS. In terms of these parameters the TLS reads

s = 1 N N i=N (1-κ)+1 x * i = 1 N N nr i=1 d + 1 N N nr j=1 d + 4α N j = d(n r + nr ) + 2αn r n r + 1 N .
(5.41) Using the relation n r + nr = κ and keeping the leading terms in N we get

s ≃ d κ + 2α (κ -nr ) 2 .
(

Returning to the original parameters nr , nl , and l we have

s ≃ κ(l + 4α(1 -κ -nl )) + 2α(κ -nr ) 2 .
(5.43)

We use this expression to evaluate the TLS in four different phases.

• Phase II This phase is characterized by the parameters l, nl , and nr given in Eq. (5.31). Inserting these expressions in Eq. (5.43) we find

s ≃ κ 2α(1 -2κ) - µ 2 + 2α κ - µ 8α 2 . ( 5.44)
Inverting this relation leads us to a quadratic equation for µ. Taking into account that µ → 0 when s → 2ακ(1 -κ) we can eliminate one solution and we are left with

µ(s) = 4 4ακ -4α 2 κ 2 -2α(2ακ(1 -4κ) -s) .
(5.45)

In the κ-µ space the upper boundaries of Phase II are given by µ = 8α min(κ, 1 -κ).

By inserting them into Eq. (5.44) we get the following boundaries in the κ-s space

s(κ) = 2ακ(1 -4κ) , 0 ≤ κ ≤ 1/2 between Phase IIIb and Phase II 2α(1 -κ)(1 -4κ) , 1/2 ≤ κ ≤ 1 between
Phase IIIa and Phase II .

(

The upper boundary given by µ = 0 transforms into s(κ) = 2ακ(1 -κ) , 0 ≤ κ ≤ 1 between Phase I and Phase II . (5.47)

The three lines given in Eqs. (5.46), (5.47) define the grey area shown in Fig. 5.1.

• Phase IIIa

In this case we insert the solutions from Eq. (5.32) into Eq. (5.43). This yields

s = 2α(1 -κ) + (1 -2κ)µ -2 √ 2(1 -κ) αµ(1 -κ) . ( 5.48)
Inverting this relation and choosing the solution that matches with the solution for Phase II at the boundary we get

µ(s) = α(2 -2κ(2(κ -2)κ + 3)) -2 √ 2 α(κ -1) 3 (2κ(α(κ -1)κ + s) -s) -2κs + s (1 -2κ) 2 .
(5.49

)
This relation is valid in the green region in Fig. 5.1. The boundary between Phase II and Phase IIIa is already given in Eq. (5.46). Instead the boundary separating Phase IIIa and Phase IV in κ-µ space is µ = 2α/(1 -κ). Inserting this in Eq. (5.48) yields

s(κ) = - 2ακ 2 1 -κ , for 1 2 ≤ κ ≤ 1,
between Phase IIIa and Phase IV . (

• Phase IIIb Using the solutions for l, nr , and nl from Eq. (5.34) gives us the expression for s in Phase IIIb

s = 2κ α -2αµκ . ( 5.51)
By inverting it we get

µ(s) = (2ακ -s) 2 8ακ 3 . ( 5.52)
The boundary between Phase IIIb and Phase II is already given in Eq. (5.46). The boundary between Phase IIIb and Phase IV in κ-µ space is described by µ = 2α/κ. We obtain the boundary in κ-s space by inserting this relation into Eq. (5.51)

s(κ) = -2ακ, 0 ≤ κ ≤ 1/2,
between Phase IIIb and Phase IV .

(5.53)

• Phase IV As in the other cases, we insert the solutions for the parameters l, nr , and nl for Phase IV from Eq. (5.36) into Eq. (5.43). We get s = -µκ 2 .

(5.54) In the center, we see a schematic depiction of the probability distribution of the truncated linear statistics P N,κ (s). Four different regimes that appear for 0 < κ ≤ 0.5 (see the text) are separated by dotted vertical lines (for 0.5 ≤ κ ≤ 1 the Phase IIIb is replaced by Phase IIIa). In the upper right panel, the associated saddle-point densities are plotted. In the upper left panel, the large deviation function ϕ κ (s) is plotted vs. s, together with the results from numerical simulation (red dots).

We invert this relation to obtain

µ(s) = - s κ 2 .
(5.55)

We already obtained both boundaries of Phase IV, and they are given in Eq. (5.50) and Eq. (5.53).

With this, we completed the phase diagram in κ-s space shown in Fig. 5.1. The next step is to use the relations µ(s) to find the rate function in each phase.

. The full rate function

The last part of the calculation consists of computing the rate function ϕ κ (s), which describes the probability distribution of the truncated linear statistics for large N . To avoid evaluating the energy at the saddle-point, we use the shortcut introduced in Ch. 2. We recall it here ϕ κ (s) = -µ(s)ds + const .

(5.56)

In the previous section, we already computed µ as a function of s at the saddle-point for all the phases. Now we insert these relations in Eq. (5.56) to find the rate function. We need to separate two cases: 0 ≤ κ ≤ 1/2 and 1/2 < κ ≤ 1. For fixed κ we encounter either Phase IIIa (1/2 < κ ≤ 1) or Phase IIIb (0 ≤ κ ≤ 1/2). at all three boundaries. The heights of the jump are given by

ϕ ′′′ κ (s → s 1 (κ) + ) -ϕ ′′′ κ (s → s 1 (κ) -) = 1 16ακ 3
(5.62)

ϕ ′′′ κ (s → s 2 (κ) + ) -ϕ ′′′ κ (s → s 2 (κ) -) = - 1 4ακ 3
(5.63)

ϕ ′′′ κ (s → s 3 (κ) + ) -ϕ ′′′ κ (s → s 3 (κ) -) = - 1 4ακ 3 .
(

This means that all four phases are separated by a third-order phase transition.

Case 1/2 ≤ κ ≤ 1

Now we pick a value of κ from the interval 1/2 < κ ≤ 1 and vary s from -∞ to ∞. In the process we cross the following phases: Phase IV, Phase IIIa, Phase II, and Phase I (see Fig. 5.1). The boundaries between phases are summarized as follows

s1 (κ) = 2ακ(1 -κ) ,
between Phase I and Phase II (5.65)

s2 (κ) = 2α(1 -κ)(1 -4κ)
, between Phase II and Phase IIIa (5.66)

s3 (κ) = - 2ακ 2 1 -κ ,
between Phase IIIa and Phase IV .

(

Using the shortcut from Eq. (5.56) we integrate the expressions for µ(s) given in Eqs. (

. This determines the rate function, given by

ϕ κ (s) =                                                            (s -2ακ(1 -κ)) 2 2κ , s1 (κ) ≤ s (I) 4 -64 3 α 2 κ 3 -8α 2 (κ -1)κ 2 -4ακ s + 2 3 √ 2α (2ακ(5κ -1) + s) 3/2 , s2 (κ) ≤ s ≤ s1 (κ) (II) 4α 2 (κ -1) 2 8κ 4 -8κ 3 + 4κ -1 + 12α(κ -1)(2κ -1)(2(κ -1)κ + 1)s 6(2κ -1) 3 + s 2 2(2κ -1) + 4 √ 2 α(κ -1) 3 2α(κ -1)κ 2 + (2κ -1)s 3/2 3α(κ -1) 3 (2κ -1) 3 , s3 (κ) ≤ s ≤ s2 (κ) (IIIa) 2α 2 3 + s 2 2κ 2 , s ≤ s3 (κ) (IV) .
(5.68)

The function is plotted in the right panel of Fig. 5.6 for κ = 0.7. Again we see that the analytical prediction (red line) agrees very well with the numerical results (black points).

For fixed κ the rate function and its first and second derivatives are continuous at all three boundaries given in Eqs. (5.65), (5.66), (5.67). Just like for 0 ≤ κ ≤ 1/2 the third derivative has a jump at all three boundaries. The discontinuities are given by

ϕ ′′′ κ (s → s1 (κ) + ) -ϕ ′′′ κ (s → s1 (κ) -) = 1 16ακ 3
(5.69) 3 (5.70)

ϕ ′′′ κ (s → s2 (κ) + ) -ϕ ′′′ κ (s → s2 (κ) -) = - 1 4α(3κ -1)
ϕ ′′′ κ (s → s3 (κ) + ) -ϕ ′′′ κ (s → s3 (κ) -) = - (1 -κ) 3 4ακ 6 .
(5.71)

Thus, the large deviation function again displays a third-order phase transition at all three boundaries. Note that for both cases, 0 ≤ κ ≤ 1/2, and 1/2 < κ ≤ 1, we also find third order phase transition if we fix s and vary κ.

. Different limiting behaviors

After deriving the main results for the rate function given in Eqs. (5.60), and (5.68), we analyze the limiting behaviors of ϕ κ (s). First, we want to check if the behavior of the rate functions near their minimum matches with the typical Gaussian fluctuations of the TLS given in Eqs. (5.9), and (5.10). Next, we verify that in the limit κ → 0 we recover the results for the distribution of the rightmost particle x max from Refs. [START_REF] Dhar | Exact extremal statistics in the classical 1d Coulomb gas[END_REF][START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF] summarized in Eqs. (5.4), (5.5), (5.6), and (5.7). Similarly, we check if the limit κ → 1 agrees with the result for the full linear statistics given in Eq. (5.2).

. Typical fluctuations

The minimum of the rate function occurs at s = 2ακ(1 -κ), which coincides with the boundary between Phase I and Phase II. By analyzing the ϕ κ (s) in Eq. (5.60) or equivalently, Eq. (5.68), we find that ϕ κ (s) has a quadratic form ϕ κ (s) ≃ (ss) 2 /(2κ) as s → s. Consequently, the PDF of s behaves, for large N near s = s as

P N,κ (s) ≃ e -N 3
2κ (s-s) 2 , s → s . (5.72) This behavior thus matches perfectly with the form of the typical fluctuations given in Eqs. (5.9) and (5.10).

.

Limit κ → 0 Since we defined the truncated linear statistics as

s = 1 N N i=N -N ′ +1 x i = 1 N N i=N (1-κ)+1
x i , (5.73) we expect that s reduced to x max /N if we set N ′ = 1 or κ = 1/N . First, we want to check which of the phases remain in this case. Setting κ = 1/N in the equations for the boundaries in Eqs. (5.57), (5.58), and (5.59), we find to the leading order

s 1 κ → 1 N = s 2 κ → 1 N = 2α N (5.74) s 3 κ → 1 N = - 2α N .
(5.75) and its analytical predictions for κ = 0.3 (left panel) and κ = 0.7 (right panel). We computed ϕ κ (s) by using the importance sampling algorithm described in Sec. 5.5 Since the boundaries s 1 (κ) and s 2 (κ) become equivalent in this limit, Phase II does not exist anymore. We are left with one phase for s > 2α/N , the second phase for -2α/N ≤ s ≤ 2α/N , and the third phase for s < -2α/N . We find the rate function by inserting κ = 1/N and s = w/N in Eq. (5.60), where w denotes the position of x max /N . This yields

ϕ κ=1/N (s = w/N ) =                          1 2N (w -2α) 2 , 2α ≤ w (2α -w) 3 24α , -2α ≤ w ≤ 2α 2α 2 3 + w 2 2 , w ≤ -2α .
(5.76)

Our aim is to compare this result with the previously known result for the probability distribution of w [START_REF] Dhar | Exact extremal statistics in the classical 1d Coulomb gas[END_REF][START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF]. The expressions for the large deviation function of w are given in Eq. (5.6) and (5.7). It was found in [START_REF] Dhar | Exact extremal statistics in the classical 1d Coulomb gas[END_REF][START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF] that the two phase boundaries are located at w = ±2α. Just like in our case, the three phases are separated by a third-order phase transition. Furthermore, inserting the expression for the TLS in the limit κ → 0 given in Eq. (5.76) in the large deviation form P N,κ (s) ∼ e -N 3 ϕκ(s) we get the same probability distribution as given in Eq. (5.5).

. Limit κ → 1

In the limit κ → 1, the truncated linear statistics becomes the full linear statistics describing the center of mass. Inserting κ = 1 in the phase boundaries from Eqs. (5.65), (5.66), and (5.67), we get s1 (κ = 1) = s2 (κ = 1) = 0, (5.77) s3 (κ = 1) → -∞ .

(5.78)

Phase II and IV do not exist in this limit. Moreover, the rate function in Phases I and IIIa reduces to

ϕ κ=1 (s) =            s 2 2 , s ≤ 0 s 2 2
, s ≥ 0 .

(5.79)

Unlike before, the third derivative over s does not display a jump at s = 0. Hence, there is no phase transition in this case. Furthermore, plugging in the rate function from Eq. (5.79) into the large deviation form P N,κ (s) ∼ e -N 3 ϕ κ=1 (s) gives us the result for the full linear statistics written in Eq. (5.2).

. Sampling rare events

In this section, we describe the numerical method that we used for obtaining the rate function shown in Fig. 5.6. Like other numerical simulations shown in Ch. 3, Ch. 4, and Ch. 5, the algorithm is based on the standard Metropolis Monte-Carlo algorithm. Starting with the energy of the 1dOCP given in Eq. ( 1.3) we use Metropolis dynamics to evolve the configurations of the charges. Given a configuration {x i }, we propose a small change {∆x i } of the configuration. The new configuration is accepted with a probability min{1, e -β∆E }, where ∆E is the energy difference between the original configuration and the proposed new configuration. This dynamics satisfies detailed balance, which ensures that the system, at long times, reaches the stationary equilibrium state with the correct Boltzmann weight ∝ e -βE . Once the system has reached equilibrium, which can be checked by looking at the energy of the system, one can compute the statistics of any observable. With this method, we can access typical fluctuations around the average value. For example, the plots showing the typical fluctuations of the bulk gap (4.5) and the full counting statistics (4.9) from Ch. 4 are obtained in this way.

