We prove a general theorem on cutoffs for symmetric simple exclusion processes on graphs with open boundaries, under the natural assumption that the graphs converge geometrically and spectrally to a compact metric measure space with Dirichlet boundary condition. Our theorem is valid on a variety of settings including, but not limited to: the $d$-dimensional grid for every integer dimension $d$; and self-similar fractal graphs and products thereof. Our method of proof is to identify a rescaled version of the density fluctuation field---the cutoff martingale---which allows us to prove the mixing time upper bound that matches the lower bound obtained via Wilson's method.
We consider the biased card shuffling and the Asymmetric Simple Exclusion Process (ASEP) on the segment. We obtain the asymptotic of their mixing times: our result show that these two continuous-time Markov chains display cutoff. Our analysis combines several ingredients including: a study of the hydrodynamic profile for ASEP, the use of monotonic eigenfunctions, stochastic comparisons and concentration inequalities.
We study the large deviation functional of the current for the Weakly Asymmetric Simple Exclusion Process in contact with two reservoirs.We compare this functional in the large drift limit to the one of the Totally Asymmetric Simple Exclusion Process, in particular to the Jensen-Varadhan functional. Conjectures for generalizing the Jensen-Varadhan functional to open systems are also stated.
We prove a general theorem on cutoffs for symmetric exclusion and interchange processes on finite graphs $G_N=(V_N,E_N)$, under the assumption that either the graphs converge geometrically and spectrally to a compact metric measure space, or they are isomorphic to discrete Boolean hypercubes. Specifically, cutoffs occur at times $\displaystyle t_N= (2γ_1^N)^{-1}\log |V_N|$, where $γ_1^N$ is the spectral gap of the symmetric random walk process on $G_N$. Under the former assumption, our theorem is applicable to the said processes on graphs such as: the $d$-dimensional discrete grids and tori fo…
We consider the reversible exclusion process with reservoirs on arbitrary networks. We characterize the spectral gap, mixing time, and mixing window of the process, in terms of certain simple statistics of the underlying network. Among other consequences, we establish a non-conservative analogue of Aldous's spectral gap conjecture, and we show that cutoff occurs if and only if the product condition is satisfied. We illustrate this by providing explicit cutoffs on discrete lattices of arbitrary dimensions and boundary conditions, which substantially generalize recent one-dimensional results. We…
We study the time that the simple exclusion process on the complete graph needs to reach equilibrium in terms of total variation distance. For the graph with n vertices and 1<<k<n/2 particles we show that the mixing time is of order (n/2)\log \min(k, \sqrt{n}), and that around this time, for any small positive epsilon the total variation distance drops from 1-epsilon to epsilon in a time window whose width is of order n (i.e. in a much shorter time). Our proof is purely probabilistic and self-contained.
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 find the total variation mixing time of the interchange process on the dumbbell graph (two complete graphs, $K_n$ and $K_m$, connected by a single edge), and show that this sequence of chains exhibits the cutoff phenomenon precisely when the smaller size $m$ goes to infinity. The mixing time undergoes a phase transition at $m\asymp \sqrt{n}$. We also state a conjecture on when exactly cutoff holds for the interchange process on general graphs. Our proofs use coupling methods, and they also give the mixing time of the simple exclusion process of $k$ labelled particles in the complete graph $…
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.
In this paper, we investigate the mixing time of the adjacent transposition shuffle for a deck of $N$ cards. We prove that around time $N^2\log N/(2π^2)$, the total variation distance to equilibrium of the deck distribution drops abruptly from $1$ to $0$, and that the separation distance has a similar behavior but with a transition occurring at time $(N^2\log N)/π^2$. This solves a conjecture formulated by David Wilson. We present also similar results for the exclusion process on a segment of length $N$ with $k$ particles.
We consider the one dimensional symmetric simple exclusion process (SSEP) with additional births and deaths restricted to a subset of configurations where there is a leftmost hole and a rightmost particle. At a fixed rate birth of particles occur at the position of the leftmost hole and at the same rate, independently, the rightmost particle dies. We prove convergence to a hydrodynamic limit and discuss its relation with a free boundary problem.
In this paper, we give a very accurate description of the way the simple exclusion process relaxes to equilibrium. Let $P_t$ denote the semi-group associated the exclusion on the circle with $2N$ sites and $N$ particles. For any initial condition $χ$, and for any $t\ge\frac{4N^2}{9π^2}\log N$, we show that the probability density $P_t(χ,\cdot)$ is given by an exponential tilt of the equilibrium measure by the main eigenfunction of the particle system. As $\frac{4N^2}{9π^2}\log N$ is smaller than the mixing time which is $\frac{N^2}{2π^2}\log N$, this allows to give a sharp description of the c…
We prove that the equilibrium fluctuations of the symmetric simple exclusion process in contact with slow boundaries is given by an Ornstein-Uhlenbeck process with Dirichlet, Robin or Neumann boundary conditions depending on the range of the parameter that rules the slowness of the boundaries.
