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Clément Erignoux

Publications and source records attributed to Clément Erignoux.

At least 19 recordsLinked to original sources
arXiv2026

Sharp freezing time estimates for the subcritical Facilitated Exclusion Process

We investigate the exact transience time of the Facilitated Exclusion Process (FEP) on the one-dimensional torus with $N$ sites. The FEP exhibits an active/inactive phase transition at critical density $1/2$, such that in the subcritical density regime $(0,1/2)$, it becomes frozen after a finite time period -- the transience time or freezing time. We first show that for the FEP starting from a Bernoulli product measure of marginal density $ρ\in (0,1/2)$, the transience time has exactly the scale of $Θ(\log^3 N)$. Secondly, we prove that in the near-critical case $ρ\simeq 1/2 - N^{-α}$ for $α\in (0,1)$, the transience time is polynomial and has a scale of $N^{1 \wedge (2α)}$. The key idea is to estimate the typical size of locally supercritical intervals of the initial distribution, which has order $\log N$ in the subcritical case and $N^{1 \wedge (2α)}$ in the near-critical case. In the subcritical case this is enough, whereas in the near-critical case we need additional dynamical decorrelation inequalities to apply this static result to estimate the freezing time.

math.PR↗
arXiv2025

Scaling Relations For The CLG's Critical Exponents

We consider, in any dimension, the constrained lattice gas introduced by Rossi et al., which is an exclusion process on a d-dimensional lattice following the additional constraint that only particles with at least one occupied neighbour can jump. In dimension d > 2, this model features self-organized criticality at some critical density of particles. Numerical simulations predict the existence of scaling exponents close to criticality, and several relations can be derived between these exponents. The goal of this article is to give a mathematical framework for these relations, which have been numerically established in a companion article.

math-ph↗
arXiv2025

Velocity trapping in the lifted TASEP and the true self-avoiding random walk

We discuss non-reversible Markov-chain Monte Carlo algorithms that, for particle systems, rigorously sample the positional Boltzmann distribution and that have faster than physical dynamics. These algorithms all feature a non-thermal velocity distribution. They are exemplified by the lifted TASEP (totally asymmetric simple exclusion process), a one-dimensional lattice reduction of event-chain Monte Carlo. We analyze its dynamics in terms of a velocity trapping that arises from correlations between the local density and the particle velocities. This allows us to formulate a conjecture for its out-of-equilibrium mixing time scale, and to rationalize its equilibrium superdiffusive time scale. Both scales are faster than for the (unlifted) TASEP. They are further justified by our analysis of the lifted TASEP in terms of many-particle realizations of true self-avoiding random walks. We discuss velocity trapping beyond the case of one-dimensional lattice models and in more than one physical dimensions. Possible applications beyond physics are pointed out.

cond-mat.stat-mech↗
arXiv2025

Asymmetric attractive zero-range processes with particle destruction at the origin

We investigate the macroscopic behavior of asymmetric attractive zero-range processes on $\mathbb{Z}$ where particles are destroyed at the origin at a rate of order $N^β$, where $β\in \mathbb{R}$ and $N\in\mathbb{N}$ is the scaling parameter. We prove that the hydrodynamic limit of this particle system is described by the unique entropy solution of a hyperbolic conservation law, supplemented by a boundary condition depending on the range of $β$. Namely, if $β\geqslant 0$, then the boundary condition prescribes the particle current through the origin, whereas if $β<0$, the destruction of particles at the origin has no macroscopic effect on the system and no boundary condition is imposed at the hydrodynamic limit.

math.PR↗
arXiv2025

Mapping hydrodynamics for the facilitated exclusion and zero-range processes

We derive the hydrodynamic limit for two degenerate lattice gases, the \emph{facilitated exclusion process} (FEP) and the \emph{facilitated zero-range process} (FZRP), both in the symmetric and the asymmetric case. For both processes, the hydrodynamic limit in the symmetric case takes the form of a diffusive Stefan problem, whereas the asymmetric case is characterized by a hyperbolic Stefan problem. Although the FZRP is attractive, a property that we extensively use to derive its hydrodynamic limits in both cases, the FEP is not. To derive the hydrodynamic limit for the latter, we exploit that of the zero-range process, together with a classical mapping between exclusion and zero-range processes, both at the microscopic and macroscopic level. Due to the degeneracy of both processes, the asymmetric case is a new result, but our work also provides a simpler proof than the one that was previously proposed for the FEP in the symmetric case in \cite{blondel2021stefan}.

math.PR↗
arXiv2025

Hydrodynamic behavior near dynamical criticality of a facilitated conservative lattice gas

We investigate a $2d$-conservative lattice gas exhibiting a dynamical active-absorbing phase transition with critical density $ρ_c$. We derive the hydrodynamic equation for this model, showing that all critical exponents governing the large scale behavior near criticality can be obtained from two independent ones. We show that as the supercritical density approaches criticality, distinct length scales naturally appear. Remarkably, this behavior is different from the subcritical one. Numerical simulations support the critical relations and the scale separation.

