Bertrand Duplantier
email: bertrand.duplantier@cea.fr.

Rémi Rhodes
email: rhodes@ceremade.dauphine.fr.

Scott Sheffield
email: sheffield@math.mit.edu.

Vincent Vargas
email: vargas@ceremade.dauphine.fr.

Critical Gaussian Multiplicative Chaos: Convergence of the Derivative Martingale

In this paper, we study Gaussian multiplicative chaos in the critical case. We show that the so-called derivative martingale, introduced in the context of branching Brownian motions and branching random walks, converges almost surely (in all dimensions) to a random measure with full support. We also show that the limiting measure has no atom. In connection with the derivative martingale, we write explicit conjectures about the glassy phase of log-correlated Gaussian potentials and the relation with the asymptotic expansion of the maximum of log-correlated Gaussian random variables.

Introduction 1.1 Overview

In the eighties, Kahane [START_REF] Kahane | Sur le chaos multiplicatif[END_REF] developed a continuous parameter theory of multifractal random measures, called Gaussian multiplicative chaos. His efforts were followed by several authors [START_REF] Allez | Lognormal -scale invariant random measures, Probability Theory and Related Fields[END_REF][START_REF] Barral | Multifractal products of cylindrical pulses[END_REF][START_REF] Bacry | Log-infinitely divisible multifractal processes[END_REF][START_REF] Fan | Sur le chaos de Lévy d'indice 0 < α < 1[END_REF][START_REF] Rhodes | -scale invariant random measures[END_REF][START_REF] Rhodes | Multidimensional multifractal random measures[END_REF][START_REF] Rhodes | KPZ formula for log-infinitely divisible multifractal random measures[END_REF] coming up with various generalizations at different scales. This family of random fields has found many applications in various fields of science, especially in turbulence and in mathematical finance. Recently, the authors in [START_REF] Duplantier | Liouville Quantum Gravity and KPZ[END_REF] constructed a probabilistic and geometrical framework for Liouville quantum gravity and the so-called Knizhnik-Polyakov-Zamolodchikov (KPZ) equation [START_REF] Knizhnik | Fractal structure of 2Dquantum gravity[END_REF], based on the two-dimensional Gaussian free field (GFF) (see [START_REF] David | Conformal Field Theories Coupled to 2-D Gravity in the Conformal Gauge[END_REF][START_REF] Francesco | 2D gravity and random matrices[END_REF][START_REF] Distler | Conformal Field Theory and 2-D Quantum Gravity or Who's Afraid of Joseph Liouville?[END_REF][START_REF] Duplantier | Liouville Quantum Gravity and KPZ[END_REF][START_REF] Ginsparg | Lectures on 2D gravity and 2D string theory[END_REF][START_REF] Knizhnik | Fractal structure of 2Dquantum gravity[END_REF][START_REF] Nakayama | Liouville Field Theory -A decade after the revolution[END_REF] and references therein). In this context, the KPZ formula has been proved rigorously [START_REF] Duplantier | Liouville Quantum Gravity and KPZ[END_REF], as well as in the closely related case of Gaussian multiplicative chaos [START_REF] Rhodes | KPZ formula for log-infinitely divisible multifractal random measures[END_REF] (see also [START_REF] Benjamini | KPZ in one dimensional random geometry of multiplicative cascades[END_REF] in the context of Mandelbrot's multiplicative cascades). This was done in the standard case of Liouville quantum gravity, namely strictly below the critical value of the GFF coupling constant γ in the Liouville conformal factor, i.e, for γ < 2 (in a chosen normalization). Beyond this threshold, the standard construction yields vanishing random measures [START_REF] Duplantier | Duality and KPZ in Liouville Quantum Gravity[END_REF][START_REF] Kahane | Sur le chaos multiplicatif[END_REF]. The issue of mathematically constructing singular Liouville measures beyond the phase transition (i.e. for γ > 2) and deriving the corresponding (non-standard dual) KPZ formula has been investigated in [START_REF] Barral | Gaussian multiplicative chaos and KPZ duality[END_REF][START_REF] Duplantier | A rigorous perspective on Liouville quantum gravity and KPZ[END_REF][START_REF] Duplantier | Duality and KPZ in Liouville Quantum Gravity[END_REF], giving the first mathematical understanding of the so-called duality in Liouville quantum gravity (see [START_REF] Alvarez-Gaumé | A proposal for strings at D > 1[END_REF][START_REF] Ambjørn | A solvable 2d gravity model with γ > 0[END_REF][START_REF] Das | New critical behavior in d = 0 large-N matrix models[END_REF][START_REF] Duplantier | Conformal fractal geometry and boundary quantum gravity[END_REF][START_REF] Durhuus | Multi-spin systems on a randomly triangulated surface[END_REF][START_REF] Jain | World-sheet geometry and baby universes in 2-D quantum gravity[END_REF][START_REF] Klebanov | Touching random surfaces and Liouville gravity[END_REF][START_REF] Klebanov | Non-perturbative Solution of Matrix Models Modified by Trace-squared Terms[END_REF][START_REF] Klebanov | Wormholes, Matrix Models, and Liouville Gravity[END_REF][START_REF] Kostov | Boundary Loop Models and and 2D Quantum Gravity[END_REF] for an account of physical motivations). However, the rigorous construction of random measures at criticality, that is for γ = 2, does not seem to ever have been carried out.

As stated above, once the Gaussian randomness is fixed, the standard Gaussian multiplicative chaos describes a random positive measure for each γ < 2 but yields 0 when γ = 2. Naively, one might therefore guess that -1 times the derivative at γ = 2 would be a random positive measure. This intuition leads one to consider the so-called derivative martingale, formally obtained by differentiating the standard measure w.r.t. γ at γ = 2, as explained below. In the case of branching Brownian motions [START_REF] Neveu | Multiplicative martingales for spatial branching processes[END_REF], or of branching random walks [START_REF] Biggins | Measure change in multitype branching[END_REF][START_REF] Kyprianou | Slow variation and uniqueness of solutions to the functional equation in the branching random walk[END_REF] (see also [START_REF] Aïdékon | The Seneta-Heyde scaling for the branching random walk[END_REF] for a recent different but equivalent construction), the construction of such an object has already been carried out mathematically. In the context of branching random walks, the derivative martingale was introduced in the study of the fixed points of the smoothing transform at criticality (the smoothing transform is a generalization of Mandelbrot's -equation for discrete multiplicative cascades; see also [START_REF] Biggins | Fixed points of the smoothing transform; the boundary case[END_REF]). Our construction will therefore appear as a continuous analogue of those works in the context of Gaussian multiplicative chaos.

Besides the 2D-Liouville Quantum Gravity framework (and the KPZ formula), many other important models or questions involve Gaussian multiplicative chaos of log-correlated Gaussian fields in all dimensions. Let us mention the glassy phase of log-correlated random potentials (see [START_REF] Arguin | Poisson-Dirichlet statistics for the extremes of a log-correlated Gaussian field[END_REF][START_REF] Carpentier | Glass transition of a particle in a random potential, front selection in nonlinear RG and entropic phenomena in Liouville and Sinh-Gordon models[END_REF][START_REF] Fyodorov | Freezing and extreme-value statistics in a random energy model with logarithmically correlated potential[END_REF][START_REF] Fyodorov | Statistical Mechanics of Logarithmic REM: Duality, Freezing and Extreme Value Statistics of 1/f Noises generated by Gaussian Free Fields[END_REF]) or the asymptotic expansion of the maximum of logcorrelated random variables (see [START_REF] Bramson | Tightness of the recentered maximum of the twodimensional discrete Gaussian Free Field[END_REF][START_REF] Ding | Extreme values for two-dimensional discrete Gaussian free field[END_REF]). In all these problems, one of the key tools is the derivative martingale at the critical point γ 2 = 2d (where d is the dimension), whose construction is precisely the purpose of this paper.

In dimension d, a standard Gaussian multiplicative chaos is a random measure that can be written formally, for any Borelian set A ⊂ R d , as

M γ (A) = A e γX(x)-γ 2 2 E[X 2 (x)] dx (1)
where X is a centered log-correlated Gaussian field:

E[X(x)X(y)] = ln + 1 |x -y| + g(x, y), with ln + (x) = max(ln x, 0) and g a continuous bounded function over R d × R d . Although such an X cannot be defined as a random function (and may be a random distribution, like the GFF) the measures can be rigorously defined all for γ 2 < 2d using a straightforward limiting procedure involving a time-indexed family of improving approximations to X [START_REF] Kahane | Sur le chaos multiplicatif[END_REF], as we will review in Section 2. By contrast, it is well known that for γ 2 2d the measures constructed by this procedure are identically zero [START_REF] Kahane | Sur le chaos multiplicatif[END_REF]. Other techniques are thus required to create similar measures beyond the critical value γ 2 = 2d [START_REF] Barral | Gaussian multiplicative chaos and KPZ duality[END_REF][START_REF] Duplantier | A rigorous perspective on Liouville quantum gravity and KPZ[END_REF][START_REF] Duplantier | Duality and KPZ in Liouville Quantum Gravity[END_REF]. Roughly speaking, the derivative martingale is defined as (recall that γ = √ 2d is the critical value)

M (A) := - ∂ ∂γ M γ (A) γ= √ 2d = A γE[X 2 (x)]-X(x) e γX(x)-γ 2 2 E[X 2 (x)] dx γ= √ 2d
.

Here we have differentiated the measure M γ in [START_REF] Aïdékon | The Seneta-Heyde scaling for the branching random walk[END_REF] with respect to the parameter γ to obtain the above expression [START_REF] Aïdékon | Convergence in law of the minimum of a branching random walk[END_REF]. Note that this is the same as (1) except for the factor γE[X 2 (x)] -X(x) . To give the reader some intuition, we remark that we will ultimately see that the main contributions to M (A) come from locations x where this factor is positive but relatively close to zero (on the order of E[X 2 (x)]) which correspond to locations x where X(x) is nearly maximal. Indeed, in what follows, the reader may occasionally wish to forget the derivative interpretation of (2) and simply view γE[X 2 (x)] -X(x) as the factor by which one rescales [START_REF] Aïdékon | The Seneta-Heyde scaling for the branching random walk[END_REF] in order to ensure that one obtains a non-trivial measure (instead of zero) when using the standard limiting procedure.

In a sense, the measures M γ (1) become more concentrated as γ 2 approaches 2d. (They assign full measure to a set of Hausdorff dimension d -γ 2 /2, which tends to zero as γ 2 → 2d.) It is therefore natural to wonder how concentrated the γ 2 = 2d measure will be. In particular, it is natural to wonder whether it possesses atoms (in which case it could in principle assign full measure to a countable set). In our context, we will answer in the negative. At the time we posted the first version of this manuscript online, this question was open in the context of discrete models as well as continuous models. However, a proof of the non-atomicity of the discrete cascade measures was posted very shortly afterward in [START_REF] Barral | Critical Mandelbrot cascades[END_REF], which uses a method independent of our proof. Since our proof is based on a continuous version of the spine decomposition, as developed in the context of branching random walks, we expect that it can be adapted to these other models as well.

Roughly speaking, the reason that establishing non-atomicity in critical models is non-trivial is that proofs of non-atomicity for (non-critical) multiplicative chaos usually rely on the existence of moments higher than 1 (see [START_REF] Daley | An introduction to the theory of point processes volume 2, Probability and its applications[END_REF]) and the scaling relations of multifractal random measures (see, e.g., [START_REF] Allez | Lognormal -scale invariant random measures, Probability Theory and Related Fields[END_REF]). At criticality, the random measures involved (cascades, branching random walks, or Gaussian multiplicative chaos) no longer possess finite moments of order 1, and the scaling relations become useless.

