Consider a one-dimensional exclusion process with finite-range translation-invariant jump rates with non-zero drift. Let the process be stationary with product Bernoulli invariant distribution at density ρ. Place a second class particle initially at the origin. For the case ρdifferent from 1/2 we show that the time spent by the second class particle at the origin has finite expectation. This strong transience is then used to prove that variances of additive functionals of local mean-zero functions are diffusive when ρis not 1/2. As a corollary to previous work, we deduce the invariance principle for these functionals. The main arguments are comparisons of H_{-1} norms, a large deviation estimate for second-class particles, and a relation between occupation times of second-class particles and additive functional variances.
We study the hydrodynamic limit for a model of symmetric exclusion processes with heavy-tailed long jumps and in contact with infinitely extended reservoirs. We show how the corresponding hydrodynamic equations are affected by the parameters defining the model. The hydrodynamic equations are characterized by a class of super-diffusive operators that are given by the regional fractional Laplacian with some additional reaction terms and various boundary conditions. This work solves some questions left open in \cite{BJGO2} about the same model.
In this article we obtain the equilibrium fluctuations of a symmetric exclusion process in $\mathbb{Z}$ with long jumps. The transition probability of the jump from $x$ to $y$ is proportional to $|x-y|^{-γ-1}$. Here we restrict to the choice $γ\geq 2$ so that the system has a diffusive behavior. Moreover, when particles move between $\mathbb{Z}_{-}^{*}$ and $\mathbb N$, the jump rates are slowed down by a factor $αn^{-β}$, where $α>0$, $β\geq 0$ and $n$ is the scaling parameter. Depending on the values of $β$ and $γ$, we obtain several stochastic partial differential equations, corresponding t…
Sharp freezing time estimates for the subcritical Facilitated Exclusion Process
Université Paris Cité（巴黎西岱大学）；Sorbonne Université（索邦大学）；Inria（法国国家数字与工程科学研究所）；Universite Claude Bernard Lyon 1（里昂第一大学）；Yonsei University（延世大学）；Seoul National University（首尔大学）
We study the equilibrium fluctuations of a tagged particle in finite-range simple exclusion processes on Z^d with biased single particle jump rates. It is known the variance of the tagged particle at time t is diffusive, that is on order O(t), in d\geq 3, and in d=1 when in addition the jump rate is nearest-neighbor, and moreover, in these cases, central limit theorems in diffusive scale have been proved. In this article, we give some partial results in the open cases in d\leq 2. Namely, we show diffusivity of the tagged particle variance at time t in the sense of some upper and lower bounds o…
We consider the asymmetric exclusion process. We start from a profile which is constant along the drift direction and prove that the density profile, under a diffusive rescaling of time, converges to the solution of a parabolic equation.
We consider a one-dimensional, weakly asymmetric, boundary driven exclusion process on the interval $[0,N]\cap Z$ in the super-diffusive time scale $N^2 ε^{-1}_N$, where $1\ll ε^{-1}_N \ll N^{1/4}$. We assume that the external field and the chemical potentials, which fix the density at the boundaries, evolve smoothly in the macroscopic time scale. We derive an equation which describes the evolution of the density up to the order $ε_N$.
We prove a strong law of large numbers and an annealed invariance principle for a random walk in a one-dimensional dynamic random environment evolving as the simple exclusion process with jump parameter $γ$. First, we establish that if the asymptotic velocity of the walker is non-zero in the limiting case "$γ= \infty$", where the environment gets fully refreshed between each step of the walker, then, for $γ$ large enough, the walker still has a non-zero asymptotic velocity in the same direction. Second, we establish that if the walker is transient in the limiting case $γ= 0$, then, for $γ$ sma…
The purpose of this article is to study the hydrodynamic limit of the symmetric exclusion process with long jumps and in contact with infinitely extended reservoirs for a particular critical regime. The jumps are given in terms of a transition probability that can have finite or infinite variance and the hydrodynamic equation is a diffusive equation, in the former case, or a fractional equation, in the latter case. In this work we treat the critical case, that is, when the variance is infinite and grows logarithmically with the size of the system. This is the case in which there is a transitio…
We consider weakly asymmetric exclusion processes whose initial density profile is a small perturbation of a constant. We show that in the diffusive time-scale, in all dimensions, the density defect evolves as the solution of a viscous Burgers equation.
