Mixing and hitting times for finite Markov chains

TL;DR

This paper establishes a close relationship between the mixing time of finite reversible Markov chains and the largest expected hitting time of certain subsets, with extensions to non-reversible and discrete-time chains.

Contribution

It introduces a new connection between mixing times and hitting times, including a construction of a special random set, extending known results to broader settings.

Findings

Mixing time is comparable to the largest expected hitting time of a subset with stationary measure at least .

Results extend to discrete-time and non-reversible Markov chains.

Similar results were independently obtained by other researchers.

Abstract

Let 0<\alpha<1/2. We show that the mixing time of a continuous-time reversible Markov chain on a finite state space is about as large as the largest expected hitting time of a subset of stationary measure at least \alpha of the state space. Suitably modified results hold in discrete time and/or without the reversibility assumption. The key technical tool is a construction of a random set A such that the hitting time of A is both light-tailed and a stationary time for the chain. We note that essentially the same results were obtained independently by Peres and Sousi [arXiv:1108.0133].