Spectral gap for the interchange process in a box

TL;DR

This paper proves that the spectral gap for the interchange and symmetric exclusion processes in a d-dimensional box approaches π²/L², supporting Aldous's conjecture that the spectral gaps are equal for these processes on any graph.

Contribution

It establishes the asymptotic behavior of the spectral gap in a box, providing evidence for Aldous's conjecture in a new setting.

Findings

Spectral gap asymptotic to π²/L² in a box

Supports Aldous's conjecture for general graphs

Uses a novel proof technique similar to prior methods

Abstract

We show that the spectral gap for the interchange process (and the symmetric exclusion process) in a $d$-dimensional box of side length $L$ is asymptotic to $\pi^2/L^2$. This gives more evidence in favor of Aldous's conjecture that in any graph the spectral gap for the interchange process is the same as the spectral gap for a corresponding continuous-time random walk. Our proof uses a technique that is similar to that used by Handjani and Jungreis, who proved that Aldous's conjecture holds when the graph is a tree.