Random Walk on the Simple Symmetric Exclusion Process
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Communications in
Mathematical Physics
Random Walk on the Simple Symmetric Exclusion Process Marcelo R. Hilário1 , Daniel Kious2 , Augusto Teixeira3 1 Departamento de Matemática, Universidade Federal de Minas Gerais, Belo Horizonte 31270-901, Brazil. 2 Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK.
E-mail: [email protected]
3 IMPA, Estrada Dona Castorina 110, Rio de Janeiro, RJ 22460-320, Brazil.
Received: 22 June 2019 / Accepted: 29 May 2020 © The Author(s) 2020
Abstract: 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. 1. Introduction Over the last decades the study of the long-term behavior of the position of a particle subject to the influence of a random environment has received great attention from the physics and mathematics community. In this context, one is usually interested in proving the existence of a well-defined limiting speed for the particle and, once the existence of such a speed is known, to characterize its fluctuations around the average position. The random environment can be either static or dynamic depending on whether it is kept fixed or evolves stochastically after the initial configuration is sampled from a given distribution. For one-dimensional static random environments, since the pioneering work of Solomon [Sol75], criteria for recurrence or transience, law of large numbers, central limit theorems, anomalous fluctuation regimes and large deviations have been obtained,
M. R. Hilário, D. Kious, A. Teixeira
see for instance [Sol75,KKS75,Sin82]. For higher dimensional static environments, important progress has also been achieved, however the knowledge is still modest when compared to the one-dimensional setting (see for instance [Szn04,BDR14] among many others). A number of important questions remain open and there is still much to be understood. We refer the reader to [HMZ12,Szn04] and, more recently [DR14
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