Mixing Times and Cutoff for the Rook's Walk

TL;DR

This paper analyzes the mixing time of the Rook's Walk Markov chain on a high-dimensional chessboard, proving it exhibits cutoff and providing precise bounds through spectral analysis and state space lumping.

Contribution

It introduces a novel lumping technique to analyze the Rook's Walk, identifies eigenvalues and eigenfunctions, and establishes cutoff behavior with sharpened bounds.

Findings

The Rook's Walk has the same mixing time as its lumped birth-death chain.

The paper identifies all eigenvalues and eigenfunctions of the projected chain.

It proves the Rook's Walk exhibits cutoff, indicating sharp convergence to stationarity.

Abstract

We study the mixing time of the Rook's Walk Markov chain on a $d$-dimensional chess board of side length $n\geq 3$, where a rook moves by first selecting an axis uniformly at random and then selecting a new position along that axis uniformly from among the $n-1$ unoccupied alternatives. Our method is to lump the state space of the Rook's Walk by Hamming distance, yielding a birth-death Markov chain. We prove that this lumped birth-death chain has the same mixing time as the Rook's Walk and identify all eigenvalues and eigenfunctions of the projected chain. We then combine the eigenfunction lower bound approach of Wilson (2004) with an $L^2$ upper bound to obtain new sharpened bounds on the mixing time of the Rook's Walk. As a consequence, we show that the Rook's Walk Markov chain exhibits cutoff.