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 @math of the underlying SSEP. Our first result is a law of large numbers (LLN) for the random walker for all densities @math except for at most two values @math . The asymptotic speed we obtain in our LLN is a monotone function of @math . Also, @math and @math 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 @math 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. Finally, we prove similar LLN and CLT results for a different environment, given by a family of independent simple symmetric random walks in equilibrium.
In @cite_31 , a (continuous-time) random walk on the SSEP was studied by means of simulations. There the authors investigated the limiting behavior as a function of three parameters: the density @math of the SSEP, the rate @math of the SSEP and the local drift @math of the random walk on occupied sites. They restrict themselves to the case where the local drift on vacant sites satisfy @math .
@cite_31
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We study branching random walks in random environment on the d- dimensional square lattice, d 1. In this model, the environment has nite range dependence, and the population size cannot decrease. We prove limit theorems (laws of large numbers) for the set of lattice sites which are visited up to a large time as well as for the local size of the population. The limiting shape of this set is compact and convex, and the local size is given by a concave growth exponent. Also, we obtain the law of large numbers for the logarithm of the total number of particles in the process. 1. Introduction and results We start with an informal description of the model we study in this paper. Particles live in Z d and evolve in discrete time. At each time, every particle is substituted by (possibly more than one) ospring which are placed in neighboring sites, independently of the other particles. The rules of ospring generation depend only on the location of the particle. The collection of those rules (so-called the environment) is itself random, it is chosen randomly before starting the process, and then it is kept xed during all the subsequent evolution of the particle system.