cond-mat.stat-mech↗
arXiv2024

Hydrodynamic limit for an open facilitated exclusion process with slow and fast boundaries

We study the symmetric facilitated exclusion process (FEP) on the finite one-dimensional lattice $\lbrace 1,\dots ,N-1\rbrace$ when put in contact with boundary reservoirs, whose action is subject to an additional kinetic constraint in order to enforce ergodicity, and whose speed is of order $N^{-θ}$ for some parameter $θ$. We derive its hydrodynamic limit as $N\to\infty$, in the diffusive space-time scaling, when the initial density profile is supercritical. More precisely, the macroscopic density of particles evolves in the bulk according to a fast diffusion equation as in the periodic case, which is now subject to boundary conditions that can be of Dirichlet, Robin or Neumann type depending on the parameter $θ$. In the Dirichlet case, the FEP exhibits a very peculiar behaviour: unlike for the classical SSEP, and due to the two-phased nature of FEP, the reservoirs impose boundary densities which do not coincide with their equilibrium densities. The proof is based on the classical entropy method, but requires significant adaptations to account for the FEP's non-product stationary states and to deal with the non-equilibrium setting.

math.PR↗
arXiv2024

Cutoff for the transience and mixing time of a SSEP with traps and consequences on the FEP

We introduce a new particle system that we call the SSEP with traps, which is non reversible, attractive, and has a transient regime. We study its \emph{transience time} $θ_K$, meaning the time after which the system is no longer in a transient state with high probability, on the ring with $K$ sites. We first show that $θ_K$ is of order $K^2 \log K$ for a system of size $K$, and more precisely that it exhibits a cutoff at time $\frac{1}{π^2} K^2 \log K$. We then show that its \emph{mixing time} also undergoes cutoff at the same time. We further define a new mapping between the SSEP with traps and the Facilitated Exclusion Process (FEP) which has attracted significant scrutiny in recent years. We expect that this mapping will be a very useful tool to study the FEP's microscopic and macroscopic behaviour. In particular, using this mapping, we show that the FEP's transience time also undergoes a cutoff at time $\frac{1}{4 π^2} N^2 \log N$. Notably, our results show that for a FEP with particle density strictly greater than $\frac12$, the transient component is exited in a diffusive time. This allows to extend the upper-bound from [Ayre Chleboun 2024] for the mixing time of the FEP with particle density $ρ> 1/2$.

math.PR↗
arXiv2023

Stationary fluctuations for the facilitated exclusion process

We derive the stationary fluctuations for the Facilitated Exclusion Process (FEP) in one dimension in the symmetric, weakly asymmetric and asymmetric cases. Our proof relies on the mapping between the FEP and the zero-range process, and extends the strategy in \cite{erignoux2022mapping}, where hydrodynamic limits were derived for the FEP, to its stationary fluctuations. Our results thus exploit works on the zero-range process's fluctuations \cite{gonccalves2010equilibrium,gonccalves2015stochastic}, but we also provide a direct proof in the symmetric case, for which we derive a sharp estimate on the equivalence of ensembles for the FEP's stationary states.

math.PR↗
arXiv2021

On the hydrodynamics of active matter models on a lattice

Active matter has been widely studied in recent years because of its rich phenomenology, whose mathematical understanding is still partial. We present some results, based on [8, 17] linking microscopic lattice gases to their macroscopic limit, and explore how the mathematical state of the art allows to derive from various types of microscopic dynamics their hydrodynamic limit. We present some of the crucial aspects of this theory when applied to weakly asymmetric active models. We comment on the specific challenges one should consider when designing an active lattice gas, and in particular underline mathematical and phenomenological differences between gradient and non-gradient models. Our purpose is to provide the physics community, as well as member of the mathematical community not specialized in the mathematical derivation of scaling limits of lattice gases, some key elements in defining microscopic models and deriving their hydrodynamic limit.

math-ph↗
arXiv2020

Stefan problem for a non-ergodic facilitated exclusion process

We consider the facilitated exclusion process, which is a nonergodic, kinetically constrained exclusion process. We show that in the hydrodynamic limit, its macroscopic behavior is governed by a free boundary problem. The particles evolve on the one-dimensional lattice according to jump rates which are degenerate, since they can vanish on non-trivial configurations and create distinct phases: indeed, configurations can be totally blocked (they cannot evolve under the dynamics), ergodic (they belong to an irreducible component), or transient (after a transitive period of time they will become either blocked or ergodic). We additionally prove that the microscopic separation into blocked/ergodic phases fully coincides with the moving interface problem given by the hydrodynamic equation.

math.PR↗
arXiv2020

Hydrodynamics for SSEP with non-reversible slow boundary dynamics: Part I, the critical regime and beyond