To explain this issue in more detail, we recall that it is proved in [START_REF] Daley | An introduction to the theory of point processes volume 2, Probability and its applications[END_REF] that a stationary random measure M over R d is almost surely non-atomic if (C stands here for the unit cube of R d ) ∀δ > 0,

n d P M (n -1 C) > δ → 0 as n → ∞. (3)
When M = M γ for 0 < γ 2 < 2d, a computable property of M γ is its power-law spectrum ξ characterized by:

E M γ (n -1 C) q D q n -ξ(q) as n → ∞, (4)
for all those q making the above expectation finite, i.e. q ∈ [0, 2d γ 2 [. It matches

ξ(q) = d + γ 2 2 q - γ 2 2 q 2 . ( 5
)
By using the Markov inequality in (3), (4) obviously yields for q ∈ [0, 2d γ 2 [

n d P M γ (n -1 C) > δ D q δ q n d-ξ(q) .
Therefore, the non-atomicity of the measure boils down to finding a q such that the powerlaw spectrum is strictly larger than d:

• In the subcritical situation γ 2 < 2d, the function ξ increases on [0, 1] from 0 to d. Such a q is necessarily larger than 1 and a straightforward computation shows that any q ∈]1, 2d γ 2 [ suffices.

• For γ 2 = 2d, the above relations ( 4) and ( 5) should remain valid only for q < 1. Therefore the subcritical strategy fails because the power-law spectrum achieves its maximum d at q = 1. It is tempting to try to replace the gauge function x → x q by something that could be more appropriate at criticality like x → x ln(1 + x) q , etc. However, the fact that the measure does not possess a moment of order 1 (see Proposition 5 below) shows that there is no way of changing the gauge so as to make ξ go beyond d.

More sophisticated machinery is thus necessary to investigate non-atomicity at criticality. Indeed, we expect the derivative martingale to assign full measure to a (random) Hausdorff set of dimension 0, indicating that the measure is in some sense just "barely" non-atomic. Let us finally mention the interesting work of [START_REF] Tecu | Random conformal welding at criticality[END_REF] where the author constructs on the unit circle (d = 1) a classical Gaussian multiplicative Chaos given by the exponential of a field X such that for each ε the covariance of X at points x and y lies strictly between (2 -ε) ln + 1 |x-y| and 2 ln + 1 |x-y| when |x -y| is sufficiently small. In some sense, his construction is a near critical construction, different from the measures constructed here. This is illustrated by the fact that the measures in [START_REF] Tecu | Random conformal welding at criticality[END_REF] possess moments of order 1 (and even belong to L log L), which is atypical for the critical multiplicative chaos associated to log-correlated random variables.

In this paper, we tackle the problem of constructing random measures at criticality for a large class of log-correlated Gaussian fields in any dimension, the covariance kernels of which are called -scale invariant kernels. This approach allows us to link the measures under consideration to a functional equation, the -equation, giving rise to several conjectures about the glassy phase of log-correlated Gaussian potentials and about the three-terms expansion of the maximum of log-correlated Gaussian variables.

Another important family of random measures is the class defined by taking X to be the Gaussian Free Field (GFF) with free or Dirichlet boundary conditions on a planar domain, as in [START_REF] Duplantier | Liouville Quantum Gravity and KPZ[END_REF] (see also [START_REF] Sheffield | Gaussian free fields for mathematicians[END_REF] for an introduction to the GFF). The measures defined in this way are also known as the (critical) Liouville quantum gravity measures, and are closely related to conformal field theory, as well as various 2-dimensional discrete random surface models and their scaling limits. Although the Gaussian free field is in some sense a log-correlated random field, it does not fall exactly into the framework of this paper, which deals with translation invariant random measures (defined on all of R 2 or R d ) that can be approximated in a particular way (via the -equation). Although some of the arguments of this paper can be easily extended to settings where the strict translation invariance requirement for X is relaxed (e.g., X is the Gaussian free field on a disk), we will still need additional arguments to show that the derivative martingale associates a unique non-atomic random positive measure to a given instance of the GFF almost surely, that this measure is independent of the particular approximation scheme used, and that this measure transforms under conformal maps in the same way as the γ < 2 measures constructed in [START_REF] Duplantier | Liouville Quantum Gravity and KPZ[END_REF]. For the sake of pedagogy, this other part of our work will appear in a companion paper. For the time being, we just announce that all the results of this paper are valid for the GFF construction.

Physics literature: history and motivation

It is interesting to pause for a moment and consider the physics literature on Liouville quantum gravity. We first remark that the non-critical case, with d = 2 and γ < 2, was treated in [START_REF] Duplantier | Liouville Quantum Gravity and KPZ[END_REF], which contains an extensive overview of the physics literature and an explanation of the relationships (some proved, some conjectural) between random measures and discrete and continuum random surfaces. Roughly speaking, when one takes a random two-dimensional manifold and conformally maps it to a disk, the image of the area measure is a random measure on the disk that should correspond to an exponential of a log-correlated Gaussian random variable (some form of the GFF). From this point of view, many of the physics results about discrete and continuum random surfaces can be interpreted as predictions about the behavior of these random measures, where the value of γ < 2 depends on the particular physical model in question.

There is also a physics literature focusing on the critical case γ = 2, which we expect to be related to the measure constructed in this paper. This section contains a brief overview of the results from this literature, as appearing in e.g. [START_REF] Brézin | Scaling violation in a field theory of closed strings in one physical dimension[END_REF][START_REF] Ginsparg | Lectures on 2D gravity and 2D string theory[END_REF][START_REF] Ginsparg | 2D gravity + 1D matter[END_REF][START_REF] Gross | One-dimensional string theory on a circle[END_REF][START_REF] Gross | A nonperturbative solution of D = 1 string theory[END_REF][START_REF] Gubser | A modified c = 1 matrix model with new critical behavior[END_REF][START_REF] Kazakov | A Matrix Model for the 2d Black Hole, in Nonperturbative Quantum Effects[END_REF][START_REF] Klebanov | Non-perturbative Solution of Matrix Models Modified by Trace-squared Terms[END_REF][START_REF] Kostov | Loop amplitudes for nonrational string theories[END_REF][START_REF] Kostov | Strings with discrete target space[END_REF][START_REF] Kostov | Multicritical phases of the O(n) model on a random lattice[END_REF][START_REF] Parisi | On the one dimensional discretized string[END_REF][START_REF] Polchinski | Critical behavior of random surfaces in one dimension[END_REF][START_REF] Sugino | Critical behavior in c = 1 matrix model with branching interactions[END_REF]. Most of the results surveyed in this section have not yet been established or understood in a mathematical sense.

The critical case γ = 2 corresponds to the value c = 1 of the so-called central charge c of the conformal field theory coupled to gravity, via the famous KPZ result [START_REF] Knizhnik | Fractal structure of 2Dquantum gravity[END_REF]:

γ = 1 √ 6 ( √ 25 -c - √ 1 -c).
Discrete critical statistical physical models having c = 1 then include one-dimensional matrix models (also called "matrix quantum mechanics" (MQM)) [START_REF] Brézin | Scaling violation in a field theory of closed strings in one physical dimension[END_REF][START_REF] Ginsparg | Lectures on 2D gravity and 2D string theory[END_REF][START_REF] Ginsparg | 2D gravity + 1D matter[END_REF][START_REF] Gross | One-dimensional string theory on a circle[END_REF][START_REF] Gubser | A modified c = 1 matrix model with new critical behavior[END_REF][START_REF] Kazakov | A Matrix Model for the 2d Black Hole, in Nonperturbative Quantum Effects[END_REF][START_REF] Klebanov | Non-perturbative Solution of Matrix Models Modified by Trace-squared Terms[END_REF][START_REF] Parisi | On the one dimensional discretized string[END_REF][START_REF] Polchinski | Critical behavior of random surfaces in one dimension[END_REF][START_REF] Sugino | Critical behavior in c = 1 matrix model with branching interactions[END_REF], the so-called O(n) loop model on a random planar lattice for n = 2 [START_REF] Kostov | Loop amplitudes for nonrational string theories[END_REF][START_REF] Kostov | Strings with discrete target space[END_REF][START_REF] Kostov | Boundary Loop Models and and 2D Quantum Gravity[END_REF][START_REF] Kostov | Multicritical phases of the O(n) model on a random lattice[END_REF], and the Q-state Potts model on a random lattice for Q = 4 [START_REF] Bernardi | Counting colored planar maps: algebraicity results[END_REF][START_REF] Daul | q-state Potts model on a random planar lattice[END_REF][START_REF] Eynard | The Potts-q random matrix model: loop equations, critical exponents, and rational case[END_REF].

(For an introduction to the above mentioned 2D statistical models, see, e.g., [START_REF] Nienhuis | Coulomb gas formulation of two-dimensional phase transitions[END_REF].)

In the continuum, a natural coupling also exists between Liouville quantum gravity and the Schramm-Loewner evolution SLE κ for γ = √ κ, rigorously established for κ < 4 [START_REF] Duplantier | Schramm-Loewner Evolution and Liouville Quantum Gravity[END_REF][START_REF] Sheffield | Conformal weldings of random surfaces: SLE and the quantum gravity zipper[END_REF]. Thus, the critical value γ = 2 corresponds to the special SLE parameter value κ = 4, above which the SLE κ curve no longer is a simple curve, but develops double points at all scales.

The standard c = 1, γ = 2 Liouville field theory [START_REF] Brézin | Scaling violation in a field theory of closed strings in one physical dimension[END_REF][START_REF] Ginsparg | Lectures on 2D gravity and 2D string theory[END_REF][START_REF] Ginsparg | 2D gravity + 1D matter[END_REF][START_REF] Gross | One-dimensional string theory on a circle[END_REF][START_REF] Kazakov | A Matrix Model for the 2d Black Hole, in Nonperturbative Quantum Effects[END_REF][START_REF] Klebanov | Non-perturbative Solution of Matrix Models Modified by Trace-squared Terms[END_REF][START_REF] Parisi | On the one dimensional discretized string[END_REF][START_REF] Polchinski | Critical behavior of random surfaces in one dimension[END_REF] involves violations of scaling by logarithmic factors. For example, the partition function (number) of genus 0 random surfaces of area A grows as [START_REF] Gross | One-dimensional string theory on a circle[END_REF][START_REF] Kazakov | A Matrix Model for the 2d Black Hole, in Nonperturbative Quantum Effects[END_REF]

Z ∝ exp(µA) A -3 (log A) -2 ,
where µ is a non-universal growth constant depending on the (planar lattice) regularization. The area exponent (-3) is universal for a c = 1 central charge, while the subleading logarithmic factor is attributed to the unusual dependence on the Liouville field ϕ (equivalent to X here) of the so-called "tachyon field" T (ϕ) ∝ ϕ e 2ϕ [START_REF] Kazakov | A Matrix Model for the 2d Black Hole, in Nonperturbative Quantum Effects[END_REF][START_REF] Klebanov | Non-perturbative Solution of Matrix Models Modified by Trace-squared Terms[END_REF][START_REF] Polchinski | Critical behavior of random surfaces in one dimension[END_REF]. Its integral over a "background" Borelian set A generates the quantum area A = A T (ϕ)dx, that we can recognize as the formal heuristic expression for the derivative measure (2) introduced above.

At c = 1, a proliferation of large "bubbles" (the so-called "baby universes" which are relatively large amounts of area cut off by relatively small bottlenecks) is generally anticipated in the bulk of the random surface [START_REF] Gross | One-dimensional string theory on a circle[END_REF][START_REF] Jain | World-sheet geometry and baby universes in 2-D quantum gravity[END_REF][START_REF] Kostov | Loop amplitudes for nonrational string theories[END_REF], or at its boundary in the case of a disk topology [START_REF] Kostov | Strings with discrete target space[END_REF][START_REF] Kostov | Multicritical phases of the O(n) model on a random lattice[END_REF]. We believe that this should correspond to the fact that the measure we construct is concentrated on a set of Hausdorff dimension zero.

