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Asymmetric conservative processes with random rates

I. Benjamini,P. A. Ferrari,C. Landim

Published 1996 in Stochastic Processes and their Applications

ABSTRACT

We study a one-dimensional nearest neighbor simple exclusion process for which the rates of jump are chosen randomly at time zero and fixed for the rest of the evolution. The ith particle's right and left jump rates are denoted pi and qi respectively; pi+ qi = 1. We fix c [epsilon] (1/2, 1) and assume that pi [epsilon] [c, 1] is a stationary ergodic process. We show that there exists a critical density [varrho]* depending only on the distribution of {{pi}} such that for almost all choices of the rates: (a) if [varrho] [epsilon] [[varrho]*, 1], then there exists a product invariant distribution for the process as seen from a tagged particle with asymptotic density [varrho]; (b) if [varrho] [epsilon] [0, [varrho]*), then there are no product measures invariant for the process. We give a necessary and sufficient condition for [varrho]* > 0 in the iid case. We also show that under a product invariant distribution, the position Xt of the tagged particle at time t can be sharply approximated by a Poisson process. Finally, we prove the hydrodynamical limit for zero range processes with random rate jumps.

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