Making Markov chains less lazy

TL;DR

This paper explores methods to improve bounds on the spectral gap of Markov chains by addressing the smallest eigenvalue, proposing alternatives to the standard lazy chain approach to achieve tighter bounds.

Contribution

It introduces an alternative approach to bounding the smallest eigenvalue, demonstrating it can significantly outperform traditional methods focused on the second-largest eigenvalue.

Findings

Using the alternative approach yields bounds on the smallest eigenvalue that are much tighter.

Examples show the potential for several orders of magnitude improvement.

The method offers a new perspective on analyzing Markov chain mixing times.

Abstract

The mixing time of an ergodic, reversible Markov chain can be bounded in terms of the eigenvalues of the chain: specifically, the second-largest eigenvalue and the smallest eigenvalue. It has become standard to focus only on the second-largest eigenvalue, by making the Markov chain "lazy". (A lazy chain does nothing at each step with probability at least 1/2, and has only nonnegative eigenvalues.) An alternative approach to bounding the smallest eigenvalue was given by Diaconis and Stroock and Diaconis and Saloff-Coste. We give examples to show that using this approach it can be quite easy to obtain a bound on the smallest eigenvalue of a combinatorial Markov chain which is several orders of magnitude below the best-known bound on the second-largest eigenvalue.