On sensitivity of uniform mixing times

TL;DR

This paper demonstrates that small bounded perturbations in edge weights can significantly increase the uniform mixing time of simple random walks on bounded degree graphs, answering longstanding questions in the field.

Contribution

It establishes that the $L_{ ext{infinity}}$-mixing time can increase by a factor of $ heta(\log \log n)$ due to bounded edge weight perturbations, resolving a conjecture of Kozma.

Findings

Mixing time can increase by a factor of $ heta(\log \log n)$ due to bounded perturbations.

The result is optimal, matching the known lower bounds.

Answers an open question and confirms a conjecture of Kozma.

Abstract

We show that the order of the $L_{\infty}$-mixing time of simple random walks on a sequence of uniformly bounded degree graphs of size $n$ may increase by an optimal factor of $\Theta( \log \log n)$ as a result of a bounded perturbation of the edge weights. This answers a question and a conjecture of Kozma.