Spectral gap of biased adjacent-transposition chains

TL;DR

This paper provides a precise lower bound on the spectral gap of biased adjacent-transposition Markov chains, resolving a longstanding conjecture and characterizing the vectors that minimize the spectral gap.

Contribution

It introduces a novel algebraic decomposition method to analyze the spectral gap and proves the minimal spectral gap occurs at the uniform probability vector.

Findings

Established a sharp lower bound on the spectral gap.

Proved the uniform probability vector minimizes the spectral gap.

Determined the multiplicity of the second-largest eigenvalue.

Abstract

We establish a sharp lower bound on the spectral gap of the biased adjacent-transposition Markov chain on the symmetric group. As a consequence, we resolve a longstanding conjecture of Fill, proving that among all regular probability vectors, the minimum spectral gap of the transition matrix is attained by the uniform probability vector. We also characterise the regular probability vectors attaining the minimum spectral gap and determine the exact multiplicity of the corresponding second-largest eigenvalue. Our proof relies on a novel algebraic decomposition of the transition matrix into elementary orthogonal projections.