Spectral gap of nonreversible Markov chains

TL;DR

This work defines a nonreversible spectral gap $\gamma$ as the second-smallest singular value of the generator $L=I-P$ with respect to the invariant measure $\mu$, and introduces the relaxation time $\tau=1/\gamma$ as a robust measure of convergence for empirical averages in $L^2$. It establishes that the $L^2$ concentration of empirical averages occurs when the number of steps $n$ satisfies $n\gg \tau$, and provides tight upper and lower bounds linking $\tau$ to the mixing time, the Cheeger constant, and the gaps of reversibilized chains. The paper develops a directed-path version of the path method and demonstrates through a broad suite of nonreversible examples—circulant random walks, toral walks, the Chung--Diaconis--Graham chain, a card-shuffling scheme, and random walks on groups—that empirical averages can converge substantially before full mixing. It also gives explicit formulas or sharp asymptotics for $\tau$ in several families, highlighting speedups achievable by nonreversibility and connecting spectral properties to isoperimetric-type quantities. Overall, the results provide a versatile framework for understanding convergence of nonreversible Markov chains and offer practical tools for assessing empirical-average convergence in complex stochastic systems.

Abstract

We define the spectral gap of a Markov chain on a finite state space as the second-smallest singular value of the generator of the chain, generalizing the usual definition of spectral gap for reversible chains. We then define the relaxation time of the chain as the inverse of this spectral gap, and show that this relaxation time can be characterized, for any Markov chain, as the time required for convergence of empirical averages. This relaxation time is related to the Cheeger constant and the mixing time of the chain through inequalities that are similar to the reversible case, and the path argument can be used to get upper bounds. Several examples are worked out. An interesting finding from the examples is that the time for convergence of empirical averages in nonreversible chains can often be substantially smaller than the mixing time.

Theorems & Definitions (37)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3