However, the introduction of higher trace terms [START_REF] Gubser | A modified c = 1 matrix model with new critical behavior[END_REF][START_REF] Klebanov | Non-perturbative Solution of Matrix Models Modified by Trace-squared Terms[END_REF][START_REF] Sugino | Critical behavior in c = 1 matrix model with branching interactions[END_REF] in the action of the c = 1 matrix model of two-dimensional quantum gravity is known to generate a "non-standard" random surface model with an even stronger concentration of bottlenecks. (See also the related detailed study of a MQM model for a c = 1 string theory with vortices in [START_REF] Kazakov | A Matrix Model for the 2d Black Hole, in Nonperturbative Quantum Effects[END_REF].) As we shall see shortly, these non-standard constructions do not seem to correspond to our model, at least not so directly. In these constructions, one encounters a new critical behavior of the random surface, with a critical proliferation of spherical bubbles connected one to another by microscopic "wormholes". This is reminiscent of the construction for c < 1, γ < 2 of the dual phase of Liouville quantum gravity [START_REF] Alvarez-Gaumé | A proposal for strings at D > 1[END_REF][START_REF] Ambjørn | A solvable 2d gravity model with γ > 0[END_REF][START_REF] Das | New critical behavior in d = 0 large-N matrix models[END_REF][START_REF] Durhuus | Multi-spin systems on a randomly triangulated surface[END_REF][START_REF] Klebanov | Touching random surfaces and Liouville gravity[END_REF][START_REF] Klebanov | Non-perturbative Solution of Matrix Models Modified by Trace-squared Terms[END_REF][START_REF] Klebanov | Wormholes, Matrix Models, and Liouville Gravity[END_REF], where the associated random measure develops atoms [START_REF] Barral | Gaussian multiplicative chaos and KPZ duality[END_REF][START_REF] Duplantier | A rigorous perspective on Liouville quantum gravity and KPZ[END_REF][START_REF] Duplantier | Duality and KPZ in Liouville Quantum Gravity[END_REF].

The partition function of the non-standard c = 1 (genus zero) random surface then scales as a function of the area A as [START_REF] Gubser | A modified c = 1 matrix model with new critical behavior[END_REF][START_REF] Kazakov | A Matrix Model for the 2d Black Hole, in Nonperturbative Quantum Effects[END_REF][START_REF] Klebanov | Non-perturbative Solution of Matrix Models Modified by Trace-squared Terms[END_REF][START_REF] Sugino | Critical behavior in c = 1 matrix model with branching interactions[END_REF] Z ∝ exp(µ A)A -3 , with an apparent suppression of logarithmic terms. This has been attributed to the appearance for c = 1 of a tachyon field of the atypical form T (ϕ) ∝ e 2ϕ [START_REF] Gubser | A modified c = 1 matrix model with new critical behavior[END_REF][START_REF] Kazakov | A Matrix Model for the 2d Black Hole, in Nonperturbative Quantum Effects[END_REF][START_REF] Klebanov | Wormholes, Matrix Models, and Liouville Gravity[END_REF]. Heuristically, this would seem to correspond to a measure of type ( 1), but we know that the latter vanishes for γ = 2. (See Proposition 19 below.) The literature about the analogous problem of branching random walks [START_REF] Aïdékon | The Seneta-Heyde scaling for the branching random walk[END_REF][START_REF] Hu | Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees[END_REF] also suggests for γ = 2 a logarithmically renormalized measure obtained by multiplying by log(1/ε) = √ t the object (see (7) below) whose limit is taken in (1), but we expect this to converge (up to constant factor) to the same measure as the derivative martingale [START_REF] Aïdékon | Convergence in law of the minimum of a branching random walk[END_REF]. In order to model the non-standard theory, it might be necessary to modify the measures introduced here by explicitly introducing "atoms" on top of them, using the procedure described in [START_REF] Barral | Gaussian multiplicative chaos and KPZ duality[END_REF][START_REF] Duplantier | A rigorous perspective on Liouville quantum gravity and KPZ[END_REF][START_REF] Duplantier | Duality and KPZ in Liouville Quantum Gravity[END_REF] for adding atoms to γ < 2 random measures. In the approach of [START_REF] Barral | Gaussian multiplicative chaos and KPZ duality[END_REF][START_REF] Duplantier | A rigorous perspective on Liouville quantum gravity and KPZ[END_REF][START_REF] Duplantier | Duality and KPZ in Liouville Quantum Gravity[END_REF], the "dual Liouville measure" corresponding to γ < 2 involves choosing a Poisson point process from η -α-1 dηM γ (dx), where α = γ 2 /4 ∈ (0, 1), and letting each point (η, x) in this process indicate an atom of size η at location x. When γ = 2 and α = 1, we can replace M γ with the M of (2) and use the same construction; in this case (since α = 1) the measure a.s. assigns infinite mass to each positive-Lebesgue-measure A ∈ B(R d ). However, one may use standard Lévy compensation to produce a random distribution, assigning a finite value a.s. to each fixed A ∈ B(R d ) with a positive atom of size η at location x corresponding to each (η, x) in the Poisson point process. We suspect that that this construction is somehow equivalent to the continuum random measure associated with the non-standard c = 1, γ = 2 Liouville random surface with enhanced bottlenecks, as described in [START_REF] Gubser | A modified c = 1 matrix model with new critical behavior[END_REF][START_REF] Kazakov | A Matrix Model for the 2d Black Hole, in Nonperturbative Quantum Effects[END_REF][START_REF] Sugino | Critical behavior in c = 1 matrix model with branching interactions[END_REF].

Finally, we note that the boundary critical Liouville quantum gravity poses similar challenges. A subtle difference in logarithmic boundary behavior is predicted between the so-called dilute and dense phases of the O(2) model on a random disk [START_REF] Kostov | Strings with discrete target space[END_REF][START_REF] Kostov | Multicritical phases of the O(n) model on a random lattice[END_REF], which thus may differ in their boundary bubble structure. It also remains an open question whether the results about the conformal welding of two boundary arcs of random surfaces to produce SLE, as described in [START_REF] Sheffield | Conformal weldings of random surfaces: SLE and the quantum gravity zipper[END_REF], can be extended to the case γ = 2.

Setup

Notations

For a Borelian set A ⊂ R d , B(A) stands for the Borelian sigma-algebra on A. All the considered fields are constructed on the same probability space (Ω, F, P). We denote by E the corresponding expectation.

-scale invariant kernels

Here we introduce the Gaussian fields that we will use throughout the papers. We consider a family of centered stationary Gaussian processes ((X t (x)) x∈R d ) t 0 where, for each t 0, the process (X t (x)) x∈R d has covariance given by:

K t (x) = E[X t (0)X t (x)] = e t 1 k(ux) u du (6)
for some covariance kernel k satisfying k(0) = 1, of class C 1 and vanishing outside a compact set (actually this latter condition is not necessary but it simplifies the presentation). We also assume that the process (X t (x) -X s (x)) x∈R d is independent of the processes (X u (x)) x∈R d u s for all s < t. Put in other words, the mapping t → X t (•) has independent increments. Such a construction of Gaussian processes is carried out in [START_REF] Allez | Lognormal -scale invariant random measures, Probability Theory and Related Fields[END_REF]. For γ 0, we consider the approximate Gaussian multiplicative chaos M γ t (dx) on R d :

M γ t (dx) = e γXt(x)-γ 2 2 E[Xt(x) 2 ] dx (7)
It is well known [START_REF] Allez | Lognormal -scale invariant random measures, Probability Theory and Related Fields[END_REF][START_REF] Kahane | Sur le chaos multiplicatif[END_REF] that, almost surely, the family of random measures (M γ t ) t>0 weakly converges as t → ∞ towards a random measure M γ , which is non-trivial if and only if γ 2 < 2d. The purpose of this paper is to investigate the phase transition, that is γ 2 = 2d. Recall that we have: Proposition 1. For γ 2 = 2d, the standard construction (7) yields a vanishing limiting measure:

lim t→∞ M √ 2d t (dx) = 0 almost surely. (8)
Let us also mention that the authors in [START_REF] Allez | Lognormal -scale invariant random measures, Probability Theory and Related Fields[END_REF] have proved that, for γ 2 < 2d, the measure M γ satisfies the following scale invariance relation, called -equation: Definition 2. Log-normal -scale invariance. The random Radon measure M γ is lognormal -scale invariant: for all 0 < ε 1, M γ obeys the cascading rule

M γ (A) A∈B(R d ) law = A e γX ln 1 ε (r)-γ 2 2 E[X ln 1 ε (r) 2 ] ε d M γ,ε (dr) A∈B(R d ) (9)
where X ln 1 ε is the Gaussian process introduced in (6) and M γ,ε is a random measure independent from X ln 1 ε satisfying the scaling relation

M γ,ε (A) A∈B(R d ) law = M γ ( A ε ) A∈B(R d ) . (10)
Intuitively, this relation means that when zooming in the measure M , one should observe the same behaviour up to an independent Gaussian factor. It has some canonical meaning since it is the exact continuous analog of the smoothing transformation intensively studied in the context of Mandelbrot's multiplicative cascades [START_REF] Durrett | Fixed points of the smoothing transformation[END_REF] or branching random walks [START_REF] Biggins | Fixed points of the smoothing transform; the boundary case[END_REF][START_REF] Liu | Fixed points of a generalized smoothing transformation and applications to the branching random walk[END_REF].

Observe that this equation perfectly makes sense for the value γ 2 = 2d. Therefore, to define a natural Gaussian multiplicative chaos at the value γ 2 = 2d, one has to look for a solution to this equation when γ 2 = 2d and conversely, each random measure candidate for being a Gaussian multiplicative chaos at the value γ 2 = 2d must satisfy this equation.

Remark 3. The main motivation for considering -scale invariant kernels is the connection between the associated random measures and the -equation. Nevertheless, we stress that our proofs can be easily adapted for any Gaussian multiplicative chaos of log-correlated Gaussian fields "à la Kahane" [START_REF] Kahane | Sur le chaos multiplicatif[END_REF]. In particular, we can construct the derivative martingale associated to exact scale invariant kernels [START_REF] Bacry | Log-infinitely divisible multifractal processes[END_REF][START_REF] Rhodes | Multidimensional multifractal random measures[END_REF].

Derivative martingale

One way to construct a solution to the -equation at the critical value γ 2 = 2d is to introduce the derivative martingale M t (dx) defined by:

M t (dx) := ( √ 2d t -X t (x))e √ 2dXt(x)-dE[Xt(x) 2 ] dx.
It is plain to see that, for each open bounded set A ⊂ R d , the family (M t (A)) t is a martingale. Nevertheless, it is not nonnegative. It is therefore not obvious that such a family converges towards a (non trivial) positive limiting random variable. The following theorem is the main result of this section:

Theorem 4.
For each bounded open set A ⊂ R d , the martingale (M t (A)) t 0 converges almost surely towards a positive random variable denoted by M (A), such that M (A) > 0 almost surely. Consequently, almost surely, the (locally signed) random measures (M t (dx)) t 0 converge weakly as t → ∞ towards a positive random measure M (dx). This limiting measure has full support and is atomless. Furthermore, the measure M is a solution to the -equation ( 9) with γ = √ 2d.

