A characterization of $L_{2}$ mixing and hypercontractivity via hitting times and maximal inequalities

TL;DR

This paper provides a probabilistic characterization of the $L_2$ mixing time and hypercontractivity for reversible Markov chains using hitting times, offering new insights and robustness results.

Contribution

It introduces a hitting time-based characterization of $L_2$ mixing time and a new extremal characterization of the Log-Sobolev constant, linking analytic and probabilistic perspectives.

Findings

$L_2$ mixing time characterized by hitting times distributions.

New extremal characterization of the Log-Sobolev constant.

Robustness of $L_2$ mixing time under bounded perturbations.

Abstract

There are several works characterizing the total-variation mixing time of a reversible Markov chain in term of natural probabilistic concepts such as stopping times and hitting times. In contrast, there is no known analog for the $L_{2}$ mixing time, $\tau_{2}$ (while there are sophisticated analytic tools to bound $ \tau_2$, in general they do not determine $\tau_2$ up to a constant factor and they lack a probabilistic interpretation). In this work we show that $\tau_2$ can be characterized up to a constant factor using hitting times distributions. We also derive a new extremal characterization of the Log-Sobolev constant, $c_{\mathrm{LS}}$, as a weighted version of the spectral gap. This characterization yields a probabilistic interpretation of $c_{\mathrm{LS}}$ in terms of a hitting time version of hypercontractivity. As applications of our results, we show that (1) for every reversible Markov chain, $\tau_2$ is robust under addition of self-loops with bounded weights, and (2) for weighted nearest neighbor random walks on trees, $\tau_2 $ is robust under bounded perturbations of the edge weights.