MIXING TIME FOR THE ASYMMETRIC SIMPLE EXCLUSION PROCESS IN A RANDOM ENVIRONMENT

We consider the simple exclusion process in the integer segment [[1, N]] with k <= N/2 particles and spatially inhomogenous jumping rates. A particle at site x is an element of [[1, Nil jumps to site x - 1 (if x >= 2) at rate 1 - omega(x) and to site x + 1 (if x <= N 1) at rate omega(x) if the target site is not occupied. The sequence omega = (omega(x))(x is an element of Z) is chosen by IID sampling from a probability law whose support is bounded away from zero and one (in other words the random environment satisfies the uniform ellipticity condition). We further assume E[log p(1)] < 0 where p(1) := (1 - omega(1))/omega(1), which implies that our particles have a tendency to move to the right. We prove that the mixing time of the exclusion process in this setup grows like a power of N. More precisely, for the exclusion process with N beta+o(1) particles where beta E [0, 1], we have in the large N asymptotic N-max(1, 1/lambda, beta+1/2 lambda) + o(1) (<=) (N,k)(tmix) <= NC+o(1), where lambda > 0 is such that E[p(1)(lambda)] = 1 (lambda = infinity if the equation has no positive root) and C is a constant, which depends on the distribution of omega. We conjecture that our lower bound is sharp up to subpolynomial correction.