Eric Luc

Large population asymptotics for interacting diffusions in a quenched random environment

Keywords: Interacting diffusion, random environment, self-averaging, fluctuations, McKean-Vlasov equation, FitzHugh-Nagumo model, Kuramoto model, stochastic partial differential equation

Large population asymptotics for interacting diffusions in a quenched random environment

Eric Luçon

Introduction

Diffusions in mean-field interaction

In this paper, we review some recent results ( [START_REF] Luc ¸on | Quenched limits and fluctuations of the empirical measure for plane rotators in random media[END_REF][START_REF] Luc ¸on | Large time asymptotics for the fluctuation SPDE in the Kuramoto synchronization model[END_REF][START_REF] Luc ¸on | Mean field limit for disordered diffusions with singular interactions[END_REF]) concerning large population asymptotics of interacting diffusions in a random environment. This class of models generalizes systems of particles in a mean-field interaction that have been intensively studied since McKean [START_REF] Mckean | Propagation of chaos for a class of non-linear parabolic equations[END_REF]. A general instance of such mean-field models may be given as follows: for any fixed T > 0, for any N ≥ 1 and m ≥ 1, consider the system of N coupled stochastic differential equations in R m dθ i,t = c(θ i,t )dt + b(θ i,t , ν N,t )dt + σ (θ i,t , ν N,t ) • dB i, t , t ∈ [0, T ], i = 1 . . . , N , [START_REF] Acebrón | The Kuramoto model: A simple paradigm for synchronization phenomena[END_REF] where ν N is the empirical measure of the particles (θ 1 , . . . , θ N ):

Eric Luc ¸on Laboratoire MAP 5 (UMR CNRS 8145), Université Paris Descartes, Sorbonne Paris Cité, 45 rue des Saints-Pères, 75270 Paris cedex 06, France e-mail: eric.lucon@parisdescartes.fr

1 ν N,t := 1 N N ∑ i=1 δ θ i,t , i = 1, . . . , N, t ∈ [0, T ] . (2)
For fixed N and t, ν N,t is an element of M 1 (R m ), the set of probability measures on R m . In the following, unless specified otherwise, M 1 is endowed with its weak topology. In [START_REF] Acebrón | The Kuramoto model: A simple paradigm for synchronization phenomena[END_REF], (B i ) i=1,...,N is a collection of independent Brownian motions in R p (p ≥ 1), (θ , ν) → b(θ , ν) ∈ R m and (θ , ν) → σ (θ , ν) ∈ R m×p are (possibly not) regular functions of (θ , ν), where ν belongs to the set of probability measures on R m . To fix ideas, the reader may think of the particular case where b or σ are of the form (θ , ν) → Γ (θ , θ )ν(d θ ) where Γ is a regular function. In such a case, the mean-field term in (1) reduces to 1 N ∑ N j=1 Γ (θ i , θ j ). Mean-field models like [START_REF] Acebrón | The Kuramoto model: A simple paradigm for synchronization phenomena[END_REF] have been quite popular in recent years in both physical and mathematical literature, as they provide the most natural way to represent the time evolution of a population of particles in all-to-all interaction. In (1), we have distinguished the local contribution c(θ i,t )dt to the dynamics (that would govern the dynamics if θ i had not been connected to the rest of the population) from the mean-field contribution of all other particles b(θ i,t , ν N,t )dt modeling the interaction with the rest of the population.

A reason for the interest in such systems is that they also provide natural particle approximations for various partial differential equations appearing in physics (e.g. granular media equation [START_REF] Bolley | Uniform convergence to equilibrium for granular media[END_REF][START_REF] Malrieu | Convergence to equilibrium for granular media equations and their Euler schemes[END_REF], porous media equation [START_REF] Blanchard | Probabilistic representation for solutions of an irregular porous media type equation[END_REF], Vicsek model for alignment of self-propelled particles [START_REF] Degond | Macroscopic limits and phase transition in a system of self-propelled particles[END_REF]), biology (neuronal models [START_REF] Delarue | Particle systems with a singular mean-field self-excitation, Application to neuronal networks[END_REF][START_REF] Baladron | Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons[END_REF][START_REF] Louca | Stationary states in infinite networks of spiking oscillators with noise[END_REF], Keller-Segel model for chemotaxis [START_REF] Godinho | Propagation of chaos for a sub-critical Keller-Segel model[END_REF]) or finance (rank-based models [START_REF] Jourdain | Propagation of chaos and Poincaré inequalities for a system of particles interacting through their CDF[END_REF][START_REF] Jourdain | Capital distribution and portfolio performance in the mean-field atlas model[END_REF]).

The problematic concerns the large population behavior of (1): under mild assumptions on the coefficients c, b and σ , one can show that the empirical measure (2) converges weakly as N → ∞ to a measure-valued solution t → ν t of a nonlin- ear Fokker-Planck equation (or McKean-Vlasov equation [START_REF] Gärtner | On the McKean-Vlasov limit for interacting diffusions[END_REF][START_REF] Mckean | Propagation of chaos for a class of non-linear parabolic equations[END_REF]). In that extent, a crucial feature of such mean-field systems is exchangeability: if, at t = 0 the law of the vector (θ 1,0 , . . . , θ N,0 ) is invariant under permutation, the same property holds for (1) at any positive time t. It is well understood ( [START_REF] Sznitman | Topics in propagation of chaos[END_REF]) that, under this assumption, the above convergence is equivalent to the notion of propagation of chaos: for any fixed k ≥ 1, the first k particles (θ 1 , . . . , θ k ) converges in law as N → ∞ to k independent copies of a nonlinear process, whose law is given by the solution to the McKean-Vlasov equation. More generally, a vast literature addresses the questions of fluctuations and large deviation of the empirical measure around its limit (see [START_REF] Fernandez | A Hilbertian approach for fluctuations on the McKean-Vlasov model[END_REF][START_REF] Dawson | Large deviations from the McKean-Vlasov limit for weakly interacting diffusions[END_REF][START_REF] Gärtner | On the McKean-Vlasov limit for interacting diffusions[END_REF][START_REF] Oelschläger | A law of large numbers for moderately interacting diffusion processes[END_REF][START_REF] Fathi | A gradient flow approach to large deviations for diffusion processes[END_REF][START_REF] Hitsuda | Tightness problem and stochastic evolution equation arising from fluctuation phenomena for interacting diffusions[END_REF] and references therein) as well as long-time behavior of such processes (uniform propagation of chaos and concentration inequalities [START_REF] Bolley | Quantitative concentration inequalities for empirical measures on non-compact spaces[END_REF][START_REF] Bolley | Uniform convergence to equilibrium for granular media[END_REF]).

Inhomogeneous interacting diffusions

General framework

The class of models we address in this paper is a generalization of (1) in the presence of a random environment. Namely, the system we consider is the following

dθ i,t = c(θ i,t , ω i )dt + 1 N N ∑ j=1 Γ (θ i,t , ω i , θ j,t , ω i )dt + dB i,t , i = 1, . . . , N, t ∈ [0, T ] ,
(3) endowed with an initial condition (θ i,0 ) i=1,...,N i.i.d. with law ζ on R m .

The difference between (3) and ( 1) is that the particles (θ 1 , . . . , θ N ) now live in a random environment, that is both the local dynamics c(•) and the mean-field term Γ (•) are perturbed by a given sequence (ω i ) i=1,...,N ∈ (R n ) N (n ≥ 1) of i.i.d. random variables, independent of the thermal noise (B 1 , . . . , B N ). This sequence models a local inhomogeneity in the system: particles are similar but not necessarily identical. The common law of the random variables

(ω i ) i≥1 is denoted as µ ∈ M 1 (R n ).
Remark 1. Note that we do not address the whole generality of the coefficients b and σ as described in [START_REF] Acebrón | The Kuramoto model: A simple paradigm for synchronization phenomena[END_REF]. In particular, we restrict ourselves for simplicity to the case p = m and σ = I. But the results exposed here should remain valid for general σ , provided further regularity and non-degeneracy assumptions are made.

From a point of view from statistical physics, the additional randomness (ω i ) i≥1 in (3) will be considered as a disorder. In this framework, there exist two ways to consider (3): one can either study the averaged (or annealed) model (where one looks at (θ 1 , . . . , θ N ) under the joint law of the noise and the disorder) or one can fix once and for all a typical realization of the disorder and consider (3) under the law of the noise only (quenched model). In this paper, we focus on the quenched model, which is more realistic from a modeling point of view. Under the same hypotheses, the asymptotic results proven here remain valid in the (technically easier) averaged framework.

The difficulty of working in a quenched environment (ω i ) i≥1 lies in the fact that the particles (θ 1 , . . . , θ N ) are no longer exchangeable. In particular, the question of propagation of chaos concerning (3) is a priori not clear. More precisely, the question we want to address is the influence of a fixed realization of the disorder on the behavior of the empirical measure of the system as

N → ∞ ν N,t = ν (ω) N,t := 1 N N ∑ i=1 δ (θ i,t ,ω i ) , t ∈ [0, T ], N ≥ 1 . (4)
As in (4), we will specify when required the dependence of the empirical measure in the specific choice of the disorder (ω) := (ω i ) i≥1 .

Synchronization models and neural networks

A first motivation for systems such as (3) comes from synchronization models in physics. The Kuramoto model [START_REF] Acebrón | The Kuramoto model: A simple paradigm for synchronization phenomena[END_REF], which will be the guiding thread of this review, is a particular case of (3) where the particles θ i reduce to one-dimensional oscillators (or rotators) on the circle S := R/2π, within a mean-field sine interaction, perturbed by random frequencies ω i ∈ R:

dθ i,t = ω i dt + K N N ∑ j=1 sin(θ j,t -θ i,t )dt + dB i,t , i = 1, . . . , N, t ∈ [0, T ] . (5)
In [START_REF] Balmforth | A shocking display of synchrony[END_REF], K > 0 is the coupling strength between rotators, and the disorder ω i is the local random frequency of rotator θ i , that may differ from one rotator to another. As N → ∞, the system is described by the following nonlinear Fokker-Planck equation (whose solution (t, θ , ω) → q t (θ , ω) is the density of rotators at time t at position θ with frequency ω):

∂ t q t = 1 2 ∂ 2 θ q t (θ , ω) -∂ θ q t (θ , ω) ω + K S×R sin( θ -θ )q t ( θ , ω)d θ µ(d ω) .
(6) The Kuramoto model was first introduced in [START_REF] Kuramoto | Self-entrainment of a population of coupled non-linear oscillators[END_REF] in order to study collective behavior of synchronizing individuals (neurons, social insects, cardiac cells) and has been since the subject of a vast literature, mostly in physics (see [START_REF] Kuramoto | Self-entrainment of a population of coupled non-linear oscillators[END_REF][START_REF] Strogatz | Stability of incoherence in a population of coupled oscillators[END_REF][START_REF] Acebrón | The Kuramoto model: A simple paradigm for synchronization phenomena[END_REF] and references therein) and more recently in mathematics.