However, if we want to numerically verify our results for the rate function of the truncated linear statistics given in Eq.(5.68), and (5.60), we need to adapt the algorithm to sample atypical configurations. This is achieved by the so-called importance sampling method [START_REF] Nadal | Phase transitions in the distribution of bipartite entanglement of a random pure state[END_REF][START_REF] Nadal | Statistical Distribution of Quantum Entanglement for a Random Bipartite State[END_REF][START_REF] Schawe | Ground-state energy of noninteracting fermions with a random energy spectrum[END_REF][START_REF] Hartmann | Large-deviation properties of largest component for random graphs[END_REF][START_REF] Hartmann | Highprecision simulation of the height distribution for the KPZ equation[END_REF][START_REF] Banerjee | Current fluctuations in noninteracting run-and-tumble particles in one dimension[END_REF][START_REF] Mori | Condensation transition in the late-time position of a run-and-tumble particle[END_REF].

In our case, we want to explore configurations that result in the value of the truncated linear statistics, which is far away from the average s = 2κα(1 -κ). If we want to explore the region on the left from the average where s < s, we first pick a value s * < s and only accept moves where s new ≤ s * . Similarly, for exploring the region to the right of s, we can choose s * > s and only accept moves with s new ≥ s * .

Focusing on s < s, the main steps of the algorithm can be summarized as follows:

• Choose an initial configuration of {x i } that satisfies s = 1

N N i=(1-κ)N +1 x i < s * .
• Propose a move of a particle and calculate s new and E new . The new position of the particle is chosen as ∆x i = d max (1 -2n), (5.80) in the third derivative. We already encountered such third-order phase transitions in the previous two chapters when deriving the probability distributions for the full counting statistics, gap, and full linear statistics. At the end of the chapter, we verify that in two limits, κ → 1 and κ → 0, we recover the previous results for the distributions of the full linear statistics and the rightmost particle.

97

Chapter 6

Going out of equilibrium: The ranked diffusion

In the previous chapters, we were investigating the equilibrium properties of the jellium model. In this chapter, we investigate out-of-equilibrium properties of a one-dimensional gas of particles interacting via repulsive linear Coulomb interactions. The interactions between particles studied in this chapter are exactly the same as in the equilibrium onedimensional one-component plasma discussed in the previous chapters, however, the particles are not confined by an outside potential. As expected, in this case, the gas expands with time. We denote the positions of the particles at time t by x 1 (t), • • • , x N (t). The position of the i-th particle evolves with time according to the following overdamped Langevin equation

dx i dt = -∂ x i W (⃗ x) + √ 2T ξ i (t) = -c N j=1 sgn(x j -x i ) + √ 2T ξ i (t) . (6.1)
Here ξ i (t)'s denote the N unit independent white noises with zero mean ⟨ξ i (t)⟩ = 0 and delta-correlator ⟨ξ i (t)ξ j (t ′ )⟩ = δ i,j δ(t -t ′ ). We denote the pairwise interaction potential by W (⃗ x) = -c i<j |x i -x j |, where ⃗ x = {x i (t)} i=1,...,N . The strength of the interaction is determined by c. Since we focus on the case where particles repel each other, we always have c > 0, and by convention, we use sgn(0) = 0. Despite the repulsive interactions, the particles are free to cross each other. Often it is useful to consider the ordered configuration of the particles, which we denote by x (i) (t). If the particles are ordered, we can express the drift felt by the (i)-th particle as

δ i = -c N j=1 sgn(x (j) -x (i) ) = -c (N + 1 -2i) . (6.2)
From this formulation of the drift, we can clearly see that the Coulomb force acting on a particle depends on the position of the particle in the ordered configuration. The strength of the force is proportional to the number of particles ahead of it minus the number of particles behind. This kind of diffusion of N one-dimensional particles, where the drift is proportional to the rank of the particle, is known as the ranked diffusion and has been previously studied in other contexts such as finance [START_REF] Banner | Atlas models of equity markets[END_REF] and mathematics [START_REF] Pal | One-dimensional brownian particle systems with rankdependent drifts[END_REF][START_REF] O'connell | Product-form invariant measures for Brownian motion with drift satisfying a skew-symmetry type condition[END_REF]. Recently, ranked diffusion with attractive and repulsive interactions was studied in Ref. [START_REF]Ranked diffusion, delta Bose gas and Burgers equation[END_REF]. In this chapter, we focus solely on the particles with repulsive interactions. We show that at late times and after rescaling of the positions, the time-dependent density profile of N charges undergoing ranked diffusion can be identified with the density profile of the particles in the equilibrium 1dOCP. A similar connection exists between the eigenvalues of certain random matrices and the heights of N non-intersecting Brownian bridges akin to the so-called Dyson's Brownian motion [START_REF] Nadal | Nonintersecting Brownian interfaces and Wishart random matrices[END_REF][START_REF] Katori | Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems[END_REF][START_REF] Johansson | Non-intersecting paths , random tilings[END_REF][START_REF] Schehr | Exact distribution of the maximal height of p vicious walkers[END_REF][START_REF] Tracy | Nonintersecting brownian excursions[END_REF].

1/N 2 ln(c 2 t) 1 t ⟨x N -x 1 ⟩ t = (N c) -2 x x x c 2 t c 2 t c 2 t ρ(x, t) ρ(x, t) ρ(x, t) x x x
Before continuing with a more detailed analysis, we present the results of the numerical simulations, which give us some intuition about the properties of the system at different times. In Fig. 6.1, we show several realizations of the dynamics governed by Eq. (6.1). We simulated the time evolution of N = 500 particles that all start from x = 0 at t = 0. The obtained trajectories are shown in the middle panel of Fig. 6.1. For short times the particles are still close to each other (see plot a) in the middle panel). We call this regime the dense regime or Regime I. Because the gas is very dense, the particles cross each other many times. When time progresses, the gas expands, and therefore it becomes more dilute. At large times, the particles are well separated, and their trajectories almost never cross. Due to the long-range interactions, they form a well-organized crystal structure (see plot c) in the middle panel of Fig. 6.1). We call this regime Regime III or the dilute regime. In between Regime I and Regime III there is Regime II (plot b) in the middle panel of Fig. 6.1). In this regime, the crossings still occur but with lower frequency than in Regime I. We can better understand the three regimes and their properties by finding the characteristic timescales at which they appear. Note that the two parameters, the temperature T and the interaction strength c, can be absorbed in the change of units. The time scale is T /c 2 , and the length scale is T /c. In the remainder of this chapter, we fix T = 1. We can explore different time scales by scaling c in various ways with N .

The first characteristic time t * 1 marks the crossover from Regime I to Regime II. We find it by considering the typical thermal diffusion length of independent particles l T ∼ √ 2T t and the total size of the gas at large times l = 2cN t. The former is connected to the diffusion and the latter to the drift. By comparing both, we can find the first characteristic time scale

t * 1 ∼ 1 N 2 T c 2 .
(6.3) Before t * 1 , in Regime I, the diffusion dominates over the drift. After t * 1 , the drift starts to dominate, and the effect of interactions becomes important. The transition from the dense regime (Regime I) to Regime II is also clearly visible by looking at the size of the system as a function of time. We define the size of the system by the distance between the leftmost and the rightmost particle. The upper right panel of Fig. 6.1 shows the time dependence of the size of the system as a function of time in ln-ln scale. The black dots represent the system size measured numerically. Since diffusion is dominant at short times, the size of the system increases as ∝ √ 2t. At later times the size of the system increases linearly with time and is given by l = 2cN t. This can be understood by looking at the drift felt by the two edge particles given by Eq. (6.2). They feel the drift ∼ cN or -cN , which means that at large times we expect the size of the system to grow as 2cN t. The crossover happens close to the first characteristic time t * 1 (in the right upper panel of Fig. 6.1 denoted by a black vertical dotted line).

The transition between Regimes II and III is marked by the second characteristic timescale t * 2 . We find it by comparing the interparticle distance a = l/N = 2ct with the typical thermal diffusion length l T ∼ √ 2T t. This yields

t * 2 ∼ T c 2 . (6.4)
When the interparticle length a is smaller or comparable to the typical thermal diffusion length l T (Regime II), the trajectories still cross each other. When the typical distance between neighboring particles becomes larger than the typical thermal diffusion length, the crossings become very rare (Regime III). In the upper left panel of Fig. 6.1, we see the summary of the three regimes. We marked the two characteristic times t * 1 and t * 2 (black vertical lines) on the axis of the logarithm of the dimensionless time c 2 t. The red dots show the values of the dimensionless time c 2 t at which we stopped each of the simulations shown in the middle panel of Fig. 6.1. Note that all values in this plot are given for T = 1.

In Fig. 6.1, we also show the averaged densities that were measured at the final time of each simulation shown in the middle panel of Fig. 6.1. Starting with Regime I (plot a) in the bottom panel) we find a bell-shaped density profile. When time increases, and we enter into Regime II (see plot b) in the bottom panel), the density starts to approach a square shape. Around the edges at [-l/2, l/2] = [-N ct, N ct], we notice the presence of the boundary layers with a width that is bigger than the interparticle distance. The shape of boundary layers is discussed in detail in Ref. [START_REF] Flack | Out of equilibrium dynamics of repulsive ranked diffusions: the expanding crystal[END_REF]. It is shown that the width of the layers is l T ∼ √ 2T t ≪ l. In Regime III (see plot c) in the bottom panel), we find a square-shaped particle density with edges at [-l/2, l/2] = [-N ct, N ct].

In the remainder of this chapter, we study the properties of the ranked diffusion at large times. We show that in this limit, there is an analogy between equilibrium onedimensional one-component plasma and the ranked diffusion.

We organize the rest of this chapter as follows. We start by analyzing the joint probability density function for N particles by using the saddle-point approximation valid for large times. We find that in this case the most probable positions of particles are equidistantly positioned between the two edges of the system that are located at ∼ ±N ct. This indicates that our solution obtained by the saddle-point approximation describes the previously introduced Regime III. To further examine the fluctuations of the particle positions, we approximate the joint probability distribution of the positions by neglecting the rational prefactor (see Eq. (6.16)), which is subleading in the large t limit. In this case, we can identify the joint probability distribution of the particles at a fixed time with the equilibrium distribution of particles in the jellium model introduced in Ch. 1 in Eq. (1.48). We demonstrate that the equilibrium results for the typical fluctuations of the bulk and the edge gaps presented in Ch. 4 correctly predict the leading order of the fluctuations of the gaps in the ranked diffusion. Using the results from Ch. 4 (see. Eq. (4.48), (4.50), (4.5), and (4.6)) we find that the leading order of the fluctuations of both edge and bulk gaps are of order O( √ t) and are given by a Gaussian distribution. We verify this by numerically simulating the dynamics given in Eq. (6.1) and comparing the obtained distribution of the gaps with the predictions. As shown in Ref. [START_REF] Flack | Out of equilibrium dynamics of repulsive ranked diffusions: the expanding crystal[END_REF] there are additional fluctuations of the positions that are of order O(T /c) and have a dynamical origin. Consequently, they can not be predicted by the equilibrium model. They are obtained by more precisely treating the rational prefactor in Eq. (1.48). However, we do not discuss them in this chapter, where we focus on the connection with the equilibrium model.

Note that the time evolution of the density at shorter times in Regimes I and II can not be accessed by the saddle-point approximation that we use to study Regime III. But it can be studied by a different approach that is based on the Burgers equation, which was recently introduced in the context of ranked diffusion in Ref. [START_REF]Ranked diffusion, delta Bose gas and Burgers equation[END_REF]. In this way, one can obtain the predictions for the time dependence of the density for large N for arbitrary time. See Section IV of Ref. [START_REF] Flack | Out of equilibrium dynamics of repulsive ranked diffusions: the expanding crystal[END_REF] for results obtained by this approach.

. The dilute regime and the connection to the equilibrium jellium model

We start this section by providing the solution for the probability density function (PDF) P (⃗ x, t) of the particles evolving according to Eq. (6.1). The PDF satisfies the following Fokker-Planck equation

∂ t P = -H FP P = i [T ∂ 2 x i -c∂ x i j sgn(x i -x j )]P . (6.5)
We consider the delta initial condition where all particles are at x = 0 at t = 0

P (⃗ x, t = 0) = i δ(x i ) . ( 6.6)
When N = 2, it is possible to solve the Fokker-Planck equation given in Eq. (6.5) directly by the Laplace transform. For details, see Section III and Appendix A of Ref. [START_REF] Flack | Out of equilibrium dynamics of repulsive ranked diffusions: the expanding crystal[END_REF]. The case of N particles is more complex. However, for the delta initial condition given in Eq. (6.6), it has been shown that it is possible to find a solution to the Fokker-Planck equation by using a mapping to the Lieb-Lininger model or the 1d delta Bose gas [START_REF]Ranked diffusion, delta Bose gas and Burgers equation[END_REF].

