Spectral gap of the KMP and other stochastic exchange models on arbitrary graphs

TL;DR

This paper develops universal bounds on the spectral gap for reversible stochastic exchange models on arbitrary graphs, including KMP, HP, and IEM, using a simple strategy that depends on graph-independent features.

Contribution

It introduces a universal bounding strategy for spectral gaps applicable to various models on arbitrary graphs, extending previous methods.

Findings

Bounds are sharp for KMP, HP, and IEM.

Bounds saturate in certain regimes for HP and IEM.

Method applies even in non-reversible contexts.

Abstract

We present a simple strategy to derive universal bounds on the spectral gap of reversible stochastic exchange models on arbitrary graphs. The Kipnis-Marchioro-Presutti (KMP) model, the harmonic process (HP), and the immediate exchange model (IEM) are all examples that fall into this class. Our upper and lower bounds depend only on two features: worst-case linear statistics and a kinetic factor, which is, in essence, graph-independent. For the three aforementioned examples, these bounds are sharp, and even saturate to an identity for HP and IEM in some log-concave regimes. The proof -- which yields bounds for eigenvalues even in the non-reversible context -- crucially exploits the rigidity of the eigenstructure of these models and quantitative contraction rates of the corresponding hidden parameter models recently introduced in [DMFG24, GRT25].