Equality in Fill's spectral gap problem

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.