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

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.