The intuition for the dynamics of the Kuramoto model is simple: in a large population, each rotator θ i obeys to the influence of its local frequency ω i which tends to desynchronize the rotators, in contradiction with the mean-field coupling that tends to make the particles rotate together. A striking result, first observed by Kuramoto and Sakaguchi ([50]), is that (6) exhibits a phase transition: if K is smaller than a critical value K c , the uniform distribution q ≡ 1 2π is the only stationary solution of (6) (there is no synchronization), whereas it coexists with nontrivial synchronized profiles for K > K c . Recent results address the question of the long-time stability of such synchronized solutions, in the case without disorder ( [START_REF] Bertini | Dynamical aspects of mean field plane rotators and the Kuramoto model[END_REF][START_REF] Bertini | Synchronization and random long time dynamics for mean-field plane rotators[END_REF][START_REF] Giacomin | Global attractor and asymptotic dynamics in the Kuramoto model for coupled noisy phase oscillators[END_REF]) and also with small disorder ( [START_REF] Giacomin | Coherence stability and effect of random natural frequencies in populations of coupled oscillators[END_REF]).

A second motivation for this work comes from the modeling of the spiking activity of neurons in a noisy environment. The FitzHugh-Nagumo model, which is a 2-dimensional simplification of the Hodgkin-Huxley model (see [START_REF] Baladron | Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons[END_REF][START_REF] Ermentrout | Mathematical foundations of neuroscience[END_REF] for further neurophysiological insights on the subject) is given as follows: θ i := (V i , w i ) and

   dV i (t) = V i (t) -V i (t) 3 3 -w i (t) + I dt, dw i (t) = a i (b i V i (t) -w i (t)) dt, i = 1, . . . , N, t ∈ [0, T ] , (7)
where a i and b i are random coefficients. The variable V i (t) denotes the voltage activity of the neuron, w i (t) is a recovery variable and I is the exterior input current.

In [START_REF] Berglund | Mixed-mode oscillations and interspike interval statistics in the stochastic Fitzhugh-Nagumo model[END_REF], the disorder ω i = (a i , b i ) ∈ R 2 plays the role of a random discrimination be- tween inhibited and excited neurons: it is well known ( [START_REF] Ermentrout | Mathematical foundations of neuroscience[END_REF]) that, depending on the values of the parameters (a i , b i ), the dynamics (7) exhibits either periodic behavior around a limit cycle (spiking activity) or convergence to a fixed point (inhibition of the neuron). If one incorporates this dynamics into (3), the mean-field term Γ (•) models connections between neurons through electrical synapses. We refer to [START_REF] Baladron | Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons[END_REF] for precise details and the exposition of more elaborate mean-field models applied to neuronal activity. Note that one difficulty of the dynamics given by ( 7) is that it is unbounded and not uniformly Lipschitz-continuous.

Existing literature on disordered interacting diffusions

The subject of diffusions in random environment has already been addressed in the literature. In particular, one should mention the seminal paper of Dai Pra and den Hollander [START_REF] Dai Pra | McKean-Vlasov limit for interacting random processes in random media[END_REF] where an averaged large deviations principle for models similar to (3) is proved, with applications to the Kuramoto model and spin-flip systems. Other interesting applications of (3) may be found in the context of statistical physics (random Curie-Weiss model [START_REF] Collet | The role of disorder in the dynamics of critical fluctuations of mean field models[END_REF], model of social interactions [START_REF] Collet | A simple mean field model for social interactions: dynamics, fluctuations, criticality[END_REF]).

Instead of putting the disorder on the particles, it would also make sense to put the disorder on the connections between particles. There is currently a growing interest in mean-field models with random connectivities (in particular with applications to neuronal models, see [START_REF] Faugeras | A large deviation principle for networks of rate neurons with correlated synaptic weights[END_REF][START_REF] Wainrib | Topological and dynamical complexity of random neural networks[END_REF][START_REF] Cabana | Large deviations, dynamics and phase transitions in large stochastic heterogeneous neural networks[END_REF][START_REF] Touboul | Limits and dynamics of stochastic neuronal networks with random heterogeneous delays[END_REF] and references therein), that is, models of the type

dθ i,t = N ∑ j=1 J i, j Γ (θ i,t , θ j,t )dt + dB i,t , i = 1, . . . , N, t ∈ [0, T ] , (8)
where (J i, j ) i, j=1,...,N is a (possibly symmetric) collection of random variables. Large population asymptotics of such models have been first studied through large deviations techniques by Guionnet and Ben Arous [START_REF] Arous | Large deviations for Langevin spin glass dynamics[END_REF][START_REF] Ben Arous | Symmetric Langevin spin glass dynamics[END_REF] in the context of spin-glass systems. The long time analysis of the associated nonlinear process appears to be significantly more difficult than for the models considered here.

Other mean-field models of neurons with integrate-and-fire dynamics have recently been studied [START_REF] Cáceres | Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states[END_REF][START_REF] Delarue | Global solvability of a networked integrateand-fire model of McKean-Vlasov type[END_REF][START_REF] Delarue | Particle systems with a singular mean-field self-excitation, Application to neuronal networks[END_REF]. For such models, existence of a solution for all time to the nonlinear Fokker Planck turns out to be problematic.

Organization of the paper

We summarize in Section 2 the main results of the paper, that is a quenched law of large numbers and a quenched central limit theorem for the empirical measure (4). In Section 3, the main lines of proof for the two results are indicated.

Main results

In the remaining of the paper, for any p ≥ 1, the euclidean norm and scalar product in R p are respectively denoted as |u| and u • v (u, v ∈ R p ). P is the law of the sequence of Brownian motions and P is the law of the sequence of the disorder. The corresponding expectations are denoted as E and E respectively.

Quenched law of large numbers

The first result concerns a quenched law of large numbers for the empirical measure ν N defined in (4).

Assumptions

We assume that the coefficients and initial condition in (3) satisfy:

1. The function (θ , ω) → c(θ , ω) is locally Lipschitz-continuous in θ (for fixed ω)
and satisfy a one-sided Lipschitz condition:

∀(θ , ω), ( θ , ω), θ -θ • c(θ , ω) -c( θ, ω) ≤ L θ -θ 2 + |ω -ω| 2 , (9)
for some constant L. The function c also satisfies a polynomial bound:

∀(θ , ω), |c(θ , ω)| ≤ C 1 + |θ | κ + |ω| ι , (10)
for some constant C > 0 and where κ ≥ 2 and ι ≥ 1.

The interaction term Γ is bounded and globally

Lipschitz on (R m × R n ) 2 .
3. For fixed θ , ω, ω, the functions θ → c(θ , ω) and θ → Γ (θ , ω, θ , ω) are twice differentiable with continuous derivatives. 4. The initial distribution ζ of the particles in (3) and the law µ of the disorder satisfy the following moment conditions:

R m |θ | κ ζ (dθ ) < ∞ and R n |ω| ι µ(dω) < ∞ , (11)
where the constants κ and ι are given in [START_REF] Billingsley | Convergence of probability measures[END_REF].

Remark 2. These assumptions differ from the hypotheses of the original proof in [START_REF] Luc ¸on | Quenched limits and fluctuations of the empirical measure for plane rotators in random media[END_REF]. Indeed, in [START_REF] Luc ¸on | Quenched limits and fluctuations of the empirical measure for plane rotators in random media[END_REF], we were mostly concerned with particles living in a compact space (as in the Kuramoto model ( 5)), in which case assumptions on c, Γ and ζ (especially the differentiability of c and Γ ) can be simplified. We refer to [START_REF] Luc ¸on | Quenched limits and fluctuations of the empirical measure for plane rotators in random media[END_REF], Section 2.3.1 for further details. Note that the present assumptions specifically include the case of polynomial coefficients (as in the FitzHugh-Nagumo case [START_REF] Berglund | Mixed-mode oscillations and interspike interval statistics in the stochastic Fitzhugh-Nagumo model[END_REF], set κ = 3 and ι = 1) which is not covered by [START_REF] Luc ¸on | Quenched limits and fluctuations of the empirical measure for plane rotators in random media[END_REF].

Law of large numbers and McKean-Vlasov equation

Theorem 1 ([39], [START_REF] Luc ¸on | Mean field limit for disordered diffusions with singular interactions[END_REF]). Under the hypotheses of Section 2.1.1,

1. there is a unique process t → ν t in C ([0, T ], M 1 (R m × R n )) such that ν 0 = ζ × µ and sup t≤T |θ | κ ∨ |ω| ι ν t (dθ , dω) < +∞, satisfying the weak McKean- Vlasov equation ν t , f = ν 0 , f + t 0 ν s , 1 2 ∆ θ f + ∇ θ f • ([Γ , ν s ] + c) ds, t ∈ [0, T ] , ( 12
)
where ν , f := f (θ , ω)ν(dθ , dω) and

[Γ , m](θ , ω) := Γ (θ , ω, θ , ω)m(d θ , d ω) . ( 13
)
2. for almost-every sequence (ω i ) i≥1 , the sequence (ν

(ω) N ) N≥1 converges as N → ∞ in C ([0, T ], M 1 (R m × R n ))
to the unique solution ν of [START_REF] Bolley | Uniform convergence to equilibrium for granular media[END_REF]. Remark 3. Note that if we take a test function f in (12) that does not depend on θ , we obtain that ν 0 , f = ν t , f , ∀t ∈ [0, T ]. In particular, the marginal distribution on ω of ν t is independent of t and equal to µ.

Since the noise in ( 3) is non-degenerate, using the regularizing properties of the heat kernel, one can prove that for any measure-valued initial condition in [START_REF] Bolley | Uniform convergence to equilibrium for granular media[END_REF], the solution of ( 12) has a regular density ν t (dθ , dω) = q t (θ , ω)dθ µ(dω) for all t > 0. Integrating by parts in [START_REF] Bolley | Uniform convergence to equilibrium for granular media[END_REF], we find that q t is a strong solution to

∂ t q t = 1 2 ∆ θ q t (θ , ω) -div θ (q t (θ , ω)c(θ , ω)) -div θ q t (θ , ω) Γ (θ , ω, θ , ω)q t ( θ , ω)d θ µ(d ω) , t > 0 . ( 14
)
For a proof of this fact and further details, we refer to [START_REF] Giacomin | Coherence stability and effect of random natural frequencies in populations of coupled oscillators[END_REF]Prop. A.1].

A consequence of Theorem 1 is that, at the level of the law of large numbers, the system only depends on the law µ of the disorder, but not on a typical realization of the (ω i ) i≥1 : there is a self-averaging phenomenon. The notion of self-averaging (or its absence) is crucial in many disordered models of statistical physics and is deeply related to the influence of the disorder on the phase transition in such systems (see e.g. [START_REF] Pastur | Absence of self-averaging of the order parameter in the Sherrington-Kirkpatrick model[END_REF][START_REF] Aizenman | Some rigorous results on the Sherrington-Kirkpatrick spin glass model[END_REF] and references therein).

Quenched central limit theorem.

The second result concerns the influence of a typical realization of the disorder (ω j ) j≥1 on the fluctuations of the empirical measure (4) around its McKean-Vlasov limit [START_REF] Bolley | Uniform convergence to equilibrium for granular media[END_REF]. The question is whether or not self-averaging also holds at the level of the fluctuations, and if not, if it is possible to quantify the dependance in the disorder of the system (3) at the level of fluctuations.

