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Register a new userQuantitative ergodicity for the symmetric exclusion process with stationary initial data
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We consider the symmetric exclusion process on the $d$-dimensional lattice with translational invariant and ergodic initial data. It is then known that as $t$ diverges the distribution of the process at time $t$ converges to a Bernoulli product measure. Assuming a summable decay of correlations of the initial data, we prove a quantitative version of this convergence by obtaining an explicit bound on the Ornstein $bar d$-distance. The proof is based on the analysis of a two species exclusion process with annihilation.
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Strong negative dependence properties have recently been proved for the symmetric exclusion process. In this paper, we apply these results to prove convergence to the Poisson and normal distributions for various functionals of the process.
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 parameters, proving a strikingly different (and richer) behavior for the simple exclusion process with open boundaries than for the process on the closed segment. Our arguments combine coupling, second class particle and censoring techniques with current estimates. A novel idea is the use of multi-species particle arguments, where the particles only obey a partial ordering.
We analyze the mixing behavior of the biased exclusion process on a path of length $n$ as the bias $beta_n$ tends to $0$ as $n to infty$. 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 $beta_n$ is of order $1/n$, and the other when $beta_n$ is order $log n/n$.
We consider the simple exclusion process on Z x {0, 1}, that is, an horizontal ladder composed of 2 lanes. Particles can jump according to a lane-dependent translation-invariant nearest neighbour jump kernel, i.e. horizontally along each lane, and vertically along the scales of the ladder. We prove that generically, the set of extremal invariant measures consists of (i) translation-invariant product Bernoulli measures; and, modulo translations along Z: (ii) at most two shock measures (i.e. asymptotic to Bernoulli measures at $pm$$infty$) with asymptotic densities 0 and 2; (iii) at most three shock measures with a density jump of magnitude 1. We fully determine this set for certain parameter values. In fact, outside degenerate cases, there is at most one shock measure of type (iii). The result can be partially generalized to vertically cyclic ladders with arbitrarily many lanes. For the latter, we answer an open question of [5] about rotational invariance of stationary measures.
A fractional Ficks law and fractional hydrostatics for the one dimensional exclusion process with long jumps in contact with infinite reservoirs at different densities on the left and on the right are derived.
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