arXiv subjects

Michel Pain

Publications and source records attributed to Michel Pain.

15 recordsLinked to original sources

arXiv2026

Fluctuations of additive martingale limits of branching Brownian motion

Consider a one-dimensional branching Brownian motion. Let $W_\infty(\beta)$ denote the limit of the additive martingale in the subcritical regime $\lvert \beta\rvert < \beta_c$ and $Z_\infty$ be the limit of the derivative martingale at criticality. Madaule (Stochastic Process. Appl. 126 (2016), no. 2, 470--502) established the following convergence \[ \frac{W_\infty(\beta)}{\beta_c-\beta}\xrightarrow[\beta\nearrow \beta_c]{\mathbb{P}} 2Z_\infty. \] The goal of this paper is twofold: firstly, we strengthen this result into an almost sure convergence; secondly, we describe the fluctuations occurring in this convergence by proving \[ \frac{1}{\beta_c-\beta}\left( \frac{W_\infty(\beta)}{\beta_c-\beta} - 2 Z_\infty +2(\beta_c-\beta)\log(\beta_c-\beta) Z_\infty\right) \xrightarrow[\beta\nearrow \beta_c]{(d)} S, \] where, conditionally on $Z_\infty$, $S$ follows a spectrally negative 1-stable distribution with scale and shift parameters proportional to $Z_\infty$. Furthermore, these results are extended to the setting of complex additive martingales and the fluctuations to a multi-dimensional convergence.

math.PR↗

arXiv2025

CLT for $\beta$-ensembles with Freud weights, application to the KLS conjecture in Schatten balls

In this paper, we are interested in the $\beta$-ensembles (or 1D log-gas) with Freud weights, namely with a potential of the form $|x|^{p}$ with $p \geq 2$. Since this potential is not of class $\mathcal{C}^{3}$ when $p \in (2,3]$, most of the literature does not apply. In this singular setting, we prove a central limit theorem for linear statistics with general test-functions. Our strategy relies on establishing an optimal local law in the spirit of [Bourgade, Mody, Pain 22'. Our results allow us to give a consistency check of the KLS conjecture for the uniform distributions on $p$-Schatten balls and the functions $f(X)=\mathrm{Tr}\left(X^r\right)^q$. While the case $p>3$, $q=1$, $r=2$ was proven in [Dadoun, Fradelizi, Gu\'edon, Zitt 23'], we address in the present paper the case $p\geq2$, $q\geq1$ and $r\geq2$ an even integer. The proofs are based on a link between the moments of norms of uniform laws on $p$-Schatten balls and the $\beta$-ensembles with Freud weights.

math.PR↗

arXiv2025

Polynomial slowdown in an angle-dependent 2d branching Brownian motion

We consider a branching Brownian motion in $\mathbb{R}^2$ in which particles independently diffuse as standard Brownian motions and branch at an inhomogeneous rate $b(\theta)$ which depends only on the angle $\theta$ of the particle. We assume that $b$ is maximal when $\theta=0$, which is the preferred direction for breeding. Furthermore we assume that $b(\theta ) = 1 - \beta \abs{\theta }^\alpha + O(\theta ^2)$, as $\theta \to 0$, for $\alpha \in (2/3,2)$ and $\beta>0.$ We show that if $M_t$ is the maximum distance to the origin at time $t$, then $(M_t-m(t))_{t\ge 1}$ is tight where $$m(t) = \sqrt{2} t - \frac{\vartheta_1}{\sqrt{2}} t^{(2-\alpha)/(2+\alpha)} - \left(\frac{3}{2\sqrt{2}} - \frac{\alpha}{2\sqrt{2}(2+\alpha)}\right) \log t. $$ and $\vartheta_1$ is explicit in terms of the first eigenvalue of a certain operator.

math.PR↗

arXiv2025

Interlacing sequences resulting from an interval split-merge dynamics and the induced probability measures

We study sequences of partitions of the unit interval into subintervals, starting from the trivial partition, in which each partition is obtained from the one before by splitting its subintervals in two, according to a given rule, and then merging pairs of subintervals at the break points of the old partition. The $n$th partition then comprises $n+1$ subintervals with $n$ break points, which inherently possess an interlacing property. The empirical distribution of these points reveals a surprisingly rich structure, even when the splitting rule is completely deterministic. We consider both deterministic and randomized splitting rules and we study from multiple angles the limiting behavior of the empirical distribution of the break points.

math.PR↗

arXiv2024

Asymptotics of the overlap distribution of branching Brownian motion at high temperature

At high temperature, the overlap of two particles chosen independently according to the Gibbs measure of the branching Brownian motion converges to zero as time goes to infinity. We investigate the precise decay rate of the probability to obtain an overlap greater than $a$, for some $a>0$, in the whole subcritical phase of inverse temperatures $\beta \in [0,\beta_c)$. Moreover, we study this probability both conditionally on the branching Brownian motion and non-conditionally. Two sub-phases of inverse temperatures appear, but surprisingly the threshold is not the same in both cases.