The resulting quantum model is integrable and can be solved via the Bethe ansatz [START_REF]Ranked diffusion, delta Bose gas and Burgers equation[END_REF][START_REF] Gaudin | The Bethe Wavefunction[END_REF][START_REF] Lieb | Exact analysis of an interacting bose gas. I. the general solution and the ground state[END_REF]. In this way, one can obtain the solution valid in the ordered sector x 1 < • • • < x N , for c > 0, t > 0, and any N . The solution reads [START_REF] Gaudin | The Bethe Wavefunction[END_REF][START_REF] Borodin | Macdonald processes[END_REF][START_REF] Tracy | The dynamics of the one-dimensional delta-function Bose gas[END_REF]]

P (⃗ x, t) = e c 4 N i,j=1 |x i -x j | e E 0 t × R dk 1 2π • • • R dk N 2π 1≤a<b≤N ik a -ik b ik a -ik b + c e -t N j=1 k 2 j +i N j=1 x j k j , (6.7)
where

E 0 = - c 2 12 (N 3 -N ) .
One can check that the solution in Eq. (6.7) is also normalized to unity. Our goal is to analyze Eq. (6.7) for late times, and therefore we rewrite it in terms of the rescaled variables x j = z j t. We will consider t → ∞ with fixed N and O(z j ) = 1. In the ordered sector where z

1 ≤ • • • ≤ z N we get P (⃗ z, t) = R dk 1 2π • • • R dk N 2π e t [ c 4 N i,j=1 |z i -z j |- N j=1 k 2 j +i N j=1 z j k j -c 2 12 N (N 2 -1)] × 1≤a<b≤N ik a -ik b ik a -ik b + c . ( 6
.8)

In the limit of large t, we can use the saddle-point approximation to obtain the dominant contribution to the integral over k j 's in Eq. (6.8). First, we need to find the values of k * j where the expression multiplying t in the argument of the exponential function in Eq. (6.8) reaches the minimal value. The values k * j are thus determined by the following equation

∂[ c 4 N i,j=1 |z i -z j | -N j=1 k 2 j + i N j=1 z j k j -c 2 12 N (N 2 -1)] ∂k j k * j = 0 .
(6.9)

After solving Eq. (6.9) we get

k * j = iz j 2 , ( 6.10)
which are on the imaginary k-axis. Since the original integrals in (6.8) are on the real k-axis, we need to deform the contours in the complex k-plane so that they pass through the saddle-points and pick up the leading contribution for large t. This can be done without crossing the poles in the prefactor in (6.8) by deforming the contours for each k j successively maintaining the condition Im

k N > • • • > Im k 1 . This gives P (⃗ z, t) ≃ 1 (4πt) N/2 e tc 4 N i,j=1 |z i -z j |-t j z 2 j 4 e -c 2 12 N (N 2 -1)t 1≤a<b≤N z b -z a z b -z a + 2c
. (6.11) Note that since z b > z a for b > a, there are no poles in the double product. The expression in Eq. (6.11) is valid for c 2 t ≫ 1 and any finite N . An interesting observation is that although there is no external confining potential in our model, the terms j z 2 j 4 in the exponential in Eq. (6.11) indicate that the particles feel an effective quadratic potential that is generated by the dynamics.

Since we are in the ordered sector where z 1 ≤ • • • ≤ z N , we can get rid of the absolute value in the exponent in Eq. (6.11). Performing the same manipulations as in Section 1.2 of Ch. 1, one finds c 4

N i,j=1 |z i -z j | = - c 2 N j=1 (N + 1 -2j)z j . ( 6.12)
Inserting this expression back in Eq. (6.11) and completing the squares yields

P (⃗ z, t) ≃ 1 (4πt) N/2 e -t 4 N j=1 (z j -c(2j-N -1)) 2 1≤a<b≤N z b -z a z b -z a + 2c , for z 1 ≤ z 2 • • • ≤ z N . (6.13)
Since we expressed the argument of the exponent in Eq. (6.13) as a sum of quadratic terms, we can conclude that the most probable values of the positions are given by .14) This result agrees with the simulations depicted in the last figures in the middle and bottom panels of Fig. 6.1. As discussed at the beginning of the chapter, in the dilute regime, the charges form an equidistantly spaced lattice with edges at x 1 = -c(N -1)t and x N = c(N -1)t. Thus, the system in this limit can be described as an expanding crystal.

x j t = z j = c(2j -N -1) . ( 6
After obtaining the most probable positions, we want to investigate the fluctuations around these values. We denote them by δx j = x j -c t(2j -N -1) .

(6.15)

By inserting δx i 's in the exponent of Eq. (6.13) we find that it reduces toj δx 2 j 4t , which would indicate that the fluctuations are independent and Gaussian with the width given by the diffusion length l T = √ 2t. However, this is not the case since there is the double product factor, and in addition, the particles are in the ordered configuration. We continue our analysis by considering the effects of the ordering, but we neglect the double prefactor, which is subleading for large t. Rewriting Eq. (6.13) without the prefactor yields

P (⃗ z, t) ∝ e -t 4 N j=1 (z j -c(2j-N -1)) 2 , for z 1 < z 2 < • • • < z N .
(6.16)

The above expression looks very similar to the one describing the joint probability distribution of the ordered particles in the equilibrium one-dimensional one-component plasma, which reads

P J [⃗ y] ∝ e -βE J [⃗ y] = e -N 2 2 N i=1 [yi-2α N (2i-N -1)] 2 , for y 1 < y 2 < • • • < y N . (6.17)
Here we used the formula for the energy of the 1dOCP derived in Ch. 1 in Eq. (1.30). We adapted the notation to match the notation in this chapter. Here we denote the configuration of the particles by ⃗ y instead of {y i }. We also added the superscript J, which denotes the quantities that correspond to the equilibrium jellium model. Comparing the expressions in Eq. (6.16) and Eq. ( 6.17) we can conclude that both probability distributions are given by the same expression if we use the following transformations

z i = 2 t N y i , x i = z i t = √ 2tN y i , α = c √ t 2 √ 2 . ( 6

.18)

The joint probability distribution in the dynamical picture given in Eq. (6.16) is valid for c 2 t ≫ 1. Thus, the third relation in Eq. (6.18) implies that we can compare the ranked diffusion to the equilibrium 1dOCP in the limit of strong interactions where α ≫ 1.

In the next step, we use the relations in Eq. (6.18) to predict the macroscopic density profile for the particles in the dynamical problem. We derived the equilibrium density profile of the jellium in Eq. (1.48) in Ch. 1. It reads

ρ J (x) = 1 4α , -2α ≤ x ≤ 2α . ( 6.19)
By transforming α and x according to Eq. (6.18), we find the density profile of the ranked diffusion model

ρ(z) ≈ 1 2c N , -c N ≤ z ≤ +c N . (6.20)
In terms of the original coordinates x i = z j t this expression reads

ρ(x, t) ≈ 1 2c N t , -c N t ≤ x ≤ +c N . (6.21)
This equation describes a gas with particles that are uniformly spread between two edges given by ± -cN t. With time the distance between edges grows linearly with the speed cN . This corresponds to Regime III introduced in the beginning and shown in the middle panel in plot c) in Fig. 6.1.

. Fluctuations of the gaps

After establishing the connection between the densities of the two models, we can use the results derived in the context of the jellium model to predict the fluctuations of the particles in the dynamical problem. More specifically, we focus on the fluctuations of the gaps between neighboring particles. We discussed the fluctuations of the gaps of the equilibrium jellium model in Sec. 4.1.2 of Ch. 4, where we derived the expression for the typical fluctuations of the middle gap. We found that in the large N limit we can describe the typical fluctuations around the average value ⟨g mid-gap ⟩ ≈ 4α/N by the following scaling form (see Eq. (4.48) and Eq. (4.50))

P J mid-gap (g, N ) ∼ N H α (gN ), H J α (z) = θ(z)[A(α)] 2 ∞ -∞ dye -1 2 [(y+z-4α) 2 +y 2 ] F α (y + 4α)F α (-y -z + 8α) . (6.22)
This expression is valid for any value of the interaction strength α. However, we showed that the connection between the densities of both systems exists when the interaction strength of the jellium model α is large (see Eq. (6.18)). Therefore, we need to find the large α limit of the expression in Eq. (6.22). We can approximate the integral over y by the saddle-point method. Before proceeding, let us recall the asymptotic behavior of the function F α (x). We defined F α (x) as a unique solution of the non-local differential equation given in Eq. (4.7). Its asymptotic behavior is described by

1 -F α (x) ∼ e -x 2 /2+o(x 2 ) , x → +∞ , F α (x) ∼ e -|x| 3 /(24α)+o(x 3 ) , x → -∞ . (6.23)
Returning to the integral in Eq. (6.22), we see that the argument of the exponential function in Eq. (6.22) reaches its minimum at y * = (4α -z)/2. By inserting y * as the argument of both functions F α we see that their product reduces to [F α (6α -z/2)] 2 . Therefore, when α is large and z < 12α we can approximate [F α (6α -z/2)] 2 ≈ 1. We will see later that z < 12α holds near the peak of the distribution for the middle gap. Since we can approximate both functions F α (x) in Eq. (6.22) by unity, the dominant contribution to the integral comes from the exponential function when y = y * = (4α -z)/2. Thus, for α ≫ 1 the typical fluctuations of the gap are described by lim α→∞

H J α (z) ∼ e -1 4 (z-4α) 2 . (6.24)
By inserting this result back into Eq. (6.22) we obtain the probability distribution of the middle gap that reads

P J mid-gap (g) ∼ e -N 2 4 (g-4α/N ) 2 . (6.25)
Hence, when α is large the typical fluctuations of the middle gap approach the Gaussian distribution with mean 4α/N and variance 2/N 2 . This means that when both α and N are large, we can express the random variable g mid-gap that describes the middle gap as for times t = 500 and t = 3000, from the numerical simulation of the Langevin equation (6.1) both for the mid-gap G RD ≡ G RD mid-gap and for the edge gap G RD ≡ G RD edge-gap . Here N = 500, c = 0.1 (note that c 2 t is the dimensionless time). It is compared with the normal Gaussian distribution e -x 2 /2 / √ 2π as predicted in (6.31) (black solid line). The inset shows ln P as a function of G2 using the same numerical data. The black solid line is normal Gaussian distribution.

g J mid-gap ≈ 4α N + √ 2 N N (0, 1) , ( 6
where N (0, 1) is a standard normal variable with zero mean and unit variance.

As we explained in Ch. 4 the fluctuations of the gap change when we move from the bulk towards the edge of the system. The distribution of the edge gap g edge = x N -x N -1 was derived in Ref. [START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF] and in the limit of large N reads

P J edge-gap (g) ∼ N h α (gN ), h J α (z) = θ(z)[A(α)] 2 ∞ -∞ dye -1 2 [(y+z-4α) 2 +y 2 ] F α (y + 4α). (6.27)
Again, we need to analyze the behavior of Eq. (6.27) when α ≫ 1. The exponent in the integrand is the same as in Eq. (6.22), and therefore it reaches its minimum at the same value y * = (4α -z)/2. At this value of y the function F α (x) reads F α (6α -z/2).

Taking into account the asymptotics from Eq. (6.23) we find that for large α we can write F α (6α -z/2) ≈ 1. Hence, the scaling function h α from Eq. (6.27) can be approximated by

lim α→∞ h J α (z) ∼ 1 √ 4π e -1 4 (z-4α) 2 . (6.28)
Inserting (6.28) back in the expression Eq. (6.27), we obtain a Gaussian distribution for the edge gap

P J edge-gap (g) ∼ e -N 2 4 (g-4α/N ) 2 . (6.29)
Hence, we can conclude that in the limit of strong interactions, we can describe both the fluctuations of the middle gap (see Eq. (6.25)) and the edge gap (see Eq. (6.29)) by the Gaussian distribution. Therefore, we can describe all of the gaps in the system by the following random variable

g J edge-gap ≈ 4α N + √ 2 N N (0, 1) . (6.30)
Now we can use this result to predict the fluctuations of the gaps in Regime III of the dynamical problem. We use the superscript 'RD' to refer to the gaps in the ranked diffusion model. Taking into account the equation Eq. (6.30) and using the mapping between both system given in Eq. (6.18) we can conclude that

g RD edge-gap = z i+1 -z i = 2 t N 4α N + √ 2 N N (0, 1) = 2c + 2 √ t N (0, 1) , ( 6.31)
where N (0, 1) is a standard normal variable with zero mean and unit variance. Returning to the original variables x i = z i t, we find

G RD mid-gap = x N/2+1 (t) -x N/2 (t) = tg RD mid-gap ≈ 2c t + 2 √ t N (0, 1) . (6.32)
We check these predictions by numerically simulating the Langevin equation in Eq. ( 6.1) and measuring the middle and edge gaps. The results of the simulation are shown in Fig. 6.2. The markers show the data from the numerical simulation, where we have rescaled the gaps as G = G RD -2ct

2 √ t
. The solid black line is the normal Gaussian distribution. We can see an almost perfect matching between the simulation and the Gaussian distribution at times t = 500 and t = 3000. This confirms that the 1dOCP very accurately describes the fluctuations of the rescaled positions of the ranked diffusion model at late times.

. Conclusion of Chapter 6

In this chapter, we have studied the out-of-equilibrium Langevin dynamics of N particles interacting via repulsive linear interaction potential, which corresponds to the onedimensional Coulomb potential. We focused on the initial condition where all particles start from x = 0 at t = 0. First, we described the main features of the three different regimes summarized in Fig. 6.1. We found that up to the first characteristic time t * 1 = T /(cN 2 ), the gas is very dense, and the particles perform independent Brownian motion. Between t * 1 and the second characteristic time t * 2 = T /c 2 the particles start to feel the long-range interactions. The size of the system starts to grow linearly with time, and the particles still cross regularly. When t > t * 2 , the system is in Regime III. In this regime, the particles are equidistantly spaced, and they almost never cross each other. We analyzed the joint probability distribution of the positions given in Eq. (6.7) in the limit t → ∞ by using the saddle-point approximation. We found that by neglecting the double product prefactor in Eq. (6.13), we can identify the particle distribution at a fixed time with the particle distribution of the equilibrium one-dimensional one-component plasma studied in the previous chapters. Furthermore, we find that the late times in the ranked diffusion problem correspond to the limit of strong interactions in the equilibrium system. By using the results for the gaps in the equilibrium 1dOCP given in Ch. 4 in the strong coupling limit, we quantify the fluctuations of the displacements in Regime III of the ranked diffusion. We show that for strong interactions, the fluctuations of the bulk and the edge gaps are Gaussian and of order O( √ t). We have tested this prediction by numerically measuring the gaps. These results are shown in Fig. 6.2. As reported in Ref. [START_REF] Flack | Out of equilibrium dynamics of repulsive ranked diffusions: the expanding crystal[END_REF], there are sub-leading non-Gaussian corrections to the fluctuations which have a purely dynamical origin. Consequently, they can not be accessed by comparison with the equilibrium model.