The purpose of this article is to provide a simple proof of the hydrodynamic and hydrostatic behavior of the SSEP in contact with slowed reservoirs which inject and remove particles in a finite size windows at the extremities of the bulk. More precisely, the reservoirs inject/remove particles at/from any point of a window of size K placed at each extremity of the bulk and particles are injected/removed to the first open/occupied position in that window. The hydrodynamic limit is given by the heat equation with non-linear Robin boundary conditions or Neumann boundary conditions, the latter being in the case when the reservoirs are too slow. The proof goes through the entropy method of Guo, Papanicolaou and Varadhan. We also derive the hydrostatic limit for this model, whose proof is based on the method developed by Landim and Tsunoda. We observe that we do not make use of correlation estimates in none of our results.

math-ph↗
arXiv2020

Nematic first order phase transition for liquid crystals in the van der Waals-Kac limit

In this paper we revisit and extend some mathematical aspects of Onsager's theory of liquid crystals that have been investigated in recent years by different communities (statistical mechanics, analysis and probability). We introduce a model of anisotropic molecules with three-dimensional orientations interacting via a Kac-type interaction. We prove that, in the limit in which the range of the interaction is sent to infinity after the thermodynamic limit, the free energy tends to the infimum of an effective energy functional à la Onsager. We then prove that, if the spherical harmonic transform of the angular interaction has a negative minimum, this effective free energy functional displays a first order phase transition as the total density of the system increases.

math-ph↗
arXiv2020

Hydrodynamic limit for a facilitated exclusion process

We study the hydrodynamic limit for a periodic $1$-dimensional exclusion process with a dynamical constraint, which prevents a particle at site $x$ from jumping to site $x\pm1$ unless site $x\mp1$ is occupied. This process with degenerate jump rates admits transient states, which it eventually leaves to reach an ergodic component, assuming that the initial macroscopic density is larger than $\frac{1}{2}$, or one of its absorbing states if this is not the case. It belongs to the class of conserved lattice gases (CLG) which have been introduced in the physics literature as systems with active-absorbing phase transition in the presence of a conserved field. We show that, for initial profiles smooth enough and uniformly larger than the critical density $\frac{1}{2}$, the macroscopic density profile for our dynamics evolves under the diffusive time scaling according to a fast diffusion equation (FDE). The first step in the proof is to show that the system typically reaches an ergodic component in subdiffusive time.

math.PR↗
arXiv2020

Equilibrium fluctuations for the disordered harmonic chain perturbed by an energy conserving noise

We investigate the macroscopic behavior of the disordered harmonic chain of oscillators, through energy diffusion. The Hamiltonian dynamics of the system is perturbed by a degenerate conservative noise. After rescaling space and time diffusively, we prove that energy fluctuations in equilibrium evolve according to a linear heat equation. The diffusion coefficient is obtained from the non-gradient Varadhan's approach, and is equivalently defined through the Green-Kubo formula. Since the perturbation is very degenerate and the symmetric part of the generator does not have a spectral gap, the standard non-gradient method is reviewed under new perspectives.

math.PR↗
arXiv2020

Hydrodynamics for SSEP with non-reversible slow boundary dynamics: Part II, below the critical regime

The purpose of this article is to provide a simple proof of the hydrodynamic and hydrostatic behavior of the SSEP in contact with reservoirs which inject and remove particles in a finite size windows at the extremities of the bulk. More precisely, the reservoirs inject/remove particles at/from any point of a window of size $K$ placed at each extremity of the bulk and particles are injected/removed to the first open/occupied position in that window. The reservoirs have slow dynamics, in the sense that they intervene at speed $N^{-θ}$ w.r.t. the bulk dynamics. In the first part of this article, we treated the case $θ>1$ for which the entropy method can be adapted. We treat here the case where the boundary dynamics is too fast for the Entropy Method to apply. We prove, using duality estimates inspired by previous work by Erignoux, Landim and Xu, that the hydrodynamic limit is given by the heat equation with Dirichlet boundary conditions, where the density at the boundaries is fixed by the parameters of the model.

math.PR↗
arXiv2019

Hydrodynamic Limit For An Active Exclusion Process

Collective dynamics can be observed among many animal species, and have given rise in the last decades to an active and interdisciplinary field of study. Such behaviors are often modeled by active matter, in which each individual is self-driven and tends to update its velocity depending on the one of its neighbors. In a classical model introduced by Vicsek and al., as well as in numerous related active matter models, a phase transition between chaotic behavior at high temperature and global order at low temperature can be observed. Even though ample evidence of these phase transitions has been obtained for collective dynamics, from a mathematical standpoint, such active systems are not fully understood yet. Significant progress has been achieved in the recent years under an assumption of mean-field interactions, however to this day, few rigorous results have been obtained for models involving purely local interactions. In this paper, as a first step towards the mathematical understanding of active microscopic dynamics, we describe a lattice active particle system, in which particles interact locally to align their velocities. We obtain rigorously, using the formalism developed for hydrodynamic limits of lattice gases, the scaling limit of this out-of-equilibrium system. This article builds on the multi-type exclusion model introduced by Quastel by detailing his proof and incorporating several generalizations, adding significant technical and phenomenological difficulties.

math-ph↗