A Class of Random Walks in Reversible Dynamic Environments: Antisymmetry and Applications to the East Model

We introduce via perturbation a class of random walks in reversible dynamic environments having a spectral gap. In this setting one can apply the mathematical results derived in Avena et al. (-Perturbed Markov processes and applications to random walks in dynamic random environments, Preprint, 2016). As first results, we show that the asymptotic velocity is antisymmetric in the perturbative parameter and, for a subclass of random walks, we characterize the velocity and a stationary distribution of the environment seen from the walker as suitable series in the perturbative parameter. We then consider as a special case a random walk on the East model that tends to follow dynamical interfaces between empty and occupied regions. We study the asymptotic velocity and density profile for the environment seen from the walker. In particular, we determine the sign of the velocity when the density of the underlying East process is not 1 / 2, and we discuss the appearance of a drift in the balanced setting given by density 1 / 2.