Non-self-averaging phenomenon in the Kuramoto model

The motivation comes from the Kuramoto model (5). To fix ideas, consider the case where the frequencies ω i in ( 5) are sampled with µ = 1 2 (δ -1 + δ 1 ): this is simply a random decomposition of (θ 1 , . . . , θ N ) between two subpopulations, one naturally rotating clockwise (ω i = +1) and the second rotating anti-clockwise (ω i = -1). One can imagine that fluctuations in the finite sample (ω 1 , . . . , ω N ) ∈ {±1} N may lead, for example, to a majority of +1 with respect to -1, so that the rotators with positive frequency induce a global rotation of the whole system in the direction of the majority. Direction and speed of rotation depend on this initial configuration of the disorder (see Fig. 1 and2). This phenomenon, noticed numerically in [START_REF] Balmforth | A shocking display of synchrony[END_REF], can be computed through the order parameters (r N,t , ψ N,t ):

r N,t e iψ N,t := 1 N N ∑ j=1 e iθ j,t = S×R e iθ ν N,t (dθ , dω), N ≥ 1, t ≥ 0 , (15)
Here r N,t ∈ [-1, 1] gives a notion of synchronization of the system (5) (e.g. r N,t = 1 if the oscillators θ j,t are all equal) and ψ N,t captures the position of the center of synchronization (see Figure 1). One can see on Figure 2a that t → ψ N,t has an approximately linear behavior whose slope depends on the sample of the disorder. Note that this disorder-induced phenomenon does not happen at the level of the nonlinear Fokker-Planck equation ( 6), but only at the level of fluctuations (the speed of rotation in Figure 2a is of order N -1/2 which vanishes as N → ∞). Consequently, in order to understand this phenomenon, one needs to make sense to a quenched central limit theorem for the empirical measure ν N .

Weak quenched convergence of the fluctuation process

We consider, for a fixed realization of the disorder (ω) = (ω i ) i≥1 the fluctuation process η N = η

(ω)

N given by η (ω)

N,t := √ N ν (ω) N,t -ν t , t ≤ T, N ≥ 1 . (16)
For fixed t ≤ T , (ω) and N ≥ 1, η

N,t is a random element of S ′ , the Schwartz space of tempered distributions on R m × R n . Remark 4. One observation about [START_REF] Collet | The role of disorder in the dynamics of critical fluctuations of mean field models[END_REF] is that the convergence of η (ω) N as N → ∞ cannot hold for a fixed realization of (ω): consider the particular case of (3) when m = 1, c(θ , ω) = ω, Γ = 0 Fig. 1: Evolution of the marginal on S of ν N in the Kuramoto model (N = 600, µ = 1 2 (δ -1 + δ 1 ), K = 6). The rotators are initially independent and uniformly distributed on S and independent of the disorder. First the dynamics leads to synchronization (t = 6) to a profile close to a nontrivial stationary solution of [START_REF] Ben Arous | Symmetric Langevin spin glass dynamics[END_REF]. Second, the center ψ N,t of this density moves to the right with an approximately constant speed; this speed of rotation turns out to be sample-dependent (see Fig. 2a). (b) Trajectories of the process η t (sin), for different realizations of the mean-value C(ω) (see [START_REF] Delarue | Particle systems with a singular mean-field self-excitation, Application to neuronal networks[END_REF]).

Fig. 2: In Fig. 2a, direction and speed of ψ N depend on the initial configuration of the frequencies. These simulations are compatible with speeds of order N -1/2 . In Fig. 2b, trajectories of the limiting fluctuation process η(sin) are almost linear and compatible with Fig. 2a. The red trajectories in both figures correspond to the averaged model where there is no disorder-dependent rotation.

θ i,t = ω i t + B i,t , t ≤ T, i = 1, . . . , N , (17)
that is simply Brownian motions with random drifts. Studying the fluctuations of the empirical measure associated to [START_REF] Collet | A simple mean field model for social interactions: dynamics, fluctuations, criticality[END_REF] requires to look at functionals of the type

√ N 1 N ∑ N i=1 ϕ(ω i ) -ϕ(ω)µ(dω)
, for regular functions ϕ. But almost-surely in (ω i ) i≥1 , the latter quantity does not converge (it only converges in law with respect to (ω i ) i≥1 ).

In order to make sense of any possible limit for η (ω) N , one needs to find a weak formulation of a notion of quenched convergence. To do so, fix (ω) and denote as H N (ω) the law of the process η (ω) N ; H N (ω) belongs to M 1 (C ([0, T ], S ′ )), the set of probability measures on continuous paths with values in S ′ . As noted in Remark 4, H N (ω) is not likely to converge weakly as N → ∞ for fixed (ω). Instead, consider the random variable

(ω) ∈ (R n ) N → H N (ω) ∈ M 1 (C ([0, T ], S ′ )) .
The purpose of Theorem 2 below is precisely to state that this random variable (with values in the big set M 1 (C ([0, T ], S ′ ))) converges in law to a random variable ω → H (ω) ∈ M 1 (C ([0, T ], S ′ )). The second point of Theorem 2 is to identify H (ω) as the law of the solution to a linear stochastic partial differential equation.

Remark 5. It is important to note that this weak notion of quenched convergence differs from an averaged convergence: it is also possible under the same hypotheses to state an averaged central limit theorem which gives a different limit from the one found in [START_REF] Delarue | Particle systems with a singular mean-field self-excitation, Application to neuronal networks[END_REF] (see [START_REF] Luc ¸on | Quenched limits and fluctuations of the empirical measure for plane rotators in random media[END_REF]). The quenched convergence still keeps track of the dependence in the disorder of the particle system as N → ∞ (see Theorem 2), whereas the averaged limit does not.

Assumptions

In addition to the assumptions of Section 2.1.1, we suppose that 1. The functions (θ , ω) → c(θ , ω) and (θ , ω, θ , ω) → Γ (θ , ω, θ , ω) are infinitely differentiable. The derivatives of Γ are uniformly bounded and the derivatives of c satisfy the same polynomial bound as [START_REF] Billingsley | Convergence of probability measures[END_REF]. 2. The following moment conditions are satisfied

R m |θ | 2(κ+γ) ζ (dθ ) < ∞ and R n |ω| 2(ι+γ) µ(dω) < ∞ , (18)
where the constants κ and ι are given by [START_REF] Billingsley | Convergence of probability measures[END_REF], for a sufficiently large constant γ (depending explicitly on m, n, κ, ι).

Quenched central limit theorem

Before stating the result, let us give some definitions: for all 0 ≤ s ≤ T , let L s be the second order differential operator defined by

L s f (θ , ω) := 1 2 ∆ θ f (θ , ω) + ∇ θ f (θ , ω) • ([Γ , ν s ](θ , ω) + c(θ , ω)) + ∇ θ f ( θ , ω) • Γ ( θ , ω, θ , ω)ν s (d θ , d ω) .
Let W the Gaussian process with covariance (for every s,t ∈ [0, T ]):

E(W t ( f 1 )W s ( f 2 )) = s∧t 0 ∇ θ f 1 (θ , ω) • ∇ θ f 2 (θ , ω)ν u (dθ , dω)du . (19)
For all f 1 , f 2 bounded and continuous on R m × R n , let

Γ 1 ( f 1 , f 2 ) := R n Cov ζ ( f 1 (•, ω), f 2 (•, ω)) µ(dω), ( 20
) = R n R m f 1 - R m f 1 (•, ω)dζ f 2 - R m f 2 (•, ω)dζ dζ µ(dω) ,
and

Γ 2 ( f 1 , f 2 ) = Cov µ R m f 1 dζ , R m f 2 dζ , ( 21
) = R n R m f 1 dζ - R m ×R n f 1 dζ dµ R m f 2 dζ - R m ×R n f 2 dζ dµ dµ .
The main theorem is the following:

Theorem 2 ([ 39
η ω t = X + C(ω) + t 0 L * s η ω s ds + W t , (22)
where, L * s is the formal adjoint operator of L s , X is a centered Gaussian process with covariance Γ 1 and where for fixed ω, C(ω) is the nontrivial deterministic mean value of the initial condition. As a random variable in ω, ω → C(ω) is a Gaussian process with covariance Γ 2 . Moreover, W is independent on the initial value (X,C).

In [START_REF] Delarue | Particle systems with a singular mean-field self-excitation, Application to neuronal networks[END_REF], the linear operator L * s is deterministic ; the only dependence in ω lies in the initial condition η ω 0 = X +C(ω), through its non trivial mean-value C(ω). We give here some intuition about the fact that C(•) precisely captures the fluctuations of the disorder in the microscopic model and show how one can understand from ( 22) the non-self-averaging behavior of the Kuramoto model described in Section 2.2.1. For N ≥ 1 and f : R m × R n → R, the initial fluctuation applied to f is given by

η N,0 ( f ) = 1 √ N N ∑ j=1 f (θ j,0 , ω j ) - R m ×R n f (θ , ω)ζ (dθ )µ(dω) , = 1 √ N N ∑ j=1 f (θ j,0 , ω j ) - R m f (θ , ω j )ζ (dθ ) + 1 √ N N ∑ j=1 R m f (θ , ω j )ζ (dθ ) - R m ×R n f (θ , ω)ζ (dθ )µ(dω) ,( 23
) := X N ( f ) + C N ( f ) .
The process X N captures the initial fluctuations of the rotators whereas C N captures the fluctuations of the disorder. It is easy to see that C N converges in law (w.r.t. the disorder) to the process C with covariance [START_REF] Delarue | Global solvability of a networked integrateand-fire model of McKean-Vlasov type[END_REF]. In the framework of the Kuramoto model with binary disorder (recall Section 2.2.1), computations show ( [START_REF] Luc ¸on | Large time asymptotics for the fluctuation SPDE in the Kuramoto synchronization model[END_REF]) that the relevant quantity for the dynamics of ( 22) is the restriction C + of the process C to the component on +1:

C +,ψ := C ψ1 ω=+1 , ψ : S → R . ( 24
)
C + is the limit in law of the microscopic process C N,+ defined by

C N,+ (ψ) := S ψ(•)dζ 1 √ N N ∑ i=1 1 (ω i =+1) - 1 2 := S ψ(•)dζ α N √ N . (25)
Here, α N is exactly the (centered) number of frequencies among (ω 1 , . . . , ω N ) that are positive, so that C N,+ captures the lack of symmetry of the disorder: α N > 0 (resp. α N < 0) represents the case of an asymmetry in favor of positive (resp. neg- ative) frequencies. Hence, Theorem 2 provides a way to study, through the process C, the influence of the asymmetry of the disorder on (3). The main lines of proof of Theorem 2 are given in Section 3.2.

Long-time behavior of the fluctuation process in the Kuramoto model

We restrict in this paragraph to the Kuramoto case (5) with binary disorder: µ = 1 2 δ -ω 0 + δ ω 0 for some ω 0 > 0. One can see numerically (see Figure 2b) that the initial asymmetry C(•) propagates from t = 0 to positive times and provide analogous trajectories to the ones observed in the microscopic model (Figure 2a). One can make this observation rigorous, at least when ω 0 is small: [START_REF] Luc ¸on | Large time asymptotics for the fluctuation SPDE in the Kuramoto synchronization model[END_REF]). In the Kuramoto model (5) in the small-disorder regime, the solutions of [START_REF] Delarue | Particle systems with a singular mean-field self-excitation, Application to neuronal networks[END_REF] are asymptotically linear and disorder dependent. More precisely, for all K > 1, there exist an Hilbert space of distributions H and ω * > 0 such that for all 0 < ω 0 < ω * , for fixed initial condition

Theorem 3 ([
η ω 0 = X + C(ω), there exists V (ω) ∈ H such that η ω t t in law -----→ = V (ω), as t → +∞ . ( 26
)
Moreover, ω → V (ω) is a Gaussian process in H with explicit (in terms of C(ω)) covariance.

