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 @math , we refer to the resulting process as asymmetric simple exclusion process (ASEP). studied the mixing time of the ASEP in the context of biased card shuffling and showed that it is asymptotically of order @math for any number of particles @math , see @cite_2 . We will follow their approach for an upper bound on the mixing time in the marginal nestling case. Labbé and Lacoin @cite_0 refined this bound for the ASEP and proved cutoff. The proof relies on an explicit calculation of the spectral gap which was independently obtained by Levin and Peres @cite_15 . They studied the mixing time of the ASEP when the bias vanishes for @math going to infinity. Recently, cutoff results were established in this regime by Labbé and Lacoin @cite_12 .
@cite_0 @cite_15 @cite_12 @cite_2
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We consider the biased card shuffling and the Asymmetric Simple Exclusion Process (ASEP) on the segment. We obtain the asymptotic of their mixing times, thus showing 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 eigenfunction, stochastic comparisons and concentration inequalities. We consider the simple exclusion process with @math particles on a segment of length @math performing random walks with transition @math to the right and @math to the left. We focus on the case where the asymmetry in the jump rates @math vanishes in the limit when @math and @math 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 @math . We show that in the former case ( @math ), the mixing time corresponds to the time needed to reach macroscopic equilibrium, like for the strongly asymmetric (i.e. constant @math ) case studied in [LL18], while the latter case ( @math ) macroscopic equilibrium is not sufficient for mixing and one must wait till local fluctuations equilibrate, similarly to what happens in the symmetric case worked out in [Lac16b]. In both cases, convergence to equilibrium is abrupt: we have a cutoff phenomenon for the total-variation distance. We present a conjecture for the remaining regime when the asymmetry is of order @math . 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.