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Hubert Lacoin

Publications and source records attributed to Hubert Lacoin.

At least 19 recordsLinked to original sources
arXiv2026

Local limit Theorem for Directed polymers in a random environment in the full weak disorder regime

For directed polymers in random environment, we provide a short and mostly self-contained proof of the local limit theorem in the full weak disorder phase, including the critical inverse temperature $β_c$. That is, the point-to-point partition function is well approximated by the product of two independent point-to-line partition functions (one starting from each endpoint of the system) multiplied by the simple random walk transition kernel. The approximation is valid in the $L^p$ sense for an optimal range of $p$. The result is new at $β=β_c$ and for $p>2$, and in all cases the proof is significantly simpler, technically and conceptually, than what can be found in the literature. As a consequence of the result we establish an invariance principle for the polymer measure with fixed end-point, valid for all $β\in [0,β_c]$.

math.PR↗
arXiv2026

The localization transition for the directed polymer in a random environment is smooth

When $d\ge 3$, the directed polymer a in random environment on $\mathbb Z^d$ is known to display a phase transition from a diffusive phase, known as \textit{weak disorder} to a localized phase, referred to as \textit{strong disorder}. This transition is encoded by the behavior of the the free energy of the model, defined by $$\mathfrak f(β):=\lim_{N\to \infty} (1/n)\log W^β_n$$ where $W^β_n$ is the normalized partition function for the directed polymer of length $n$. More precisely weak disorder corresponds to $\mathfrak f(β)=0$ and strong disorder to $\mathfrak f(β)<0$. Monotonicity and continuity of $\mathfrak f$ imply that there exists $β_c\in [0,\infty]$ such that weak disorder is equivalent to $β\in [0,β_c]$. Furthermore $β_c>0$ if and only if $d\ge 3$. We prove that this transition is infinitely smooth in the sense that $\mathfrak f$ grows slower than any power function at the vicinity of $β_c$, that is $$ \lim_{β\downarrow β_c }\frac{\log |\mathfrak f(β)|}{\log (β-β_c)}=\infty.$$

math.PR↗
arXiv2025

Strong disorder and very strong disorder are equivalent for directed polymers

We show that if the normalized partition function $W^β_n$ of the directed polymer model on $\mathbb Z^d$ converges to zero, then it does so exponentially fast. This implies that there exists a critical value $β_c$ for the inverse temperature such that the normalized partition function has a non-degenerate limit for all $β\in [0,β_c]$ -- weak disorder holds -- while for $β\in (β_c,\infty)$ it converges exponentially fast to zero -- very strong disorder holds. This solves a twenty-years-old conjecture formulated by Comets, Yoshida, Carmona and Hu. Our proof requires a technical assumption on the environment, namely, that it is bounded from above.

math.PR↗
arXiv2025

Coincidence of critical points for directed polymers for general environments and random walks

For the directed polymer in a random environment (DPRE), two critical inverse-temperatures can be defined. The first one, $β_c$, separates the strong disorder regime (in which the normalized partition function $W^β_n$ tends to zero) from the weak disorder regime (in which $W^β_n$ converges to a nontrivial limit). The other, $\bar β_c$, delimits the very strong disorder regime (in which $W^β_n$ converges to zero exponentially fast). It was proved previously that $β_c=\bar β_c$ when the random environment is upper-bounded for the DPRE based on the simple random walk. We extend this result to general environment and arbitrary reference walk. We also prove that $β_c=0$ if and only the $L^2$-critical point is trivial.

math.PR↗
arXiv2025

The Random Walk Pinning Model II: Upper bounds on the free energy and disorder relevance

This article investigates the question of disorder relevance for the continuous-time Random Walk Pinning Model (RWPM) and completes the results of our companion paper. The RWPM considers a continuous time random walk $X=(X_t)_{t\geq 0}$, whose law is modified by a Gibbs weight given by $\exp(β\int_0^T \mathbf{1}_{\{X_t=Y_t\}} dt)$, where $Y=(Y_t)_{t\geq 0}$ is a quenched trajectory of a second (independent) random walk and $β\geq 0$ is the inverse temperature. The random walk $Y$ has the same distribution as $X$ but a jump rate $ρ\geq 0$, interpreted as the disorder intensity. For fixed $ρ\ge 0$, the RWPM undergoes a localization phase transition as $β$ crosses a critical threshold $β_c(ρ)$. The question of disorder relevance then consists in determining whether a disorder of arbitrarily small intensity $ρ$ changes the properties of the phase transition. We focus our analysis on the case of transient $γ$-stable walks on $\mathbb{Z}$, i.e. random walks in the domain of attraction of a $γ$-stable law, with $γ\in (0,1)$. In the present paper, we show that disorder is relevant when $γ\in (0,\frac23]$, namely that $β_c(ρ)>β_c(0)$ for every $ρ>0$. We also provide lower bounds on the critical point shift, which are matching the upper bounds obtained in our companion paper. Interestingly, in the marginal case $γ= \frac23$, disorder is always relevant, independently of the fine properties of the random walk distribution. When $γ\in (\frac23,1)$, our companion paper proves that disorder is irrelevant (in particular $β_c(ρ)=β_c(0)$ for $ρ$ small enough). We provide here an upper bound on the free energy in the regime $γ\in (\frac 2 3,1)$ that highlights the fact that although disorder is irrelevant, it still has a non-trivial effect on the phase transition, at any $ρ>0$.