We provide a complete description of the equilibrium fluctuations for diffusive symmetric exclusion processes with long jumps in contact with infinitely extended reservoirs and prove that they behave as generalized Ornstein-Uhlenbeck processes with various boundary conditions, depending mainly on the strength of the reservoirs. On the way, we also give a general statement about uniqueness of the Ornstein-Uhlenbeck process originated by the microscopic dynamics of the underlying interacting particle systems and adapt it to our study.
We prove a nonequilibirum central limit theorem for the position of a tagged particle in the one-dimensional nearest-neighbor symmetric simple exclusion process under diffusive scaling starting from a Bernoulli product measure associated to a smooth profile $ρ_0:\bb R\to [0,1]$.
In this paper, we investigate the mixing time of the simple exclusion process on the circle with $N$ sites, with a number of particle $k(N)$ tending to infinity, both from the worst initial condition and from a typical initial condition. We show that the worst-case mixing time is asymptotically equivalent to $(8π^2)^{-1}N^2\log k$, while the cutoff window, is identified to be $N^2$. Starting from a typical condition, we show that there is no cutoff and that the mixing time is of order $N^2$.
We prove density and current fluctuations for two examples of symmetric, interacting particle systems with anomalous diffusive behavior: the zero-range process with long jumps and the zero-range process with degenerated bond disorder. As an application, we obtain subdiffusive behavior of a tagged particle in a simple exclusion process with variable diffusion coefficient.
We consider one-dimensional exclusion processes with long jumps given by a transition probability of the form $p_n(\cdot)=s(\cdot)+γ_na(\cdot)$, such that its symmetric part $s(\cdot)$ is irreducible with finite variance and its antisymmetric part is absolutely bounded by $s(\cdot).$ We prove that under diffusive time scaling and strength of asymmetry $\sqrt n γ_n \to_{n\to\infty} b\neq 0$, the equilibrium density fluctuations are given by the unique energy solution of the stochastic Burgers equation.
We consider a finite range symmetric exclusion process on the integer lattice in any dimension. We interpret it as a non-elliptic time-dependent random conductance model by setting conductances equal to one over the edges with end points occupied by particles of the exclusion process and to zero elsewhere. We prove a law of large number and a central limit theorem for the random walk driven by such a dynamical field of conductances by using the Kipnis-Varhadan martingale approximation. Unlike the tagged particle in the exclusion process, which is in some sense similar to this model, this rando…
In this article we analyse the hydrodynamical behavior of the symmetric exclusion process with long jumps and in the presence of a slow barrier. The jump rates for fast bonds are given by a transition probability $p(\cdot)$ which is symmetric and has finite variance, while for slow bonds the jump rates are given $p(\cdot)αn^{-β}$ (with $α>0$ and $β\geq 0$), and correspond to jumps from $\mathbb{Z}_{-}^{*}$ to $\mathbb N$. We prove that: if there is a fast bond from $\mathbb{Z}_{-}^{*}$ and $\mathbb N$, then the hydrodynamic limit is given by the heat equation with no boundary conditions; other…
Given a doubly infinite sequence of positive numbers {c_k: k in Z} satisfying a LLN with limit A, we consider the nearest-neighbor simple exclusion process on Z where c_k is the probability rate of jumps between k and k+1. If A is infinite we require an additional minor technical condition. By extending a method developed by K. Nagy, we show that the diffusively rescaled process has hydrodynamic behavior described by the heat equation with diffusion constant 1/A. In particular, the process has diffusive behavior for finite A and subdiffusive behavior for infinite A.
In this article, we consider a one-dimensional symmetric exclusion process in weak contact with reservoirs at the boundary. In the diffusive time-scaling the empirical measure evolves according to the heat equation with Robin boundary conditions. We prove the associated dynamical large deviations principle.