The proof of Theorem 3 relies on a spectral analysis (based on perturbation argument from the case without disorder, [START_REF] Bertini | Dynamical aspects of mean field plane rotators and the Kuramoto model[END_REF]) of the linear operator L * s governing [START_REF] Delarue | Particle systems with a singular mean-field self-excitation, Application to neuronal networks[END_REF]. Following the intuition of the finite-dimensional Jordan block x y ′ = 0 1 0 0 • x y (whose solutions are obviously linear), one of the key ingredients to the proof of Theorem 3 is to prove a similar property for the unbounded operator L * s : there exist u and v such that L * s u = 0 and L * s v = u. We refer to [START_REF] Luc ¸on | Large time asymptotics for the fluctuation SPDE in the Kuramoto synchronization model[END_REF] for more details.

Conclusion and perspectives

Disordered mean-field such as (3) (and especially the Kuramoto model) are not selfaveraging at the level of fluctuations: the dynamics of the quenched fluctuations of ( 4) is disorder-dependent. However, in order to derive rigorously the exact speed of rotation of synchronized solutions in the Kuramoto model described in Figure 2, it would be necessary to study [START_REF] Balmforth | A shocking display of synchrony[END_REF] on larger time scales, as in [START_REF] Bertini | Synchronization and random long time dynamics for mean-field plane rotators[END_REF]. This is currently under investigation. Another difficulty is that, although both law of large numbers and central limit theorem are valid in a rather general setting, investigating the long-time behavior of the limiting objects ν and η is often very difficult. In that sense, one of the reasons for the popularity of the Kuramoto model is that the stationary solutions of the nonlinear Fokker-Planck equation ( 6) are explicitly computable ( [START_REF] Sakaguchi | Phase transitions and their bifurcation analysis in a large population of active rotators with mean-field coupling[END_REF]). A key point in the analysis of the stability of such solutions ( [START_REF] Bertini | Dynamical aspects of mean field plane rotators and the Kuramoto model[END_REF][START_REF] Bertini | Synchronization and random long time dynamics for mean-field plane rotators[END_REF][START_REF] Giacomin | Global attractor and asymptotic dynamics in the Kuramoto model for coupled noisy phase oscillators[END_REF]) is that the Kuramoto model without disorder is reversible ( [START_REF] Bertini | Dynamical aspects of mean field plane rotators and the Kuramoto model[END_REF]), whereas reversibility is lost for many other neuronal models (e.g. FitzHugh-Nagumo [START_REF] Berglund | Mixed-mode oscillations and interspike interval statistics in the stochastic Fitzhugh-Nagumo model[END_REF]). The results presented here should be applicable to other models of disordered diffusions, provided sufficient information is known about characterization and linear stability of stationary states (see for example [START_REF] Louca | Stationary states in infinite networks of spiking oscillators with noise[END_REF]).

There is currently a growing interest in generalizations of (3) where the interactions depend on the topology of non-mean-field networks or on the distance between particles. The motivation comes from the biological observation that neurons do not interact in a mean-field way (see [START_REF] Wainrib | Topological and dynamical complexity of random neural networks[END_REF] and references therein). As far as weighted interactions are considered, models of moderately interacting diffusions ( [START_REF] Oelschläger | A law of large numbers for moderately interacting diffusion processes[END_REF][START_REF] Jourdain | Propagation of chaos and fluctuations for a moderate model with smooth initial data[END_REF]) have been one of the first attempts to go beyond pure mean-field models. Several papers in physics [START_REF] Omelchenko | Loss of coherence in dynamical networks: Spatial chaos and chimera states[END_REF][START_REF] Gupta | One-dimensional lattice of oscillators coupled through power-law interactions: Continuum limit and dynamics of spatial fourier modes[END_REF][START_REF] Maródi | Synchronization of oscillators with long range interaction: Phase transition and anomalous finite size effects[END_REF] have also considered models of oscillators within P-nearest neighbors or with power-law interaction. Those models are known to exhibit anomalous dynamical properties (chimera states). The mean-field limit in this framework has been rigorously shown in [START_REF] Luc ¸on | Mean field limit for disordered diffusions with singular interactions[END_REF], showing in particular anomalous speed of convergence to the nonlinear Fokker-Planck equation. The exact central limit theorem in this case is currently under investigation.

Generalizing the result of Dai Pra and den Hollander concerning possible quenched large deviations for (3) is of course a natural perspective for this work and is the object of an ongoing work.

3 Sketchs of proofs

Law of large numbers

We give in this paragraph the main lines of the proof of Theorem 1. The difficulty here is that we allow the coefficient c in (3) to have polynomial growth (recall Section 2.1.1); in particular, one needs to have a priori controls on the moments of any accumulation point of ( 4). The well-posedness of the McKean-Vlasov equation [START_REF] Bolley | Uniform convergence to equilibrium for granular media[END_REF] under the hypotheses of Section 2.1.1 can be seen as a consequence of [START_REF] Luc ¸on | Mean field limit for disordered diffusions with singular interactions[END_REF]. The existence of a least one solution ν (satisfying the required moment conditions) to ( 12) is established in [START_REF] Luc ¸on | Mean field limit for disordered diffusions with singular interactions[END_REF] via a fixed-point procedure on the nonlinear process associated to [START_REF] Bolley | Uniform convergence to equilibrium for granular media[END_REF] using arguments from Sznitman [START_REF] Sznitman | Topics in propagation of chaos[END_REF] and the uniqueness comes from the fact that for any such solution ν, the empirical measure ν N necessarily converges to ν as N → ∞, in terms of an adequate Wasserstein metric. Note that we cannot directly use this last result for the quenched convergence of ν N , since the convergence in [START_REF] Luc ¸on | Mean field limit for disordered diffusions with singular interactions[END_REF] is averaged w.r.t. the disorder. Remark 6. The framework of [START_REF] Luc ¸on | Mean field limit for disordered diffusions with singular interactions[END_REF] concerns the more general case of diffusions within weighted spatial interactions (P-nearest neighbor with parameter R ∈]0, 1] and power law with exponent α ≥ 0, see [START_REF] Luc ¸on | Mean field limit for disordered diffusions with singular interactions[END_REF], Section 1.2.2 for detailed definitions of these models). It is easy to see that one retrieves the full mean-field setting for R = 1 or α = 0.

As far as the quenched convergence of the empirical measure ( 4) is concerned, one can proceed as follows. The result for the initial condition is clear: since (θ i,0 , ω i ) 1≤i≤N are i.i.d. random variables with law ζ ⊗ µ, the initial empirical measure ν N,0 converges almost surely in (ω) to ν 0 (dθ , dω) = ζ (dθ )µ(dω), as N → ∞.

An application of Ito's formula to (3) (for any regular function (θ , ω) → f (θ , ω)) leads to the following semi-martingale representation for ν N :

ν N,t , f = ν N,0 , f + t 0 ν N,s , 1 2 ∆ θ f + ∇ θ f • c ds + t 0 ν N,s , ∇ θ f • Γ (•, •, θ , ω)ν N,s (d θ , d ω) ds + M N,t ( f ) , (27)
where M N,t ( f

) := 1 N ∑ N j=1 t 0 ∇ θ f (θ j,s , ω j )
• dB j,s is a martingale. Using usual tight- ness criteria ( [START_REF] Méléard | Sur les convergences étroite ou vague de processus à valeurs mesures[END_REF][START_REF] Roelly-Coppoletta | A criterion of convergence of measure-valued processes: application to measure branching processes[END_REF]) based on Aldous criterion ( [START_REF] Billingsley | Convergence of probability measures[END_REF]) for real valued continuous processes, it is easy to derive from [START_REF] Gärtner | On the McKean-Vlasov limit for interacting diffusions[END_REF] the tightness of

(ν N ) in C ([0, T ], (M 1 (R m × R n ), v))
, where v is the vague topology (i.e. the coarsest topology that makes the evaluations ν → ν , f continuous for every f continuous with compact support).

Using the one-sided Lipschitz continuity of c, one can prove that, almost surely in (ω), sup N≥1 sup 1≤i≤N E |θ i,t | κ < ∞. Using this estimate and a localization argument ( [START_REF] Luc ¸on | Quenched limits and fluctuations of the empirical measure for plane rotators in random media[END_REF], Lemma 3.4) one obtains that for any accumulation point ν of (ν N ) N≥1 ,

R m ×R n |θ | κ ∨ |ω| ι ν(dθ , dω) < +∞ . ( 28
)
The tightness of (ν N ) N≥1 in the weak topology and its convergence to [START_REF] Bolley | Uniform convergence to equilibrium for granular media[END_REF] follows from the a priori estimate [START_REF] Giacomin | Coherence stability and effect of random natural frequencies in populations of coupled oscillators[END_REF] and the fact that M N,t ( f ) in ( 27) vanishes as N → ∞.

Fluctuations

We now turn to the proof of Theorem 2. The principal tool used here is Hilbertian techniques for measure-values processes developed by Fernandez, Méléard and Jourdain [START_REF] Fernandez | A Hilbertian approach for fluctuations on the McKean-Vlasov model[END_REF][START_REF] Jourdain | Propagation of chaos and fluctuations for a moderate model with smooth initial data[END_REF] for similar models without disorder. An application of Ito's formula to (3) leads to a semi-martingale decomposition of the fluctuation process η (ω) N of the following form: for all regular functions f , for every sequence (ω), for all t ≤ T :

η (ω) N,t , f = η (ω) N,0 , f + t 0 η (ω) N,s , L N,s ( f ) ds + W (ω) N,t ( f ) , (29)
where, L N,s is an unbounded linear operator defined by

L N,s f (θ , ω) := 1 2 ∆ θ f (θ , ω) + ∇ θ f (θ , ω) • ([Γ , ν N,s ](θ , ω) + c(θ , ω)) + ∇ θ f ( θ , ω) • Γ ( θ , ω, θ , ω)ν s (d θ , d ω), θ ∈ R m , ω ∈ R n .( 30
)
and W

(ω) N,t ( f ) is a real continuous martingale with quadratic variation process

W (ω) N ( f ) t = t 0 ν (ω) N,s , |∇ θ f (θ , ω)| 2 ds .
Equation ( 29) is nothing else than a discrete version of the SPDE [START_REF] Delarue | Particle systems with a singular mean-field self-excitation, Application to neuronal networks[END_REF]. The natural procedure is then to show that (ω) → H N (ω) is tight and identify the limit. Since the identification of the limit is standard, we focus on the main difficulty, that is the tightness result.

The nonlinear process

The core of the proof consists in introducing the nonlinear process ( [START_REF] Jourdain | Propagation of chaos and fluctuations for a moderate model with smooth initial data[END_REF][START_REF] Malrieu | Convergence to equilibrium for granular media equations and their Euler schemes[END_REF][START_REF] Sznitman | Topics in propagation of chaos[END_REF][START_REF] Sznitman | Nonlinear reflecting diffusion process, and the propagation of chaos and fluctuations associated[END_REF]) associated to the McKean-Vlasov equation [START_REF] Bolley | Uniform convergence to equilibrium for granular media[END_REF], that is the diffusion θ whose finite dimensional laws are precisely given by ν t , t ≥ 0. This notion was first introduced by Sznitman [START_REF] Sznitman | Topics in propagation of chaos[END_REF] for systems without disorder. In the context of disordered diffusions, the nonlinear process may be defined as the solution ( θt , ω) t∈[0,T ] to

   θt = θ 0 + t 0 c( θs , ω)ds + t 0 [Γ , ν s ]( θs , ω)ds + B t , ω ∼ µ, ν t = L ( θt , ω), ∀t ∈ [0, T ] . (31)
Note that in (31), θ depends on ν, which itself is the law of θ , so the existence of such a nonlinear process is unclear.

