Asymptotics of the Spectral Gap for the Interchange Process on Large Hypercubes

TL;DR

This paper proves that on large hypercubes, the spectral gap of the interchange process asymptotically matches that of the random walk, supporting Aldous's conjecture and impacting quantum ferromagnet studies.

Contribution

It establishes the asymptotic equivalence of spectral gaps for the interchange process and random walk on large hypercubes, extending previous results to higher dimensions.

Findings

Spectral gaps are asymptotically equivalent as n approaches infinity.

Supports Aldous's conjecture for large hypercubes.

Implications for spectral gap analysis in quantum ferromagnets.

Abstract

We consider the interchange process (IP) on the $d$-dimensional, discrete hypercube of side-length $n$. Specifically, we compare the spectral gap of the IP to the spectral gap of the random walk (RW) on the same graph. We prove that the two spectral gaps are asymptotically equivalent, in the limit $n \to \infty$. This result gives further supporting evidence for a conjecture of Aldous, that the spectral gap of the IP equals the spectral gap of the RW on all finite graphs. Our proof is based on an argument invented by Handjani and Jungreis, who proved Aldous's conjecture for all trees. This also has implications for the spectral gap of the quantum Heisenberg ferromagnet.