Since M t (dx) is not uniformly non-negative when t < ∞, there are several complications involved in establishing its convergence to a non-negative limit (let alone the nontriviality of the limit). We have to introduce some further tools to study its convergence. These tools have already been introduced in the context of discrete multiplicative cascade models in order to study the corresponding derivative martingale (see [START_REF] Biggins | Measure change in multitype branching[END_REF]).

We denote by F t the sigma algebra generated by {X s (x); s t, x ∈ R d }. Given a Borelian set A ⊂ R d and parameters t, β > 0, we introduce the random variables

Z β t (A) = A ( √ 2dt -X t (x) + β)1I {τ β >t} e √ 2dXt(x)-dE[Xt(x) 2 ] dx Z β t (A) = A ( √ 2dt -X t (x))1I {τ β >t} e √ 2dXt(x)-dE[Xt(x) 2 ] dx,
where, for each x ∈ A, τ β is the (F t ) t -stopping time defined by

τ β = inf{u > 0, X u (x) - √ 2du > β}.
What is the relation between Z β t (A) and M t (A)? Roughly speaking, we will show that the convergence of M t (A) as t → ∞ towards a non-trivial object boils down to proving the convergence of Z β t (A) towards a non-trivial object: we will prove that the difference Z β t (A) -Z β t (A) almost surely goes to 0 as t → ∞ and that Z β t (A) coincides with M t (A) for β large enough. In particular, we will prove that Z β t (A) converges towards a random variable Z β (A) which itself converges as β → ∞ to the limit of M t (A) (as t → ∞). The details and proofs are gathered in the appendix.

As a direct consequence of our method of proof, we get the following properties of M (dx): Proposition 5. The positive random measure M (dx) possesses moments of order q for all q 0. It does not possess moments of order 1.

Proof. As a direct consequence of the fact that the measure M satisfies the -equation, it possesses moments of order q for all q 0. This is a straightforward adaptation of the corresponding theorem in [START_REF] Barral | Moments, continuité, et analyse multifractale des martingales de Mandelbrot[END_REF] (see also [START_REF] Benjamini | KPZ in one dimensional random geometry of multiplicative cascades[END_REF] for a proof in English). Since Z β (dx) increases towards M as β goes to infinity, we have M (dx)

Z β (dx) for any β. Since Z β t is a uniformly integrable martingale, we have E[Z β (A)] = E[Z β 0 (A)] = β|A|, we deduce that E[M (A)] = +∞ for every bounded open set A.

Conjectures

In this section, we present a few results we can prove about the -equation and some conjectures related to these results.

About the -equation

Consider the -equation in great generality, that is: Figure 1: Height landscape of the derivative martingale measure plotted with a logarithmic scale color-bar, showing that the measure is very "peaked" (for t = 12, a multiplicative factor of about 10 8 stands between extreme values, i.e., between warm and cold colors). Definition 6. Log-normal -scale invariance. A random Radon measure M is lognormal -scale invariant if for all 0 < ε 1, M obeys the cascading rule

M (A) A∈B(R d ) law = A e ωε(r) M ε (dr) A∈B(R d ) (11)
where ω ε is a stationary stochastically continuous Gaussian process and M ε is a random measure independent from X ε satisfying the scaling relation

M ε (A) A∈B(R d ) law = M ( A ε ) A∈B(R d ) . (12)
Observe that, in comparison with ( 9) and ( 10), we do not require the scaling factor to be ε d . As stated in [START_REF] Barral | Limiting laws of supercritical branching random walks[END_REF] and [START_REF] Bacry | Log-infinitely divisible multifractal processes[END_REF], it is proved in [START_REF] Allez | Lognormal -scale invariant random measures, Probability Theory and Related Fields[END_REF] that E[e ωε(r) ] = ε d as soon as the measure possesses a moment of order 1 + δ for some δ > 0. Roughly speaking, it remains to investigate situations when the measure does not possess a moment of order 1 and we will see that the scaling factor is then not necessarily ε d .

Inspired by the discrete multiplicative cascade case (see [START_REF] Durrett | Fixed points of the smoothing transformation[END_REF]), our conjecture is that all the non-trivial ergodic solutions to this equation belong to one of the families we will describe below.

First we conjecture that there exists a α ∈]0, 1] such that

E[e αωε(r) ] = ε d .
Assuming this, it is proved in (see [START_REF] Allez | Lognormal -scale invariant random measures, Probability Theory and Related Fields[END_REF][START_REF] Rhodes | -scale invariant random measures[END_REF]) that the Gaussian process αω e -t has a covariance structure given by ( 6). More precisely, there exists some continuous covariance kernel k with k(0) = 1 and γ 2 2d such that

Cov αω e -t (0), αω e -t (x) = γ 2 e t 1 k(ux) u du
We can then rewrite the process ω as

ω e -t (x) = γ α X t (x) - γ 2 2α t - d α t.
where (X t ) t is the family of Gaussian fields introduced in Section 2. We now consider four cases, depending on the values of α and γ (cases 2,3,4 are conjectures):

1. If α = 1 and γ 2 < 2d then the law of the solution M is the standard Gaussian multiplicative chaos M γ (see [START_REF] Barral | Multifractal products of cylindrical pulses[END_REF]) up to a multiplicative constant. This case has been treated in [START_REF] Allez | Lognormal -scale invariant random measures, Probability Theory and Related Fields[END_REF].

2. If α = 1 and γ 2 = 2d, then the law of the solution M is that of the derivative martingale that we have constructed in this paper (Theorem 4), up to a multiplicative constant.

3. If α < 1 and γ 2 < 2d, then M is an atomic Gaussian multiplicative chaos as constructed in [START_REF] Barral | Gaussian multiplicative chaos and KPZ duality[END_REF] up to a multiplicative constant. More precisely, the law can be constructed as follows:

(a) sample a standard Gaussian multiplicative chaos

M (dx) = e γX(x)-γ 2 2 E[X(x) 2 ] dx.
The measure M is perfectly defined since γ 2 < 2d.

(b) sample an independently scattered random measure N whose law, conditioned on M , is characterized by

∀q 0, E[e -qN (A) | M ] = e -q α M (A) .
Then the law of M is that of N up to a multiplicative constant.

4. If α < 1 and γ 2 = 2d, then M is an atomic Gaussian multiplicative chaos of a new type. More precisely, the law can be constructed as follows:

(a) sample the derivative Gaussian multiplicative chaos

M (dx) = ( √ 2dE[X(x) 2 ] -X(x))e √ 2dX(x)-dE[X(x) 2 ] dx.
The measure M is constructed as prescribed by Theorem 4.

(b) sample an independently scattered random measure N whose law, conditioned on M , is characterized by

∀A ∈ B(R d ), ∀q 0, E[e -qN (A) |M ] = e -q α M (A) .
Then the law of M is that of N up to a multiplicative constant.

Notice that the results of our paper together with [START_REF] Allez | Lognormal -scale invariant random measures, Probability Theory and Related Fields[END_REF][START_REF] Barral | Gaussian multiplicative chaos and KPZ duality[END_REF] allow to prove existence of all the random measures described above. Therefore it remains to complete the uniqueness part of this statement.

Remark 7. The α < 1, γ 2 < 2d case above has been used in [START_REF] Barral | Gaussian multiplicative chaos and KPZ duality[END_REF][START_REF] Duplantier | A rigorous perspective on Liouville quantum gravity and KPZ[END_REF][START_REF] Duplantier | Duality and KPZ in Liouville Quantum Gravity[END_REF] to give a mathematical understanding of the duality in Liouville Quantum Gravity: this corresponds to taking special values of the couple (α, γ). More precisely, we choose some parameter γ2 > 2d. If the measure M γ was well defined, it would satisfy the scaling relation:

M γ (A) A∈B(R d ) law = A e γX ln 1 ε (r)- γ2 2 E[X ln 1 ε (r) 2 ] ε d M γ,ε (dr) A∈B(R d ) (13)
where M γ,ε is a random measure independent from X ε satisfying the scaling relation

M γ,ε (A) A∈B(R d ) law = M γ ( A ε ) A∈B(R d ) . (14)
Nevertheless, we know that M γ yields a vanishing measure. The idea is thus to use the -equation to determine what the unique solution of this scaling relation is. Writing γ = 2d γ < 2d and α = 2d γ2 , it is plain to see that

E e γX ln 1 ε (r)- γ2 2 E[X ln 1 ε (r) 2 ] ε d α = ε d .
Therefore we are in situation 4, which yields a natural candidate for Liouville duality [START_REF] Barral | Gaussian multiplicative chaos and KPZ duality[END_REF][START_REF] Duplantier | A rigorous perspective on Liouville quantum gravity and KPZ[END_REF][START_REF] Duplantier | Duality and KPZ in Liouville Quantum Gravity[END_REF].

Another construction of solutions to the critical -equation

Recall that the measures M γ for γ < 2 are obtained as limits of (1) as X varies along approximations to a limit field. The measure constructed in Theorem 4 is defined analogously except that one replaces (1) with (2), which is minus the derivative of (1) at γ = √ 2d. If we could exchange the order of the differentiation and the limit-taking, we would conclude that the measure constructed in Theorem 4 is equal to

- ∂ ∂γ [M γ ] γ= √ 2d = lim γ→ √ 2d 1 √ 2d -γ M γ .
We will not fully justify this order exchange here, but we will establish a somewhat weaker result. Namely, we show that one can at least obtain some solution to theequation as a limit of this general type. This construction is inspired by a similar construction for discrete multiplicative cascades in [START_REF] Durrett | Fixed points of the smoothing transformation[END_REF]. More precisely, we have the following (proved in Section A.2): Proposition 8. There exist two increasing sequence (λ n ) n and (γ n ) n , with γ 2 n < 2d and γ 2 n → 2d as n → ∞, such that

λ n M γn (dx) law → M c (dx),
where M c is a positive random measure satisfying [START_REF] Barral | Gaussian multiplicative chaos and KPZ duality[END_REF].

The following conjecture is a consequence of the uniqueness conjecture for theequation exposed in Section 4.1 above: Conjecture 9. The construction of Proposition 8 gives the same measure as the one described in Section 3 (up to some multiplicative constant). Moreover, the sequence (λ n ) n can be chosen as

λ n = 1 √ 2d-γn (in dimension d).

Glassy phase of log-correlated Gaussian potentials

The glassy phase of log-correlated Gaussian potentials is concerned with the behaviour of measures beyond the critical value γ 2 > 2d. More precisely, for γ 2 > 2d, consider the measure

M γ t (dx) = e γXt(x)-γ 2 2 E[Xt(x) 2 ] dx.
The limiting measure, as t → ∞, vanishes as proved in [START_REF] Kahane | Sur le chaos multiplicatif[END_REF]. Therefore, it is natural to look for a suitable family of normalizing factors to make this measure converge. With the arguments used in subsection B.1 to compare with the results obtained in [START_REF] Barral | Limiting laws of supercritical branching random walks[END_REF][START_REF] Madaule | Convergence in law for the branching random walk seen from its tip[END_REF], we can rigorously prove:

Proposition 10. The renormalized family

t 3γ 2 √ 2d e t γ √ 2 - √ d 2 M γ t (dx) t 0
is tight. Furthermore, every converging subsequence is non trivial.

The above proposition can be obtained using the results in [START_REF] Barral | Limiting laws of supercritical branching random walks[END_REF][START_REF] Madaule | Convergence in law for the branching random walk seen from its tip[END_REF] and Section B.1 (tightness statement). The main result in [START_REF] Bramson | Tightness of the recentered maximum of the twodimensional discrete Gaussian Free Field[END_REF] about the behaviour of the maximum of the discrete GFF implies that every converging subsequence is non trivial.

