Intersection and mixing times for reversible chains

TL;DR

This paper investigates the relationship between intersection times and mixing times in reversible Markov chains, providing bounds, equivalences, and explicit formulas for various classes of chains including trees and regular graphs.

Contribution

It establishes bounds and equivalences between intersection and mixing times for reversible chains, and derives explicit formulas for expected intersection times in specific graph structures.

Findings

Total variation mixing time is bounded by expected intersection time.

On trees, intersection time and mixing time are equivalent.

Explicit eigenvalue-based expression for intersection time in transitive chains.

Abstract

Suppose X and Y are two independent irreducible Markov chains on n states. We consider the intersection time, which is the first time their trajectories intersect. We show for reversible and lazy chains that the total variation mixing time is always upper bounded by the expected intersection time taken over the worst starting states. For random walks on trees we show the two quantities are equivalent. We obtain an expression for the expected intersection time in terms of the eigenvalues for reversible and transitive chains. For such chains we also show that it is up to constants the geometric mean of n and E[I], where I is the number of intersections up to the uniform mixing time. Finally for random walks on regular graphs we obtain sharp inequalities that relate the expected intersection time to maximum hitting time and mixing time.