In this chapter, we focused on Regime III, where we established the connection with the equilibrium jellium model. However, the ranked diffusion model exhibits many interesting properties that were not discussed here. For example, one could investigate the possibility that fluctuations in Regime II, where crossings of particles play an important role, can be captured by the equilibrium jellium model with smaller interaction strengths. Another possibility is to investigate the effects of different initial conditions. In Ref. [START_REF] Flack | Out of equilibrium dynamics of repulsive ranked diffusions: the expanding crystal[END_REF], we show that in the case of two particles, the choice of the initial condition affects the sub-leading non-Gaussian fluctuations. It remains to be seen how this translates to the case with many particles.

Conclusion

In this thesis, we studied the properties of one-dimensional charged particles interacting via long-range repulsive Coulomb interactions. We mainly focused on the one-dimensional one-component plasma at equilibrium. In this case, the effects of repulsive interactions are balanced by a confining harmonic potential. By using the Coulomb gas technique, we derived probability distributions of various observables in the limit of a large number of particles. We demonstrated that the 1dOCP is an useful laboratory for the study of long-range interactions. Despite its simplicity which allows us to obtain many exact results, it possesses various interesting properties connected to the long-range nature of the interactions.

In Ch. 1 we defined the jellium model and presented some of its well-known properties such as the average positions of the particles and the partition function. We also discussed two possible generalizations of the model: the Coulomb gases and the Riesz gases.

In Ch. 2, we explained the Coulomb gas method, which we used in Ch. 3, Ch. 4, and Ch. 5 to derive analytical results describing large deviations of different observables. First, we presented the discrete formulation of the method. In the second part of the chapter, we presented the Coulomb gas method by using the continuous framework valid for a large number of particles.

In Ch. 3 we explored the probability distribution of the general full linear statistics s = (1/N ) N i=1 f (x i ) in the limit of large N . By using the Coulomb gas technique, we derived a surprisingly compact formula for the variance, which reads Var(s)

≈ 1 4αN 3 2α -2α [f ′ (x)] 2 dx.
We also demonstrated that for large N , the full probability distribution of s admits a large deviation form P(s, N ) ∼ e -N 3 ϕ(s) . We provided an expression for the rate function ϕ(s) that captures typical and large fluctuations of the full linear statistics. We tested the validity of both results, the variance formula and the rate function, by considering different choices of the test functions f (x). For f (x) = x, f (x) = x 2 , we explicitly calculated the rate functions and found that they match with our formula for general f (x). We further tested the validity of the formula for the variance by considering f (x) = x 3 and f (x) = |x|. We found that our formula predicts the correct variance, even though some assumptions made during the derivation are not satisfied in these two cases. In future works, one could find a general formula for the variance and the rate function that does not assume single support of the saddle-point density, which was already done for Hermitian matrix models in Ref. [START_REF] Pastur | Limiting laws of linear eigenvalue statistics for Hermitian matrix models[END_REF]. There is another open question concerning the full rate function of cases similar to f (x) = x 3 where the effective potential is only locally confining. While our general formula for the variance correctly describes small fluctuations around the average, it would be interesting to find a description of large deviations.

In Ch. 4, we studied the probability distributions of two observables in the bulk of the system: the middle gap and the full counting statistics. In Section 4.1, we used the saddle-point approximation to show that for large N the probability distribution of the gap between two middle particles admits a large deviation form P bulk, gap (g, N ) ∼ e -N 3 ψgap(g) with the rate function ψ gap (g) = g 3 /(96α). This expression captures the large deviations of the middle gap that are of order O(1). The typical fluctuations of order O(1/N ) are instead described by the scaling form that we computed without resorting to the saddlepoint approximation. The results for the fluctuations of the middle gap are especially interesting if we compare them with the previously known results for the probability distribution of the gap near the edge given in Ref. [START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF]. We found that both large and typical fluctuations get modified if we move from the edge to the bulk of the system. Most notably, the tails of the distribution of the edge gap are Gaussian and independent on the interaction strength α. In the case of the bulk gap, the large deviation tails are not Gaussian anymore, and they are dependent on the interaction strength α. We attributed these changes to the fact that the bulk gaps are more strongly affected by the long-range interactions between the particles. In the future, it could be interesting to more closely examine these effects and find the crossover between the bulk gap and the edge gap distributions. In Section 4.2, we analyzed the probability distribution of the full counting statistics, which measures the number of particles inside of an interval [-L, L]. We considered only intervals where the edges of the interval are deep inside the bulk. We again used the saddle-point approximation to obtain the large deviation form that

reads P FCS (N I = κ I N, N ) ∼ e -N 3 ψ(κ I ) where ψ(z) = 2 3 α 2 z -L 2α 3
. Interestingly, the rate function displays a jump in the third derivative at the minimum that occurs at the average value of the full counting statistics NI = L/(2α). While the rate function captures the atypical fluctuations of order O(N ), it does not describe the typical fluctuations that are of order O(1). We found that they are described by a scaling form that we compute explicitly. The typical fluctuations are independent of the size of the interval 2L and the number of particles N . This indicates that the fluctuations of the number statistics in the jellium model are strongly suppressed due to the long-range interactions. In fact, they are suppressed more than in the log-gas, where the variance grows logarithmically with the average number of particles in the interval. We only investigated the full counting statistics when the interval is deep in the bulk of the system. In the future, it would be interesting to see how the probability distribution changes when the edges of the interval approach the edges of the system. In fact, in the case of the log-gas, the authors of Ref. [START_REF] Marino | Number statistics for β -ensembles of random matrices: Applications to trapped fermions at zero temperature[END_REF][START_REF] Marino | Phase transitions and edge scaling of number variance in gaussian random matrices[END_REF] investigated how the variance of the full counting statistics changes when we move the edges of the interval from the bulk to the edge of the system. The authors identified three distinct regimes as a function of the size of the interval. While the variance of the FCS grows logarithmically with the size of the interval in bulk, it decreases abruptly as L approaches the edges of the system.

In Ch. 5 we presented results connected to the truncated linear statistics s. It is defined as the rescaled sum of N ′ = κN rightmost particles, where 0 ≤ κ ≤ 1. This observable is proportional to the center of mass of the N ′ rightmost particles. It is especially interesting since it interpolates between two limits. On the one hand, when κ → 0, it reduces to the position of the rightmost particle. On the other hand, when κ → 1, it describes the full linear statistics, which was computed in Refs. [START_REF] Dhar | Exact extremal statistics in the classical 1d Coulomb gas[END_REF][START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF]. We extended these results by computing the full probability distribution of the truncated linear statistics for 0 ≤ κ ≤ 1. In this way, we obtained the connection between both limits κ → 0 and κ → 1. We found that the atypical fluctuations of the TLS are of order O(1) and can be described by the large deviation form ∼ e -N 3 ϕκ(s) . By using the Coulomb gas technique, we derived the exact expression for the rate function ϕ κ (s) in the limit of large N . Surprisingly, we found a rich phase diagram consisting of five phases in the κ-s space. The rate function ϕ κ (s) is given by different functional forms in each of the phases. Furthermore, we showed that at all phase boundaries, the rate function displays a jump in the third derivative, which indicates a third-order phase transition of the underlying Coulomb gas. We compare the analytical expressions for the rate function with numerical simulations, and we observe good agreement. In order to numerically compute the tails of the distribution, we used the importance sampling algorithm, which allowed us to sample very rare configurations. We also analyzed the rate function for κ → 0 and κ → 1. We found that the asymptotic results agree with the previously known expressions for the distribution of the rightmost particle and the full linear statistics. An obvious extension of this work would be to find the probability distribution of the truncated linear statistics for a general function f (x) as it was accomplished for the case of logarithmic interactions [START_REF] Grabsch | General truncated linear statistics for the top eigenvalues of random matrices[END_REF][START_REF] Grabsch | Truncated Linear Statistics Associated with the Eigenvalues of Random Matrices II. Partial Sums over Proper Time Delays for Chaotic Quantum Dots[END_REF][START_REF] Grabsch | Truncated linear statistics associated with the top eigenvalues of random matrices[END_REF].

Finally, in Ch. 6, we investigated thermal diffusion at the temperature T of N onedimensional particles that are not confined by a harmonic potential. Just like in the equilibrium case, the particles interact via linear repulsive pairwise Coulomb potential. The drift felt by each particle is the result of the Coulomb forces and is proportional to the position of the particle in the ordered configuration. Such a process is also known as the ranked diffusion. We focus on the initial condition where all particles start from the same point and find three different regimes that emerge as a function of time. In the beginning, the particles are still close to each other, and they cross each other very often. We name this regime Regime I or the dense regime. After the first characteristic time t * 1 ∼ T /(N 2 c 2 ), the system enters into Regime II. The crossover between Regime I and Regime II is clearly visible by looking at the system size that grows as √ 2N t in Regime I and as 2cN t in Regime II. The dilute regime or Regime III begins after the second characteristic time t * 2 ∼ T /c 2 . For t ≫ t * 2 , the particles form a well-organized crystal-like structure. They are far apart from each other and almost never cross. After describing the main properties of different regimes, we analyzed the formula for the joint probability density function of the positions P (x, t) in the large time limit by using saddle-point approximation. We found that in this limit, we can identify the time-dependent particle density to the one describing the equilibrium jellium model studied in the previous chapters. We used this analogy to obtain the leading order fluctuations of the bulk and edge gaps in the dynamical problem. We showed that both are Gaussian and of order O( √ t). There are many open questions that could be investigated in the future. For example, in Ref. [START_REF] Flack | Out of equilibrium dynamics of repulsive ranked diffusions: the expanding crystal[END_REF], we show that for N = 2 different initial conditions affect the subleading non-Gaussian fluctuations of the positions. One could investigate these effects for general N . Another possibility is to study further the connection between the equilibrium jellium model and the ranked diffusion. We showed that Regime III corresponds to the equilibrium model with strong interactions. However, there is a possibility that weaker interactions could capture the behavior of the dynamical system in Regime II.

In all of the chapters, we studied one-dimensional particles that interact via linear Coulomb potential. However, this is just one realization of more general Riesz gas, where the interactions decay potentially as a function of the interparticle distance ∼ |x j -x i | -k . An ambitious goal would be to extend any of our results to general Riesz gases.

Résumé en français

De nombreux phénomènes naturels sont régis par des interactions qui diminuent lentement avec la distance. Les forces gravitationnelles et électrostatiques sont des exemples évidents de ces interactions. En raison de leur omniprésence, les systèmes d'interaction à longue portée ont été étudiés par diverses communautés scientifiques, telles que l'astrophysique [START_REF] Padmanabhan | Statistical mechanics of gravitating systems[END_REF][START_REF] Chavanis | Statistical Mechanics of Twodimensional Vortices and Collisionless Stellar Systems[END_REF][START_REF] Chavanis | Phase transitions in self-gravitating systems and bacterial populations with a screened attractive potential[END_REF][START_REF] Hertel | A soluble model for a system with negative specific heat[END_REF][START_REF] Lynden-Bell | The gravo-thermal catastrophe in isothermal spheres and the onset of red-giant structure for stellar systems[END_REF][START_REF] Marcos | Collisional relaxation of two-dimensional self-gravitating systems[END_REF], la physique des plasmas [START_REF] Nicholson | Introduction to Plasma Theory[END_REF][START_REF] Elskens | Microscopic Dynamics of Plasmas and Chaos[END_REF], les lasers à électrons libres [START_REF] Antoniazzi | Free electron laser as a paradigmatic example of systems with long-range interactions[END_REF][START_REF] Barré | Statistical theory of high-gain free-electron laser saturation[END_REF], l'hydrodynamique [START_REF] Miller | Statistical mechanics of Euler equations in two dimensions[END_REF][START_REF] Ellis | Nonequivalent statistical equilibrium ensembles and refined stability theorems for most probable flows[END_REF][START_REF] Robert | Statistical equilibrium states for two-dimensional flows[END_REF][START_REF] Kulkarni | Hydrodynamics of cold atomic gases in the limit of weak nonlinearity, dispersion, and dissipation[END_REF] et la physique des atomes froids [START_REF] Chalony | Long-range onedimensional gravitational-like interaction in a neutral atomic cold gas[END_REF][START_REF] Zhang | Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator[END_REF]. La caractéristique déterminante d'un système d'interaction à longue portée est un potentiel d'interaction. Celui-ci varie comme une puissance de l'inverse de la distance interparticulaire avec un exposant plus petit que la dimension de l'espace d'intégration. Il s'avère que les systèmes avec de telles interactions se comportent de manière très différente des systèmes avec des interactions qui décroissent rapidement avec la distance interparticulaire. Une des conséquences les plus notables des interactions à longue portée est peut-être que l'énergie devient non additive. Cela signifie que l'énergie de deux systèmes en contact ne peut pas être calculée simplement en additionnant les deux énergies. L'absence d'additivité entraîne de nombreuses propriétés inhabituelles des systèmes avec des interactions à longue portée, notamment le comportement non extensif de l'énergie ainsi que la possibilité d'une chaleur spécifique négative [START_REF] Loladze | Effective negative specific heat by destabilization of metastable states in dipolar systems[END_REF][START_REF] Staniscia | Negative specific heat in the canonical statistical ensemble[END_REF][START_REF] Thirring | Systems with negative specific heat[END_REF], l'inéquivalence d'ensemble [START_REF] Lederhendler | Long-Range correlations and ensemble inequivalence in a generalized ABC model[END_REF][START_REF] Barré | Inequivalence of ensembles in a system with long-range interactions[END_REF][START_REF] Bouchet | Classification of phase transitions and ensemble inequivalence, in systems with long range interactions[END_REF][START_REF] Leyvraz | Ensemble inequivalence in systems with long-range interactions[END_REF][START_REF] Baldovin | Physical interpretation of the canonical ensemble for long-range interacting systems in the absence of ensemble equivalence[END_REF] et une relaxation anormalement lente vers l'équilibre [START_REF] Gupta | Quasistationarity in a model of classical spins with long-range interactions[END_REF][START_REF] Van Den | Relaxation timescales and decay of correlations in a long-range interacting quantum simulator[END_REF][START_REF] Gupta | Slow relaxation in long-range interacting systems with stochastic dynamics[END_REF][START_REF] Patelli | Linear response theory for longrange interacting systems in quasistationary states[END_REF]. Si certaines des caractéristiques particulières de certains systèmes spécifiques ont été comprises il y a longtemps, une compréhension théorique plus générale de la physique statistique des systèmes avec des interactions à longue portée a été développée plus récemment [START_REF] Campa | Statistical mechanics and dynamics of solvable models with long-range interactions[END_REF][START_REF] Campa | Physics of Long-Range Interacting Systems[END_REF][START_REF] Dauxois | Dynamics and Thermodynamics of Systems with Long-Range Interactions: An Introduction[END_REF][START_REF] Defenu | Longrange interacting quantum systems[END_REF][START_REF] Mukamel | Statistical mechanics of systems with long range interactions[END_REF].

