A version of Aldous' spectral-gap conjecture for the zero range process

TL;DR

This paper establishes a bound on the spectral-gap of zero range processes based on the spectral-gap of a single particle, extending previous results to arbitrary geometries and providing exact estimates for regular graphs.

Contribution

It introduces a new inequality that decouples geometry from kinetics, generalizing spectral-gap estimates beyond complete graphs and tori.

Findings

Spectral-gap of zero range process can be controlled by single particle spectral-gap.

Extends spectral-gap estimates to arbitrary geometries.

Determines the order of magnitude of spectral-gap on regular graphs.

Abstract

We show that the spectral-gap of a general zero range process can be controlled in terms of the spectral-gap of a single particle. This is in the spirit of Aldous' famous spectral-gap conjecture for the interchange process. Our main inequality decouples the role of the geometry (defined by the jump matrix) from that of the kinetics (specified by the exit rates). Among other consequences, the various spectral-gap estimates that were so far only available on the complete graph or the $d$-dimensional torus now extend effortlessly to arbitrary geometries. As an illustration, we determine the exact order of magnitude of the spectral-gap of the rate-one $\textbf{ZRP}$ on any regular graph and, more generally, for any doubly stochastic jump matrix.