math.PR↗

arXiv2024

Branching Brownian motion versus Random Energy Model in the supercritical phase: overlap distribution and temperature susceptibility

In comparison with Derrida's REM, we investigate the influence of the so-called decoration processes arising in the limiting extremal processes of numerous log-correlated Gaussian fields. In particular, we focus on the branching Brownian motion and two specific quantities from statistical physics in the vicinity of the critical temperature. The first one is the two-temperature overlap, whose behavior at criticality is smoothened by the decoration process - unlike the one-temperature overlap which is identical - and the second one is the temperature susceptibility, as introduced by Sales and Bouchaud, which is strictly larger in the presence of decorations and diverges, close to the critical temperature, at the same speed as for the REM but with a different multiplicative constant. We also study some general decorated cases in order to highlight the fact that the BBM has a critical behavior in some sense to be made precise.

math.PR↗

arXiv2022

Height of weighted recursive trees with sub-polynomially growing total weight

Weighted recursive trees are built by adding successively vertices with predetermined weights to a tree: each new vertex is attached to a parent chosen at random with probability proportional to its weight. In the case where the total weight of the tree at step $n$ grows polynomially in $n$, we obtained in (Pain-Sénizergues 2022) an asymptotic expansion for the height of the tree, which falls into the university class of the maximum of branching random walks. In this paper, we consider the case of a total weight growing sub-polynomially in $n$ and obtain asymptotics for the height of the tree in several regimes, showing that universality is broken and exhibiting new behaviors.

math.PR↗

arXiv2022

Optimal local law and central limit theorem for $β$-ensembles

In the setting of generic $β$-ensembles, we use the loop equation hierarchy to prove a local law with optimal error up to a constant, valid on any scale including microscopic. This local law has the following consequences. (i) The optimal rigidity scale of the ordered particles is of order $(\log N)/N$ in the bulk of the spectrum. (ii) Fluctuations of the particles satisfy a central limit theorem with covariance corresponding to a logarithmically correlated field; in particular each particle in the bulk fluctuates on scale $\sqrt{\log N}/N$. (iii) The logarithm of the electric potential also satisfies a logarithmically correlated central limit theorem. Contrary to much progress on random matrix universality, these results do not proceed by comparison. Indeed, they are new for the Gaussian $β$-ensembles. By comparison techniques, (ii) and (iii) also hold for Wigner matrices.

math.PR↗

arXiv2021

Correction terms for the height of weighted recursive trees

Weighted recursive trees are built by adding successively vertices with predetermined weights to a tree: each new vertex is attached to a parent chosen randomly proportionally to its weight. Under some assumptions on the sequence of weights, the first order for the height of such trees has been recently established in [Electron. J. Probab. 26 (2021), Paper No. 80] by one of the authors. In this paper, we obtain the second and third orders in the asymptotic expansion of the height of weighted recursive trees, under similar assumptions. Our methods are inspired from those used to prove similar results for branching random walks. Our results also apply to a related model of growing trees, called the preferential attachment tree with additive fitnesses.

math.PR↗

arXiv2021

1-stable fluctuations in branching Brownian motion at critical temperature II: general functionals

Let $\mu_t$ denote the critical derivative Gibbs measure of branching Brownian motion at time $t$. It has been proved by Madaule (Stochastic Process. Appl. 126 (2016), no. 2, 470--502) and Maillard and Zeitouni (Ann. Inst. Henri Poincar\'e Probab. Stat. 52 (2016), no. 3, 1144--1160) that $\mu_t$ converges weakly to the random measure $Z_\infty \sqrt{2/\pi} x^2 e^{-x^2/2} \boldsymbol 1_{x >0} d x$, where $Z_\infty$ is the limit of the derivative martingale. In this paper, we are interested in the fluctuations that occur in this convergence and prove for a large class of functions $F$ that \begin{align*} \sqrt{t} \left( \int_{\mathbb R} F d \mu_t - Z_\infty \int_0^\infty F(x) \sqrt{\frac{2}{\pi}} x^2 e^{-x^2/2} d x - \frac{c(F) \log t}{\sqrt{t}} Z_\infty \right) \to S(F), \end{align*} in law, as $t\to\infty$, where $c(F)$ is a constant depending on $F$ and, given $Z_\infty$, $S(F)$ has an explicit 1-stable distribution. Moreover, we extend this result to a functional convergence, and we identify precisely the particles responsible for the fluctuations. In particular, this proves the following result for the critical additive martingale $(W_t)_{t\geq 0}$: \[ \sqrt{t} \left( \sqrt{t} W_t - \sqrt{\frac{2}{\pi}} Z_\infty \right) \xrightarrow[t\to\infty]{} C Z_\infty, \quad \text{in law}, \] where here $C$ is a Cauchy variable independent of $Z_\infty$, confirming a conjecture by Mueller and Munier (Phys. Rev. E 90 (2014), 042143) in the physics literature.