Proposition 1 ([39]

). There is pathwise existence and uniqueness in [START_REF] Gupta | One-dimensional lattice of oscillators coupled through power-law interactions: Continuum limit and dynamics of spatial fourier modes[END_REF].

Proof. The proof follows ideas from Sznitman [START_REF] Sznitman | Topics in propagation of chaos[END_REF], Th 1.1, p.172. The point is to use a Picard iteration in the space of probabilities on C ([0, T ], R m × R n ) endowed with an appropriate Wasserstein metric. ⊓ ⊔

The key point in the proof of the fluctuation theorem (see [START_REF] Fernandez | A Hilbertian approach for fluctuations on the McKean-Vlasov model[END_REF]) is to build a coupling between the particle-system (θ i ) 1≤i≤N given by ( 3) and a well-chosen collection of independent nonlinear processes. Namely, for all i = 1, . . . , N, consider the nonlinear process θi defined by (31), with the same initial value as θ i , with the same inhomogeneity ω i , driven by the same Brownian motion B i . Then one has Proposition 2 ( [START_REF] Luc ¸on | Quenched limits and fluctuations of the empirical measure for plane rotators in random media[END_REF]).

E sup t≤T θ i,t -θi,t 2 ≤ Z N (ω 1 , . . . , ω N ) , (32)
where the random variable

(ω) → Z N (ω) is such that lim A→∞ lim sup N→∞ P (NZ N (ω) > A) = 0 . (33)
Proof. By (3) and ( 31) and using the one-sided Lipschitz continuity of c, one has

E sup s≤t θ i,s -θi,s 2 ≤ C t 0 sup 1≤ j≤N E sup u≤s θ j,u -θ j,u 2 ds + t 0 E [Γ , νN,s -ν s ]( θi,s , ω i ) 2 ds ,
where νN is the empirical measure of the nonlinear processes νN,t :=

1 N ∑ N i=1 δ ( θi,t ,ω i ) . Define T i, j,s := Γ ( θi,s , ω i , θ j,s , ω j ) -Γ (θ i,s , ω i , θ , ω)ν s (d θ , d ω), so that E [Γ , νN,s -ν s ]( θi,s , ω i ) 2 = 1 N 2 E N ∑ j=1 T i, j,s 2 + ∑ k =l T i,k,s • T i,l,s ≤ 1 N 2   CN + E    ∑ k =i,l =i k =l T i,k,s • T i,l,s       ,
where we used that Γ is bounded. Suppose for a moment that there is no disorder. Taking conditional expectation w.r.t. ( θr , r = l) in the last term and using the exchangeability of the particles, we obtain that this term is zero, leading, by Gronwall's Lemma, to an upper bound in [START_REF] Hitsuda | Tightness problem and stochastic evolution equation arising from fluctuation phenomena for interacting diffusions[END_REF] of the form C/N (see [START_REF] Fernandez | A Hilbertian approach for fluctuations on the McKean-Vlasov model[END_REF], Lemma 3.2). Since we work here in a frozen environment (i.e. we do not integrate w.r.t. the disorder and lose the exchangeability of the particles), this additional term becomes nontrivial and not bounded for fixed (ω). This fact precisely motivates the weak formulation of the quenched convergence introduced in Section 2.2.2. Proof of (33) can be found in [START_REF] Luc ¸on | Quenched limits and fluctuations of the empirical measure for plane rotators in random media[END_REF]. ⊓ ⊔

Weighted Sobolev spaces

Proposition 2 is the key result in order to show that the random variable H N defined in Section 2.2.2 is tight. The second tool we use is the introduction of weighted-Sobolev norms that are specifically adapted to the analysis of (29) (see [START_REF] Fernandez | A Hilbertian approach for fluctuations on the McKean-Vlasov model[END_REF] for a previous similar approach). Namely, for every integer j, α ≥ 0, we consider the space of all real functions f defined on R m × R n differentiable up to order j such that

f j,α :=    ∑ |k 1 |+|k 2 |≤ j R m ×R n D k 1 θ D k 2 ω f (θ , ω) 2 1 + |θ | α 2 1 + |ω| α 2 dθ dω    1/2 < ∞ ,
where, if k = (k 1 , . . . , k m ) and θ = (θ (1) , . . . , θ (m) ), we define |k| := ∑ m i=1 k i and D k θ h(θ ) := ∂ k 1 θ (1) . . . ∂ k m θ (m) h(θ ). Let W j,α 0 be the completion of C ∞ c (R m × R n ) for this norm; (W j,α 0 , • j,α ) is a Hilbert space and W -j,α 0 is its dual space. The only thing that differs from the usual Sobolev norm is the weight which is here to control the possibly unbounded coefficient c in (3) and the unbounded disorder. In this framework, using Proposition 2, one can show that under the assumptions of Theorem 2, Proposition 3. There exist well-chosen Sobolev indices ( j 1 , α 1 ) and ( j 2 , α 2 ) such that for fixed (ω) and N ≥ 1, for all T > 0,

Tightness result

In the case without disorder (or in the averaged case), the random variables A N and B N would simply be replaced by constants. Using the estimates of Proposition 3 and standard tightness criteria for continuous processes with values in Hilbert spaces (see Joffe and Métivier [START_REF] Joffe | Weak convergence of sequences of semimartingales with applications to multitype branching processes[END_REF]), proving that the fluctuation process is tight is straightforward ( [START_REF] Fernandez | A Hilbertian approach for fluctuations on the McKean-Vlasov model[END_REF]). In the quenched model, A N and B N are not bounded for fixed (ω).

We need to work harder to prove that the random variable (ω) → H N (ω) is tight in M 1 (C ([0, T ], S ′ )).

Theorem 4. For all ε > 0, there exists a relatively compact subset K ε in the set M 1 (C ([0, T ], S ′ )) such that lim sup

N→∞ P ({(ω) ; H N (ω) ∈ K c ε }) ≤ ε . ( 36
)
The idea of the proof of Theorem 4 is that, based on Proposition 3, for all ε > 0, one can choose A(ε) sufficiently large so that the quantities in [START_REF] Jourdain | Propagation of chaos and Poincaré inequalities for a system of particles interacting through their CDF[END_REF] and [START_REF] Jourdain | Propagation of chaos and fluctuations for a moderate model with smooth initial data[END_REF] are smaller than ε with large probability. K ε is then constructed as a set of elements in M 1 (C ([0, T ], S ′ )) satisfying a Joffe and Métivier criterion (hence relatively compact) with parameters (depending on ε) chosen in such a way that H N (ω) belongs to K ε with high probability. We leave the technical details to [START_REF] Luc ¸on | Quenched limits and fluctuations of the empirical measure for plane rotators in random media[END_REF], Theorem 4.10.

  (a) Trajectories of the center of synchronization ψ N for different realizations of the disorder (ω 1 , . . ., ω N ) (µ = 1 2 (δ -0.5 + δ 0.5 ), K = 4, N = 400).

1 . 2 -j 1 ,α 1 ≤ 34 ) 3 . 2 0to W j 1 2 -j 2 ,α 2 <

 12113432122 the imbedding W -j 1 ,α 1 0 ֒→ W -j 2 ,α 2 0 is of Hilbert-Schmidt type, 2. the process η (ω) N is bounded in W -j 1 ,α A N (ω 1 , . . . , ω N ) ,(The linear operator L N,s defined in[START_REF] Godinho | Propagation of chaos for a sub-critical Keller-Segel model[END_REF] is continuous from W j 2 ,α B N (ω 1 , . . . , ω N ) . (35) Moreover, as random variables in (ω), A N and B N satisfy lim A→∞ lim sup N→∞ P (A N > A) = 0, and lim A→∞ lim sup N→∞ P (B N > A) = 0 .

Acknowledgements One of the articles reviewed here ( [41]) was written at the Technische Universität of Berlin and was supported by the BMBF, FKZ 01GQ 1001B.


Olivier Julien

Christophe Macabiau

Emmanuel Bertrand

Analysis of Galileo E1 OS unbiased BOC/CBOC tracking Techniques for Mass Market Applications

Keywords: component; Galileo, Tracking, ASPeCT, DET I

BOC modulation is used for Galileo E1 OS signals. The Offset Carrier (OC) modulation is interesting to transmit the Galileo signal because it offers a high degree of spectral separation with signals centered in the used band and a better robustness against noise, multipath and interference. However, BOC modulation introduces several peaks to the autocorrelation function, potentially biasing the pseudorange measurements. Several techniques were proposed in the last ten years to overcome this problem. The aim of this paper is to present an analysis of the suitability of unbiased BOC/CBOC tracking techniques of the Galileo E1 OS signal implemented in mass market receivers by reviewing their models and by testing them.

INTRODUCTION

While with the GPS L1 C/A signals the BPSK modulation is used, BOC modulation is applied to Galileo E1 OS signals.

The Offset Carrier (OC) modulation is interesting to transmit the Galileo signal because it provides a simple and effective way of moving signal energy away from the band centre and consequently allows the use of a same frequency band several GNSS. Moreover it offers a high degree of spectral separation with signals centered in the used band and a better robustness against noise, multipath and interference. However, BOC modulation introduces several peaks to the autocorrelation function, so we must make sure the measurement is made on the main peak in order not to bias the pseudorange measurements.

Several techniques were proposed in the last ten years to overcome this problem. In the frame of the FP7 HIMALAYA project, we are analyzing the potential of some of these techniques to fight the biased measurements for mass market applications with severe implementation constraints.

The aim of this paper is to present an analysis of the suitability of unbiased BOC/CBOC tracking techniques of the Galileo E1 OS signal implemented in mass market receivers by reviewing their models and by testing them.

The paper focuses on two techniques that both resist to the bias due to the BOC signal : ASPeCT and DET.

Autocorrelation

Side-Peak Cancellation Technique (ASPeCT)'s principle is to remove the side-peaks of the sine-BOC(n,n) autocorrelation function in order to be able to reliably obtain unambiguous measurements. The fundamental concept of the double estimation technique (DET) is to track separately and independently the subcarrier and code delays using different delays. It is based on the fact that the time delay of the received signal is the same in both the code and the sub-carrier. Both methods do not rely on peak discrimination to determine valid tracking states.

The second part of the article reviews the performance of these techniques for CBOC tracking in presence of acquisition bias, in presence of noise and multipath.