We now formulate a conjecture about the limiting law of this family. Assuming that the above renormalized family converges in law (so we strengthen tightness into convergence), it turns out that the limit M γ of this renormalized family necessarily satisfies the following -equation

M γ (dx) = e γX ln 1 ε (x)-d 2 γE[X ln 1 ε (x) 2 ] ε d 2 γ M γ dx ε
where M γ is a random measure with the same law as M γ and independent of the process

(X t (x)) x∈R d . Setting α = √ 2d
γ ∈]0, 1[, this equation can be rewritten as

M γ (dx) = e √ 2d α X ln 1 ε (x)-d α E[X ln 1 ε (x) 2 ] ε d/α M γ dx ε .
Therefore, assuming that the conjectures about uniqueness of the -equation are true, we have that:

Conjecture 11.

t 3γ 2 √ 2d e t γ √ 2 - √ d 2 M γ t (dx) law → c γ N α (dx), as t → ∞ ( 15
)
where c γ is a positive constant depending on γ and the law of N α is given, conditioned on the derivative martingale M , by an independently scattered random measure the law of which is characterized by

∀A ∈ B(R d ), ∀q 0, E[e -qNα(A) |M ] = e -q α M (A) .
In particular, physicists are interested in the behaviour of the Gibbs measure associated to M γ t (dx) on a ball B. It is the measure renormalized by its total mass:

G γ t (dx) = M γ t (dx) M γ t (B)
.

From ( 15), we deduce

G γ t (dx) law → N α (dx) N α (B) , as t → ∞. (16)
The size reordered atoms of the latter object form a Poisson-Dirichlet process as conjectured by physicists [START_REF] Carpentier | Glass transition of a particle in a random potential, front selection in nonlinear RG and entropic phenomena in Liouville and Sinh-Gordon models[END_REF] and proved rigorously in [START_REF] Arguin | Poisson-Dirichlet statistics for the extremes of a log-correlated Gaussian field[END_REF]. Nevertheless, we point out that this conjecture is more powerful than the Poisson-Dirichlet result since it also makes precise the spatial localization of the atoms. We stress that this result has been proved in the case of branching random walks [START_REF] Barral | Limiting laws of supercritical branching random walks[END_REF], built on the work [START_REF] Madaule | Convergence in law for the branching random walk seen from its tip[END_REF].

About the maximum of the log-correlated Gaussian random variables

It is proved in [START_REF] Bramson | Tightness of the recentered maximum of the twodimensional discrete Gaussian Free Field[END_REF] (in fact d = 2 in [START_REF] Bramson | Tightness of the recentered maximum of the twodimensional discrete Gaussian Free Field[END_REF] but this is general) that the family sup

x∈[0,1] d X t (x) - √ 2dt + 3 2 √ 2d ln t t 0
is tight. One can thus conjecture by analogy with the branching random walk case ( [START_REF] Aïdékon | Convergence in law of the minimum of a branching random walk[END_REF]):

Conjecture 12. sup x∈[0,1] d X t (x) - √ 2dt + 3 2 √ 2d ln t → G d , in law as t → ∞
where the distribution of G d is given in terms of the distribution of the limit M ([0, 1] d ) of the derivative martingale. More precisely, there exists some constant c > 0 such that:

E[e -qG d ] = 1 c q Γ(1 + q √ 2d )E M ([0, 1] d ) -q √ 2d (17)
Here we give a heuristic derivation of identity 17 using the conjectures of the above subsections. By performing an inversion of limits: (γ ↔ t and conjecturing

ln cγ γ → ln c as γ → ∞): E[e -qG d ] = lim γ→+∞ lim t→+∞ E exp -qγ -1 ln t 3γ 2 √ 2d e t γ √ 2 - √ d 2 M γ t ([0, 1] d ) = lim γ→+∞ E c γ N α= √ 2d γ ([0, 1] d )] -q γ = 1 c q Γ(1 + q √ 2d )E M ([0, 1] d ) -q √ 2d
where, for x > 0, Γ(x) = ∞ 0 t x-1 e -t dt is the standard Gamma function. Therefore G d can be viewed as a modified Gumbel law. Otherwise stated, we conjecture:

lim t→∞ P sup x∈[0,1] d X t (x) - √ 2dt + 3 2 √ 2d ln t u = E exp[-c √ 2d e - √ 2d u M ([0, 1] d )] .
We point out that we recover in a heuristic and alternative way the result proved rigorously in [START_REF] Aïdékon | Convergence in law of the minimum of a branching random walk[END_REF] for branching random walks.

A. Proofs

A.1 Proofs of results from Section 3

We follow the notations of Section 3. We first investigate the convergence of (Z β t (A)) t 0 :

Proposition 13. The process (Z β t (A)) t 0 is a continuous positive F t -martingale and thus converges almost surely towards a positive random variable denoted by Z β (A).

Proof. Proving that (Z β t (A)) t 0 is a martingale boils down to proving, for each x ∈ A, that

E[( √ 2dt-X t (x)+β)1I {τ β >t} e √ 2dXt(x)-dE[Xt(x) 2 ] |F s ] = ( √ 2ds-X s (x)+β)1I {τ β >s} e √ 2dXs(x)-dE[Xs(x) 2 ] .
Let us first stress that, for each x ∈ A, the process (X t (x)) t 0 is a Brownian motion. Furthermore, we can use the (weak) Markov property of the Brownian motion to get

E[( √ 2dt -X t (x) + β)1I {τ β >t} e √ 2dXt(x)-dE[Xt(x) 2 ] |F s ] = 1I {τ β >s} e √ 2dXs(x)-dE[Xs(x) 2 ] F √ 2ds -X s (x) + β
where

F (y) = E[( √ 2d(t -s) -X t-s (x) + y)1I {τ (X•(x)- √ 2d•)>t-s} e √ 2dX t-s (x)-dE[X t-s (x) 2 ] ]
and, for a stochastic process Y , τ (Y ) is defined by

τ (Y ) = inf{u > 0; Y u > y}.
Using the Girsanov transform yields

F (y) = E[(- √ 2dX t-s (x) + y)1I {τ ( √ 2dX•(x))>t-s} ].
Hence we get

F (y) = E[(- √ 2dX t-s (x) + y)1I {τ ( √ 2dX•(x))>t-s} ] = E[(- √ 2dX t-s∧τ (X•(x)) (x) + y)] = y
by the optional stopping theorem. This completes the proof.

Proposition 14. Assume that A is a bounded open set. Then, the martingale (Z β t (A)) t 0 is regular.

Proof. Without loss of generality, we may assume k(u) = 0 for |u| > 1 since k has a compact support (so we just assume that the smallest ball centered at 0 containing the support of k has radius 1 instead of R for some R > 0). We may also assume that A ⊂ B(0, 1/2): indeed, any bigger bounded set can be recovered with finitely many balls with radius less than 1 2 . Finally, we will also assume that x • ∇k(x) 0. This condition need not be true over the whole R d . Nevertheless, it must be valid in a neighborhood of 0 (and even x • ∇k(x) < 0 if x = 0) in order not to contradict the fact that k is positive definite and non constant. Therefore, even if it means considering a smaller set A, we may (and will) assume that this condition holds.

Write for x ∈ R d :

f β t (x) = ( √ 2dt -X t (x) + β)1I {τ β >t} e √ 2dXt(x)-dE[Xt(x) 2 ] .
Define then the analog of the rooted random measure in [START_REF] Duplantier | Liouville Quantum Gravity and KPZ[END_REF] (also called the "Peyrière probability measure" in this context [START_REF] Kahane | Sur le chaos multiplicatif[END_REF])

Θ β t = 1 |A|β f β t (x)dx dP.
It is a probability measure on B(A)⊗F t . We denote by Θ β t (•|G) the conditional expectation of Θ β t given some sub-σ-algebra G of B(A) ⊗ F t . If y is a B(A) ⊗ F t -measurable random variable on A × Ω, we denote by Θ β t (•|y) the conditional expectation of Θ β t given the σ-algebra generated by y.

We first observe that

Θ β t (•|x) = 1 β f β t (x) dP.
Therefore, under Θ β t (•|x), the process (X t (x) -√ 2dt -β) s t has the law of (-β s ) s t where (β s ) s t is a 3d-Bessel process starting from β. Let us now recall the following result (see [START_REF] Motoo | Proof of the iterated logarithm through diffusion equation[END_REF]): Theorem 15. Let X be a 3d-Bessel process on R + started from x 0 with respect to the law P x .

1. Suppose that φ ↑ ∞ such that ∞ 1 φ(t) 3 t e -1 2 φ(t) 2 dt < +∞. Then P x X t > √ tφ(t) i.o. as t ↑ +∞ = 0. 2. Suppose that ψ ↓ 0 such that ∞ 1 ψ(t) t dt < +∞. Then P x X t < √ tψ(t) i.o. as t ↑ +∞ = 0.
Therefore, we can choose R large enough such that the set

B = ∀t 0; √ t R(ln(2 + t)) 2 β t R(1 + t ln(1 + t))
has a probability arbitrarily close to 1, say 1 -ε.

We can now prove the uniform integrability of (Z β t (A)) t , i.e.

lim δ→∞ lim sup t→∞ E[Z β t (A)1I {Z β t (A)>δ} ] = 0.
Observe that

E[Z β t (A)1I {Z β t (A)>δ} ] = β|A|Θ β t (Z β t (A) > δ). Therefore it suffices to prove that lim δ→∞ lim sup t→∞ Θ β t (Z β t (A) > δ) = 0.
We have:

Θ β t (Z β t (A) > δ) = 1 |A| A Θ β t (Z β t (A) > δ|x) dx = 1 |A| A Θ β t Θ β t Z β t (A) > δ|x, (X s (x)) s t |x dx ε + 1 |A| A Θ β t Θ β t Z β t (A) > δ|x, (X s (x)) s t , B |x dx ε + 1 |A| A Θ β t Θ β t Z β t (B(x, e -t )) > δ 2 |x, (X s (x)) s t , B |x dx + 1 |A| A Θ β t Θ β t Z β t (B(x, e -t ) c ) > δ 2 |x, (X s (x)) s t , B |x dx def = ε + Π 1 + Π 2 .
We are now going to estimate Π 1 , Π 2 . Observe that the two quantities roughly reduces to expressions like (K is a ball or its complementary)

Θ β t F K f β t (w) dw |x, (X s (x)) s t , B .
To carry out our computations, we thus have to compute the law of the process (X s (w)) s t knowing that of the process (X s (x)) s t . To that purpose, we will use the following lemma whose proof is left to the reader since it follows from a standard (though not quite direct) computation of covariances for Gaussian processes:

Lemma 16. For w = x and all s 0 , the law of the process (X s (w)) s s 0 can be decomposed as: X s (w) = P x,w s + Z x,w s where:

-P x,w s = -s 0 g x,w (u)X u (x) du+K s (x-w)X s (x) is measurable with respect to the σ-algebra generated by (X s (x)) s s 0 and g x,w (u) = K u (x -w), -the process (Z x,w s ) 0 s s 0 is a centered Gaussian process independent of (X s (x)) 0 s s 0 with covariance kernel q x,w (s, s

) def = E[Z x,w s Z x,w s ] = s ∧ s - s∧s 0 (K u (x -w)) 2 du.
We first estimate Π 2 with the above lemma. It is enough to estimate properly the quantity

Π 2 = Θ β t Z β t (B(x, e -t ) c ) > δ 2 |x, (X s (x)) s t , B . (18)
Notice that:

Π 2 2 δ B(x,e -t ) c Θ β t f β t (w)|x, (X s (x)) s t , B dw. ( 19
)
For each w ∈ B(x, e -t ) c , i.e. such that |w-x| > e -t , let us define s 0 = ln 1 |x-w| . Notice that s 0 is the time at which the evolution of (X s (w) -X s 0 (w)) s 0 s t becomes independent of the process (X s (x)) 0 s t . Under Θ β t , the process (X s (w)) s 0 s t can be rewritten as:

