Spectral Gap Inequality for Long-Range Random Walks

TL;DR

This paper establishes the order of the spectral gap for long-range random walks within stable law domains, providing a comparison principle that also bounds gaps in related processes.

Contribution

It introduces a comparison principle for spectral gaps that applies to long-range random walks and related particle systems, offering sharp bounds in finite boxes.

Findings

Spectral gap of order O(n^α) for long-range random walks in stable law domains.

A comparison principle for spectral gaps that is broadly applicable.

Sharp bounds on spectral gaps for exclusion and zero-range processes with long jumps.

Abstract

We show that the spectral gap of a random walk on the domain of normal attraction of an $\alpha$-stable law is of order $\mathcal O(n^{\alpha})$ when restricted to boxes of size $n$. The proof is based on a comparison principle that may be of independent interest. The comparison principle also allows to derive a sharp bound on the spectral gap of exclusion and zero-range processes with long jumps when restricted to finite boxes in terms of the gap on the complete graph.