Spectral Geometry of Non-Reversible Markov Chains

Abstract

This paper explores the extension of spectral geometric concepts to non-reversible Markov chains. While the spectral theory for reversible chains is well-developed, leveraging the self-adjoint nature of their generators to relate geometric properties of the state space (like conductance) to spectral properties (like the spectral gap), the non-reversible case presents significant challenges. The generators of non-reversible chains are non-self-adjoint operators, leading to complex spectra, non-orthogonal eigenvectors, and the failure of classical variational principles. This work investigates modern tools to overcome these difficulties. We focus on the role of pseudospectra in characterizing the transient behavior and stability of non-normal dynamics, which eigenvalues alone fail to capture. Furthermore, we examine the generalization of Cheeger's inequality to directed graphs, providing a link between the combinatorial structure of the graph and the spectral properties of the non-reversible generator. By analyzing the real part of the spectrum, we establish bounds on convergence rates to the stationary distribution. The theoretical framework is illustrated with a detailed analysis of a biased random walk on a cycle, showcasing how complex eigenvalues and pseudospectral effects manifest. The implications of this analysis for understanding mixing times in non-equilibrium systems are discussed, highlighting the richer and more complex dynamics exhibited by non-reversible processes compared to their reversible counterparts.