We consider the one-dimensional symmetric simple exclusion process with a slow bond. In this model, whilst all the transition rates are equal to one, a particular bond, the \emph{slow bond}, has associated transition rate of value $N^{-1}$, where $N$ is the scaling parameter. This model has been considered in previous works on the subject of hydrodynamic limit and fluctuations. In this paper, assuming uniqueness for weak solutions of hydrodynamic equation associated to the perturbed process, we obtain dynamical large deviations estimates in the diffusive scaling. The main challenge here is the…
In this paper we give a new proof of the second order Boltzmann-Gibbs principle. The proof does not impose the knowledge on the spectral gap inequality for the underlying model and it relies on a proper decomposition of the antisymmetric part of the current of the system in terms of polynomial functions. In addition, we fully derive the convergence of the equilibrium fluctuations towards 1) a trivial process in case of supper-diffusive systems, 2) an Ornstein-Uhlenbeck process or the unique energy solution of the stochastic Burgers equation, in case of weakly asymmetric diffusive systems. Exam…
We construct a nearest-neighbour interacting particle system of exclusion type, which illustrates a transition from slow to fast diffusion. More precisely, the hydrodynamic limit of this microscopic system in the diffusive space-time scaling is the parabolic equation $\partial_tρ=\nabla (D(ρ)\nabla ρ)$, with diffusion coefficient $ D(ρ)=mρ^{m-1} $ where $ m\in(0,2] $, including therefore the fast diffusion regime in the range $ m\in(0,1) $, and the porous {medium} equation for $ m\in(1,2) $. The construction of the model is based on the generalized binomial theorem, and interpolates continuous…
We consider the (discrete) parabolic Anderson model $\partial u(t,x)/\partial t=Δu(t,x) +ξ_t(x) u(t,x)$, $t\geq 0$, $x\in \mathbb{Z}^d$, where the $ξ$-field is $\mathbb{R}$-valued and plays the role of a dynamic random environment, and $Δ$ is the discrete Laplacian. We focus on the case in which $ξ$ is given by a properly rescaled symmetric simple exclusion process under which it converges to an Ornstein--Uhlenbeck process. Scaling the Laplacian diffusively and restricting ourselves to a torus, we show that in dimension $d=3$ upon considering a suitably renormalised version of the above equati…
We study the hydrodynamic and the hydrostatic behavior of the Simple Symmetric Exclusion Process with \emph{slow boundary}. The term \emph{slow boundary} means that particles can be born or die at the boundary sites, at a rate proportional to $N^{-θ}$, where $θ> 0$ and $N$ is the scaling parameter. In the bulk, the particles exchange rate is equal to $1$. In the hydrostatic scenario, we obtain three different linear profiles, depending on the value of the parameter $θ$; in the hydrodynamic scenario, we obtain that the time evolution of the spatial density of particles, in the diffusive scaling…
Let $Λ$ be a connected closed region with smooth boundary contained in the $d$-dimensional continuous torus $\bb T^d$. In the discrete torus $N^{-1} \bb T^d_N$, we consider a nearest neighbor symmetric exclusion process where occupancies of neighboring sites are exchanged at rates depending on $Λ$ in the following way: if both sites are in $Λ$ or $Λ^\complement$, the exchange rate is one; If one site is in $Λ$ and the other one is in $Λ^\complement$ and the direction of the bond connecting the sites is $e_j$, then the exchange rate is defined as $N^{-1}$ times the absolute value of the inner p…
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-ab…
Fix a strictly increasing right continuous with left limits function $W: \bb R \to \bb R$ and a smooth function $Φ: [l,r] \to \bb R$, defined on some interval $[l,r]$ of $\bb R$, such that $0<b \le Φ'\le b^{-1}$. We prove that the evolution, on the diffusive scale, of the empirical density of exclusion processes, with conductances given by $W$, is described by the weak solutions of the non-linear differential equation $\partial_t ρ= (d/dx)(d/dW) Φ(ρ)$. We derive some properties of the operator $(d/dx)(d/dW)$ and prove uniqueness of weak solutions of the previous non-linear differential equatio…
We introduce a two-dimensional, distribution-valued field which we call the quadratic field associated to the one-dimensional Ornstein-Uhlenbeck process. We show that the stationary quadratic fluctuations of the simple exclusion process, when rescaled in the diffusive scaling, converge to this quadratic field. We show that this quadratic field evaluated at the diagonal corresponds to the Wick-renormalized square of the Ornstein-Uhlenbeck process, and we use this new representation in order to prove some small and large-time properties of it.
We consider the symmetric simple exclusion process with slow boundary first introduced in [Baldasso {\it et al.}, Journal of Statistical Physics, 167(5), 2017]. We prove a law of large number for the empirical measure of the process under a longer time scaling instead of the usual diffusive time scaling.
