Spectral gap and curvature of monotone Markov chains

TL;DR

This paper establishes a precise relationship between the spectral gap and Ollivier-Ricci curvature for monotone Markov chains, providing new formulas and explicit calculations for complex non-reversible processes.

Contribution

It proves the equivalence of spectral gap and optimal Ollivier-Ricci curvature for monotone Markov chains and introduces a practical local variation formula.

Findings

Spectral gap equals optimal Ollivier-Ricci curvature for monotone chains

New local variation expression simplifies curvature and spectral gap analysis

Explicit curvature and spectral gap for non-reversible exclusion processes

Abstract

We prove that the absolute spectral gap of any monotone Markov chain coincides with its optimal Ollivier-Ricci curvature, where the word `optimal' refers to the choice of the underlying metric. Moreover, we provide a new expression in terms of local variations of increasing functions, which has several practical advantages over the traditional variational formulation using the Dirichlet form. As an illustration, we explicitly determine the optimal curvature and spectral gap of the non-conservative exclusion process with heterogeneous reservoir densities on any network, despite the lack of reversibility.