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Spectral gap for the signed interchange process with arbitrary sets

Gil Alon, Subhajit Ghosh

TL;DR

This work determines the spectral-gap behavior for a signed interchange process on the hyperoctahedral group $B_n$ with generators consisting of transpositions and all diagonal elements. The main result shows that the gap $\psi_{B_n}(w)$ for any nonnegative combination $w=w_T+w_N$ of transpositions and diagonal generators is captured by the minimum of $\psi_{B_n}(w,\sigma)$ over a small explicit family of irreps $\mathcal{F}_n$, with a parallel result for $w_T+w_N^-$ over $\mathcal{F}_n^{-}$. The authors prove a key lemma via restriction to $S_n$ and the Aldous-Caputo-Liggett-Richthammer theorem, and they demonstrate the necessity of these irreps by explicit constructions of weight choices that isolate each irrep as the unique minimizer. This settles a conjecture of Cesi and extends spectral-gap results from transposition-only generators to arbitrary diagonal generators, enriching the understanding of representation-theoretic control in Markov chains on groups.

Abstract

In 2020, F. Cesi introduced a random walk on the hyperoctahedral group $B_n$ and analysed its spectral gap when the allowed generators are transpositions and diagonal elements corresponding to singletons. In this paper we extend the allowed generators to transpositions and any diagonal elements, and characterise completely the set of representations from which the spectral gap arises. This settles a conjecture posed in Cesi's paper.

Spectral gap for the signed interchange process with arbitrary sets

TL;DR

This work determines the spectral-gap behavior for a signed interchange process on the hyperoctahedral group with generators consisting of transpositions and all diagonal elements. The main result shows that the gap for any nonnegative combination of transpositions and diagonal generators is captured by the minimum of over a small explicit family of irreps , with a parallel result for over . The authors prove a key lemma via restriction to and the Aldous-Caputo-Liggett-Richthammer theorem, and they demonstrate the necessity of these irreps by explicit constructions of weight choices that isolate each irrep as the unique minimizer. This settles a conjecture of Cesi and extends spectral-gap results from transposition-only generators to arbitrary diagonal generators, enriching the understanding of representation-theoretic control in Markov chains on groups.

Abstract

In 2020, F. Cesi introduced a random walk on the hyperoctahedral group and analysed its spectral gap when the allowed generators are transpositions and diagonal elements corresponding to singletons. In this paper we extend the allowed generators to transpositions and any diagonal elements, and characterise completely the set of representations from which the spectral gap arises. This settles a conjecture posed in Cesi's paper.

Paper Structure

This paper contains 4 sections, 9 theorems, 57 equations.

Table of Contents

  1. Representations of $B_n$
  2. Proof of Theorems \ref{['thm:main']} and \ref{['thm:stronger_main']}
  3. Proof of Theorem \ref{['thm:required_irreps']}
  4. Concluding remarks

Key Result

Theorem 1

Let $w=w_T+w_N$ be an element of the group ring $\mathbb{R}[B_n]$, where $w_T$ is a nonnegative combination of transpositions $(ij)\in S_n$, and $w_N$ is a nonnegative combination of the elements $s_{\{i\}}$, for $1\leq i \leq n$. Then we have

Theorems & Definitions (18)

  • Theorem : F. Cesi, 2020
  • Conjecture : F. Cesi, 2020
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • Definition 5
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 8 more