II. GALILEO E1 OS SIGNAL MODEL

A. Galileo E1 Signal Model

() () ( ) () ( ) ( ) ( ) ( ) () k n T R k s R k d P k I D D D I I f L CBOC L CBOC D + - = - + ϕ τ τ ε πε ε ε cos sinc 4 ), ( sec ), ( () () ( ) () ( ) ( ) ( ) ( ) () k n T R k s R k d P k Q D D D Q I f L CBOC L CBOC D + - = - + ϕ τ τ ε πε ε ε sin sinc 4 ), ( sec ), ( () () ( ) () ( ) ( ) ( ) ( ) () k n T R k s R k d P k I P P P I I f L CBOC L CBOC P + - = - + ϕ τ τ ε πε ε ε cos sinc 4 ) ), ( sec ), ( () () ( ) () ( ) ( ) ( ) ( ) () k n T R k s R k d P k Q P P P Q I f L CBOC L CBOC P + - = - + ϕ τ τ ε πε ε ε sin sinc 4
), ( sec ), ( with . where

• represents the total power of the incoming signal at the correlator input (and thus includes a certain number of losses due to the front-end), • the subscripts and are for the incoming "data component" and "pilot component" respectively, • the subscripts is for the local spreading sequence, • and are the equivalent front-end filter impulse response and transfer function respectively, • is the Fourier transform of the signal ,

•

is the correlation function between the incoming spreading sequence and the locally generated spreading sequence taking into account the effect of the front-end filter ,

• is the code delay error between the incoming code delay and the one estimated by the receiver, • is the phase error (in the middle of the correlation interval) between the incoming carrier and the locally generated carrier, • is the frequency error between the incoming carrier and the locally generated carrier, • is the correlation time, • are the noise components of the two discriminator outputs associated with the data component respectively; are the noise components of the 2 discriminator outputs associated with the pilot component respectively. All these noise components are all uncorrelated, and:

• is the equivalent double-sided thermal noise PSD at the correlator input. In the context of this study, it also includes the interference components that can be equivalently represented by a white noise component at the correlator input.

• is the loss of noise power due to the front-end filter.

As a first remark, it is important to understand that the variance of the noise on the data and pilot correlator output might be different due to the different CBOC subcarriers used on each component (

D P β β ≠ since ) ' ' , 11 / 1 , 1 , 6 ( ) ' ' , 11 / 1 , 1 , 6 ( + = ≠ - = CBOC L CBOC L D P
). This is however not the case if a BOC receiver is used since in this case the same replica is used on the data and pilot components (BOC [START_REF] Ec | European GNSS (Galileo) Open Service -Signal[END_REF][START_REF] Ec | European GNSS (Galileo) Open Service -Signal[END_REF]). This is shown in Figure 1 for different values of front-end filter bandwidth.

III. OPPORTUNITY OF CBOC FOR MASS MARKET RECEIVERS

Galileo E1 OS signal will use the CBOC(6,1,1/11) modulation. It is interesting to highlight the differences in performance between a BOC receiver (narrow-band) and a CBOC receiver (wide band). It is known that a CBOC receiver requires more complexity due to the need for a higher sampling rate, and due to the need for a more complex 4-level local replica. Regarding acquisition performances, a BOC receiver requires a wider code delay acquisition bin and thus has a faster acquisition time. To be able to have the same kind of performance, the CBOC receiver would need a narrower code delay acquisition bin, and a corresponding increased number of correlators to compensate for this. This could be acceptable, however, pending the fact that the information on the signal characteristics (Doppler, time) is fairly accurate.

Regarding code tracking, a CBOC receiver exhibits performances that are significantly higher than those of a BOC receiver. This is due to the fact that the CBOC autocorrelation function is very sharp, and to the fact that a narrower earlylate spacing can be used (due to the wider front-end filter bandwidth). This is quite significant regarding the capability of the receiver to provide proper measurements (with limited impact from the thermal noise and multipath). To lower the complexity of the CBOC receiver, two techniques using only 1-bit local replicas can be used. The two methods exhibit interesting performances. The simplest one, TM61 [START_REF] Julien | 1st CNES-ESA Workshop on GALILEO Signals and Signal Processing[END_REF], degrades the resistance to thermal noise, but improves the resistance to multipath. Its resistance to false lock still needs some work. The second method [START_REF] Julien | 1-bit Processing of Composite BOC (CBOC) Signals and Extension to Time-Multiplexed BOC (TMBOC) Signals[END_REF], which requires twice as many correlators exhibits excellent performance due to its adaptive possibility. It can be adapted to best fight thermal noise, or multipath, or interference. Both BOC and CBOC receivers will have to include a specific tracking module to fight against biased measurements inherent to a BOC modulation. However, this threat is the same for both the BOC and CBOC receivers (the CBOC receiver does not seem to have more false lock points). Regarding phase tracking, the CBOC receiver will have better performance than the BOC receiver since the correlation losses are lower for the former. As a conclusion, if the main requirement for the receiver is the complexity, then it seems that the choice of a narrow band BOC receiver would be indicated. This is very important also for acquisition performance with a simple receiver (acquisition time and detection threshold) that is critical for mass market receiver. That being said, it appears that a CBOC receiver would significantly improve the quality of the measurements. This would however have to be confirmed with tests in a real environment.

IV. CONSIDERED UNAMBIGUOUS BOC TRACKING TECHNIQUES

A variety of new modulation schemes including the BOC has been introduced. BOC signals have several advantages over traditional BPSK signals such as an increased robustness against multipath and improved tracking performance. However, they are characterized by autocorrelation functions with multiple peaks which have secondary peaks which are not much smaller than the main (central) peak and that may lead to false code lock. In the case of the Galileo E1 OS signal, this can lead to a bias around 150 m and has thus to be taken into account in the design of the tracking loops. This has led to the design of various BOC tracking algorithms. In the frame of this study 2 widely spread techniques were investigated: -Autocorrelation Side-Peak Cancellation Technique (ASPeCT) -Double Estimator (DE)

The purpose of these methods is to solve the problem of incorrectly locking onto secondary peaks.

V. AUTOCORRELATION SIDE-PEAK CANCELLATION TECHNIQUE

ASPeCT's principle is to remove the side-peaks of the BOC(1,1) autocorrelation function in order to be able to reliably obtain unambiguous measurements [START_REF] Julien | ASPeCT: an Unambiguous Sine-BOC(n,n) Acquisition/Tracking Technique for Navigation Applications[END_REF]. To do so, it is interesting to calculate the correlation of the sine-BOC(n,n) modulated spreading code (PRN code × square-wave subcarrier) with the PRN code only. Without considering frontend filtering, it is equal to [START_REF] Julien | New Unambiguous BOC(n,n) Tracking Technique, NaviTec Conference[END_REF]:

() + - - =
() () () τ β τ τ 2 / 2 P B B ASPeCT R R R - =
where the bar represents the effect of the front-end filter on the correlation functions Figure 3 shows the ASPeCT modified correlation function using an experimental β value of 1.4 for a BOC(1,1) signal using a 6 MHz front-end filter bandwidth (double-sided). There is no side-peak remaining in the modified correlation function close to the BOC(1,1) autocorrelation side-peaks. . This is very important when considering the DLL stability for low C/N0. However, ASPeCT stability domain remains significantly lower than for the BPSK(1) signal tracking, which constitutes the main advantage of the SSL technique. appears to create a false lock point around ±0.95 chips. This is, however, not a real threat as there is no energy at that location on the ASPeCT-modified correlation function (

() ( ) ( ) [ ] ( ) ( ) [ ] ( ) ( ) ( ) 2
Q I d Q Q Q I I I Q Q Q I I I D + + - + - - - + - = β β ε τ
( ) 0 95 . 0 ≈ ASPeCT R
) as already seen in Figure 2.22. As a consequence, no false lock will occur even for high C/N0.

It is important to note that the optimal choice of β depends upon the front-end filter and the early-late correlator chip spacing.

The DLL architecture adapted to ASPeCT is shown in Figure 5. The fundamental concept of the Dual Estimation Technique (DET) is to track separately and independently the subcarrier and code delays using different delay estimates. It is based on the fact that the time delay of the received signal is the same in both the code and the sub-carrier. The following presentation is based on the tracking of a BOC signal. However, [START_REF] Hodgart | A Receiver of Binary Offset Carrier (MBOC) Modulated Signals[END_REF] claims that the method can be extended to CBOC tracking.

The DET uses three tracking loops (architecture shown in [START_REF] Hodgart | A Receiver of Binary Offset Carrier (MBOC) Modulated Signals[END_REF]) -a delay-locked loop (DLL) tracks the code phase of the received signal based on the generation of a Prompt version of the sub-carrier multiplied by an Early-Minus-Late version of the spreading code. Thus, this can be linked with a BPSK tracking: not very accurate but robust to tracking ambiguity.

a sub-carrier locked loop (SLL) tracks the sub-carrier phase of the received signal based on the generation of an Early-Minus-Late version of the sub-carrier multiplied by a Prompt version of the spreading code. Thus this can be linked with a sub-carrier tracking: accurate but not robust to tracking ambiguity a third loop, either a frequency locked loop (FLL) or a phase locked loop (PLL) tracks the carrier frequency and / or phase of the received signal.

Assuming that the tracking of the code is done perfectly by the DLL, the SLL will rely on the sub-carrier correlation function shown on Figure 2.29 (case of BOC(1,1) tracking). It can be seen that the subcarrier correlation function is ambiguous with peaks every 0.5 chips (including down-pointing peaks). It can also be seen that the slope of the correlation function is steeper than that of the BOC [START_REF] Ec | European GNSS (Galileo) Open Service -Signal[END_REF][START_REF] Ec | European GNSS (Galileo) Open Service -Signal[END_REF]. It can thus be thought that SLL tracking will be more accurate that the BOC(1,1) tracking. This however, might be compensated by the noise on the DLL.

Assuming that the SLL tracks perfectly the subcarrier, the DLL will be based on the code correlation function shown on Figure 6 (case of BOC(1,1) tracking). It can be seen that the code correlation function is unambiguous, but with a slope flatter than that of the BOC [START_REF] Ec | European GNSS (Galileo) Open Service -Signal[END_REF][START_REF] Ec | European GNSS (Galileo) Open Service -Signal[END_REF]. Consequently, the DLL should provide an unambiguous tracking. The ambiguity of the SLL estimate can thus be solved through the noisier but unambiguous DLL estimate. An example is shown in [START_REF] Hodgart | A Receiver of Multiplexed Binary Offset Carrier (MBOC) Modulated Signals[END_REF]. The SLL settles to integer subchip offsets from the correct, valid delay estimate. The delay estimate derived from the DLL allows correction of the SLL delay estimate wherever the SLL has settled. Therefore, the corrected delay estimate of the DET, is instantaneously corrected for tracking slips of integer subchips and the integrity of the receiver not compromised. One of the advantages of the DET is that it can recover from a large initial error since it relies on an unambiguous DLL that is comparable to a DLL tracking a BPSK. [START_REF] Hodgart | A Receiver of Multiplexed Binary Offset Carrier (MBOC) Modulated Signals[END_REF] states that there are no false-lock or invalid tracking states when operating the DET. The method is said to function with the cos-BOC(15,2.5), which is a very difficult BOC to track due to the high number of side-peaks compared to the BOC(1,1) or CBOC signals.

This method is said to maintain reliable and robust tracking in high noise conditions (24 dB-Hz) and with strong multipath interference (half-amplitude) [START_REF] Hodgart | A Receiver of Multiplexed Binary Offset Carrier (MBOC) Modulated Signals[END_REF], although it is not demonstrated. The authors mention that the tracking jitter of the corrected DET delay estimate is equivalent to that achieved by the BJ algorithm (standard early-minus-late discriminators), preserving the precision of the BOC modulated signal. This seems in line with the correlation function analysis made earlier. In addition, it appears that multipath mitigation techniques such as narrow correlators and double-delta correlation can be easily implemented into the DET correlator structure, in particular into the SLL.

