Random Walk on the Simple Symmetric Exclusion Process

Abstract

AbstractWe investigate the long-term behavior of a random walker evolving on top of the simple symmetric exclusion process (SSEP) at equilibrium, in dimension one. At each jump, the random walker is subject to a drift that depends on whether it is sitting on top of a particle or a hole, so that its asymptotic behavior is expected to depend on the density $$\rho \in [0, 1]$$ ρ ∈ [ 0 , 1 ] of the underlying SSEP. Our first result is a law of large numbers (LLN) for the random walker for all densities $$\rho $$ ρ except for at most two values $$\rho _-, \rho _+ \in [0, 1]$$ ρ - , ρ + ∈ [ 0 , 1 ] . The asymptotic speed we obtain in our LLN is a monotone function of $$\rho $$ ρ . Also, $$\rho _-$$ ρ - and $$\rho _+$$ ρ + are characterized as the two points at which the speed may jump to (or from) zero. Furthermore, for all the values of densities where the random walk experiences a non-zero speed, we can prove that it satisfies a functional central limit theorem (CLT). For the special case in which the density is 1/2 and the jump distribution on an empty site and on an occupied site are symmetric to each other, we prove a LLN with zero limiting speed. We also prove similar LLN and CLT results for a different environment, given by a family of independent simple symmetric random walks in equilibrium.