Mixing times for the rook's walk via path coupling

Abstract

The mixing time of a convergent Markov chain measures the number of steps required for the state distribution to be within a prescribed distance of the stationary distribution. In this paper, we illustrate the strength of the probabilistic technique called coupling and its extension, path coupling, to bound the mixing time of Markov chains. The application studied is the rook’s walk on an $n^d$-chessboard, for which the mixing time has recently been studied using the spectral method. Our path-coupling result improves the previously obtained spectral bounds and includes an asymptotically tight upper bound in $n$ for the two-dimensional case.