La famille des gaz de Riesz [START_REF] Riesz | Intégrales de Riemann-Liouville et potentiels[END_REF] confinés constitue une plate-forme utile pour étudier le comportement des systèmes à longue portée. Il s'agit de l'un des modèles les plus généraux décrivant des particules interagissant par le biais d'interactions répulsives qui varient avec la distance sous la forme d'une loi de puissance ∼ |x i -x j | -k . En faisant varier le paramètre k, nous pouvons changer la nature des interactions de courte à longue portée, ce qui nous permet d'observer différents effets qui émergent lorsque les interactions commencent à décroître de plus en plus lentement avec la distance. Cependant, obtenir des résultats exacts qui s'appliquent à k en général est assez complexe. Par exemple, la solution pour le profil de densité du gaz de Riesz dans la limite d'un grand nombre de particules a été calculée il y a seulement quelques années [START_REF] Agarwal | Harmonically Confined Particles with Long-Range Repulsive Interactions[END_REF][START_REF] Hardin | Large Deviation Principles for Hypersingular Riesz Gases[END_REF]. Ce résultat a ensuite été étendu en calculant le profil de densité moyenne du gaz de Riesz en présence d'un mur [START_REF] Kethepalli | Harmonically confined long-ranged interacting gas in the presence of a hard wall[END_REF]. Dans les deux cas, ils ont trouvé des comportements significativement différents de la densité pour différentes valeurs de k. D'autres résultats importants comprennent la caractérisation complète des fluctuations de la particule la plus à droite [START_REF] Kethepalli | Edge fluctuations and third-order phase transition in harmonically confined longrange systems[END_REF] et l'expression de la fonction de corrélation des paires [START_REF] Beenakker | Pair correlation function of the one-dimensional Riesz gas[END_REF]. Très récemment, les fluctuations dynamiques dans le gaz de Riesz ont également été étudiées [START_REF] Dandekar | Dynamical fluctuations in the Riesz gas[END_REF][START_REF] Huse | Spatiotemporal spread of perturbations in power-law models at low temperatures: Exact results for classical out-of-time-order correlators[END_REF]. Il est intéressant de noter que les gaz de Riesz ne sont pas étudiés uniquement par la communauté des physiciens. Il convient de noter que l'étude des gaz de Riesz est également un domaine de recherche actif en mathématiques [START_REF] Hardin | Large Deviation Principles for Hypersingular Riesz Gases[END_REF][START_REF] Leblé | Large deviation principle for empirical fields of Log and Riesz gases[END_REF][START_REF] Lewin | Coulomb and Riesz gases: The known and the unknown[END_REF][START_REF] Chafaï | Aspects of Coulomb gases[END_REF][START_REF] Dereudre | Number-Rigidity and beta-Circular Riesz gas[END_REF][START_REF] Boursier | Decay of correlations and thermodynamic limit for the circular Riesz gas[END_REF][START_REF] Boursier | Optimal local laws and CLT for the circular Riesz gas[END_REF].

Bien qu'ils soient très intéressants, les résultats analytiques pour les k généraux sont, dans de nombreux cas, encore inaccessibles. Il est souvent plus réaliste de se concentrer sur des valeurs spécifiques de k. En fait, pour certaines valeurs spéciales de k, le gaz de Riesz correspond à certains modèles physiques renommés, qui ont fait l'objet d'études approfondies. Le plasma unidimensionnel à une composante (1dOCP) (k = -1) [START_REF] Lenard | Exact statistical mechanics of a one-dimensional system with Coulomb forces[END_REF][START_REF] Prager | The One-Dimensional Plasma[END_REF][START_REF] Baxter | Statistical mechanics of a one-dimensional Coulomb system with a uniform charge background[END_REF][START_REF] Choquard | One-Dimensional Coulomb Systems BT -Physics in One Dimension[END_REF] et le gaz logarithmique de Dysons (k → 0) [START_REF] Mehta | Random Matrices and the Statistical Theory of Spectra[END_REF][START_REF] Forrester | Log-Gases and Random Matrices[END_REF][START_REF] Dyson | Statistical theory of the energy levels of complex systems. I[END_REF][START_REF] Dyson | Statistical theory of the energy levels of complex systems. II[END_REF] sont deux exemples marquants de modèles à longue portée appartenant aux gaz de Riesz. Un modèle célèbre à courte portée qui appartient à la famille des gaz de Riesz est le système intégrable de Calogero-Moser qui est retrouvé pour k = 2 [START_REF] Calogero | Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials[END_REF][START_REF] Calogero | Exactly solvable one-dimensional many-body problems[END_REF][START_REF] Moser | Three integrable Hamiltonian systems connected with isospectral deformations[END_REF][START_REF] Diejen | Calogero-Moser-Sutherland Models[END_REF][START_REF] Sutherland | Exact results for a quantum many-body problem in one dimension[END_REF].

L'étude du gaz logarithmique de Dyson a permis de mieux comprendre le comportement des systèmes à longue portée. La popularité de ce modèle peut être largement attribuée à son lien avec la théorie des matrices aléatoires. Comme l'ont établi Dyson et Mehta [START_REF] Dyson | Statistical theory of the energy levels of complex systems. I[END_REF][START_REF] Dyson | Statistical theory of the energy levels of complex systems. II[END_REF][START_REF] Dyson | Statistical theory of the energy levels of complex systems. III[END_REF][START_REF] Mehta | Statistical theory of the energy levels of complex systems. IV[END_REF][START_REF] Mehta | Statistical Theory of the Energy Levels of Complex Systems. V[END_REF], il existe une correspondance entre la distribution de probabilité conjointe des valeurs propres des matrices aléatoires invariantes et les positions des particules confinées dans une dimension interagissant par l'intermédiaire d'un potentiel logarithmique répulsif. En reformulant les questions qui se posent dans la théorie des matrices aléatoires en termes de log-gaz, on peut utiliser une variété d'outils de la mécanique statistique pour étudier les propriétés des grandes matrices aléatoires. Une méthode qui exploite ce lien est connue sous le nom de méthode du gaz de Coulomb. Un avantage particulier de cette méthode est qu'elle permet d'étudier les fluctuations atypiques de différentes observables dans la limite d'un grand nombre de particules. Au cours des deux dernières décennies, la technique des gaz de Coulomb a permis d'obtenir de nombreux nouveaux résultats concernant les grands écarts de différentes quantités dans les ensembles de log-gaz ou, de manière équivalente, dans les ensembles de matrices aléatoires. Quelques exemples sont les queues de grand écart de la distribution de probabilité de la plus grande valeur propre, qui correspond à la particule la plus à droite dans l'image du gaz logarithmique [START_REF] Majumdar | Top eigenvalue of a random matrix: large deviations and third order phase transition[END_REF][START_REF] Vivo | Large deviations of the maximum eigenvalue in Wishart random matrices[END_REF][START_REF] Majumdar | Large deviations of the maximum eigenvalue for Wishart and Gaussian random matrices[END_REF][START_REF] Majumdar | Large deviations of the top eigenvalue of large Cauchy random matrices[END_REF][START_REF] Nadal | A simple derivation of the Tracy-Widom distribution of the maximal eigenvalue of a Gaussian unitary random matrix[END_REF], les résultats liés aux grands écarts des statistiques linéaires [START_REF] Krajenbrink | Linear statistics and pushed Coulomb gas at the edge of β-random matrices: Four paths to large deviations[END_REF][START_REF] Grabsch | Distribution of spectral linear statistics on random matrices beyond the large deviation function -Wigner time delay in multichannel disordered wires[END_REF][START_REF] Grabsch | Capacitance and charge relaxation resistance of chaotic cavities -Joint distribution of two linear statistics in the Laguerre ensemble of random matrices[END_REF][START_REF] Chen | Distribution of linear statistics in random matrix models (metalic conductance fluctuations)[END_REF][START_REF] Cunden | Universality of the weak pushedto-pulled transition in systems with repulsive interactions[END_REF][START_REF] Cunden | A shortcut through the Coulomb gas method for spectral linear statistics on random matrices[END_REF], les statistiques linéaires tronquées [START_REF] Grabsch | General truncated linear statistics for the top eigenvalues of random matrices[END_REF][START_REF] Grabsch | Truncated Linear Statistics Associated with the Eigenvalues of Random Matrices II. Partial Sums over Proper Time Delays for Chaotic Quantum Dots[END_REF][START_REF] Grabsch | Truncated linear statistics associated with the top eigenvalues of random matrices[END_REF], et les statistiques de comptage [START_REF] Dean | Extreme value statistics of eigenvalues of Gaussian random matrices[END_REF][START_REF] Dean | Large deviations of extreme eigenvalues of random matrices[END_REF][START_REF] Majumdar | How many eigenvalues of a Gaussian random matrix are positive?[END_REF][START_REF] Fogler | Probability of an eigenvalue number fluctuation in an interval of a random matrix spectrum[END_REF][START_REF] Marino | Number statistics for β -ensembles of random matrices: Applications to trapped fermions at zero temperature[END_REF][START_REF] Marino | Phase transitions and edge scaling of number variance in gaussian random matrices[END_REF]. L'approche du gaz de Coulomb nous permet d'obtenir la forme des grands écarts d'une observable donnée en résolvant le problème d'optimisation déterminé par le gaz de Coulomb contraint. Dans de nombreux cas, le système contraint présente différentes phases qui correspondent à différents régimes de la distribution de probabilité [START_REF] Majumdar | Top eigenvalue of a random matrix: large deviations and third order phase transition[END_REF][START_REF] Nadal | A simple derivation of the Tracy-Widom distribution of the maximal eigenvalue of a Gaussian unitary random matrix[END_REF][START_REF] Cunden | A shortcut through the Coulomb gas method for spectral linear statistics on random matrices[END_REF][START_REF] Allez | Index distribution of the Ginibre ensemble[END_REF][START_REF] Cunden | Universal covariance formula for linear statistics on random matrices[END_REF][START_REF] Cunden | Universality of the third-order phase transition in the constrained Coulomb gas[END_REF][START_REF] Cunden | Joint statistics of quantum transport in chaotic cavities[END_REF][START_REF] Nadal | Phase transitions in the distribution of bipartite entanglement of a random pure state[END_REF][START_REF] Nadal | Statistical Distribution of Quantum Entanglement for a Random Bipartite State[END_REF][START_REF] Colomo | Third-order phase transition in random tilings[END_REF].

Le deuxième modèle à longue portée qui appartient à la famille des gaz de Riesz est le plasma unidimensionnel à un seul composant. Dans ce cas, le potentiel d'interaction croît linéairement avec la distance interparticulaire, ce qui signifie que les effets à longue portée sont encore plus forts que dans le cas du gaz logarithmique. Le 1dOCP est un représentant particulièrement intéressant des gaz de Riesz unidimensionnels puisque le potentiel d'interaction linéaire correspond à l'interaction électrostatique de Coulomb en une dimension. Les propriétés statistiques du 1dOCP ont été étudiées pour la première fois dans les années soixante par Lenard, Prager, Baxter et Choquard [START_REF] Lenard | Exact statistical mechanics of a one-dimensional system with Coulomb forces[END_REF][START_REF] Prager | The One-Dimensional Plasma[END_REF][START_REF] Baxter | Statistical mechanics of a one-dimensional Coulomb system with a uniform charge background[END_REF][START_REF] Choquard | One-Dimensional Coulomb Systems BT -Physics in One Dimension[END_REF]. Depuis lors, différents aspects de la question ont été étudiés [START_REF] Dean | Effects of dielectric disorder on van der Waals interactions in slab geometries[END_REF][START_REF] Téllez | Screening like charges in one-dimensional Coulomb systems: Exact results[END_REF][START_REF] Téllez | Like-charge attraction in one-and two-dimensional Coulomb systems[END_REF][START_REF] Varela | Like-charge attraction at short distances in a chargeasymmetric two-dimensional two-component plasma: Exact results[END_REF][START_REF] Varela | One-dimensional colloidal model with dielectric inhomogeneity[END_REF][START_REF] Varela | Configurational and energy landscape in onedimensional Coulomb systems[END_REF][START_REF] Varela | Relaxation dynamics of two interacting electrical double-layers in a 1D Coulomb system[END_REF][START_REF] Chafaï | At the edge of a one-dimensional jellium[END_REF][START_REF] Rojas | Universal behavior of the full particle statistics of one-dimensional Coulomb gases with an arbitrary external potential[END_REF]. Très récemment, les auteurs de Ref. [START_REF] Dhar | Exact extremal statistics in the classical 1d Coulomb gas[END_REF][START_REF] Dhar | Extreme statistics and index distribution in the classical 1d Coulomb gas[END_REF] ont adapté avec succès la technique des gaz de Coulomb pour étudier les grandes déviations de la particule la plus à droite, le nombre de particules sur l'axe positif et l'écart entre deux particules près du bord du système. Cependant, comparé à l'abondance des résultats décrivant les grandes déviations dans le gaz logarithmique, il y a beaucoup d'observables dans le plasma unidimensionnel à une composante qui n'ont pas encore été explorés de ce point de vue. Le but de cette thèse est de présenter de nouveaux résultats qui complètent les connaissances existantes sur les propriétés du plasma unidimensionnel à une composante. Nous utilisons la technique du gaz de Coulomb pour étudier les grandes déviations des statistiques linéaires complètes, le bulk gap, les statistiques de comptage complètes et les statistiques linéaires tronquées. Dans certains cas, nous complétons cette approche par d'autres techniques qui nous permettent de sonder les fluctuations typiques. Les recherches qui ont abouti aux résultats présentés dans cette thèse ont été menées pendant les trois années de mon doctorat sous la direction de Satya N. Majumdar et Grégory Schehr, en collaboration avec d'autres chercheurs. Certains des résultats obtenus sous la supervision de Sergei Nechaev et d'autres collaborateurs ne sont pas présentés dans ce manuscrit. Nous avons pris cette décision parce que les sujets étudiés divergent de manière significative du thème central de cette thèse. Par souci d'exhaustivité, nous incluons la prépublication [START_REF] Flack | Generalized Devil ' s staircase and RG flows[END_REF] basée sur ce travail dans l'annexe B de la thèse.

