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

TL;DR

This paper studies a class of random walks in reversible dynamic environments, focusing on their antisymmetry properties, velocity characterization, and applications to the East model, revealing how environment density influences drift and velocity.

Contribution

It introduces a perturbative framework for random walks in reversible environments, characterizes their velocity and stationary distribution, and applies these results to the East model.

Findings

Asymptotic velocity is antisymmetric in the perturbation parameter.

Velocity and environment distribution are expressed as series in the perturbative parameter.

The sign of the velocity depends on the environment density, with drift appearing at density 1/2.

Abstract

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 http://arxiv.org/abs/1602.06322. 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.