No theoretical derivation or simulations providing the code tracking standard deviation (in thermal noise) using DET as a function of the C/N0 was found in the literature. The multipath error envelope using DET was also not found in the literature. It is difficult to assess the effect of the DLL tracking noise on the SLL. In particular, this might be quite important at low C/N0. [START_REF] Hodgart | A Receiver of Binary Offset Carrier (MBOC) Modulated Signals[END_REF][START_REF] Hodgart | A Receiver of Multiplexed Binary Offset Carrier (MBOC) Modulated Signals[END_REF] proposes a model of the correlator outputs based on the assumption that the correlation function of a BOC incoming signal with a replica composed of the code delayed by 1 ε multiplied by a subcarrier delayed by 2 ε is approximated by:

( ) ( ) ( ) 2 1 2 1 , ε ε ε ε Λ × = BPSK R R Where • BPSK R
is the autocorrelation of a BPSK(1) signal (as shown in Figure 6), and • Λ is a periodical triangular function with magnitude 1 and period the sub-carrier period (as shown in Figure 6). This model is valid when the DLL and SLL tracking errors are very small. It is however quite clear that it is not valid for large errors as represented in Figure 6 in the case of an infinite front-end filter bandwidth. In particular, it seems that even small errors in the DLL tracking can significantly impact the SLL (with an uneven tracking).

Still, the combination of the DLL and SLL will be absorbed by the neighboring higher peak, which means that the tracking should unavoidably go towards an unambiguous tracking for the DLL and a potentially ambiguous tracking for the SLL. However, it can be anticipated that strong tracking errors in either the DLL or SLL will have a strong impact on the overall tracking accuracy. This is true also for tracking in low C/N0 conditions.

Figure 7 shows the true DET correlation function taking into account the use of a 6 MHz (Double-Sided) front-end filter. It can be seen that in this case the model from [START_REF] Hodgart | A Receiver of Binary Offset Carrier (MBOC) Modulated Signals[END_REF] cannot be used.

It seems that the only way to effectively test this method in terms of resistance to thermal noise and interference is through an independent testing.

The DET method, as presented in [START_REF] Hodgart | A Receiver of Binary Offset Carrier (MBOC) Modulated Signals[END_REF][START_REF] Hodgart | A Receiver of Multiplexed Binary Offset Carrier (MBOC) Modulated Signals[END_REF], uses a coherent discriminator for both the SLL and the DLL. However, it seems reasonable to think that an adaptation of the DET tracking algorithm using a non-coherent discriminator would be more adapted for a mass-market receiver. With this architecture, the DET method requires 3 complex correlators, thus 1 less than ASPeCT and BJ.

It is important to note that this method seems conceptually very similar to the patent proposed in [START_REF] Blunt | Demonstration of BOC(15, 2.5) acquisition and tracking with a prototype hardware receiver[END_REF]. This method is patented by SSTL for BOC and CBOC use [START_REF] Hodgart | A Receiver of Binary Offset Carrier (MBOC) Modulated Signals[END_REF]. The simulations are based on the reception of BOC signal, which is an approximation of a BOC receiver receiving a CBOC signal, as seen earlier. In the frame of the simulations, it is assumed that only the pilot signal is tracked. Also, it is assumed that the secondary code has been acquired. As a consequence, code tracking will be based upon a 100-ms correlation time while the phase/frequency tracking will be based on a 20-ms correlation time (to take into account the oscillator drift and signal dynamics although they are not simulated herein).

The receiver is assumed to have a 9-MHz (double-sided) front-end filter. The correlator spacing is assumed to be close to the inverse of the one-sided front-end filter bandwidth, or approximately 0.14 chips. For ASPeCT's DLL and for DET's DLL and SLL, this same correlator spacing will be used.

Regarding ASPeCT, the value of β is chosen equal to 1.3. This value is a confortable value that allows an unambiguous DP discriminator.

A 3rd order PLL aided by a 2nd order FLL will be used. The PLL is a true PLL using the Q discriminator [START_REF] Fenton | Split C/A Code Receiver[END_REF] and uses an equivalent loop bandwidth equal to 15 Hz. The FLL will use the differential 4-quadrant arctangent discriminator with a loop bandwidth equal to 18 Hz.

The FLL-aided PLL will aid the 1st order DLL. The DLL will use a DP discriminator and will have a 1 Hz equivalent loop bandwidth.

A. Bias Test

It has been shown in [START_REF] Fenton | Split C/A Code Receiver[END_REF] that ASPeCT was unambiguous by nature. Thus, this section investigates the unambiguous property of DET. Figure 8 shows the response of DET to an initial error of 0.5 chips for two C/No values. It can be seen that DET is able to converge towards the true code delay. It can be noticed that this convergence is fairly slow (more than 5 seconds), which is due to fact that the correlation functions used by the DLL and SLL are distorted by large tracking errors, as already highlighted in Figure 6. This might be a problem if the dynamic of the signal is large during the acquisition phase.

B. C/N0 Test

The theoretical formulation of code delay tracking can be found in the literature for traditional BOC tracking and ASPeCT. However, such expression cannot be found in the literature for DET. The main goal of this section is thus to find this expression and test it thought simulations using the receiver parameters found above.

As already explained, assuming that the DLL tracks perfectly the code, the SLL can be considered as functioning based upon the correlation function shown in Figure 6 (assuming an infinite front-end filter). This correlation function can be used to adapt the generalized code tracking error variance. It means that the DET tracking error standard deviation can be roughly (assuming no front-end filter and a perfect DLL) estimated by: + × = , which is also interesting. Because ASPeCT has performance very similar to traditional BOC(1,1) tracking, this applies also to ASPeCT.

Figure 9 shows the results of the simulations done to test the code tracking error standard deviation of ASPeCT and DET. It can be seen that DET follows very well the theoretical formula displayed above. This means that slight errors of the DLL do not significantly degrade the SLL behavior, at least regarding the tested configuration (indeed, for a wider early-late spacing, the DLL error would be larger and would degrade more significantly the SLL). It can also be seen that DET slightly outperforms ASPeCT against thermal noise.

C. Multipath Test

This test is meant at assessing the multipath envelope for both DET and ASPeCT. This is done through simulations by injecting a multipath with a slowly growing delay, which will also modify its relative phase compared to the direct path. For this simulations, to ensure that the tracking loops reach their stability point for each multipath delay value, the equivalent loop bandwidth of the DLL (and SLL for DET) were set to 40 Hz, and no thermal noise was generated.

Figure 10 shows the corresponding multipath envelope using a multipath with an amplitude that is half of the direct signal. It can be seen that here again, DET has a better resistance to multipath than ASPeCT for shorter delay multipath. On the other hand, ASPeCT behaves better than DET for long delay multipath.

VIII. CONCLUSION

It appears that DET has an overall better performance than ASPeCT in the tests that we have conducted. This is true for tracking conditions, although this difference is fairly slim and might not be noticeable in operational conditions. DET has the great advantage of having a very large pull-in and of recovering from initial code delay errors greater than 0.5 chips. More tests in operational conditions should be performed to confirm this first assessment of the ASPeCT and DET techniques.

Figure 1 -

 1 Figure 1 -Correlator Ouput Noise Power Degradation According to the Local Replica.

B

  It can be seen that P B R / consists of two triangles perfectly located on the side-peaks of the BOC(1,1) autocorrelation function, and with exactly the same magnitude. As a consequence, the idea on which ASPeCT is based is to form a synthesized correlation function by subtracting 2/ P B R from 2 BR to remove the undesired side-peaks as shown in Figure2.

Figure 2 -

 2 Figure 2 -BOC(1,1)/ BOC(1,1), BOC(1,1)/PRN Squared Normalized Correlation Functions and ASPeCT Modified Correlation Function Without Front-End FilterHowever, in reality, one has to take into account the impact of the front-end filter on each correlation function, and as a consequence, on the alignment of the peaks and their respective magnitudes. Figure2.22 shows the impact of a 6 MHz front-end filter bandwidth (double-sided) on the BOC(1,1) signal. It underlines that a narrow front-end filter could misalign both correlation functions' peaks, inducing the appearance of very small peaks around ±0.6 chips. This can translate into false lock points for high C/N0 (only peaks of the squared correlation function pointing upward can lead to stable false lock points). This problem has to be taken into account in the design of the code delay discriminator. It is thus recommended to use a coefficient β in the combination of the two squared correlation functions in order to eliminate any small remaining peak. This can be modeled as:

Figure 3 -

 3 Figure 3 -BOC(1,1)/ BOC(1,1), BOC(1,1)/PRN Squared Normalized Correlation Functions and ASPeCT Modified Correlation Function With a 6 MHz Front-End Filter (Double Sided) and In order to focus on the tracking performance, a modified version of the normalized DP discriminator based on ASPeCT-modified correlation function is proposed:

Q

  Figure 4 shows ASPeCT DP discriminator output for a BOC(1,1) signal using a 6 MHz (double-sided) front-end filter for 1 = β

Figure 4 -

 4 Figure 4 -ASPeCT Dot Product Proposed Discriminator Output with and for an Early-Late Spacing of 0.2 Chips and a 6 MHz Front End Bandwidth (Double Sided) Finally, 4 . 1 = βappears to create a false lock point around ±0.95 chips. This is, however, not a real threat as there is no energy at that location on the ASPeCT-modified correlation function (

Figure 5 -

 5 Figure 5 -DLL Architecture adapted to ASPeCT

Figure 6 -

 6 Figure 6 -Correlation Functions Used in DET

Figure 7 -

 7 Figure 7 -True Normalized DET Correlation Function (Top), DET Correlation Function as Modeled by [6] (Middle), and Difference between the two (Top)

Figure 8 -

 8 Figure 8 -DET Code Delay Tracking Error for an Initial Error of 0.5 Chip and a C/No of 45 dB-Hz (Top) and 30 dB-Hz

σ

  This means that under our hypothesis, DET should improve lower the BOC(1,1) tracking standard deviation by 15

Figure 9 -

 9 Figure 9 -ASPeCT and DET Tracking Code Tracking Standard Deviation as a Function of the C/No

Figure 10 -

 10 Figure 10 -DET and ASPeCT Code Delay Tracking Error in Presence of Multipath (half the Amplitude of the Direct Signal).

Code Delay (Chip) Normalized Correlation Function SubCarrier Code BOC(1,1)


Kevin Elis

Alexandre Chabory

Jérôme Sokoloff

Sylvain Bolioli

published or not. The documents may come

2D interaction of a gaussian beam with a dichroic surface for the modeling of quasi-optical systems


Alexandre Chabory
email: alexandre.chabory@enac.fr

Jérôme Sokoloff

Sylvain Bolioli

Kevin Elis

Application of Gaussian-Beam based Techniques to the Quasi-Optical Systems of Radiofrequency Radiometers

come

Application of gaussian beam based techniques to the quasi-optical systems of radiofrequency radiometers

I. INTRODUCTION

In mm-wave radiometry, quasi-optical systems are used to guide, filter, split or mix the input signals. Such systems are generally constituted by the combination of different elements, e.g. mirrors, lenses, dichroic filters, or horn antennas. The number and the size of these elements render their modeling difficultly amenable by classical methods in acceptable computation times. For such systems, even physical optics may become computationally too costly.

