Sensitivity of mixing times, an example

TL;DR

This paper constructs a family of graphs where the mixing time of a random walk is logarithmic in the graph size, but quasi-isometries can drastically alter the ratio of size to mixing time, showing sensitivity in mixing times.

Contribution

It demonstrates that mixing times are highly sensitive to quasi-isometries, providing explicit graph examples where this ratio can grow arbitrarily slowly.

Findings

Mixing time is logarithmic in graph size for the constructed graphs.

Quasi-isometries can significantly change the ratio of graph size to mixing time.

Shows the sensitivity of mixing times to geometric transformations.

Abstract

We construct a family of growing finite bounded degree rooted graphs, $G_n$, in which the mixing time for simple random walk, starting at the root, is order $\log |G_n|$. Yet after a quasi - isometry, the ratio of $|G_n|$ over the mixing time grows arbitrarily slow to infinity.