math.PR↗
arXiv2025

The Random Walk Pinning Model I: Lower bounds on the free energy and disorder irrelevance

The Random Walk Pinning Model (RWPM) is a statistical mechanics model in which the trajectory of a continuous time random walk $X=(X_t)_{t\geq 0}$ is rewarded according to the time it spends together with a moving catalyst. More specifically for a system of size $T$, the law of $X$ is tilted by the Gibbs factor $\exp(β\int_0^T \mathbf{1}_{\{X_t=Y_t\}} dt)$, where $β\geq 0$ is the inverse temperature. The moving catalyst $Y=(Y_t)_{t\ge 0}$ is given by the quenched trajectory of a second continuous-time random walk, with the same distribution as $X$ but a different jump rate $ρ\geq 0$, interpreted as the disorder intensity. For fixed $ρ\ge 0$, the RWPM undergoes a localization phase transition when $β$ passes a critical value $β_c(ρ)$. We thoroughly investigate the question of disorder relevance to determine whether a disorder of arbitrarily small intensity affects the features of the phase transition. We focus our analysis on the case of transient $γ$-stable walks on $\mathbb{Z}$, i.e. random walks in the domain of attraction of a $γ$-stable law, with $γ\in (0,1)$. In the present paper, we derive lower bounds for the free energy, which results in either a proof of disorder irrelevance or upper bounds on the critical point shift. More precisely, when $γ\in(\frac23,1)$, our estimates imply that that $β_c(ρ)=β_c(0)$ and $ρ$ is small, showing disorder irrelevance. When $γ\in (0,\frac23]$ our companion paper shows that $β_c(ρ)>β_c(0)$ for every $ρ>0$, showing disorder relevance: we derive here upper bounds on the critical point shift, which are matching the lower bounds obtained in our companion paper. For good measure, our analysis also includes the case of the simple random walk of $\mathbb{Z}^d$ (for $d\ge 3$) for which no upper bound on the critical point shift was previously known.

math.PR↗
arXiv2025

The tail distribution of the partition function for directed polymer in the weak disorder phase

We investigate the upper tail distribution of the partition function of the directed polymer in a random environment on $\mathbb Z^d$ in the weak disorder phase. We show that the distribution of the infinite volume partition function $W^β_{\infty}$ displays a power-law decay, with an exponent $p^*(β)\in [1+\frac{2}{d},\infty)$. We also prove that the distribution of the suprema of the point-to-point and point-to-line partition functions display the same behavior. On the way to these results, we prove a technical estimate of independent interest: the $L^p$-norm of the partition function at the time when it overshoots a high value $A$ is comparable to $A$. We use this estimate to extend the validity of many recent results that were proved under the assumption that the environment is upper bounded.

math.PR↗
arXiv2024

Cutoff phenomenon in nonlinear recombinations

We investigate a quadratic dynamical system known as nonlinear recombinations. This system models the evolution of a probability measure over the Boolean cube, converging to the stationary state obtained as the product of the initial marginals. Our main result reveals a cutoff phenomenon for the total variation distance in both discrete and continuous time. Additionally, we derive the explicit cutoff profiles in the case of monochromatic initial distributions. These profiles are different in the discrete and continuous time settings. The proof leverages a pathwise representation of the solution in terms of a fragmentation process associated to a binary tree. In continuous time, the underlying binary tree is given by a branching random process, thus requiring a more elaborate probabilistic analysis.

