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We introduce via perturbation a class of random walks in reversible dynamicenvironments having a spectral gap. In this setting one can apply themathematical results derived in http://arxiv.org/abs/1602.06322. As firstresults, we show that the asymptotic velocity is antisymmetric in theperturbative parameter and, for a subclass of random walks, we characterize thevelocity and a stationary distribution of the environment seen from the walkeras suitable series in the perturbative parameter. We then consider as a specialcase a random walk on the East model that tends to follow dynamical interfacesbetween empty and occupied regions. We study the asymptotic velocity anddensity profile for the environment seen from the walker. In particular, wedetermine the sign of the velocity when the density of the underlying Eastprocess is not 1/2, and we discuss the appearance of a drift in the balancedsetting given by density 1/2.

A class of random walks in reversible dynamic environment: antisymmetry and applications to the East model