Gaussian beams constitute a possible alternative. Historically, they have firstly been applied assuming that the system has an axial symmetry [START_REF] Goldsmith | Quasi optical techniques at millimeter and submillimeter wavelengths[END_REF]. The incident field is approximated by a single gaussian beam. When passing through the elements of the system, this beam is transformed into reflected/transmitted beams which characteristics are determined by means of the ABCD laws. Since then, an important number of gaussian-beam based techniques have been developed in the context of quasi-optical systems [START_REF] Imbriale | Recent trends in the analysis of quasioptical systems[END_REF], but also to treat other problems as radomes [START_REF] Chabory | Computation of electromagnetic scattering by multilayer dielectric objects using gaussian beam based techniques[END_REF], or indoor propagation [START_REF] Fluerasu | gaussian beam based simulation of indoor radio propagation channels[END_REF]. In most cases, modeling approaches based on gaussian beams are constituted by two key components: the expansion and the tracking. Expansion techniques allow for the representation of fields as discrete sums of elementary beams. Tracking techniques deal with the interaction/propagation of elementary beams through the elements of the system, e.g. the interaction of beams with dielectric and/or metallic interfaces.

In this article, we consider quasi-optical systems including horn antennas, metallic reflectors and/or dielectric lenses. We choose an appropriate expansion to describe the radiation of horn antennas from a discrete set of gaussian beams. We employ a gaussian beam shooting to deal with the tracking of the beams in the system.

Section 2 presents the expansion principles and the gaussian beam shooting algorithm. In Section 3, numerical tests are performed in order to evaluate the capabilities (accuracy/computation time) of this approach. Finally in Section 4, we present an experimental validation.

II. METHOD

A. Expansion

A large family of gaussian beam expansions already exists in asymptotics [START_REF] Maciel | Gabor-based narrow waisted gaussian beam algorithm for transmission of aperture-excited 3D vector fields through arbitrarily shaped 3D dielectric layers[END_REF]- [START_REF] Chabory | Physically based expansion on conformal gaussian beams for the radiation of curved aperture in dimension 2[END_REF]. Our choice is based on two criteria. Firstly, we want to be free from the paraxial approximation in order to model correctly side lobes diverging with an angle greater than 20 degrees with respect to the horn axis. Secondly, we want to use as few beams as possible to reduce the computation time. Due to these criteria, we employ the pragmatic expansion developed in [START_REF] Chabory | Computation of electromagnetic scattering by multilayer dielectric objects using gaussian beam based techniques[END_REF], with which a field known on a moderately curved surface is expressed as a set of equally spaced gaussian beams. Note that to reduce the number of beams, the expansion characteristics are chosen from the physical properties of the configuration. Numerical experiments have demonstrated the efficiency of this approach. However, limitations exist that concern the incidence of the initial field, and the curvature of the surface, both of which must remain moderate. For the application to horn antennas, we consider a spherical expansion surface for which these limitations are not reached.

B. Gaussian beam shooting

Once the radiation of the horn antenna is described by a discrete sum of gaussian beams, the evolution of each elementary beam must be computed when passing through the quasi-optical system. In [START_REF] Deschamps | Ray Technics in Electromagnetics[END_REF] Deschamps has demonstrated that the fields reflected and/or transmitted at a dielectric/metallic interface illuminated by an incident gaussian beam can be approximated by gaussian beams. Thus, we track each elementary beam of the expansion as follows. When the beam propagation axis encounters an interface, it is transformed into reflected and/or transmitted gaussian beams, which are tracked in turn. The tracking stops either when the beam power becomes neglectable, or when the beam leaves the system.

III. IMPLEMENTATION AND VALIDATIONS

A. Implementation

The method presented in the previous section is implemented in a Scilab program, named QOSGB (quasi-optical systems via Gaussian beams), which is able to model systems made of mirrors, lenses, and horn antennas. This program is splitted in two parts. In the first part, the expansion and tracking are performed, so as to represent the field in any part of the system as a combination of Gaussian beams. In the second part, beams are sumed in order to obtain numerical values for the fields. Note that the paraxial formulation of gaussian beams is used for observation points inside/near the system, whereas the far-field formulation is employed for farfield patterns [START_REF] Chabory | Computation of electromagnetic scattering by multilayer dielectric objects using gaussian beam based techniques[END_REF]. One of the goal of this article is to carefully justify the capabilities of the method we propose to model quasi-optical systems in comparison to other techniques. Firstly, we analyze the efficiency of the expansion. We consider a conical horn antenna of length 17.8mm and of diameter 10mm at a frequency of 150GHz. The horn antenna is approximated by its equivalent aperture [START_REF] Balanis | Antenna theory, analysis and design[END_REF]. The expansion surface is a sphere centered at the aperture center and of radius 40mm. We start our analysis with a distance between elementary beams of d = 1.5λ. According to [START_REF] Chabory | Computation of electromagnetic scattering by multilayer dielectric objects using gaussian beam based techniques[END_REF], the beams are symmetric and we choose their waist size equal to d.

B. Numerical tests of the expansion

The expansion yields 1022 gaussian beams in 13.78s. In Fig 2, we observe the electric field obtained from the sum of the elementary beams. The general behaviour represents accurately the horn radiation. We have verified that the difference between this result and a reference solution is below -40dB except very near the aperture.

In Fig. 3(a), we compare the far-field pattern in the E-plane obtained for different values of d. For larger values of d, we observe that the accuracy is slightly deteriorated. Concerning the computational cost, when d is set to 2λ and 3λ, the number of beams becomes 594 and 291, which corresponds to computation times of 3.73s and 1.62s, respectively. Small values of d lead to use more expansion beams, and consequently to a better accuracy and a longer computation time. As illustrated in Fig. 3(b), the same remarks hold when the horn is corrugated.

Finally, the expansion treats accurately the radiation of horn antennas with a few hundred elementary beams, so as to yield an inexpensive computational cost.

C. Numerical tests on a complete quasi-optical system

We consider a complete quasi-optical system working at 150GHz with three elements: the preceding corrugated conical horn, a dielectric lens, and a mirror. This configuration is displayed in Fig. 4. The lens has a relative permittivity of 2.6 and a thickness of 6mm. Its two interfaces are spheric with radius 37mm. Both are covered by an anti-reflective coating of thickness 0.394mm and relative permittivity 1.61. The mirror is a parabolic reflector of focal 100mm. In Fig. 5(a), we display the total electric field obtained by means of QOSGB in the plane y = 0. In Fig. 5(b), we depict the total electric field obtained by Feko in the same configuration. This result is considered as a reference solution because Feko provides a numerical solution free from any physical approximation. We note a very good agreement between QOSGB and Feko. This is confirmed by Fig. 5(c), in which we observe the difference between both results that remains everywhere below -30dB. Besides, the total fields are plotted at y = 0 and x = 40mm in Fig. 6. Even after the interactions with the lens and the reflector, QOSGB fits with FEKO. These confrontations demonstrate the abilities of QOSGB to predict accurately the field inside and near the system.

We now analyze the far-field pattern of this system. In Fig. 7 and Fig. 8, we display the gain with respect to the observation angle in the E-and H-planes. The results obtained with Feko and QOSGB are very similar. Indeed, the level of the crosspolarization, sidelobes, and maximal gains are very close. For example, the maximal gain values are of 30.49dB and 30.19dB with Feko and QOSGB, respectively. Furthermore, both methods predict a boresight error of 0.3 o in the E-plane.

On a PC running at 2.4GHz, the first part of QOSGB has been performed in 13.1s. This time is relatively short, it comprises the expansion and the gaussian beam shooting. For the result of Fig. 5, the second part of QOSGB requires 12.65s. It corresponds to the computation of the field at 100 × 100 observation points from all the beams generated in the expansion and shooting. To obtain the result at the same observation points, the computation time of Feko using the parralelized multilevel fast multipole method of moments (MLFMM) is of order 46.18 hours. Thus, in this configuration, QOSGB maintains the accuracy of the method of moments with a greatly reduced computation time. Note also that if we include any fourth element in the system, the computation was not amenable with FEKO on this PC.

IV. EXPERIMENTAL VALIDATION

We now consider the experimental setup of Fig. 9 working at 10GHz. In this system, we keep the horn and the lens of the previous sections, except for the size, which is ten times larger. The distance between the horn aperture and the lens center is of 525mm. Note also that the horn is not corrugated, and vertically polarized. In order to measure the vertical electric field, we place a vertical dipole that can move in a volume beyond the lens.

In Fig. 10, we represent the measured electric field in a vertical plane. In this figure, the reference frame center x = y = 0 is placed at the horn aperture. We note the presence of perturbations on both sides of the system. In Fig. 11(a) and (b), we compute this configuration with Feko and QOSGB. We observe that the general behaviour of the field is well reproduced. More particularly, the field spatial spread and the minimum value near x = 0, z = 0.75m are similar. Nevertheless, there are differences in the focal zone position, and in the field outside the central region, i.e. for |x| > 0.1m. These differences may be explained by side effects on the lens edge. Indeed, the lens used in the measurements is bordered by a plexyglass support, which has not been included in the simulation. Besides, we have chosen to not model edge effects in QOSGB because this program is designed for mm-waves quasi-optical systems, in which side effects are generally neglectable. Nevertheless, note that approaches exist to deal with side effects with gaussian beams [START_REF] Hillairet | Uniform analytic scattered fields of a PEC plate illuminated by a vector paraxial gaussian beam[END_REF], [START_REF] Chou | Novel gaussian beam method for the rapid analysis of large reflector antennas[END_REF].

In order to confirm the importance of the edge diffraction by the lens in this configuration, we display with QOSGB the incident field that arrives at the lens from the horn antenna in Fig 12 . On the edge, the incident field is only 10dB lower than its maximal value on the lens, which validates our hypothesis.

V. CONCLUSION

In this article, the aim was to propose an asymptotic modeling technique based on gaussian beams, in the context of quasi-optical systems. To this end, we have chosen an expansion method to describe the radiation of horn antennas with a few hundred gaussian beams. We have numerically verified that this expansion achieves a good trade-off between computation time and accuracy. After the expansion, the beams are tracked in the system by means of a shooting algorithm. In a comprehensive test case, we have shown that this method maintains the accuracy of the method of moments with a greatly reduced computation time. Finally, the method has been confronted to measurements realized with a setup containing an horn antenna and a lens. The comparisons have been satisfying, except for the diffraction by the lens edge that is not accounted in our approach.

For future works, other elements as dichroic surfaces shall be incorporated in QOSGB in order to be able to model more general types of quasi-optical systems. This would require research works to investigate the interaction of gaussian beams with periodic surfaces.

Fig. 1 .

 1 Fig. 1. Expansion principle

Fig. 2 .

 2 Fig. 2. Electric field obtained from the expansion (dB), the expansion surface is plotted in white

Fig. 3 .

 3 Fig. 3. Gain pattern (dB) of the horn antenna in the E-plane for d = 1.5λ (black with markers), d = 2λ (blue with • markers), and d = 3λ (red with markers)

Fig. 4 .

 4 Fig. 4. Configuration of the quasi-optical system

Fig. 5 .Fig. 6 .

 56 Fig. 5. Total electric field in the quasi-optical system (dB)

Fig. 7 .Fig. 8 .

 78 Fig. 7. Gain pattern in the E-plane (dB): Feko (red with markers), QOSGB (blue with markers), co-polarization (solid line), cross-polarization (dashed line)

Fig. 9 .

 9 Fig. 9. Picture of the experimental setup

Fig. 11 .Fig. 12 .

 1112 Fig. 11. Vertical component of the electric field (dB)

VI. ACKNOWLEDGMENTS

The authors wish to thank the antenna service of CNES for supporting this work under contract 81672-00.