Proof of Aldous' spectral gap conjecture

TL;DR

This paper proves Aldous' spectral gap conjecture, demonstrating that the spectral gaps of the random walk and random transposition processes are equal on any graph, using a recursive and electric network reduction approach.

Contribution

It introduces a novel electric network reduction method and extends previous proofs to general graphs, confirming the conjecture universally.

Findings

Spectral gaps of random walk and transposition processes are equal on all graphs.

Electric network reduction effectively simplifies the spectral gap comparison.

The proof employs coset decompositions of permutation matrices.

Abstract

Aldous' spectral gap conjecture asserts that on any graph the random walk process and the random transposition (or interchange) process have the same spectral gap. We prove the conjecture using a recursive strategy. The approach is a natural extension of the method already used to prove the validity of the conjecture on trees. The novelty is an idea based on electric network reduction, which reduces the problem to the proof of an explicit inequality for a random transposition operator involving both positive and negative rates. The proof of the latter inequality uses suitable coset decompositions of the associated matrices on permutations.