On the precision of the spectral profile

TL;DR

This paper analyzes the spectral profile bound for continuous-time reversible random walks, demonstrating its near-precision up to a log log factor and proving that this factor cannot be improved.

Contribution

It provides a precise characterization of the spectral profile bound's accuracy, establishing its tightness up to a log log factor in reversible settings.

Findings

Spectral profile bound is accurate up to a log log factor.

The log log factor in the bound is proven to be optimal.

The results apply to the uniform mixing time of reversible random walks.

Abstract

We examine the spectral profile bound of Goel, Montenegro and Tetali for the uniform mixing time of continuous-time random walk in reversible settings. We find that it is precise up to a log log factor, and that this log log factor cannot be improved.