We consider the exclusion process on segments of the integers in a site-dependent random environment. We assume to be in the ballistic regime in which a single particle has positive linear speed. Our goal is to study the mixing time of the exclusion process when the number of particles is linear in the size of the segment. We investigate the order of the mixing time depending on the support of the environment distribution. In particular, we prove for nestling environments that the order of the mixing time is different than in the case of a single particle.
For general (edge-)weighted graphs @math , one can consider the exclusion process in which particles jump to a neighbor site according to the rates given by the weights on the corresponding edge. This is known as the varying speed model of the exclusion process. Oliveira showed that the mixing time of the exclusion process is in @math , where @math denotes the mixing time of the random walk on @math @cite_9 . Recently, this result was improved by Hermon and Pymar @cite_13 .
@cite_9 @cite_13
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Oliveira conjectured that the order of the mixing time of the exclusion process with @math -particles on an arbitrary @math -vertex graph is at most that of the mixing-time of @math independent particles. We verify this up to a constant factor for @math -regular graphs when each edge rings at rate @math in various cases: (1) when @math or (2) when @math the spectral-gap of a single walk is @math and @math or (3) when @math for some constant @math . In these cases our analysis yields a probabilistic proof of a weaker version of Aldous' famous spectral-gap conjecture (resolved by ). We also prove a general bound which is at worst @math , which is within a @math factor from Oliveira's conjecture when @math . As applications we get new mixing bounds: (a) @math for expanders, (b) order @math for the hypercube @math and (c) order @math for vertex-transitive graphs of moderate growth and for supercritical percolation on a fixed dimensional torus. We prove an upper bound for the e-mixing time of the symmetric exclusion process on any graph G, with any feasible number of particles. Our estimate is proportional to TRW(G)ln(|V| e), where |V| is the number of vertices in G, and TRW(G) is the 1 4-mixing time of the corresponding single-particle random walk. This bound implies new results for symmetric exclusion on expanders, percolation clusters, the giant component of the Erdos–Renyi random graph and Poisson point processes in Rd. Our technical tools include a variant of Morris’s chameleon process. We analyze the mixing behavior of the biased exclusion process on a path of length n as the bias ( _n ) tends to 0 as (n ). We show that the sequence of chains has a pre-cutoff, and interpolates between the unbiased exclusion and the process with constant bias. As the bias increases, the mixing time undergoes two phase transitions: one when ( _n ) is of order 1 n, and the other when ( _n ) is order ( n n ). By symmetry, P has eigenvalues 1 = I03 > I381 > ?> I 31xI- 1 2 -1. This paper develops methods for getting upper and lower bounds on 8i3 by comparison with a second reversible chain on the same state space. This extends the ideas introduced in Diaconis and Saloff-Coste (1993), where random walks on finite groups were considered. The bounds involve geometric properties such as the diameter and covering number of an associated graph along the lines of Diaconis and Stroock (1991). The main application gives a sharp upper bound on the second eigenvalue of the symmetric exclusion process. Thus, let S0 be a connected undirected graph with n vertices. For simplicity, we assume in this introduction that SW is regular. To start, r unlabelled particles are placed in an initial configuration, 1 < r < n. At each step, a particle is chosen at random; then one of the neighboring sites of this particle is chosen at random. If the neighboring site is unoccupied, the chosen particle is moved there; if the neighboring site is occupied, the system stays as it was. This is a reversible Markov chain on the r-sets of 1, 2, . . ., n with uniform stationary distribution. Liggett (1985) gives background and motivation (he focuses on infinite systems). Fill (1991) gives bounds on the second eigenvalue of the labeled exclusion process on the finite circle ZZn 1 We study this chain by comparison with a second Markov chain on r-sets that proceeds by picking a particle at random, picking an unoccupied site at random (not necessarily a neighboring site) and moving the particle to the unoccupied site. This is a well studied chain (the Bernoulli-Laplace model for diffusion). Its eigenvalues are known. We show that the comparison techniques apply to give upper bounds on the eigenvalues of the exclusion