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Hubert Lacoin

Publications and source records attributed to Hubert Lacoin.

At least 19 recordsLinked to original sources
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 $\exp(\beta \int_0^T \mathbf{1}_{\{X_t=Y_t\}} dt)$, where $Y=(Y_t)_{t\geq 0}$ is a quenched trajectory of a second (independent) random walk and $\beta \geq 0$ is the inverse temperature. The random walk $Y$ has the same distribution as $X$ but a jump rate $\rho \geq 0$, interpreted as the disorder intensity. For fixed $\rho\ge 0$, the RWPM undergoes a localization phase transition as $\beta$ crosses a critical threshold $\beta_c(\rho)$. The question of disorder relevance then consists in determining whether a disorder of arbitrarily small intensity $\rho$ changes the properties of the phase transition. We focus our analysis on the case of transient $\gamma$-stable walks on $\mathbb{Z}$, i.e. random walks in the domain of attraction of a $\gamma$-stable law, with $\gamma\in (0,1)$. In the present paper, we show that disorder is relevant when $\gamma \in (0,\frac23]$, namely that $\beta_c(\rho)>\beta_c(0)$ for every $\rho>0$. 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 $\gamma = \frac23$, disorder is always relevant, independently of the fine properties of the random walk distribution. When $\gamma \in (\frac23,1)$, our companion paper proves that disorder is irrelevant (in particular $\beta_c(\rho)=\beta_c(0)$ for $\rho$ small enough). We provide here an upper bound on the free energy in the regime $\gamma\in (\frac 2 3,1)$ that highlights the fact that although disorder is irrelevant, it still has a non-trivial effect on the phase transition, at any $\rho>0$.

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 $\exp(\beta \int_0^T \mathbf{1}_{\{X_t=Y_t\}} dt)$, where $\beta \geq 0$ 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 $\rho\geq 0$, interpreted as the disorder intensity. For fixed $\rho\ge 0$, the RWPM undergoes a localization phase transition when $\beta$ passes a critical value $\beta_c(\rho)$. 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 $\gamma$-stable walks on $\mathbb{Z}$, i.e. random walks in the domain of attraction of a $\gamma$-stable law, with $\gamma\in (0,1)$. 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 $\gamma \in(\frac23,1)$, our estimates imply that that $\beta_c(\rho)=\beta_c(0)$ and $\rho$ is small, showing disorder irrelevance. When $\gamma\in (0,\frac23]$ our companion paper shows that $\beta_c(\rho)>\beta_c(0)$ for every $\rho>0$, 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 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 $$\mathfrak f(\beta):=\lim_{N\to \infty} (1/n)\log W^{\beta}_n$$ where $W^{\beta}_n$ is the normalized partition function for the directed polymer of length $n$. More precisely weak disorder corresponds to $\mathfrak f(\beta)=0$ and strong disorder to $\mathfrak f(\beta)<0$. Monotonicity and continuity of $\mathfrak f$ implies that there exists $\beta_c\in [0,\infty]$ such that weak disorder is equivalent to $\beta\in [0,\beta_c]$. Furthermore $\beta_c>0$ 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 $\beta_c$, that is $$ \lim_{\beta \downarrow \beta_c }\frac{\log |\mathfrak f(\beta)|}{\log (\beta-\beta_c)}=\infty.$$

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, $\beta_c$, separates the strong disorder regime (in which the normalized partition function $W^{\beta}_n$ tends to zero) from the weak disorder regime (in which $W^{\beta}_n$ converges to a nontrivial limit). The other, $\bar \beta_c$, delimits the very strong disorder regime (in which $W^{\beta}_n$ converges to zero exponentially fast). It was proved previously that $\beta_c=\bar \beta_c$ 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 $\beta_c=0$ if and only the $L^2$-critical point is trivial.

math.PR↗
arXiv2024

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 $W^{\beta}_{\infty}$ displays a power-law decay, with an exponent $p^*(\beta)\in [1+\frac{2}{d},\infty)$. 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↗
arXiv2024

Strong disorder and very strong disorder are equivalent for directed polymers

We show that if the normalized partition function $W^{\beta}_n$ 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 $\beta_c$ for the inverse temperature such that the normalized partition function has a non-degenerate limit for all $\beta\in [0,\beta_c]$ -- weak disorder holds -- while for $\beta\in (\beta_c,\infty)$ 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↗
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 $e^{\sqrt{2d}X}\mathrm{d} \mu$ where $X$ is a $\log$-correlated Gaussian field and $\mu$ is a locally finite measure on $\mathbb R^d$. Our construction generalizes the one performed in the case where $\mu$ is the Lebesgue measure. It requires that the measure $\mu$ is sufficiently spread out, namely that for $\mu$ almost every $x$ we have $$ \int_{B(0,1)}\frac{\mu(\mathrm{d} y)}{|x-y|^{d}e^{\rho\left(\log \frac{1}{|x-y|} \right)}}<\infty, $$ for any compact set where $\rho:\mathbb R_+\to \mathbb R_+$ can be chosen to be any lower envelope function for the $3$-Bessel process (this includes $\rho(x)=x^{\alpha}$ with $\alpha\in (0,1/2)$). 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 $\mu$ 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 $e^{\gamma X} \mathrm{d} x$ where $X$ is a log correlated Gaussian field on $\mathbb R^d$ and $\gamma=\alpha+i\beta$ 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 $\gamma\in \mathcal P_{\mathrm{I/II}}$ and $\gamma\in \mathcal{P}'_{\mathrm{II/III}}$ where $$ \mathcal P_{\mathrm{I/II}}:= \{ \alpha+i \beta \ : \alpha,\beta \in \mathbb R \ ; |\alpha|>|\beta| \ ; \ |\alpha|+|\beta|=\sqrt{2d} \}, $$ and $$ \mathcal{P}'_{\mathrm{II/III}}:= \{ \alpha+i \beta \ : \alpha,\beta \in \mathbb R \ ; \ |\alpha|= \sqrt{d/2} \ ; \ |\beta|>\sqrt{2d} \},$$ We prove that if $X$ is replaced by an approximation $X_\epsilon$ obtained via mollification, then $e^{\gamma X_\epsilon} \mathrm{d} x$, when properly rescaled, converges when $\epsilon\to 0$. The limit does not depend on the mollification kernel. When $\gamma\in \mathcal P_{\mathrm{I/II}}$, the convergence holds in probability and in $L^p$ for some value of $p\in [1,\sqrt{2d}/\alpha)$. When $\gamma\in \mathcal{P}'_{\mathrm{II/III}}$ 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 stochastic heat equation with multiplicative L\'evy noise: Existence, moments, and intermittency