X s (w) = X s 0 (w) + W s-s 0
where W is a standard Brownian motion independent of the processes (X s (x)) 0 s t and (X s (w)) 0 s s 0 . This can be checked by a straightforward computation of covariance. Therefore we get:

Θ β t (f β t (w)|x, (X s (x)) s t ) = 1 β E ( √ 2dt -X t (w) + β)1I {sup [0,t] Xu(w)- √ 2du β} e √ 2dXt(w)-dt |x, (X s (x)) s t = 1 β E ( √ 2ds 0 + √ 2d(t -s 0 ) -X s 0 (w) -W t-s 0 + β)1I {sup [0,s 0 ] Xu(w)- √ 2du) β} 1I {sup [s 0 ,t] Xs 0 (w)+ √ 2ds 0 +W u-s 0 - √ 2d(u-s 0 )) β} e √ 2dXs 0 (w)-ds 0 e √ 2dW t-s 0 -d(t-s 0 ) |x, (X s (x)) s t = 1 β E ( √ 2ds 0 -X s 0 (w) + β)1I {sup [0,s 0 ] Xu(w)- √ 2du) β} e √ 2dXs 0 (w)-ds 0 |x, (X s (x)) s t ,
by the stopping time theorem. From Lemma 16, we deduce:

Θ β t (f β t (w)|x, (X s (x)) s t ) = 1 β E ( √ 2ds 0 -P x,w s 0 -Z x,w s 0 + β)1I {sup [0,s 0 ] P x,w s 0 +Z x,w s 0 - √ 2du β} e √ 2d(P x,w s 0 +Z x,w s 0 )-ds 0 |x, (X s (x)) s t 1 β E ( √ 2ds 0 -P x,w s 0 -Z x,w s 0 + β) 2 + 1 e √ 2d(P x,w s 0 +Z x,w s 0 )-ds 0 |x, (X s (x)) s t = 1 β ( √ 2d(s 0 -q x,w (s 0 , s 0 )) -P x,w s 0 + β) 2 + q x,w (s 0 , s 0 ) e √ 2dP x,w s 0 -d(s 0 -qx,w(s 0 ,s 0 )) . ( 20
)
We make two observations. First, we point out that the quantity q x,w (s 0 , s 0 ) is bounded by a constant only depending on k since q x,w (s 0 , s 0 ) =s 0 -

s 0 0 (K u (x -w)) 2 du = s 0 0 1 -k(e u (x -w)) 2 du = 1 |x-w| 1 -k y x -w |x -w| 2 1 y dy C
where C can be defined as sup z∈B(0,1)

1-k(z) 2
|z| . So the quantity q x,w (s 0 , s 0 ) won't play a part in the forthcoming computations.

Second, we want to express the random variable P x,w s 0 as a function of the Bessel process (X u (x) -√ 2du -β) u in order to use the fact that we can control the paths of this latter process (we will condition by the event B). Therefore we set

Y x,w s 0 = - s 0 0 g x,w (u)(X u (x) - √ 2du -β) du (21)
= -

s 0 0 g x,w (u)X u (x) du - √ 2dK s 0 (x -w) + β k(e s 0 (x -w)) -k(x -w) =P x,w s 0 - √ 2dK s 0 (x -w) + β k(e s 0 (x -w)) -k(x -w) .
Therefore we can write: Y x,w s 0 = P x,w s 0 -√ 2ds 0 + θ x,w (s 0 ) for some function θ x,w that is bounded independently of x, w, t since k is bounded over R d . Plugging these estimates into [START_REF] Daley | An introduction to the theory of point processes volume 2, Probability and its applications[END_REF], we obtain:

Θ β t (f β t (w)|x, (X s (x)) s t , B) = 1 β (θ x,w (s 0 ) -Y x,w s 0 ) 2 + q x,w (s 0 , s 0 ) e √ 2dY x,w s 0 +ds 0 +dqx,w(s 0 ,s 0 )- √ 2dθx,w(s 0 ) 1 β (C -Y x,w s 0 ) 2 + C) e √ 2dY x,w s 0 +ds 0 +C (22)
for some constant C that does not depend on x, w, t, A. Now we plug the exact expression of g x,w :

g x,w (u) = d i=1 (x -w) i e u ∂ i k(e u (x -w))
into the definition (21) of Y x,w s 0 :

Y x,w s 0 = ln 1 |x-w| 0 d i=1 (x -w) i e u ∂ i k(e u (x -w)) √ 2du + β -X u (x) du = 1 |x-w| y x -w |x -w| • ∇k y x -w |x -w| √ 2d ln y |x -w| + β -X ln y |x-w| (x) dy
Moreover the constraint for the Bessel process, valid on B, ∀u 0,

√ u R(ln(2 + u)) 2 β -X u (x) + √ 2du R(1 + u ln(1 + u)) (23)
implies that (here we use the relation

x • ∇k(x) 0) Y x,w s 0 R 1 |x-w| y x -w |x -w| • ∇k y x -w |x -w| ) 1 + ln y |x -w| ln 1 + ln y |x -w| dy (24)
Y x,w s 0 R 1 |x-w| y x -w |x -w| • ∇k y x -w |x -w| ln y |x-w| ln 2 + ln y |x-w| 2 dy. (25)
Using rough estimates yields

-C R 1 + ln 1 |x -w| ln 1 + ln 1 |x -w| du Y x,w s 0 -C R ln 1 |x-w| ln 2 + ln 1 |x-w| 2 (26)
for some constant C R depending on R and on the function x → x • ∇k(x). Plugging these estimates into ( 22) yields (the constant C may change, depending on the value of C R )

Θ β t (f β t (w)|x, (X s (x)) s t , B) e C β|x -w| d G(ln 1 |x -w| ) (27)
where

G(y) = 1 + y ln 1 + y 2 e - √ 2dC √ y ln 2+y 2
Finally, by gathering the above estimates ( 18), ( 19) and ( 27) and then making successive changes of variables, we obtain (V d stands for the area of the unit sphere of R d ):

Π 2 = 1 |A| A Θ β t Θ β t Z β t (B(x, e -t ) c ) > δ 2 |x, (X s (x)) s t , B |x dx = 1 |A| A Θ β t Π 2 |x dx 2 |A|δ A B(x,e -t ) c e C β|x -w| d G(ln 1 |x -w| ) dxdw 2V d δ 1 e -t e C βr d G(ln 1 r )r d-1 dr 2V d e C δβ t 0 G(u) du.
Since G is integrable, this quantity is obviously bounded by a quantity that goes to 0 when δ becomes large uniformly with respect to t. This concludes estimating Π 2 . We now estimate Π 1 . Once again, it is enough to estimate the quantity

Π 1 = Θ β t Z β t (B(x, e -t )) > δ 2 |x, (X s (x)) s t , B , (28)
which is less than

Π 1 2 δ B(x,e -t ) Θ β t f β t (w)|x, (X s (x)) s t , B dw. (29)
This time, for |w -x| < e -t , there is no need to "cut" the process (X s (w)) s t at level s = ln 1 |x-w| . We can directly use Lemma 16 to get:

Θ β t (f β t (w)|x, (X s (x)) s t , B) = 1 β E ( √ 2dt -P x,w t -Z x,w t + β)1I {sup [0,t] P x,w t +Z x,w t - √ 2du β} e √ 2d(P x,w t +Z x,w t )-dt |x, (X s (x)) s t , B 1 β E ( √ 2dt -P x,w t -Z x,w t + β) 2 + 1 e √ 2d(P x,w t +Z x,w t )-dt |x, (X s (x)) s t , B = 1 β ( √ 2d(t -q x,w (t, t)) -P x,w t + β) 2 + q x,w (t, t) e √ 2dP x,w t -d(t-qx,w(t,t)) .
Once again, the quantity q x,w (t, t) is bounded by a constant only depending on k (not on t). Second, for s t, we define the process

Y x,w s = - s 0 g x,w (u)(X u (x) - √ 2du -β) du + K S (x -w)(X s (x) - √ 2ds -β),
which turns out to be equal to

Y x,w s =P x,w s - √ 2ds + θ x,w (s)
for some function θ x,w that is bounded independently of x, w, s, t, A. We deduce:

Θ β t (f β t (w)|x, (X s (x)) s t , B) = 1 β (θ x,w (t) -Y x,w t ) 2 + q x,w (t, t) e √ 2dY x,w t +dt+dqx,w(t,t)- √ 2dθx,w(t) 1 β (C -Y x,w t ) 2 + C) e √ 2dY x,w t +dt+C (30)
for some constant C that does not depend on x, w, t. Once again on B, the Bessel process evolves in the strip [START_REF] David | Conformal Field Theories Coupled to 2-D Gravity in the Conformal Gauge[END_REF], implying that the process Y x,w is bound to live in the strip (we stick to the previous notations)

-C R 1 + t ln 1 + t du Y x,w t -C R √ t ln 2 + t 2 (31)
for some constant C R . Plugging these estimates into (30) yields (the constant C may change, depending on the value of C R )

Θ β t (f β t (w)|x, (X s (x)) s t , B) e C β G(t)e dt (32)
where the function G is still defined by

G(t) = 1 + t ln 1 + t 2 e - √ 2dC √ t ln 2+t 2 .
Notice that this estimate differs from that obtained for Π 2 because of the e dt factor. It will be absorbed by the volume of the ball B(x, e -t ) that we will integrate over. Finally, by using [START_REF] Durhuus | Multi-spin systems on a randomly triangulated surface[END_REF], we obtain:

Π 1 = 1 |A| A Θ β t Θ β t Z β t (B(x, e -t )) > δ 2 |x, (X s (x)) s t , B |x dx = 1 |A| A Θ β t Π 2 |x dx 2 |A|δ A B(x,e -t ) e C β G(t)e dt dxdw 2 δ e C β G(t)
Since G is bounded, this quantity is obviously bounded by a quantity that goes to 0 when δ becomes large uniformly with respect to t. This concludes estimating Π 1 . The proof is complete.

We are now in position to prove:

Theorem 17.
For each bounded open set A ⊂ R d , the martingale (M t (A)) t 0 converges almost surely towards a positive random variable denoted by M (A), such that M (A) > 0 almost surely. Consequently, almost surely, the (locally signed) random measures (M t (dx)) t 0 converge weakly as t → ∞ towards a positive random measure M (dx), which has full support and is atomless. Furthermore, the measure M is a solution to the -equation ( 9) with γ = √ 2d.

Proof. We first observe that the martingale (Z β t (A)) t 0 possesses almost surely the same limit as the process ( Z β t (A)) t 0 because

|Z β t (A) -Z β t (A)| = β A 1I {τ β >t} e √ 2dXt(x)-dE[Xt(x) 2 ] dx βM √ 2d t (A) (33)
and the last quantity converges almost surely towards 0 since M √ 2d t (dx) almost surely converges towards 0 as t goes to ∞ (see Proposition 19 below). Using Proposition 19, we have almost surely: sup

t∈R + max x∈A X t (x) - √ 2dt < +∞, which obviously implies ∀t, M t (A) = Z β t (A)
for β (random) large enough. Since the family of random measures (Z β t (dx)) t 0 are positive and (Z β t (A)) t 0 almost surely converges for every bounded open set A, it is plain to deduce that, almost surely, the family (M t (dx)) t 0 weakly converges towards a positive random measure.

