Large Deviation Principle for One-Dimensional Random Walk in Dynamic Random Environment: Attractive Spin-Flips and Simple Symmetric Exclusion

#### L. Avena, F. Redig, F. den Hollander

2010, v.16, Issue 1, 139-168

ABSTRACT

Consider a one-dimensional shift-invariant attractive spin-flip system in equilibrium, constituting a dynamic random environment, together with a nearest-neighbor random walk that on occupied sites has a local drift to the right but on vacant sites has a local drift to the left. In [L. Avena, F. den Hollander and F. Redig, Law of large numbers for a class of random walks in dynamic random environments. EURANDOM Report 2009-032] we proved a law of large numbers for dynamic random environments satisfying a space-time mixing property called cone-mixing. If an attractive spin-flip system has a finite average coupling time at the origin for two copies starting from the all-occupied and the all-vacant configuration, respectively, then it is cone-mixing. In the present paper we prove a large deviation principle for the empirical speed of the random walk, both quenched and annealed, and exhibit some properties of the associated rate functions. Under an exponential space-time mixing condition for the spin-flip system, which is stronger than cone-mixing, the two rate functions have a unique zero, i.e., the slow-down phenomenon known to be possible in a static random environment does not survive in a fast mixing dynamic random environment. In contrast, we show that for the simple symmetric exclusion dynamics, which is not cone-mixing (and which is not a spin-flip system either), slow-down does occur.

Keywords: dynamic random environment,random walk,quenched vs. annealedlarge deviation principle,slow-down

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