Sensitivity of mixing times of Cayley graphs

TL;DR

This paper demonstrates that the total variation mixing time of Cayley graphs is not invariant under quasi-isometries, providing explicit constructions where mixing times diverge significantly despite bounded metric distortions.

Contribution

It introduces examples of Cayley graphs and other graphs where mixing times are highly sensitive to metric distortions, challenging assumptions of invariance under quasi-isometry.

Findings

Mixing time ratios can diverge to infinity under bounded metric distortions.

Unbounded degree Cayley graphs can exhibit non-invariance of mixing times.

Edge weight modifications can asymptotically alter mixing times from worst to best starting points.

Abstract

We show that the total variation mixing time is not quasi-isometry invariant, even for Cayley graphs. Namely, we construct a sequence of pairs of Cayley graphs with maps between them that twist the metric in a bounded way, while the ratio of the two mixing times goes to infinity. The Cayley graphs serving as an example have unbounded degrees. For non-transitive graphs we construct bounded degree graphs for which the mixing time from the worst starting point for one graph is asymptotically smaller than the mixing time from the best starting point of the random walk on a network obtained by increasing some of the edge weights from 1 to $1+o(1)$.