Mixing times and moving targets

TL;DR

This paper investigates the relationship between mixing times and hitting times of moving targets in finite Markov chains, revealing equivalences in certain structures and counterexamples in others, thus resolving an open problem.

Contribution

It establishes the equivalence of mixing times and maximum hitting times for moving targets in finite Markov chains and provides a counterexample on a transitive graph.

Findings

Maximum hitting time of moving targets equals that of stationary targets on the d-dimensional torus.

Constructs a transitive graph where these two hitting times differ.

Resolves an open question of Aldous and Fill regarding a 'cat and mouse' game.

Abstract

We consider irreducible Markov chains on a finite state space. We show that the mixing time of any such chain is equivalent to the maximum, over initial states $x$ and moving large sets $(A_s)_s$, of the hitting time of $(A_s)_s$ starting from $x$. We prove that in the case of the $d$-dimensional torus the maximum hitting time of moving targets is equal to the maximum hitting time of stationary targets. Nevertheless, we construct a transitive graph where these two quantities are not equal, resolving an open question of Aldous and Fill on a "cat and mouse" game.