Bounding spectral gaps of Markov chains: a novel exact multi-decomposition technique

TL;DR

This paper introduces an exact multi-decomposition method to compute lower bounds on spectral gaps of finite reversible Markov chains, linking dynamic convergence measures with static equilibrium properties.

Contribution

The paper presents a novel exact technique for calculating spectral gap bounds, applicable to complex models like the Backgammon model and contingency tables.

Findings

Successfully evaluated the absorption time of the Backgammon model.

Connected spectral gaps with static equilibrium quantities.

Demonstrated applicability to difficult probability problems.

Abstract

We propose an exact technique to calculate lower bounds of spectral gaps of discrete time reversible Markov chains on finite state sets. Spectral gaps are a common tool for evaluating convergence rates of Markov chains. As an illustration, we successfully use this technique to evaluate the ``absorption time'' of the ``Backgammon model'', a paradigmatic model for glassy dynamics. We also discuss the application of this technique to the ``Contingency table problem'', a notoriously difficult problem from probability theory. The interest of this technique is that it connects spectral gaps, which are quantities related to dynamics, with static quantities, calculated at equilibrium.