. Panorama de la thèse

Cette thèse est structurée comme suit. Dans Ch. 1 nous définissons le plasma unidimensionnel à une composante ou le modèle du jellium. Nous présentons également les gaz de Coulomb et de Riesz. Dans la dernière partie du chapitre, nous résumons quelques propriétés d'équilibre bien connues du modèle jellium.

Dans Ch. 2 nous introduisons la technique des gaz de Coulomb, que nous utilisons dans les chapitres suivants pour trouver la forme de grand déviations. Nous présentons les formulations discrètes et continues de la méthode.

Les chapitres 3, 4, 5 et 6 sont basés sur les résultats publiés dans quatre articles différents fournis à la fin de chaque chapitre.

Dans le chapitre 3, nous nous concentrons sur les statistiques linéaires complètes. En utilisant la technique du gaz de Coulomb, nous dérivons l'expression de la fonction de taux des statistiques linéaires générales. En analysant le comportement de la fonction de taux obtenue autour de son minimum, nous trouvons une formule exacte pour la variance. Nous continuons à vérifier la validité de ce résultat en calculant explicitement les fonctions de taux pour différents choix de fonctions de test (f (x) = x, f (x) = x 2 , f (x) = x 3 , et f (x) = |x|). En développant les fonctions de taux autour de leurs minima, nous obtenons la variance pour les choix spécifiques de f (x). Nous constatons que les résultats correspondent à la formule générale que nous avons dérivée dans la première partie du chapitre.

Dans Ch. 4 nous nous concentrons sur les distributions de probabilité de deux observables : l'écart entre deux particules voisines qui sont profondes dans la masse du Au centre, nous voyons une représentation de la distribution de probabilité des statistiques linéaires tronquées P N,κ (s). Quatre régimes différents qui apparaissent pour 0 < κ ≤ 1/2 sont séparés par des lignes verticales en pointillés. Dans le coin supérieur droit, les densités de points de selle associées sont représentées. Dans le coin supérieur gauche, la fonction de grande déviation ϕ κ (s) (voir Eq. ( 21)) est représentée en fonction de s, ainsi que les résultats de la simulation numérique (points rouges). système et la statistique de comptage complet (FCS), qui mesure le nombre de particules à l'intérieur d'un intervalle donné. En utilisant la technique du gaz de Coulomb, nous dérivons d'abord la forme de grande déviation décrivant les fluctuations atypiquement grandes. Nous continuons en trouvant les fluctuations typiques qui ne sont pas capturées par la forme de grande déviation. Dans la deuxième partie du chapitre, nous nous concentrons sur le FCS. Là encore, nous commençons par dériver la forme de grande déviation décrivant les fluctuations atypiques. Ensuite, nous dérivons la forme d'échelle qui capture le comportement des fluctuations typiques. À la fin du chapitre, nous discutons du lien entre les deux observables, le bulk gap et le FCS.

Dans le Ch. 5, nous étudions la statistique linéaire tronquée, qui est donnée par la somme des N ′ particules les plus à droite. Etant donné que le système est composé de N particules, nous désignons la fraction des particules incluses dans la somme par κ = N ′ /N . Nous utilisons à nouveau la méthode des gaz de Coulomb pour dériver la fonction de taux pour toutes les valeurs possibles de κ ∈ [0, 1]. Le résultat principal de ce chapitre est l'expression explicite de la fonction de taux, qui a des formes différentes pour 0 ≤ κ ≤ 1/2 et 1/2 < κ ≤ 1. Nous montrons que pour un κ fixe, la fonction de taux présente quatre régimes différents qui sont présentés dans la Fig. 6.3. Nous analysons également les comportements asymptotiques de la fonction de taux et les comparons aux résultats précédemment connus pour la position de la particule la plus à droite (κ → 0) et les statistiques linéaires complètes (κ → 1). À la fin du chapitre, nous décrivons l'algorithme que nous utilisons pour calculer numériquement la distribution de probabilité complète des statistiques linéaires tronquées. Enfin, dans le Ch. 6, nous nous éloignons de l'équilibre et explorons la dynamique de Langevin des particules en une dimension qui interagissent via un potentiel d'interaction de Coulomb unidimensionnel répulsif. Ce processus est également connu sous le nom "Repulsive ranked diffusion". Tout d'abord, nous présentons le modèle et discutons des propriétés des trois régimes qui émergent à différentes échelles de temps et sont représentés dans Fig. 6.4. Nous poursuivons en analysant la fonction de densité de probabilité des positions à des moments tardifs. Nous montrons qu'aux temps tardifs, il existe une analogie entre le modèle dynamique et le système jellium à l'équilibre. À la fin du chapitre, nous utilisons ce lien pour prédire les fluctuations typiques des lacunes du milieu et du bord dans le problème de la diffusion classée.

Figure 1 :

 1 Figure 1: Left panel: The phase diagram in the κ-s plane for α = 1. The different phases have different functional forms for the large deviation function ϕ κ (s). For 0 < κ ≤ 1/2, there are four different phases I, II, IIIb and IV. Similarly, for κ > 1/2 we also find four phases I, II, IIIa and IV. Right panel:In the center, we see a depiction of the probability distribution of the truncated linear statistics P N,κ (s). Four different regimes that appear for 0 < κ ≤ 1/2 are separated by dotted vertical lines. In the upper right corner, the associated saddle-point densities are plotted. In the upper left corner, the large deviation function ϕ κ (s) (see Eq. (21)) is plotted as a function of s, together with the results from numerical simulation (red dots).

Figure 2 :

 2 Figure 2: Upper panel: examples of the trajectories x i (t) vs t of N = 500 particles evolving via the Langevin equation in Eq.[START_REF] Leyvraz | Ensemble inequivalence in systems with long-range interactions[END_REF]. One can identify three different regimes determined by the value of the dimensionless time c 2 t. Bottom panel: corresponding densities of particles ρ(x, t f ) at the final time t = t f for each of the three top figures. They are obtained by averaging over 10 4 realizations of the noise, and using 100 bins to construct the histograms. Here for convenience we chose t f = 50 and varied c.

Figure 1 . 1 :

 11 Figure 1.1: Schematic depiction of the 1dOCP. The red dots represent the charges confined to a one-dimensional line interacting via repulsive linear interactions. The dashed curve shows the harmonic trap that pulls the particles toward the origin.

Figure 1 . 2 :

 12 Figure 1.2: This figure shows interaction potentials (left panel) and the absolute value of interaction forces (right panel) for Riesz gases with different k ∈ {-1.8, -1, -0.5, 2}.r is the distance between two particles. The force is given by the first derivative of the potential. We can clearly see the difference between k = 2, which falls into the shortranged regime, k = -1.8, which is strongly long-ranged, and k = -0.5 which belongs to the weakly long-ranged regime. The Orange colored line represents two-particle interaction for the 1dOCP. In this case, the force is just a constant.

Figure 2 . 1 :

 21 Figure 2.1: This figure shows the comparison of the probability distributions of S Ngiven by the exact solution in Eq. (2.5) (blue circles), Gaussian approximation in Eq. (2.8) (magenta line), and the large deviation approximation in Eq. (2.7) (green line). On the vertical axis, we plot the logarithm of probability, and the horizontal axis shows the number of coins with value 1, which equals to sN , where N = 1000 in this case. The main plot shows the full range of sN , from 0 to 1000. We can see that for rare events, the Gaussian approximation starts to fail. The inset shows just a narrow region around N/2, where both approximations agree well with the exact solution.

Figure 3 . 1 :

 31 Figure 3.1: A schematic drawing of the density ρ * µ (x) with single support. The edges of the support are denoted by L 1 and L 2 .

s and s = 4α 2 3 .Figure 3 . 2 :

 332 Figure 3.2: Top left: Plot of the effective potential for f (x) = x (Eq. (3.32)) for µ = 0 (green solid line) and µ = -1 (red solid line). Bottom left: Plot of the saddle-point density for f (x) = x and two different values of s = s = 0 and s = 1, corresponding to µ = 0 and µ = -1 (see Eq. (3.33)). For f (x) = x, the average unperturbed value of s is s = 0 (this is shown by the red curve). When s changes from s = 0 to s = 1, the new density (shown by the green curve) gets shifted by s = 1. Top right: The plot of the effective potential for f (x) = x 2 given in Eq. (3.37) for µ = 0.006 (red solid line) and µ = -0.442 (green solid line). Bottom right: Plot of the saddle-point density for f (x) = x 2 and two different values of s = s = 4α 2 /3 and s = 1 corresponding to µ = 0.006 and µ = -0.442 (see Eq. (3.39)). For f (x) = x 2 , the average unperturbed value of s is s = 4α 2 /3, where the density is flat over [-2α, 2α] (this is shown by the red curve). When s changes from s = s to s = 1, the new density (shown by the green curve), while remaining flat over [-√ 3, + √ 3], gets rescaled. In the figures, we chose α = 1/10.

Figure 3 . 3 :

 33 Figure 3.3: Plot of the effective potential V eff (x) given in Eq.(3.42). We set µ = 1/10. The potential displays a local minimum at x = 0 and a local maximum at x = -1/(3µ). The height of the maximum is 1/(54µ). The inset shows the saddle-point density ρ * µ (x) calculated from Eq. (3.44) (solid black line) compared with Monte-Carlo simulation (red dots). For simulation, we used N = 1000 particles and set µ = 0.1 and α = 1/10. We averaged over 4 × 10 7 realizations.

Figure 3 . 4 :

 34 Figure 3.4: A plot of |b -b MC | vs N where b = 144 5 α 4 (with α = 0.01) is the theoretical prediction for N 3 Var(s) for f (x) = x 3 and b MC is the Monte-Carlo value of N 3 Var(s).

. 43 )

 43 Despite the non-confining nature of the effective potential, we can try to use our results to calculate the saddle-point density and the variance. We use Eqs. (3.13), (3.14), (3.15) and (3.18) to get the shape of the saddle-point density given by

. 46 )

 46 We have verified the validity of this prediction by Monte-Carlo simulations, as shown in Fig.3.4 where we plot the difference between the theoretical value of b =144 5 α 4 and the Monte-Carlo value b MC = N 3 Var(s) MC as a function of increasing N . It decreases to zero as N → ∞, thus verifying the theoretical prediction. Why does our equation for the variance (3.30) predict the correct result even though the effective potential is not confining? The answer is hiding in the observation that the variance describes the behavior when the linear statistics s is close to the average value s

Figure 3 . 5 :

 35 Figure 3.5: Effective potentials V eff (x) = x 2 /2 + µ|x| vs x for three values of µ. From left to right the values of µ are -1, 0, and 1. If µ < 0 the potential has two minima and for µ > 0 just one. The transition point is at µ = 0.

. 50 )Figure 3 . 6 :

 5036 Figure 3.6: Left: Plot of the average density obtained by Monte-Carlo simulation (red dots) for α = 1/10 and N = 1000 for s = 0.6, compared with the saddle-point prediction in Eq. (3.52). In this case, since s = 0.6 > s = α = 1/10, the density has two disjoint supports with parameters L = s + α = 0.7 and a = s -α = 0.5. Right: Plot of the Monte-Carlo simulations for the average density (red dots) for α = 1/10 and N = 1000 for s = 0.025, compared with the saddle-point prediction in Eqs. (3.50) and (3.51). In this case since s = 0.025 < s = 1/10, the density has a single support with an additional δ-peak at x = 0.

Figure 3 . 7 :

 37 Figure 3.7: A plot of |b -b MC | vs N where b = 1 (with α = 0.01) is the theoretical prediction for N 3 Var(s) for f (x) = |x| and b MC is the Monte-Carlo value of N 3 Var(s).

