We 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 ρ∈ [0 , 1] of the underlying SSEP. Our first result is a law of large numbers (LLN) for the random walker for all densities ρ except for at most two values ρ-, ρ+∈ [0 , 1]. The asymptotic speed we obtain in our LLN is a monotone function of ρ. Also, ρ- and 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.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics