We prove the non-equilibrium fluctuations for the one-dimensional symmetric simple exclusion process with a slow bond. This generalizes a result of T. Franco, A. Neumann and P. Gonçalves (2013), which dealt with the equilibrium fluctuations. The foundation stone of our proof is a precise estimate on the correlations of the system, and that is by itself one of the main novelties of this paper. To obtain these estimates, we first deduce a spatially discrete PDE for the covariance function and we relate it to the local times of a random walk in a non-homogeneous environment via Duhamel's principle. Projection techniques and coupling arguments reduce the analysis to the problem of studying the local times of the classical random walk. We think that the method developed here can be applied to a variety of models, and we provide a discussion on this matter.
We consider the one dimensional symmetric simple exclusion process with a slow bond. In this model, particles cross each bond at rate $N^2$, except one particular bond, the slow bond, where the rate is $N$. Above, $N$ is the scaling parameter. This model has been considered in the context of hydrodynamic limits, fluctuations and large deviations. We investigate moderate deviations from hydrodynamics and obtain a moderate deviation principle.
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 derive the non-equilibrium fluctuations of one-dimensional symmetric simple exclusion processes in contact with slowed stochastic reservoirs which are regulated by a factor $n^{-θ}$. Depending on the range of $θ$ we obtain processes with various boundary conditions. Moreover, as a consequence of the previous result we deduce the non-equilibrium stationary fluctuations by using the matrix ansatz method which gives us information on the stationary measure for the model. The main ingredient to prove these results is the derivation of precise bounds on the two-point space-time correlation funct…
In this paper we consider a symmetric simple exclusion process (SSEP) on the $d$-dimensional discrete torus $\mathbb{T}^d_N$ with a spatial non-homogeneity given by a slow membrane. The slow membrane is defined here as the boundary of a smooth simple connected region $Λ$ on the continuous $d$-dimensional torus $\mathbb{T}^d$. In this setting, bonds crossing the membrane have jump rate $α/N^β$ and all other bonds have jump rate one, where $α>0$, $β\in[0,\infty]$, and $N\in \mathbb{N}$ is the scaling parameter. In the diffusive scaling we prove that the hydrodynamic limit presents a dynamical ph…
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 prove a large deviations principle for the empirical measure of the one dimensional symmetric simple exclusion process in contact with reservoirs. The dynamics of the reservoirs is slowed down with respect to the dynamics of the system, that is, the rate at which the system exchanges particles with the boundary reservoirs is of order $n^{-θ}$, where $n$ is number of sites in the system, $θ$ is a non negative parameter, and the system is taken in the diffusive time scaling. Two regimes are studied here, the subcritical $θ\in(0,1)$ whose hydrodynamic equation is the heat equation with Dirichl…
We study the hydrodynamic limit of SSEP with slow boundaries on hypercubes in dimension at least two. The hydrodynamic limit equation is shown to be a heat equation with three different types of boundary conditions according to the slowness of the boundary dynamics.The proof is based on Yau's relative entropy method.
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…
We consider the continuous time symmetric random walk with a slow bond on $\mathbb Z$, which rates are equal to $1/2$ for all bonds, except for the bond of vertices $\{-1,0\}$, which associated rate is given by $αn^{-β}/2$, where $α\geq 0$ and $β\in [0,\infty]$ are the parameters of the model. We prove here a functional central limit theorem for the random walk with a slow bond: if $β<1$, then it converges to the usual Brownian motion. If $β\in (1,\infty]$, then it converges to the reflected Brownian motion. And at the critical value $β=1$, it converges to the snapping out Brownian motion (SNO…
We prove the height functions for a class of non-integrable and non-stationary particle systems converge to the KPZ equation, thereby making progress on the universality of the KPZ equation. The models herein are ASEP [4] with a mesoscopic family of slow bonds, thus we partially extend [16] to non-stationary models and add to the almost empty set of non-integrable, non-stationary interacting particle systems for which universality is established. To do this, we develop further the strategy of [41, 42] introduce a method to establish a novel principle that builds upon the classical hydrodynamic…
We consider the weakly asymmetric simple exclusion process in the presence of a slow bond and starting from the invariant state, namely the Bernoulli product measure of parameter $ρ\in(0,1)$. The rate of passage of particles to the right (resp. left) is $\frac1{\vphantom{n^β}2}+\frac{a}{2n^{\vphantomβγ}}$ (resp. $\frac1{\vphantom{n^β}2}-\frac{a}{2n^{\vphantomβγ}}$) except at the bond of vertices $\{-1,0\}$ where the rate to the right (resp. left) is given by $\fracα{2n^β}+\frac{a}{2n^{\vphantomβγ}}$ (resp. $\fracα{2n^β}-\frac{a}{2n^{\vphantomβγ}}$). Above, $α>0$, $γ\geq β\geq 0$, $a\geq 0$. Fo…
Totally Asymmetric Simple Exclusion Process (TASEP) on $\mathbb{Z}$ is one of the classical exactly solvable models in the KPZ universality class. We study the "slow bond" model, where TASEP on $\mathbb{Z}$ is imputed with a slow bond at the origin. The slow bond increases the particle density immediately to its left and decreases the particle density immediately to its right. Whether or not this effect is detectable in the macroscopic current started from the step initial condition has attracted much interest over the years and this question was settled recently (Basu, Sidoravicius, Sly (2014…
We address the question of how a localized microscopic defect, especially if it is small with respect to certain dynamic parameters, affects the macroscopic behavior of a system. In particular we consider two classical exactly solvable models: Ulam's problem of the maximal increasing sequence and the totally asymmetric simple exclusion process. For the first model, using its representation as a Poissonian version of directed last passage percolation on $\mathbb R^2$, we introduce the defect by placing a positive density of extra points along the diagonal line. For the latter, the defect is pro…
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…
