Spectral gap of the symmetric inclusion process

TL;DR

This paper establishes bounds for the spectral gap of the symmetric inclusion process on finite graphs, linking it to the spectral gap of the underlying random walk, and extends results to related diffusion systems.

Contribution

It provides universal bounds for the spectral gap of the symmetric inclusion process and connects these bounds to the spectral gap of the graph's random walk, extending Aldous' conjecture.

Findings

Bounds for spectral gap in terms of random walk spectral gap

Matching bounds in the log-concave reversible measure regime

Extension to Brownian energy process via duality

Abstract

We consider the symmetric inclusion process on a general finite graph. Our main result establishes universal upper and lower bounds for the spectral gap of this interacting particle system in terms of the spectral gap of the random walk on the same graph. In the regime in which the gamma-like reversible measures of the particle systems are log-concave, our bounds match, yielding a version for the symmetric inclusion process of the celebrated Aldous' spectral gap conjecture originally formulated for the interchange process. Finally, by means of duality techniques, we draw analogous conclusions for an interacting diffusion-like unbounded conservative spin system known as Brownian energy process.