Equality in Fill's spectral gap problem

TL;DR

This paper proves a strengthened version of Fill's spectral gap conjecture for the adjacent-transposition chain on symmetric groups, identifying conditions under which the spectral gap is minimized and analyzing eigenvalue multiplicities.

Contribution

It establishes that the spectral gap is minimized when the parameter vector has a neutral label, confirming a conjecture of Fill and extending previous results.

Findings

Spectral gap minimized if and only if the parameter vector has a neutral label.

Eigenvalue multiplicity equals the number of neutral labels, except in specific cases.

Confirms Fill's conjecture with a stronger characterization of minimizers.

Abstract

We study the adjacent-transposition chain on the symmetric group $\mathfrak{S}_n$ with a regular parameter vector $\vec{p} = (p_{i,j})_{i\neq j}$. Fill's spectral gap conjecture, recently resolved in the affirmative by Greaves-Zhu, states that among all regular parameter vectors, the spectral gap of the transition matrix is minimized by the uniform vector $p_{i,j}= 1/2$ for all $i\neq j$. We prove the stronger statement that among all regular parameter vectors, the spectral gap is minimized if and only if $\vec{p}$ has a neutral label, i.e., there exists $c \in [n]$ such that $p_{c,i} = 1/2$ for all $i\neq c$. Moreover, in this case, we show that the multiplicity of the second largest eigenvalue is equal to the number of neutral labels, unless the number of neutral labels is $n-2$ or $n$, in which case the multiplicity is $n-1$. This confirms a conjecture of Fill.