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A proof of Fill's spectral gap conjecture

Gary R. W. Greaves, Haoran Zhu

Abstract

We prove a quantitative lower bound on the spectral gap of the adjacent-transposition chain on the symmetric group with a general probability vector. As a consequence, among all regular probability vectors, the spectral gap of the transition matrix is minimised by the uniform probability vector, i.e., $p_{i,j}\equiv {\frac 1 2}$ for all $i \ne j$. A second consequence is a uniform polynomial bound on the inverse spectral gap in the regular case. This resolves a longstanding conjecture known as Fill's Gap Problem.

A proof of Fill's spectral gap conjecture

Abstract

We prove a quantitative lower bound on the spectral gap of the adjacent-transposition chain on the symmetric group with a general probability vector. As a consequence, among all regular probability vectors, the spectral gap of the transition matrix is minimised by the uniform probability vector, i.e., for all . A second consequence is a uniform polynomial bound on the inverse spectral gap in the regular case. This resolves a longstanding conjecture known as Fill's Gap Problem.

Paper Structure

This paper contains 6 sections, 11 theorems, 41 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Transitions and projections
  3. Elementary transition matrices
  4. Elementary transitions as orthogonal projections
  5. A lemma on pairs of orthogonal projections
  6. Proof of the main theorem

Key Result

Theorem 1.1

For $n\ge 2$, we have

Theorems & Definitions (17)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4: Fill's spectral gap conjecture
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 7 more