Large deviation principle for one-dimensional random walk in dynamic random environment: attractive spin-flips and simple symmetric exclusion

摘要

Consider a one-dimensional shift-invariant attractive spin-flip system inequilibrium, constituting a dynamic random environment, together with anearest-neighbor random walk that on occupied sites has a local drift to theright but on vacant sites has a local drift to the left. In previous work weproved a law of large numbers for dynamic random environments satisfying aspace-time mixing property called cone-mixing. If an attractive spin-flipsystem has a finite average coupling time at the origin for two copies startingfrom the all-occupied and the all-vacant configuration, respectively, then itis cone-mixing. In the present paper we prove a large deviation principle for the empiricalspeed of the random walk, both quenched and annealed, and exhibit someproperties of the associated rate functions. Under an exponential space-timemixing 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 knownto be possible in a static random environment does not survive in a fast mixingdynamic random environment. In contrast, we show that for the simple symmetricexclusion dynamics, which is not cone-mixing (and which is not a spin-flipsystem either), slow-down does occur.