We consider the simple exclusion process with $k$ particles on a segment of length $N$ performing random walks with transition $p>1/2$ to the right and $q=1-p$ to the left. We focus on the case where the asymmetry in the jump rates $b=p-q>0$ vanishes in the limit when $N$ and $k$ tend to infinity, and obtain sharp asymptotics for the mixing times of this sequence of Markov chains in the two cases where the asymmetry is either much larger or much smaller than $(\log k)/N$. We show that in the former case ($b \gg (\log k)/N$), the mixing time corresponds to the time needed to reach macroscopic e…
We study mixing times of the symmetric and asymmetric simple exclusion process on the segment where particles are allowed to enter and exit at the endpoints. We consider different regimes depending on the entering and exiting rates as well as on the rates in the bulk, and show that the process exhibits pre-cutoff and in some cases cutoff. Our main contribution is to study mixing times for the asymmetric simple exclusion process with open boundaries. We show that the order of the mixing time can be linear or exponential in the size of the segment depending on the choice of the boundary paramete…
We investigate the fluctuations of cumulative density of particles in the asymmetric simple exclusion process with respect to the stationary distribution (also known as the steady state), as a stochastic process indexed by $[0,1]$. In three phases of the model and their boundaries within the fan region, we establish a complete picture of the scaling limits of the fluctuations of the density as the number of sites goes to infinity. In the maximal current phase, the limit fluctuation is the sum of two independent processes, a Brownian motion and a Brownian excursion. This extends an earlier resu…
We show that the joint probability generating function of the stationary measure of a finite state asymmetric exclusion process with open boundaries can be expressed in terms of joint moments of Markov processes called quadratic harnesses. We use our representation to prove the large deviations principle for the total number of particles in the system. We use the generator of the Markov process to show how explicit formulas for the average occupancy of a site arise for special choices of parameters. We also give similar representations for limits of stationary measures as the number of sites t…
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 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.
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 study the one-dimensional asymmetric simple exclusion process on the lattice $\{1, \dots,N\}$ with creation/annihilation at the boundaries. The boundary rates are time dependent and change on a slow time scale $N^{-a}$ with $a>0$. We prove that at the time scale $N^{1+a}$ the system evolves quasi-statically with a macroscopic density profile given by the entropy solution of the stationary Burgers equation with boundary densities changing in time, determined by the corresponding microscopic boundary rates. We consider two different types of boundary rates: the "Liggett boundaries" that corre…
The misanthrope process is a class of stochastic interacting particle systems, generalizing the simple exclusion process. It allows each site of the lattice to accommodate more than one particle. We consider a special case of the one dimensional misanthrope process whose probability distribution is completely equivalent to the ordinary simple exclusion process under the periodic boundary condition. By imposing open boundaries, high- and low-density domains can coexist in the system, which we investigate by Monte Carlo simulations. We examine finite-size corrections of density profiles and corr…
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）
In this paper, we will study the long time behavior of the simple symmetric exclusion process in the ``channel'' $Λ_N=[1,N]\cap \mathbb{N}$ with reservoirs at the boundaries. These reservoirs are also systems of particles which can be exchanged with the particles in the channel. The size $M$ of each reservoir is much larger than the one of the channel, i.e. $M=N^{1+α}$ for a fixed number $α>0$. Based on the size of the channel and the holding time at each reservoir, we will investigate some types of rescaling time.
The asymmetric exclusion process is an idealised stochastic model of transport, whose exact solution has given important insight into a general theory of nonequilibrium statistical physics. In this work, we consider a totally asymmetric exclusion process with multiple species of particles on a one-dimensional lattice in contact with reservoirs. We derive the exact nonequilibrium phase diagram for the system in the long time limit. We find two new phenomena in certain regions of the phase diagram: when the density of a species becomes zero throughout the system, and dynamical localisation when …
We consider an interacting particle system in the interval $[1,N]$ with reservoirs at the boundaries. While the dynamics in the channel is the simple symmetric exclusion process, the reservoirs are also particle systems which interact with the given system by exchanging particles. In this paper we study the case where the size of each reservoir is the same as the size of the channel . We will prove that the hydrodynamic limit equation is the heat equation with boundary conditions which relate first and second spatial derivatives at the boundaries for which we will prove the existence and uniqu…
