Annals of Applied Probability

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

Jonathan Hermon and Justin Salez

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We show that the spectral gap of a general zero range process can be controlled in terms of the spectral gap for a single particle. This is in the spirit of Aldous’ famous spectral-gap conjecture for the interchange process, now resolved by Caputo et al. 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 zero-range process on any regular graph and, more generally, for any doubly stochastic jump matrix.

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Ann. Appl. Probab., Volume 29, Number 4 (2019), 2217-2229.

Received: October 2018
First available in Project Euclid: 23 July 2019

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Comparison Dirichlet form spectral gap mixing time zero range process particle system expanders


Hermon, Jonathan; Salez, Justin. A version of Aldous’ spectral-gap conjecture for the zero range process. Ann. Appl. Probab. 29 (2019), no. 4, 2217--2229. doi:10.1214/18-AAP1449.

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