We consider the weakly asymmetric exclusion process on a bounded interval with particle reservoirs at the endpoints. The hydrodynamic limit for the empirical density, obtained in the diffusive scaling, is given by the viscous Burgers equation with Dirichlet boundary conditions. We prove the associated dynamical large deviations principle.
Fix a smooth function $Φ: [l,r] \to \bb R$, defined on some interval $[l,r]$ of $\bb R$, such that $0<b \le Φ'\le b^{-1}$. We prove that the evolution, on the diffusive scale, of the empirical density of exclusion processes in $\bb Z^d$, with conductances given by special class of functions $W$, is described by the weak solutions of the non-linear parabolic partial differential equation $\partial_t ρ= \sum_{k=1}^d (d/dx_k)(d/dW_k)Φ(ρ)$. We also derive some properties of the operator $\sum^d_{k=1}(d/dx_k)(d/dW_k)$.
We study exclusion processes on the integer lattice in which particles change their velocities due to stickiness. Specifically, whenever two or more particles occupy adjacent sites, they stick together for an extended period of time, and the entire particle system is slowed down until the ``collision'' is resolved. We show that under diffusive scaling of space and time such processes converge to what one might refer to as a sticky reflected Brownian motion in the wedge. The latter behaves as a Brownian motion with constant drift vector and diffusion matrix in the interior of the wedge, and ref…
We consider the weakly asymmetric exclusion process on a bounded interval with particles reservoirs at the endpoints. The hydrodynamic limit for the empirical density, obtained in the diffusive scaling, is given by the viscous Burgers equation with Dirichlet boundary conditions. In the case in which the bulk asymmetry is in the same direction as the drift due to the boundary reservoirs, we prove that the quasi-potential can be expressed in terms of the solution to a one-dimensional boundary value problem which has been introduced by Enaud and Derrida \cite{de}. We consider the strong asymmetri…
For a sequence of i.i.d. random variables $\{ξ_x : x\in \bb Z\}$ bounded above and below by strictly positive finite constants, consider the nearest-neighbor one-dimensional simple exclusion process in which a particle at $x$ (resp. $x+1$) jumps to $x+1$ (resp. $x$) at rate $ξ_x$. We examine a quenched nonequilibrium central limit theorem for the position of a tagged particle in the exclusion process with bond disorder $\{ξ_x : x\in \bb Z\}$. We prove that the position of the tagged particle converges under diffusive scaling to a Gaussian process if the other particles are initially distribute…
We look at a superposition of symmetric simple exclusion and Glauber dynamics in the discrete torus in dimension 1. For this model, we prove that the fluctuations around the hydrodynamic limit are described, in the diffusive scale, by an infinite-dimensional Ornstein-Uhlenbeck process. Our proof technique is an adaptation of Yau's Relative Entropy Method that is robust enough to be adapted to other exclusion models. To cut the technical details to a minimum, we assume that the process starts from a product measure with a custom-chosen density, for which the solution of the hydrodynamic equatio…
We investigate the evolution of the random interfaces in a two dimensional Potts model at zero temperature under Glauber dynamics for some particular initial conditions. We prove that under space-time diffusive scaling the shape of the interfaces converges in probability to the solution of a non-linear parabolic equation. This Law of Large Numbers is obtained from the Hydrodynamic limit of a coupling between an exclusion process and an inhomogeneous one dimensional zero range process with asymmetry at the origin.