  (3.55) around s = s with our formula in Eq. (3.30). Expanding the rate function around its minimum gives us Var(s) ≈ b N 3 with b = 1 , (3.57) which is in agreement with Eq. (3.30) when f (x) = |x|. Additionally, we compared the formula in Eq. (3.30) with Monte-Carlo simulations as shown in Fig. 3.7. We plot the difference between the theoretical value of b = 1 and the Monte-Carlo value b MC = N 3 Var(s)

Article 1 An

 1 exact formula for the variance of linear statistics in the one-dimensional jellium model A. Flack, S. N. Majumdar and G. Schehr, J. Phys. A: Math. and Theor. 56, 105002

Figure 4 . 2 :

 42 Figure 4.2: Examples of functions F α (x) for α ∈ {0.01, 0.1, 1} by numerically solving Eq. (4.7). The corresponding values of the eigenvalue A(α) are: {15.737, 1.132, 0.437}. We can clearly see that for x → -∞ the function approaches zero and for x → ∞ the function goes to one (see Eq. (4.8)).

Fig. 4 .

 4 2 we show the numerically obtained function F α (x) for three values of α ∈ {0.01, 0.1, 1}. Returning to the typical fluctuations of the edge gap in the first line of Eq. (4.5), we see that the typical fluctuations of the edge gap are of order O(1/N ), and the scaling function h α (z) has an asymptotic Gaussian tail given by h α (z) ∼ e -z 2 2 .

Figure 4 . 3 :

 43 Figure 4.3: Schematic representation of a configuration of charges where the charges inside the interval [-L, +L] are shown by red-filled circles. Those to the left of -L are shown in blue, while those to the right of +L are shown in green. We denote the number of charges in the three intervals respectively by N L , N I and N R . The location of the particles that are at the edges of these intervals are also marked.

Fig. 4 . 3 ,

 43 we schematically show an ordered configuration of the particles together with the edges of the interval ±L. The FCS, as defined in Eq. (4.11) measures the number of particles inside [-L, L], which are presented as red circles in Fig.4.3. We know that the equilibrium density of the 1dOCP is uniform (see Eq. (1.32) and Eq. (1.48)), and therefore we can quickly determine the average number of particles in an interval [-L, L].

Figure 4 . 4 :

 44 Figure 4.4: Density of the 1dOCP with the additional constraint that the positions of the two central particles are x N/2 = -g/2 and x N/2+1 = +g/2 leading to a mid-gap g in the bulk. On the left panel, we provide a schematic depiction of the ansatz for the saddlepoint density in Eq. (4.19), which consists of two symmetric flat blocks over [-a, -g/2] and [+g/2, +a] with height h in each block and, in addition, two δ-peaks of amplitude Y located at ±g/2. As argued in the text (see Eq. (4.28)), the saddle-point density has parameter values a = a * = 2α, h = h * = 1/(4α) and Y = Y * = g/(8α). On the right panel, we show the result of Monte-Carlo simulations with N = 1000 particles and gap of size g = 0.1, with coupling parameter α = 0.1. This would correspond to a * = 0.2, h * = 2.5 and Y * = 0.125. The simulation confirms the saddle-point ansatz depicted schematically on the left panel.

  [START_REF] Lederhendler | Long-Range correlations and ensemble inequivalence in a generalized ABC model[END_REF]. We decide to eliminate Y , which results in the saddle-point energy Ẽ * expressed in terms of two unknown parameters a and h. Their values are determined by minimizing the energy Ẽ

Figure 4 . 5 :

 45 Figure 4.5: We measured the distribution of the middle gap P gap,bulk (g, N ) by performing Monte-Carlo simulations for N = 50, 100, 500 and N = 1000, for fixed α = 1. The data collapse in a scaling form P gap,bulk (g, N ) ≈ H α=1 (g N ) as predicted by the theory (see Eq. (4.48)). The numerically obtained scaling function H α (z) (shown with symbols) is in excellent agreement with the theoretical prediction in Eq. (4.50) with α = 1 (solid line).The inset shows the same data, however, we plot it with a logarithmic scale on the H α (z) axis.

  us begin with the overpopulated regime where the fraction of charges inside the interval [-L, L] is greater than the average value. Before computing the saddle-point density for this regime, it is instructive to look at the simulation of the jellium model with the fixed number of particles κ I > κI in the interval [-L, L]. The numerical results are presented in the left panel of Fig. 4.6. We can see that the constraint on the number of particles in the interval causes the support of the density to split in three intervals: [-2α, -b], [-L, L], and [b, 2α]. The charges in the two outer regions [-2α, -b] and [b, 2α] remained distributed equidistantly. The overpopulated middle region acquired two δ-peaks at the edges of the interval. It seems that the extra charge in this interval came from the two regions [-b, -L] and [L, b] located just outside of the interval [-L, L]. These observations lead us to the following ansatz for the saddle-point density

Figure 4 . 6 :

 46 Figure 4.6: Monte-Carlo simulations for the charge density with N = 1000 particles, α = 0.1 and L = 0.05. In this case κ = L/(2α) = 0.25. Left panel: Overpopulated regime κ I = 0.4 > κ = 0.25, which confirms the ansatz in Eq. (4.55). Right panel: Underpopulated regime κ I = 0.1 < κ = 0.25, confirming the ansatz in Eq. (4.62).

Figure 4 . 7 :

 47 Figure 4.7: Left panel: Schematic depiction of the ansatz for the saddle-point density in Eq. (4.55) for the overpopulated regime, i.e. κ I > κ. Right panel: Schematic depiction of the ansatz for the saddle-point density in Eq. (4.62) for the underpopulated regime, i.e. κ I < κ. The vertical lines at x = ±L in both figures represent a cartoon of the δ-peaks.

Figure 4 . 8 :

 48 Figure 4.8: The absolute value of the covariance |C N L ,N R (N )| in Eq. (4.75) between N L and N R , is plotted on a semi-log plot as a function of N , for the choice of parameters α = 0.1 and L = 0.05. The straight line with a negative slope confirms the exponential decay of the covariance function with increasing N .

Figure 4 . 9 :

 49 Figure 4.9: We verify by means of Monte-Carlo simulations that P FCS (N I , N ) indeed satisfies the scaling form in Eq. (4.80). The data for four values of N = 50, 250, 500 and 1000 collapse onto a single scaling curve, as shown by the symbols. This is compared with the theoretical prediction for the scaling function U α (z) in Eq. (4.81) shown by the solid line. In Eq. (4.81), the scaling function U α (z) was evaluated by computing F α (z) numerically from Eq. (4.7). The values of the parameters are α = 0.1 and L = 0.05. The inset shows the same data in semi-log plot. The agreement is excellent.

  ) are described by the large deviation form given in Eq. (4.30). Typical fluctuations of the gap are of order O(1/N ) and are captured by the scaling form in (4.48). The FCS displays large fluctuations of order O(N ) given in Eq. (4.69) and typical fluctuations of order O(1) characterized by the scaling form in Eq. (4.80).

Figure 5 . 1 :

 51 Figure 5.1: The phase diagram in the κ-s plane for α = 1. For 0 < κ ≤ 1/2, there are four different phases I, II, IIIb and IV, separated respectively by the lines s 1 (κ), s 2 (κ) and s 3 (κ) given in Eqs. (5.57)-(5.59) represented by black solid lines. Similarly, for 1/2 ≤ κ ≤ 1, there are also four phases I, II, IIIa and IV, separated respectively by the phase boundaries s1 (κ), s2 (κ) and s3 (κ) given in Eqs. (5.65)-(5.67) represented by black solid lines.

Figure 5 . 2 :

 52 Figure 5.2: The density profile associated to the minimum energy configuration for µ < 0. The density is flat over two disjoint intervals, noted by left (L) and right (R), which are separated by a gap -µ > 0.

Figure 5 . 3 :

 53 Figure 5.3: A schematic representation of the saddle-point configuration of the charges for µ > 0. The "tower" at the location d represents a delta-function which contains N nl orange particles and N nr blue particles. The number of red particles to the left of the delta-function is N n l , while the number of green particles to the right of the delta function is denoted by N n r . We denote the locations of the rightmost and the leftmost particles respectively by r and l.

Figure 5 . 4 :

 54 Figure 5.4: Phase diagram in the κ-µ plane for α = 1. We found five different phases for the jellium model: Phase I (blue), Phase II (grey), Phase IIIa (green), Phase IIIb (yellow), and Phase IV (red) with different minimum energy configurations for a fixed µ. For 0 < κ ≤ 1/2, the phase boundaries are given respectively µ = 0, µ = 8ακ and µ = 2α/κ (from bottom to top). In contrast, for 1/2 ≤ κ ≤ 1, the corresponding phase boundaries are given by µ = 0, µ = 8α(1 -κ) and µ = 2α/(1 -κ) (from bottom to top). At all of the boundaries separating any two phases, the rate function ϕ κ (s) displays a discontinuity in the third derivative, which indicates a third-order phase transition between different phases (see Eq. (5.62)-(5.64), and Eqs. (5.69)-(5.71)). On the upper and right edges of the figure, we see the associated density profiles where the color of the borders of the windows correspond to the colors of the phases. In order to transform this phase diagram shown in the κ-µ space into the phase diagram in κ-s space (see Fig. 5.1) we use the relations µ(s) derived in Sec. 5.2.

Figure 5

 5 Figure 5.5: In the center, we see a schematic depiction of the probability distribution of the truncated linear statistics P N,κ (s). Four different regimes that appear for 0 < κ ≤ 0.5 (see the text) are separated by dotted vertical lines (for 0.5 ≤ κ ≤ 1 the Phase IIIb is replaced by Phase IIIa). In the upper right panel, the associated saddle-point densities are plotted. In the upper left panel, the large deviation function ϕ κ (s) is plotted vs. s, together with the results from numerical simulation (red dots).

Figure 5 . 6 :

 56 Figure 5.6: Comparison between the numerically computed large deviation function ϕ κ (s) and its analytical predictions for κ = 0.3 (left panel) and κ = 0.7 (right panel). We computed ϕ κ (s) by using the importance sampling algorithm described in Sec. 5.5

Figure 6 . 1 :

 61 Figure 6.1: Upper panel left: summary of the values of the dimensionless final time c 2 t used in the figures a), b), and c) below. The characteristic times t * i (Eq. (6.3)) and t * 2 Eq. (6.4)) with T = 1 are represented by vertical lines separating the regimes. Upper panel right: size of the gas as a function of time (in a ln-ln scale), which shows a crossover between Regimes I and II. The time t * 1 = T /(N c) 2 is indicated by a vertical dashed line (for T = 1). Symbols are the results of numerical simulations. The pink solid line represents the prediction in Regime I, l T = √ 2 T t (here T = 1). The green solid line corresponds to the prediction in regime II, l = 2N ct. Middle panel, figures a-c: examples of the trajectories x i (t) vs t of N = 500 particles evolving via the Langevin equation in Eq. (6.1). One can identify three different regimes determined by the value of the dimensionless time c 2 t. Bottom panel, figures a-c: corresponding densities of particles ρ(x, t f ) at the final time t = t f for each of the three top figures. They are obtained by averaging over 10 4 realizations of the noise and using 100 bins to construct the histograms. Here for convenience, we chose t f = 50 and varied c. These three values fall in each of the three regimes I-III discussed in the text.

Figure 6 . 2 :

 62 Figure 6.2: Plot of the distribution of the centered and scaled gap G = G RD -2ct 2 √ t

Figure 6 . 3 :

 63 Figure 6.3: Panneau gauche : Diagramme de phase dans le plan κ-s pour α = 1. Les différentes phases ont des formes fonctionnelles différentes pour la fonction de grande déviation ϕ κ (s). Pour 0 < κ ≤ 1/2, il y a quatre phases différentes I, II, IIIb et IV. De même, pour κ > 1/2, nous trouvons également quatre phases I, II, IIIa et IV. Panneau droit :Au centre, nous voyons une représentation de la distribution de probabilité des statistiques linéaires tronquées P N,κ (s). Quatre régimes différents qui apparaissent pour 0 < κ ≤ 1/2 sont séparés par des lignes verticales en pointillés. Dans le coin supérieur droit, les densités de points de selle associées sont représentées. Dans le coin supérieur gauche, la fonction de grande déviation ϕ κ (s) (voir Eq. (21)) est représentée en fonction de s, ainsi que les résultats de la simulation numérique (points rouges).

Figure 6 . 4 :

 64 Figure 6.4: Panneau supérieur : exemples de trajectoires x i (t) en fonction de t de N = 500 particules évoluant via l'équation de Langevin[START_REF] Leyvraz | Ensemble inequivalence in systems with long-range interactions[END_REF]. On peut identifier trois régimes différents déterminés par la valeur du temps sans dimension c 2 t. Panneau inférieur : densités correspondantes de particules ρ(x, t f ) au temps final t = t f pour chacune des trois figures du haut. Elles sont obtenues en calculant la moyenne sur 10 4 réalisations du bruit, et en utilisant 100 bins pour construire les histogrammes. Ici, pour des raisons de commodité, nous avons choisi t f comme valeur de référence. Par commodité, nous avons choisi t f = 50 et fait varier c.

  2.15)). The saddle-point configuration {x * i } or the saddle-point density profile ρ * µ (x), which minimizes the energy of the constrained gas or the effective action (see Eqs. (2.16), and (2.35)) is determined by solving the saddle-point equations given in Eqs. (2.18), (2.22), in the discrete formulation and Eq. (2.36) in the continuous case. After finding the solution for {x * i } or ρ * µ (x), we need to evaluate the energy βE µ [{x *i } or the action functional S[ρ * µ (x), µ, µ 0 ] at the saddlepoint. We also present an alternative path for obtaining the rate function (see Eqs.(2.25), and (2.41)), which includes integrating the relation µ * (s). We find µ * as a function of s by evaluating the linear statistics s at the saddle-point and inverting the relation.