Let us prove that the support of M is R d . We first write the relation, for s < t,

Z β t (dx) =( √ 2ds -X s (x) + β)1I {τ β >t} e √ 2dXt(x)-dE[Xt(x) 2 ] dx (34)
+ (

√ 2d(t -s) -X t (x) + X s (x) + β)1I {τ β >t} e √ 2dXt(x)-dE[Xt(x) 2 ] dx.
By using the same arguments as throughout this section, we pass to the limit in this relation as t → ∞ and then β → ∞ to get

M (dx) =e √ 2dXs(x)-dE[Xs(x) 2 ] M ,s (dx) (35)
where M ,s is defined as

M ,s (dx) = lim β→∞ lim t→∞ Z β,s t (dx)
and Z β,s t (dx) is almost surely defined as the weak limit of

Z β,s t (A) = A √ 2d(t-s)-X t (x)+X s (x)+β 1I {τ β s >t} e √ 2d(Xt(x)-Xs(x))-d E[Xt(x) 2 ]-E[Xs(x) 2 ] dx where τ β s = inf{u > 0; X u+s (x) -X s (x) - √ 2du > β}.
Let us stress that we have used the fact that the measure

( √ 2ds -X s (x) + β)1I {τ β >t} e √ 2dXt(x)-dE[Xt(x) 2 ] dx
goes to 0 (it is absolutely continuous w.r.t. to M √ 2d t (dx)) when passing to the limit in [START_REF] Eynard | The Potts-q random matrix model: loop equations, critical exponents, and rational case[END_REF]. Therefore M is a solution to -equation [START_REF] Barral | Gaussian multiplicative chaos and KPZ duality[END_REF]. From [START_REF] Falconer | The geometry of fractal sets[END_REF], it is plain to deduce that the event {M (A) = 0} (A open non-empty set) belongs to the asymptotic sigmaalgebra generated by the field {(X t (x)) x ; t 0}. Therefore it has probability 0 or 1 by the 0 -1 law of Kolmogorov. Since we have already proved that it is not 0, this proves that P(M (A) = 0) = 0 for any non-empty open set A.

Finally, we prove that the measure is atomless. The proof is based on the computations made during the proof of Proposition [START_REF] Bernardi | Counting colored planar maps: algebraicity results[END_REF]. We will explain how to optimize these computations to obtain the atomless property. Of course, we could have done that directly in the proof of Proposition ( 14) but we feel that it is more pedagogical to separate the arguments. Let us roughly explain how we will proceed. Clearly, it is sufficient to prove that the positive random measure

Z β (dx) = lim t→∞ Z β t (dx).
does not possess atoms. To that purpose, by stationarity, it is enough to prove that (see [START_REF] Daley | An introduction to the theory of point processes volume 2, Probability and its applications[END_REF]Corollary 9.3 VI]) ∀δ > 0, lim

n n d P Z β (I n ) > δ = 0
where I n is the cube [0, 1 n ] d . From now on, we stick to the notations of Proposition 14. We have to prove that:

∀δ > 0, lim n lim sup t Θ β t Z β t (I n ) > δ = 0.
Therefore, let δ > 0 and ε > 0 be two fixed positive real numbers. We choose R and the associated event B of probability 1 -ε as in Proposition 14. We have:

lim sup t Θ β t Z β t (I n ) > δ ε + lim sup t Π 1 + lim sup t Π 2 .
First note that lim sup t Π 1 = 0; we also have the following bound for lim sup t Π 2 :

lim sup t Π 2 2V d e C δβ ∞ n ln 2 G(u) du,
which goes to 0 as n goes to ∞. In conclusion, we get:

lim n lim sup t Θ β t Z β t (I n ) > δ ε,
which is the desired result.

A.2 Proof of result from Section 4

Here, we prove Proposition 8. For notational simplicity, we further assume that the dimension d is equal to 1 and that k(u) = 0 for all |u| > 1. Generalization to all other situations is straightforward.

Let C be the interval [0, 1]. Let us denote by φ(•, γ) the Laplace transform of M γ (C)

φ(λ, γ) = E[e -λM γ (C) ].
Since P(M γ (C) > 0) = 1 the range of the mapping λ ∈ R + → φ(λ, γ) is the whole interval ]0, 1]. Choose a strictly increasing sequence (γ n ) n converging towards

√ 2. Choose a sequence (λ n ) n such that φ(λ n , γ n ) = 1 2 . ( 36
)
Let us denote by M c (C) a random variable taking values in [0, +∞] such that λ n M γn (C) → M c (C) vaguely as n → ∞ (eventually up to a subsequence). Let us define the function

ϕ(θ) = E[e -θM c (C) , M c (C) < ∞]
for θ > 0 and ϕ(0) = 1. Then φ(θλ n , γ n ) → ϕ(θ) for all θ so that, in particular, ϕ(1) = 1 2 . Let us choose ε small enough in order to have ln 1 ε even integer larger than 4. Because of (9), we have

φ(θλ n , γ n ) = E exp -θλ n C e γnX ln 1 ε (r)- γ 2 n 2 E[X ln 1 ε (r) 2 ] M γn,ε (dr) Let us denote by C k the interval [ k ln 1 ε , k+1 ln 1 ε ] for k ∈ A ε def = {0, . . . , ln 1 ε -1}.
By the Cauchy-Schwartz inequality and stationarity, we have

φ(θλ n , γ n ) E exp -2θλ n k∈Aε even C k e γnX ln 1 ε (r)- γ 2 n 2 E[X ln 1 ε (r) 2 ] M γn,ε (dr)
By the Kahane convexity inequality and because the mapping x → e -sx is convex for any s ∈ R, we deduce

φ(θλ n , γ n ) E exp -2θλ n k∈Aε even C k e √ 2X ln 1 ε (0)-E[X ln 1 ε (0) 2 ] M γn,ε (dr) = E exp -2θλ n e √ 2X ln 1 ε (0)-E[X ln 1 ε (0) 2 ] k∈Aε even M γn,ε (C k ) .
Because the sets C k are separated by a distance of at least 1 ln(1/ε) , the random variables (M γn,ε (C k )) k∈Aε even are i.i.d. with common law εM γn (C) because of [START_REF] Barral | Gaussian multiplicative chaos and KPZ duality[END_REF]. We deduce:

φ(θλ n , γ n ) E φ 2θλ n εe √ 2X ln 1 ε (0)-E[X ln 1 ε (0) 2 ] , γ n 1 2 ln 1 ε .
By taking the limit as n → ∞, we deduce

ϕ(θ) E ϕ 2θεe √ 2X ln 1 ε (0)-E[X ln 1 ε (0) 2 ] 1 2 ln 1 ε .
By letting θ go to 0, we deduce

ϕ(0 + ) ϕ(0 + ) 1 2 ln 1 ε .
Because 1 2 ln 1 ε 2, we are left with two options: either ϕ(0 + ) = 0 or ϕ(0 + ) 1. But ϕ(0 + )

1 because e -θx 1 for all x 0. Furthermore ϕ(0 + ) ϕ(1) = 1 2 . Therefore ϕ(0 + ) = 1 and M c (C) < +∞ almost surely. M c (C) is not trivial because ϕ(1) = 1 2 . We have proved that the sequence (λ n M γn (C)) n is tight and that the limit of every converging subsequence is non trivial.

Of course, we can carry out the same job for every smaller dyadic interval. But the normalizing sequence may depend on the size of the interval. Let us prove that it does not. To this purpose, it is enough to establish that for every dyadic interval C k of size 2 -k . The left-hand side is obvious because M γn (C k ) M γn (C). By using (9) with ε = 2 -k and the Kahane convexity inequality, we deduce:

lim sup n E exp -λ n M γn (C k ) lim sup n E exp -λ n M γn (C)2 -k e √ 2X k ln 2 (0)-E[X k ln 2 (0) 2 ] = E ϕ 2 -k e √ 2X k ln 2 (0)-E[X k ln 2 (0) 2 ] .
The last quantity is strictly less than 1. Indeed, if not, then ϕ 2 -k e

√ 2dX k ln 2 (0)-2d 2 E[X k ln 2 (0) 2 ] = 1 almost surely, that is ϕ(θ) = 1 for all θ, hence a contradiction.
To sum up, the sequence (λ n M γn (C)) n is tight for all dyadic intervals. By the Tychonoff theorem and the Caratheodory extension theorem, we can extract a subsequence and find a random measure M c (dx) such that (λ n M γn (C 1 ), . . . , λ n M γn (C p )) n converges in law towards (M c (C 1 ), . . . , M c (C p )) n for all dyadic intervals C 1 , . . . , C p . Finally, by multiplying both sides of (9) by λ n and passing to the limit as n → ∞, we deduce:

M c (A) A∈B(R) law = A e √ 2X ln 1 ε (r)-E[X ln 1 ε (r) 2 ] M c,ε (dr) A∈B(R) (37)
where

M c,ε (A) A∈B(R) law = ε M c ( A ε ) A∈B(R) . (38)

B. Auxiliary results

We first state the classical "Kahane's convexity inequalities" (originally written in [START_REF] Kahane | Sur le chaos multiplicatif[END_REF] , see also [START_REF] Allez | Lognormal -scale invariant random measures, Probability Theory and Related Fields[END_REF] for a proof):

Lemma 18. Let F, G : R + → R be two functions such that F is convex, G is concave and

∀x ∈ R + , |F (x)| + |G(x)| M (1 + |x| β ),
for some positive constants M, β, and σ be a Radon measure on the Borelian subsets of R d . Given a bounded Borelian set A, let (X r ) r∈A , (Y r ) r∈A be two continuous centered Gaussian processes with continuous covariance kernels k X and k Y such that

∀u, v ∈ A, k X (u, v) k Y (u, v). Then E F A e Xr-1 2 E[X 2 r ] σ(dr) E F A e Yr-1 2 E[Y 2 r ] σ(dr) E G A e Xr-1 2 E[X 2 r ] σ(dr) E G A e Yr-1 2 E[Y 2
r ] σ(dr) .

If we further assume

∀u ∈ A, k X (u, u) = k Y (u, u)
then we recover Slepian's comparison lemma: for each increasing function F : R + → R:

E F sup x∈A Y x E F sup x∈A X x .

B.1 Chaos associated to cascades

We use Kahane

Notice that ∀s, t ∈ T, q n (t,

s) = u d ln 2 ln 1 d(t, s) ∨ 2 -dn (40)
and

q n (t, s) → u d ln 2 ln 1 d(t, s)
as n → ∞.

We define the centered Gaussian process

∀t ∈ T, X n (t) = n k=1 Y k (t)
with covariance kernel q n . Let us denote by σ the uniform measure on T , ie σ(I n (t)) = 2 -dn . We set

M u n = T e Xn(t)-1 2 E[Xn(t) 2 ] σ(dt).
This corresponds to the lognormal multiplicative cascades framework. The martingale (M u n ) n converges towards a non trivial limit if and only if u < 2d ln 2. The boundary case corresponds to u = 2d ln 2. It is proved in [START_REF] Kahane | Sur certaines martingales de B. Mandelbrot[END_REF] that, for u = 2d ln 2, lim n M u n (dx) = 0 almost surely.

It turns out that the 2 d -adic tree can be naturally embedded in the unit cube of R d by iteratively dividing a cube into 2 d cubes with equal size length. Notice that the uniform measure on the tree is then sent to the Lebesgue measure by this embedding. We also stress that the dyadic distance on the cube [0, 1] d is greater than the Euclidean distance on that cube:

∀s, t ∈ [0, 1] d , |t -s| √ d d(t, s) 1 d .
This allows many one-sided comparison results between lognormal cascades and Gaussian multiplicative chaos. So, taking u = 2d ln 2 in the kernel q n of (40), we claim for all s , s ∈ [0, 1] d , ∀n ∈ N:

q n (s, s ) -C 2dK n ln 2 (s -s ) (41)
for some constant C > 0 that does not depend on n (only on k).