We study the stochastic heat equation (SHE) $\partial_t u = \frac12 \Delta u + \beta u \xi$ driven by a multiplicative L\'evy noise $\xi$ with positive jumps and amplitude $\beta>0$, 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 L\'evy 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 $\beta\to0$. 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 L\'evy measure, at any disorder intensity $\beta>0$, 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

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 $1-\omega_x$ and to site $x+1$ (if $x \le N-1$) at rate $\omega_x$ if the target site is not occupied. The sequence $\omega=(\omega_x)_{ x \in \mathbb{Z}}$ 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 $\mathbb{E}[ \log \rho_1 ]<0$ where $\rho_1:= (1-\omega_1)/\omega_1$, 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 $N^{\beta+o(1)}$ particles where $\beta\in [0,1)$, we have in the large $N$ asymptotic $$ N^{\max\left(1,\frac {1}{\lambda}, \beta+ \frac 1 {2\lambda}\right)+o(1)} \le t_{\mathrm{Mix}}^{N,k} \le N^{C+o(1)}$$ where $\lambda>0$ is such that $\mathbb{E}[\rho_1^{\lambda}]=1$ ($\lambda=\infty$ if the equation has no positive root) and $C$ is a constant which depends on the distribution of $\omega$. 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↗
arXiv2020

A universality result for subcritical Complex Gaussian Multiplicative Chaos

In the present paper, we show that (under some minor technical assumption) Complex Gaussian Multiplicative Chaos defined as the complex exponential of a $\log$-correlated Gaussian field can be obtained by taking the limit of the exponential of the field convoluted with a smoothing Kernel. We consider two types of chaos: $e^{γX}$ for a log correlated field $X$ and $γ=α+iβ$, $α, β\in \mathbb R$ and $e^{αX+iβY}$ for $X$ and $Y$ two independent fields with $α, β\in \mathbb R$. Our result is valid in the range $$ \mathcal O_{\mathrm{sub}}:=\{ α^2+β^2<d \} \cup \{ |α|\in (\sqrt{d/2},\sqrt{2d} ) \text{ and } |β|< \sqrt{2d}-|α| \},$$ which, up to boundary, is conjectured to be optimal.

math.PR↗
arXiv2020

Solid-On-Solid interfaces with disordered pinning

We investigate the localization transition for a simple model of interface which interacts with an inhomonegeous defect plane. The interface is modeled by the graph of a function $ϕ: \mathbb Z^2 \to \mathbb Z$,and the disorder is given by a fixed realization of a field of IID centered random variables$(ω_x)_{x\in \mathbb Z^2}$. The Hamiltonian of the system depends on three parameters $α,β>0$ and $h\in \mathbb R$ which determine respectively the intensity of nearest neighbor interaction the amplitude of disorder and the mean value of the interaction with the substrate, and is given by the expression $$\mathcal H(ϕ):= β\sum_{x\sim y} |ϕ(x)-ϕ(y)|- \sum_{x} (αω_x+h){\bf 1}_{\{ϕ(x)=0\}}.$$ We focus on the large-$β$/rigid phase phase of the Solid-On-Solid (SOS) model. In that regime, we provide a sharp description of the phase transition in $h$ from a localized phase to a delocalized one corresponding respectivelly to a positive and vanishing fraction of points with $ϕ(x)=0$. We prove that the critical value for $h$ corresponds to that of the annealed model and is given by $h_c(α)= -\log \mathbb E[e^{αω}]$, and that near the critical point, the free energy displays the following critical behavior $$F_β(α,h_c+u )\stackrel{u\to 0+}{\sim} \max_{n\ge 1} \left\{θ_1 e^{-4βn} u- \frac{1}{2}θ^2_1 e^{-8βn} \frac{\mathrm{Var}\left[e^{αω}\right]}{\mathbb E \left[ e^{αω} \right]^2}\right\}.$$ The positive constant $θ_1(β)>0$ is defined by the asymptotic probability of spikes for the infinite volume SOS with $0$ boundary condition $θ_1(β):=\lim_{n\to \infty} e^{4βn}\mathbf P_β (ϕ({\bf 0})=n)$ ...

math.PR↗
arXiv2020

Mixing time of the adjacent walk on the simplex

By viewing the $N$-simplex as the set of positions of $N-1$ ordered particles on the unit interval, the adjacent walk is the continuous time Markov chain obtained by updating independently at rate 1 the position of each particle with a sample from the uniform distribution over the interval given by the two particles adjacent to it. We determine its spectral gap and prove that both the total variation distance and the separation distance to the uniform distribution exhibit a cutoff phenomenon, with mixing times that differ by a factor $2$. The results are extended to the family of log-concave distributions obtained by replacing the uniform sampling by a symmetric log-concave Beta distribution.

math.PR↗