  .2 in the top left corner. The shape of the saddle-point density determined by Eq. (3.13) together with (3.14), (3.15) and (3.18) reads

  .16)

		Y		Y		10
							8
						ρ * (x)	4 6
	h -a	-g/2	0	g/2	a	x	2 0	0.2 -2α	0.1 -g/2	0.0	g/2	0.1	0.2 2α	x

Acknowledgements

Article 2 Gap probability and full counting statistics in the one-dimensional one-component plasma

A. Flack, S. N. Majumdar and G. Schehr, J. Stat. Phys. 5, 053211 (2022).

Abstract:

We consider the 1d one-component plasma (OCP) in thermal equilibrium, consisting of N equally charged particles on a line, with pairwise Coulomb repulsion and confined by an external harmonic potential. We study two observables: (i) the distribution of the gap between two consecutive particles in the bulk and (ii) the distribution of the number of particles N I in a fixed interval I = [-L, +L] inside the bulk, the so-called fullcounting-statistics (FCS). For both observables, we compute, for large N , the distribution of the typical as well as atypical large fluctuations. We show that the distribution of the typical fluctuations of the gap g are described by the scaling form P gap,bulk (g, N ) ∼ N H α (g N ), where α is the interaction coupling and the scaling function H α (z) is computed explicitly. It has a faster than Gaussian tail for large z: H α (z) ∼ e -z 3 /(96α) as z → ∞. Similarly, for the FCS, we show that the distribution of the typical fluctuations of N I is described by the scaling form P FCS (N I , N ) ∼ 2α U α [2α(N I -NI )], where NI = L N/(2α) is the average value of N I and the scaling function U α (z) is obtained explicitly. For both observables, we show that the probability of large fluctuations are described by large deviation forms with respective rate functions that we compute explicitly. Our numerical Monte-Carlo simulations are in good agreement with our analytical predictions.

Note that these parameters are not independent. The number of particles coming from the right block is fixed to κN , which means that n r + nr = κ , n l + nl = 1 -κ .

(5.25)

Because we assume that the particles outside of the δ-peak are spaced equidistantly, we have two more relations

(5.26) By using four independent relations given in Eqs. (5.25), and (5.26) we can reduce the number of unknown parameters from seven to three. We choose to keep l, nr , and nl .

Expressing the ansatz shown in Fig. 5.3 in therms of these three parameters we can rewrite the energy of the constrained jellium from Eq. (5.15) as

where we kept the leading terms in N . We want to find the saddle-point values l * , n * r , and n * l that minimize the energy F (l, nl , nr ). Taking the derivatives with respect to each of them and setting them to zero gives us the system of three saddle-point equations

These equations are equivalent of Eq. (5.20). However, by using the ansatz we reduced the number of unknown parameters from N to three. We use Mathematica to solve the system of equations (5.28), (5.29), and (5.30). By imposing the constraints 0 ≤ nl ≤ 1 -κ and 0 ≤ nr ≤ κ, we find four physical solutions. Therefore, we obtain four different phases (II, IIIa, IIIb, IV) of the constrained 1dOCP.

• Phase II In this case the solution is given, for all 0 < κ ≤ 1, by .31) This solution corresponds to a configuration with equidistantly spaced particles and a δ-peak that is located somewhere in the bulk of the system (see inset with gray border in Fig. 5.4). The two relations 0 ≤ nl ≤ 1 -κ and 0 ≤ nr ≤ κ determine the upper boundaries of Phase II given by µ = 8ακ (0 ≤ κ ≤ 1/2) and µ = 8α(1 -κ) (1/2 < κ ≤ 1). Together with the lower boundary µ > 0 they define a triangle in the κ-µ plane shown by gray color in Fig. 5.4.

Case 0 ≤ κ ≤ 1/2

If we choose a value of κ from the interval 0 ≤ κ ≤ 1/2 and vary s from -∞ to ∞ we encounter four phases: Phase IV, Phase IIIb, Phase II and Phase I (see Fig. 5.1). We cross three boundaries that we computed in the previous section and we summarize them here

between Phase I and Phase II (5.57) (1 -4κ) , between Phase II and Phase IIIb (5.58)

between Phase IIIb and Phase IV .

(5.59)

We proceed by integrating the expressions µ(s) given in Eqs. (5.55), (5.52), (5.45), and (5.39). This yields

We fix the constant from Eq. (5.56) by the condition ϕ κ (s = s) = 0 where s = 2ακ(1 -κ) .

(5.61)

This fixes the constant for the rate function in Phase I. Other constants can be determined by matching the values at the borders. In the left panel of Fig. 5.6 we plot the rate function ϕ κ (s) for κ = 0.3 (red line). We compare it with numerical simulations, and we see a perfect agreement. The dotted lines in the figure denote the values of s 1 (κ), s 2 (κ), and s 3 (κ) given in Eqs. (5.57), (5.58), and (5.59). Figure 5.5 shows the full probability distribution obtained by inserting the rate function into the large deviation form P N,κ (s) ∼ e -N 3 ϕκ(s) . The vertical lines again denote the phase boundaries. In the left inset of Fig. 5.5, we show the corresponding rate function ϕ κ (s) together with the numerical data points. The right inset depicts the saddle-point configurations typical for each of the phases.

We can check that the first and second derivatives of the rate function ϕ κ (s) are continuous at all three phase boundaries. However, the third derivative displays a jump where n is drawn from a uniform distribution between 0 and 1. The parameter d max is a real number and needs to be set in such a way that the acceptance ratio between the accepted moves and all proposed moves is around 1/2.

• If s new > s * we immediately reject the move. If s new ≤ s * we accept the move with the probability p = min(e -β(Enew-E) , 1) .

(5.81)

• We repeat the previous steps until we reach the equilibrium, this usually takes 10 7 -10 8 steps.

• We continue with the same process and sample s every 100 steps in order to construct P N,κ (s).

With this method, we can explore small regions around s * . In order to cover a large interval of values s, we need to repeat the whole process for different s * . The result of above algorithm is not P N,κ (s = S) but rather the conditional probability

(5.82)

The quantity we want to compute is therefore given by

Since P N,κ (s) ≃ e -N 3 ϕκ(s) , we need to take the logarithm and divide by -N 3 to obtain the large deviation function

(5.84) However, we do not know the value of the last term ln P N,κ (s ≤ s * ), which is different for each value s * . To get rid of these constants terms, we can look at the derivative of the large deviation function dϕ κ /ds. We approximate each histogram centered around s * by a linear function. Thus, the slope gives us the approximation of the derivative dϕ κ /ds at the value s = s * . Once we have the derivatives of the rate function, we use numerical integration to recover the rate function ϕ κ (s).

. Conclusion of Chapter 5

In this chapter, we presented the derivation of the results for the full probability distribution of the truncated linear statistics. It is an observable that describes the center of mass of the κN rightmost particles, where κ ∈ [0, 1]. The main results are the analytical expressions for the rate function given in Eqs. (5.68) and (5.60). We compare these results with numerical simulations, and we see an excellent agreement. In contrast to typical fluctuations of order O(N -3/2 ) around the mean value of the TLS, s = 2ακ(1 -κ), the large deviations of order O(1) are non-Gaussian. Interestingly, we find five different phases for the constrained Coulomb gas. They are summarized in the phase diagram depicted in Fig. 5.1. We show that at all phase boundaries, the rate function displays a discontinuity

Abstract:

In this paper, we study the probability distribution of the observable

representing the ordered positions of N particles in a 1d one-component plasma, i.e., N harmonically confined charges on a line, with pairwise repulsive 1d Coulomb interaction |x i -x j |. This observable represents an example of a truncated linear statistics -here proportional to the center of mass of the N ′ = κ N (with 0 < κ ≤ 1), rightmost particles. It interpolates between the position of the rightmost particle (in the limit κ → 0) and the full center of mass (in the limit κ → 1). We show that, for large N , s fluctuates around its mean ⟨s⟩ and the typical fluctuations are Gaussian, of width O(N -3/2 ). The atypical large fluctuations of s, for fixed κ, are instead described by a large deviation form P N,κ (s) ≃ exp -N 3 ϕ κ (s) , where the rate function ϕ κ (s) is computed analytically. We show that ϕ κ (s) takes different functional forms in five distinct regions in the (κ, s) plane separated by phase boundaries, thus leading to a rich phase diagram in the (κ, s) plane. Across all the phase boundaries the rate function ϕ κ (s) undergoes a third-order phase transition. This rate function is also evaluated numerically using a sophisticated importance sampling method, and we find a perfect agreement with our analytical predictions.

Article 4 Out-of-equilibrium dynamics of repulsive ranked diffusions: The expanding crystal

.

Abstract:

We study the non-equilibrium Langevin dynamics of N particles in one dimension with Coulomb repulsive linear interactions. This is a dynamical version of the so-called jellium model (without confinement) also known as ranked diffusion. Using a mapping to the Lieb-Liniger model of quantum bosons, we obtain an exact formula for the joint distribution of the positions of the N particles at time t, all starting from the origin. A saddle-point analysis shows that the system converges at large time to a linearly expanding crystal. Properly rescaled, this dynamical state resembles the equilibrium crystal in a timedependent effective quadratic potential. This analogy allows to study the fluctuations around the perfect crystal, which, to leading order, are Gaussian. There are however deviations from this Gaussian behavior, which embody long-range correlations of purely dynamical origin, characterized by the higher order cumulants of, e.g., the gaps between the particles, that we calculate exactly. We complement these results using a recent approach by one of us in terms of a noisy Burgers equation. In the large N limit, the mean density of the gas can be obtained at any time from the solution of a deterministic viscous Burgers equation. This approach provides a quantitative description of the dense regime at shorter times. Our predictions are in good agreement with numerical simulations for finite and large N .

Appendix A

From discrete to continuous description

This appendix is dedicated to a more detailed handling of the transition from the discrete description of the system to a continuous one. This process is composed of two tasks:

• Substitute the energy function of the jellium E[{x i }] by energy functional E[ρ(x)].

• Exchange the multiple integrals over x i in the partition function by a functional integral over the macroscopic density of charges ρ(x)

Let us start with the energy reformulation. The discrete version of the energy function is

Our aim is to express it in terms of the average macroscopic density profile

In the large N limit, this observable converges to a N independent form, and therefore, we drop the subscript N and simplify the notation

The first term of energy that describes the external quadratic potential can be written in terms of ρ(x) as

The second interacting term can be approximated by

Note that in the case of linear interaction, there is no issue with the self-interaction when x = y. However, in the case of the log gas, one needs to pay attention to handle it correctly, as discussed in [START_REF] Dyson | Statistical theory of the energy levels of complex systems. II[END_REF][START_REF] Dean | Extreme value statistics of eigenvalues of Gaussian random matrices[END_REF].

Combining both terms, we have

Equipped with the energy functional, we can try to write the partition function in terms of the macroscopic density profile. This means we need to replace the integration measure over the positions {x i } by a functional integral over ρ(x)

Thus the partition function is computed as

with J[ρ(x)] being the Jacobian coming from the change of variables from {x i } to ρ(x).

A more intuitive interpretation of this term stems from the observation that by replacing the microscopic description with a continuous function, we lose some information about multiple microscopic configurations that result in the same function ρ(x). This loss of information equal to the entropic contribution. It is possible to compute the Jacobian of the transformation, which is done, for example, in Ref. [START_REF] Dean | Extreme value statistics of eigenvalues of Gaussian random matrices[END_REF]. However, following Ref. [START_REF] Grabsch | Random Matrix Theory in Statistical Physics : Quantum Scattering and Disordered Systems[END_REF] we derive these terms by counting the number of microscopic configurations that contribute to one macroscopic profile of the density of charges. Let us imagine we have N particles that are at positions {x i }. We can build a histogram that represents this configuration. The width of each bin is set to δx, and the number of points inside j-th interval is n j . Our task is to count the number of possible permutations of particles that keep the histogram intact.

We can imagine filling the histogram with particles step by step:

• In the fist bin we put n 1 particles out of N . We have N n 1 options.

• We fill the second bin with n 2 particles out of N -n 1 options, which can be done in N -n 1 n 2 ways.

• After repeating the same steps, we reach the last bin, where we put n N particles out of

To count the number of permutations of particles that preserve the histogram, we multiply options at each step and expand the binomial symbols to get

After taking the logarithm of the above expression, we can apply Stirling's approximation valid for large

The macroscopic density profile is connected to the fraction of the particles in the each bin .11) Inserting this in Eq. (A.10) we have

Assuming that the width of the bins is very small δx → 0, we can exchange summation with integration lim

Which leads us to

with S being the Shannon entropy.As we can see, the loss of information due to the change of variables from {x i } to ρ(x) is really the entropy of the density field ρ(x).

If we return to the original expression for the partition function in Eq. (A.8), we can now write down the full expression

As we can see, the entropic term is linear in N and thus far smaller than the leading term that scales as N 3 in our case. Abstract: We discuss a two-parameter renormalization group (RG) flow when parameters are organized in a single complex variable, τ , with modular properties. Throughout the work we consider a special limit when the imaginary part of τ characterizing the disorder strength tends to zero. We argue that generalized Riemann-Thomae (gRT) function and the corresponding generalized Devil's staircase emerge naturally in a variety of physical models providing a universal behavior. In 1D we study the Anderson-like probe hopping in a weakly disordered lattice, recognize the origin of the gRT function in the spectral density of the probe and formulate specific RG procedure which gets mapped onto the discrete flow in the fundamental domain of the modular group SL (2, Z). In 2D we consider the generalization of the phyllotaxis crystal model proposed by L. Levitov and suggest the explicit form of the effective potential for the probe particle propagating in the symmetric and asymmetric 2D lattice of defects. Analyzing the structure of RG flow equations in the vicinity of saddle points we claim emergence of BKT-like transitions at imτ → 0. We show that the RG-like dynamics in the fundamental domain of SL(2, Z) for asymmetric lattices asymptotically approaches various "metallic ratios" (among which the "Silver ratio" is one of the examples). For a Hubbard model of particles on a ring interacting with long-ranged potentials we investigate the dependence of the ground state energy on the potential and demonstrate by combining numerical and analytical tools the emergence of the generalized Devil's staircase. Also we conjecture a bridge between a Hubbard model and a phyllotaxis.