We are now in position to prove:

Proposition 19. For γ 2 = 2d, the standard construction yields a vanishing limiting measure:

lim t→∞ M √ 2d t = 0 almost surely. (42)
Furthermore, for all a ∈ [0, 1 2 [ and any bounded open set A, almost surely:

sup t 0 (sup x∈A X t (x) - √ 2dt + a √ 2d ln(t + 1)) < ∞ (43)
Proof. We consider X n with covariance given by ( 40) for u = ln 2; by a slight abuse of notation, we consider that X n is defined on the unit cube by the natural embedding.

The family (M γ t ) is a positive martingale. Therefore it converges almost surely. We just have to prove that the limit is zero. We will apply Kahane's concentration inequalities (Lemma 18). Let us denote by Z a standard Gaussian random variable independent of the process (X t (x)) t,x . From [START_REF] Gross | One-dimensional string theory on a circle[END_REF], the covariance kernel of the centered Gaussian process X n is less than that of the Gaussian process √ CZ + X n ln 2 . By applying Lemma 18 to some bounded concave function F : R + → R and n ∈ N, we obtain (we stick to the notations introduced just above) For the second statement, we fix a ∈ [0, 1 2 [ and we consider the case d = 1 and k(x) = (1 -|x|) + for simplicity (this is no restriction since every C 1 kernel k with k(0) = 1 is greater or equal to some (1 -|x| L ) + for L > 0.) In this case, one can represent the variables X s (x) as integrals of truncated cones with respect to a Gaussian measure (see subsection B.2 below for a quick reminder or [START_REF] Barral | Multifractal products of cylindrical pulses[END_REF][START_REF] Bacry | Log-infinitely divisible multifractal processes[END_REF] for details). The cone representation ensures that we have the following decomposition (see subsection B.2): Lemma 20. We fix n and cut [0, 1] into 2 n intervals. We have the following decomposition for X s ln 2 (x) for all s ∈ [n, n + 1] and x ∈ I i,n := [ i 2 n , i+1 2 n [:

E F e √ CZ-
X s ln 2 (x) = X i,n + Y i,n s (x),
with the following properties:

• There exists a constant C > 0 (independent of n) such that:

E[X i,n X j,n ] = n ln 2 -(1 -1 2 n ), if i = j E[X i,n X j,n ] E[X n ( i 2 n )X n ( j 2 n )] -C, if i = j
• For all i, the process (Y i,n s (x)) s∈[n,n+1],x∈I i,n is continuous and independent of X i,n .

• For all i, j, s, s ∈ [n, n + 1] and x ∈ I i,n , x ∈ I j,n :

E[Y i,n s (x)Y j,n s (x )] 0

• For all i, j, s ∈ [n, n + 1] and x ∈ I i,n :

E[Y i,n s (x)X j,n ] 0
We introduce a standard Gaussian variable Z independent from the process (X s ln 2 (x)) x and a standard Gaussian i.i.d. sequence (Z i ) 0 i 2 n -1 . We also introduce a sequence of independent processes (Y i,n s (x)) s∈[n,n+1],x∈I i,n independent from X n and such that for all i the process (Y

√ CZ i + Y i,n s (x))(X n ( i 2 n ) + √ CZ i + Y i,n s (x ))]
and for i = j, x ∈ I i,n , x ∈ I j,n and s, s ∈ [n, n + 1]:

E[(X i,n + 1 -

1 2 n + CZ + Y i,n s (x))(X j,n + 1 - 1 2 n + CZ + Y i,n s (x ))] E[X i,n X j,n ] + 1 - 1 2 n + C E[X n ( i 2 n )X n ( j 2 n )] = E[(X n ( i 2 n ) + √ CZ i + Y i,n s (x))(X n ( j 2 n ) + √ CZ j + Y j,n s (x ))] 29
Now, let β > 1 and r < 1 be such that βr < 1 and ( , where in the last line we have used Theorem 1.6 in [START_REF] Hu | Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees[END_REF]. This entails the desired result by the Borel-Cantelli lemma.

B.2 Reminder about the cone construction

The cone construction is based on Gaussian independently scattered random measures (see [START_REF] Rajput | Spectral representations of infinitely divisible processes[END_REF] for further details). We consider a Gaussian independently scattered random measure µ distributed on the measurable space (R × R + , B(R × R + )), that is a collection of Gaussian random variables (µ(A), A ∈ B(R × R + )) such that: 1) For every sequence of disjoint sets (A n ) n in B(R × R + ), the random variables (µ(A n )) n are independent and

µ n A n = n µ(A n ) a.s.,
2) for any measurable set A in B(R × R + ), µ(A) is a Gaussian random variable whose characteristic function is given by

E(e iqµ(A) ) = e -q 2 i 2 n x i+1 2 n R i,n s (x) A i,n 2 -n 2 -s 1
It is then straightforward to check the claims of Lemma 20 by using the properties of the measure µ. The process (Y i,n (x)) s∈[n,n+1],x∈I i,n is independent of X i,n since the sets (R i,n s (x)) s∈[n,n+1],x∈I i,n are all disjoint of the triangle A i,n . We also have E[Y i,n (x)Y j,n (x )] 0 since this covariance is just given by the Γ-measure of the set R i,n s (x) ∩ R i,n s (x ). The same argument holds to prove E[Y i,n (x)X j,n ] 0.

  -λnM γn (C k ) ] lim sup n E[e -λnM γn (C k ) ] < 1.

2 n - 1 i=0 e β( √ 2

 12 Xn( i 2 n )-2n ln 2) ) r ] C β,r (n + 1) aβr E[(

  convexity inequalities (see proposition[START_REF] Brézin | Scaling violation in a field theory of closed strings in one physical dimension[END_REF] to compare the small moments of the Gaussian multiplicative chaos with those of a dyadic lognormal Mandelbrot's multiplicative cascade. Let us briefly recall the construction of lognormal Mandelbrot's multiplicative cascades. We consider the 2 d -adic tree T = ({1, 2} d ) N For each n, we denote by (Y n (t)) t∈T a centered Gaussian process indexed by T with covariance kernel p n . We assume that the processes (Y n ) n are independent. We set ∀s, t ∈ T, q n (t, s) =

	n
	p k (t, s)
	k=1

* .

For t ∈ T , we denote by π k (t) (k ∈ N * ) the k-th component of t. We equip T with the ultrametric distance

∀s, t ∈ T, d(t, s) = 2 -dn , where n = sup{N ∈ N; ∀k N, π k (t) = π k (s)}. with the convention that n = 0 if the set {N ∈ N; ∀k N, π k (t) = π k (s)} is empty. Let us define ∀s, t ∈ T, p n (t, s) = u if d(t, s) 2 -nd 0 if d(t, s) > 2 -nd . .

The kernel p n is therefore constant over each of the 2 dn cylinders defined by the prescription of the first n coordinates (in what follows, we will denote by I n (t) that cylinder containing t).

  Now we further assume that F is increasing. Because of the dominated convergence theorem, the right-hand side goes to F (0) as n → ∞. So does the left-hand side. This shows that M √ 2d n ln 2 ([0, 1] d ) goes to 0 in probability as n → ∞. Since we already know that the martingale M ([0, 1] d ) converges almost surely as t → ∞, this completes the proof of the first statement.

		1 2 C M	√ n ln 2 ([0, 1] d ) 2d	E F	T	e	√	2d Xn(t)-d E[Xn(t) 2 ] dt .	(44)
	√	2d
	t

  (x)) s∈[n,n+1],x∈I i,n has same law as (Y i,n s (t)) s∈[n,n+1],x∈I i,n . By Lemma 18, we have the following for all x: Indeed, we have the following if i = j, x, x ∈ I i,n and s, s ∈ [n, n + 1]:

	i,n
	P ( sup 0 i 2 n -1	sup s∈[n,n+1]	sup x∈I i,n	(X i,n + 1 -	1 2 n + CZ + Y i,n s (x) -	√	2n ln 2) x)
	P ( sup 0 i 2 n -1	sup s∈[n,n+1]	sup x∈I i,n	(X n (	i 2 n ) +	√	CZ i + Y	i,n s (x) -	√	2n ln 2) x)
	E[(X i,n + 1 -	1 2 n + CZ + Y i,n s (x))(X i,n + 1 -	1 2 n + CZ + Y i,n s (x ))]
	= n ln 2 + C + E[Y i,n s (x)Y i,n s (x )]
	= E[(X n (	i 2 n ) +

s

  3 2 -a)βr > 1. We have:

	P ( sup 0 i 2 n -1	sup s∈[n,n+1]	sup x∈I i,n	( √	2 X n (	i 2 n ) +	√ 2CZ i +	√	2Y	i,n s (x) -2n ln 2 + a ln(n + 1)) 1)
	= P ( sup 0 i 2 n -1 ( √	2 X n (	i 2 n ) +	√	2CZ i +	√	2	s∈[n,n+1],x∈I i,n sup	Y	i,n s (x) -2n ln 2 + a ln(n + 1)) 1)
	(n + 1) aβr e -βr E[( 2 n -1	e	β( √	2 Xn( i 2 n )+	√	2C Z i + √	2 sup s∈[n,n+1],x∈I i,n Y	i,n s (x)-2n ln 2) ) r ]
				i=0
	(n + 1) aβr e -βr E[E[( 2 n -1	e	β( √	2 Xn( i 2 n )+	√	2C Z i + √	2 sup s∈[n,n+1],x∈I i,n Y	i,n s (x)-2n ln 2) ) r |X n ]]
				i=0
	(n + 1) aβr e -βr E[(E[ 2 n -1	e	β( √	2 Xn( i 2 n )+	√	2C Z i + √	2 sup s∈[n,n+1],x∈I i,n Y	i,n
				i=0
			β( √	2C Z i + √	2 sup s∈[n,n+1],x∈I i,n Y	i,n

s (x)-2n ln 2) |X n ]) r ] (n + 1) aβr e -βr E[e s (x)) ] r × E[(

Γ(A)where the control measure Γ is given by Γ(dx, dy) = 1 y 2 dxdy.

Acknowledgements

The authors wish to thank J. Barral, J.P. Bouchaud, K. Gawedzki, I. R. Klebanov, I. K. Kostov and O. Zindy for fruitful discussions and comments that have led to the final version of this manuscript.

Partially supported by grant ANR-08-BLAN-0311-CSD5 and by the MISTI MIT-France Seed Fund. Partially supported by grant ANR-11-JCJC CHAMU. Partially supported by NSF grants DMS 064558 and OISE 0730136, and by the MISTI MIT-France Seed Fund. Partially supported by grant ANR-11-JCJC CHAMU.

We can then define the stationary Gaussian process (ω l (x)) x∈R for 0 < l 1 by

where A l (x) is the triangle like subset A l (x) := {(u, y) ∈ R × R * + : l y 1, -y/2 xu y/2}. The covariance kernel of the stationary Gaussian process ω l is given by 0

x

which can also be rewritten as:

Therefore the process ω e -t has the same law as X t . This approach is called the cone construction. Now we explain how to use the cone construction to prove Lemma 20, that is to decompose the process X s ln 2 = ω 2 -s for s ∈ [n, n + 1]. So we choose i ∈ N such that 0 i 2 n -1. We call A i,n the common part to all the cone like subsets A 2 -s (x) for s ∈ [n, n + 1] and x ∈ I i,n :

For s ∈ [n, n + 1] and x ∈ I i,n , we define the set R i,n s (x) as:

Then we set Y i,n s (x) = µ(R i,n s (x)) and X i,n = µ(A i,n ). . In particular, we find E[X i,n X j,n ] = n ln 2 + ln 1 |i -j| + 1 + |i -j| + 1 2 n -1.