math.PR↗
arXiv2023

Critical Gaussian Multiplicative Chaos for singular measures

Given $d\ge 1$, we provide a construction of the random measure - the critical Gaussian Multiplicative Chaos - formally defined $e^{\sqrt{2d}X}\mathrm{d} μ$ where $X$ is a $\log$-correlated Gaussian field and $μ$ is a locally finite measure on $\mathbb R^d$. Our construction generalizes the one performed in the case where $μ$ is the Lebesgue measure. It requires that the measure $μ$ is sufficiently spread out, namely that for $μ$ almost every $x$ we have $$ \int_{B(0,1)}\frac{μ(\mathrm{d} y)}{|x-y|^{d}e^{ρ\left(\log \frac{1}{|x-y|} \right)}}<\infty, $$ for any compact set where $ρ:\mathbb R_+\to \mathbb R_+$ can be chosen to be any lower envelope function for the $3$-Bessel process (this includes $ρ(x)=x^α$ with $α\in (0,1/2)$). We prove that three distinct random objects converge to a common limit which defines the critical GMC: the derivative martingale, the critical martingale, and the exponential of the mollified field. We also show that the above criterion for the measure $μ$ is in a sense optimal.

math.PR↗
arXiv2023

Convergence for Complex Gaussian Multiplicative Chaos on phase boundaries

The complex Gaussian Multiplicative Chaos (or complex GMC) is informally defined as a random measure $e^{γX} \mathrm{d} x$ where $X$ is a log correlated Gaussian field on $\mathbb R^d$ and $γ=α+iβ$ is a complex parameter. The correlation function of $X$ is of the form $$ K(x,y)= \log \frac{1}{|x-y|}+ L(x,y),$$ where $L$ is a continuous function. In the present paper, we consider the cases $γ\in \mathcal P_{\mathrm{I/II}}$ and $γ\in \mathcal{P}'_{\mathrm{II/III}}$ where $$ \mathcal P_{\mathrm{I/II}}:= \{ α+i β\ : α,β\in \mathbb R \ ; |α|>|β| \ ; \ |α|+|β|=\sqrt{2d} \}, $$ and $$ \mathcal{P}'_{\mathrm{II/III}}:= \{ α+i β\ : α,β\in \mathbb R \ ; \ |α|= \sqrt{d/2} \ ; \ |β|>\sqrt{2d} \},$$ We prove that if $X$ is replaced by an approximation $X_ε$ obtained via mollification, then $e^{γX_ε} \mathrm{d} x$, when properly rescaled, converges when $ε\to 0$. The limit does not depend on the mollification kernel. When $γ\in \mathcal P_{\mathrm{I/II}}$, the convergence holds in probability and in $L^p$ for some value of $p\in [1,\sqrt{2d}/α)$. When $γ\in \mathcal{P}'_{\mathrm{II/III}}$ the convergence holds only in law. In this latter case, the limit can be described a complex Gaussian white noise with a random intensity given by a critical real GMC. The regions $\mathcal P_{\mathrm{I/II}}$ and $ \mathcal{P}'_{\mathrm{II/III}}$ correspond to phase boundary between the three different regions of the complex GMC phase diagram. These results complete previous results obtained for the GMC in phase I and III and only leave as an open problem the question of convergence in phase II.

math.PR↗
arXiv2022

Critical Gaussian Multiplicative Chaos revisited

We present new, short and self-contained proofs of the convergence (with an adequate renormalization) of four different sequences to the critical Gaussian Multiplicative Chaos:(a) the derivative martingale (b) the critical martingale (c) the exponential of the mollified field (d) the subcritical Gaussian Multiplicative Chaos.

math.PR↗
arXiv2021

The continuum directed polymer in Lévy Noise

We present in this paper the construction of a continuum directed polymer model in an environment given by space-time Lévy noise. One of the main objectives of this construction is to describe the scaling limit of discrete directed polymer in an heavy-tail environment and for this reason we put special emphasis on the case of $α$-stable noises with $α\in (1,2)$. Our construction can be performed in arbitrary dimension, provided that the Lévy measure satisfies specific (and dimension dependent) conditions. We also discuss a few basic properties of the continuum polymer and the relation between this model and the Stochastic Heat Equation with multiplicative Lévy noise.

math.PR↗
arXiv2021

The stochastic heat equation with multiplicative Lévy noise: Existence, moments, and intermittency

We study the stochastic heat equation (SHE) $\partial_t u = \frac12 Δu + βu ξ$ driven by a multiplicative Lévy noise $ξ$ with positive jumps and amplitude $β>0$, in arbitrary dimension $d\geq 1$. We prove the existence of solutions under an optimal condition if $d=1,2$ and a close-to-optimal condition if $d\geq3$. Under an assumption that is general enough to include stable noises, we further prove that the solution is unique. By establishing tight moment bounds on the multiple Lévy integrals arising in the chaos decomposition of $u$, we further show that the solution has finite $p$th moments for $p>0$ whenever the noise does. Finally, for any $p>0$, we derive upper and lower bounds on the moment Lyapunov exponents of order $p$ of the solution, which are asymptotically sharp in the limit as $β\to0$. One of our most striking findings is that the solution to the SHE exhibits a property called strong intermittency (which implies moment intermittency of all orders $p>1$ and pathwise mass concentration of the solution), for any non-trivial Lévy measure, at any disorder intensity $β>0$, in any dimension $d\geq1$.