math.PR↗

arXiv2020

Two-temperatures overlap distribution for the 2D discrete Gaussian free field

In this paper, we prove absence of temperature chaos for the two-dimensional discrete Gaussian free field using the convergence of the full extremal process, which has been obtained recently by Biskup and Louidor. This means that the overlap of two points chosen under Gibbs measures at different temperatures has a nontrivial distribution. Whereas this distribution is the same as for the random energy model when the two points are sampled at the same temperature, we point out here that they are different when temperatures are distinct: more precisely, we prove that the mean overlap of two points chosen under Gibbs measures at different temperatures for the DGFF is strictly smaller than the REM's one. Therefore, although neither of these models exhibits temperature chaos, one could say that the DGFF is more chaotic in temperature than the REM.

math.PR↗

arXiv2019

An exactly solvable continuous-time Derrida--Retaux model

To study the depinning transition in the limit of strong disorder, Derrida and Retaux (2014) introduced a discrete-time max-type recursive model. It is believed that for a large class of recursive models, including Derrida and Retaux' model, there is a highly non-trivial phase transition. In this article, we present a continuous-time version of Derrida and Retaux model, built on a Yule tree, which yields an exactly solvable model belonging to this universality class. The integrability of this model allows us to study in details the phase transition near criticality and can be used to confirm the infinite order phase transition predicted by physicists. We also study the scaling limit of this model at criticality, which we believe to be universal.

math.PR↗

arXiv2018

1-stable fluctuations in branching Brownian motion at critical temperature I: the derivative martingale

Let $(Z_t)_{t\geq 0}$ denote the derivative martingale of branching Brownian motion, i.e.\@ the derivative with respect to the inverse temperature of the normalized partition function at critical temperature. A well-known result by Lalley and Sellke [\textit{Ann. Probab.}, 15(3):1052--1061, 1987] says that this martingale converges almost surely to a limit $Z_\infty$, positive on the event of survival. In this paper, our concern is the fluctuations of the derivative martingale around its limit. A corollary of our results is the following convergence, confirming and strengthening a conjecture by Mueller and Munier [\textit{Phys. Rev. E}, 90:042143, 2014]: \[ \sqrt{t} \left( Z_\infty - Z_t + \frac{\log t}{\sqrt{2πt}} Z_\infty \right) \xrightarrow[t\to\infty]{} S_{Z_\infty}, \quad \text{in law}, \] where $S$ is a spectrally positive 1-stable Lévy process independent of $Z_\infty$. In a first part of the paper, a relatively short proof of (a slightly stronger form of) this convergence is given based on the functional equation satisfied by the characteristic function of $Z_\infty$ together with tail asymptotics of this random variable. We then set up more elaborate arguments which yield a more thorough understanding of the trajectories of the particles contributing to the fluctuations. In this way, we can upgrade our convergence result to functional convergence. This approach also sets the ground for a follow-up paper, where we study the fluctuations of more general functionals including the renormalized critical additive martingale. All proofs in this paper are given under the hypothesis $E[L(\log L)^3] < \infty$, where the random variable $L$ follows the offspring distribution of the branching Brownian motion. We believe this hypothesis to be optimal.

math.PR↗

arXiv2017

The near-critical Gibbs measure of the branching random walk

Consider the supercritical branching random walk on the real line in the boundary case and the associated Gibbs measure $ν_{n,β}$ on the $n^\text{th}$ generation, which is also the polymer measure on a disordered tree with inverse temperature $β$. The convergence of the partition function $W_{n,β}$, after rescaling, towards a nontrivial limit has been proved by A\"ıdékon and Shi in the critical case $β= 1$ and by Madaule when $β>1$. We study here the near-critical case, where $β_n \to 1$, and prove the convergence of $W_{n,β_n}$, after rescaling, towards a constant multiple of the limit of the derivative martingale. Moreover, trajectories of particles chosen according to the Gibbs measure $ν_{n,β}$ have been studied by Madaule in the critical case, with convergence towards the Brownian meander, and by Chen, Madaule and Mallein in the strong disorder regime, with convergence towards the normalized Brownian excursion. We prove here the convergence for trajectories of particles chosen according to the near-critical Gibbs measure and display continuous families of processes from the meander to the excursion or to the Brownian motion.

math.PR↗

arXiv2016

Velocity of the $L$-branching Brownian motion

We consider a branching-selection system of particles on the real line that evolves according to the following rules: each particle moves according to a Brownian motion during an exponential lifetime and then splits into two new particles and, when a particle is at a distance $L$ of the highest particle, it dies without splitting. This model has been introduced by Brunet, Derrida, Mueller and Munier in the physics literature and is called the $L$-branching Brownian motion. We show that the position of the system grows linearly at a velocity $v_L$ almost surely and we compute the asymptotic behavior of $v_L$ as $L$ tends to infinity: $v_L = \sqrt{2} - π^2 / 2 \sqrt{2} L^2 + o(1/L^2)$, as conjectured by Brunet, Derrida, Mueller and Munier. The proof makes use of results by Berestycki, Berestycki and Schweinsberg concerning branching Brownian motion in a strip.

math.PR↗