Hydrodynamic limit for an open facilitated exclusion process with slow and fast boundaries
École Normale Supérieure (ENS) de Lyon（里昂高等师范学校）；Inria DRACULA, ICJ UMR5208, CNRS, École Centrale de Lyon, INSA Lyon, Université Claude Bernard Lyon 1, Université Jean Monnet（法国国家信息与自动化研究所DRACULA团队，ICJ联合研究实验室，法国国家科学研究中心，里昂中央理工学院，里昂国立应用科学学院，里昂第一大学，让·莫内大学）；Université Claude Bernard Lyon 1, ICJ UMR5208, CNRS, École Centrale de Lyon, INSA Lyon, Université Jean Monnet（里昂第一大学，ICJ联合研究实验室，法国国家科学研究中心，里昂中央理工学院，里昂国立应用科学学院，让·莫内大学）；GSSI（阿奎拉全球社会科学研究生院）
We consider continuous-time random walks on a random locally finite subset of $\mathbb{R}^d$ with random symmetric jump probability rates. The jump range can be unbounded. We assume some second--moment conditions and that the above randomness is left invariant by the action of the group $\mathbb{G}=\mathbb{R}^d$ or $\mathbb{G}=\mathbb{Z}^d$. We then add a site-exclusion interaction, thus making the particle system a simple exclusion process. We show that, for almost all environments, under diffusive space-time rescaling the system exhibits a hydrodynamic limit in path space. The hydrodynamic e…
We consider the exclusion process in the one-dimensional discrete torus with $N$ points, where all the bonds have conductance one, except a finite number of slow bonds, with conductance $N^{-β}$, with $β\in[0,\infty)$. We prove that the time evolution of the empirical density of particles, in the diffusive scaling, has a distinct behavior according to the range of the parameter $β$. If $β\in [0,1)$, the hydrodynamic limit is given by the usual heat equation. If $β=1$, it is given by a parabolic equation involving an operator $\frac{d}{dx}\frac{d}{dW}$, where $W$ is the Lebesgue measure on the …
Consider a system of particles performing nearest neighbor random walks on the lattice $\ZZ$ under hard--core interaction. The rate for a jump over a given bond is direction--independent and the inverse of the jump rates are i.i.d. random variables belonging to the domain of attraction of an $\a$--stable law, $0<\a<1$. This exclusion process models conduction in strongly disordered one-dimensional media. We prove that, when varying over the disorder and for a suitable slowly varying function $L$, under the super-diffusive time scaling $N^{1 + 1/α}L(N)$, the density profile evolves as the solut…
We consider the Symmetric Simple Exclusion Process (SSEP) on the segment with two reservoirs of densities $p, q \in (0,1)$ at the two endpoints. We show that the system exhibits cutoff with a diffusive window, thus confirming a conjecture of Gantert, Nestoridi, and Schmid in \cite{Gantert2020}. In particular, our result covers the regime $p \neq q$, where the process is not reversible and there is no known explicit formula for the invariant measure. Our proof exploits the information percolation framework introduced by Lubetzky and Sly, the negative dependence of the system, and an anticoncent…
We show a diffusive upper bound on the transition probability of a tagged particle in the symmetric simple exclusion process. The proof relies on optimal spectral gap estimates for the dynamics in finite volume, which are of independent interest. We also show off-diagonal estimates of Carne-Varopoulos type.
A diffusive lattice gas is characterized by the diffusion coefficient depending only on the density. The Green-Kubo formula for diffusivity can be represented as a variational formula, but even when the equilibrium properties of a lattice gas are analytically known the diffusion coefficient can be computed only in the exceptional situation when the lattice gas is gradient. In the general case, minimization over an infinite-dimensional space is required. We propose an approximation scheme based on minimizing over finite-dimensional subspaces of functions. The procedure is demonstrated for one-d…
A particle system with a single locally-conserved field (density) in a bounded interval with different densities maintained at the two endpoints of the interval is under study here. The particles interact in the bulk through a long range potential parametrized by $β\ge 0$ and evolve according to an exclusion rule. It is shown that the empirical particle density under the diffusive scaling solves a quasi-linear integro-differential evolution equation with Dirichlet boundary conditions. The associated dynamical large deviation principle is proved. Furthermore, for $β$ small enough, it is also de…
We consider a flip dynamics for directed (1+d)-dimensional lattice paths with length L. The model can be interpreted as a higher dimensional version of the simple exclusion process, the latter corresponding to the case d=1. We prove that the mixing time of the associated Markov chain scales like L^2\log L up to a d-dependent multiplicative constant. The key step in the proof of the upper bound is to show that the system satisfies a logarithmic Sobolev inequality on the diffusive scale L^2 for every fixed d, which we achieve by a suitable induction over the dimension together with an estimate f…
Kinetically constrained lattice gases (KCLG) are interacting particle systems on the integer lattice $\mathbb Z^d$ with hard core exclusion and Kawasaki type dynamics. Their peculiarity is that jumps are allowed only if the configuration satisfies a constraint which asks for enough empty sites in a certain local neighborhood. KCLG have been introduced and extensively studied in physics literature as models of glassy dynamics. We focus on the most studied class of KCLG, the Kob Andersen (KA) models. We analyze the behavior of a tracer (i.e. a tagged particle) at equilibrium. We prove that for a…