math.PR↗
arXiv2021

Mixing time and cutoff for one dimensional particle systems

We survey recent results concerning the total-variation mixing time of the simple exclusion process on the segment (symmetric and asymmetric) and a continuum analog, the simple random walk on the simplex with an emphasis on cutoff results. A Markov chain is said to exhibit cutoff if on a certain time scale, the distance to equilibrium drops abruptly from $1$ to $0$. We also review a couple of techniques used to obtain these results by exposing and commenting some elements of proof.

math.PR↗
arXiv2021

The scaling limit of the directed polymer with power-law tail disorder

In this paper, we study the so-called intermediate disorder regime for a directed polymer in a random environment with heavy-tail. Consider a simple symmetric random walk $(S_n)_{n\geq 0}$ on $\mathbb{Z}^d$, with $d\geq 1$, and modify its law using Gibbs weights in the product form $\prod_{n=1}^{N} (1+βη_{n,S_n})$, where $(η_{n,x})_{n\ge 0, x\in \mathbb{Z}^d}$ is a field of i.i.d. random variables whose distribution satisfies $\mathbb{P}(η>z) \sim z^{-α}$ as $z\to\infty$, for some $α\in(0,2)$. We prove that if $α< \min(1+\frac{2}{d},2)$, when sending $N$ to infinity and rescaling the disorder intensity by taking $β=β_N \sim N^{-γ}$ with $γ=\frac{d}{2α}(1+\frac{2}{d}-α)$, the distribution of the trajectory under diffusive scaling converges in law towards a random limit, which is the continuum polymer with Lévy $α$-stable noise constructed in the companion paper arXiv:2007.06484.

math.PR↗
arXiv2021

Mixing time for the asymmetric simple exclusion process in a random environment

We consider the simple exclusion process in the integer segment $ [1, N]$ with $k\le N/2$ particles and spatially inhomogenous jumping rates. A particle at site $x\in [ 1, N]$ jumps to site $x-1$ (if $x\ge 2$) at rate $1-ω_x$ and to site $x+1$ (if $x \le N-1$) at rate $ω_x$ if the target site is not occupied. The sequence $ω=(ω_x)_{ x \in \mathbb{Z}}$ is chosen by IID sampling from a probability law whose support is bounded away from zero and one (in other words the random environment satisfies the uniform ellipticity condition). We further assume $\mathbb{E}[ \log ρ_1 ]<0$ where $ρ_1:= (1-ω_1)/ω_1$, which implies that our particles have a tendency to move to the right. We prove that the mixing time of the exclusion process in this setup grows like a power of $N$. More precisely, for the exclusion process with $N^{β+o(1)}$ particles where $β\in [0,1)$, we have in the large $N$ asymptotic $$ N^{\max\left(1,\frac {1}λ, β+ \frac 1 {2λ}\right)+o(1)} \le t_{\mathrm{Mix}}^{N,k} \le N^{C+o(1)}$$ where $λ>0$ is such that $\mathbb{E}[ρ_1^λ]=1$ ($λ=\infty$ if the equation has no positive root) and $C$ is a constant which depends on the distribution of $ω$. We conjecture that our lower bound is sharp up to sub-polynomial correction.

math.PR↗
arXiv2020

Convergence in law for Complex Gaussian Multiplicative Chaos in phase III

Gaussian Multiplicative Chaos (GMC) is informally defined as a random measure $e^{γX} \mathrm{d} x$ where $X$ is Gaussian field on $\mathbb R^d$ (or an open subset of it) whose correlation function is of the form $ K(x,y)= \log \frac{1}{|y-x|}+ L(x,y),$ where $L$ is a continuous function $x$ and $y$ and $γ=α+iβ$ is a complex parameter. In the present paper, we consider the case $γ\in \mathcal P'_{\mathrm{III}}$ where $$ \mathcal P'_{\mathrm{III}}:= \{ α+i β\ : α,γ\in \mathbb R , \ |α|<\sqrt{d/2}, \ α^2+β^2\ge d \}.$$ We prove that if $X$ is replaced by the approximation $X_\varepsilon$ obtained by convolution with a smooth kernel, then $e^{γX_\varepsilon} \mathrm d x$, when properly rescaled, has an explicit non-trivial limit in distribution when $\varepsilon$ goes to zero. This limit does not depend on the specific convolution kernel which is used to define $X_{\varepsilon}$ and can be described as a complex Gaussian white noise with a random intensity given by a real GMC associated with parameter $2α$.

math.PR↗