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Cutoff for the inversion walk on tournaments and the state space of restricted inversions

Jiangdong Ai

Abstract

Given a labelled tournament on $[n]$, \emph{inverting} a vertex subset $X$ means reversing every edge with both endpoints in $X$. Alon, Powierski, Savery, Scott, and Wilmer~\cite{AlonPowierskiSaveryScottWilmer2024} asked for the mixing time of the Markov chain that repeatedly inverts a uniformly random subset of $[n]$. We show that this \emph{inversion walk} exhibits total-variation cutoff at time $n$. More precisely, there is a universal constant $C>0$ such that for all $c\ge 0$, $ d_n(n+c)\le C\,2^{-c}, $ while for all $s\in\{0,1,\dots,n\}$, $ d_n(n-s)\ge 1-2^{\,n+s\logtwo n-\binom{s}{2}}. $

Cutoff for the inversion walk on tournaments and the state space of restricted inversions

Abstract

Given a labelled tournament on , \emph{inverting} a vertex subset means reversing every edge with both endpoints in . Alon, Powierski, Savery, Scott, and Wilmer~\cite{AlonPowierskiSaveryScottWilmer2024} asked for the mixing time of the Markov chain that repeatedly inverts a uniformly random subset of . We show that this \emph{inversion walk} exhibits total-variation cutoff at time . More precisely, there is a universal constant such that for all , while for all ,
Paper Structure (13 sections, 9 theorems, 39 equations)

This paper contains 13 sections, 9 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Upper tail.
  3. Lower tail.
  4. Preliminaries
  5. Upper tail: spectral decay after time $n$
  6. Lower tail: inversion balls and a rank tail inequality
  7. Cutoff at time $n$
  8. Restricted inversions
  9. Parity invariants
  10. Boundary cases
  11. The main regime $2\le k\le n-2$
  12. The case $k=2$: the hypercube
  13. Discussion and open problems

Key Result

Theorem 1.1

There exists a universal constant $C>0$ such that for all $n$: In particular, $d_n(t)\to 1$ for $t\le n-(\sqrt{2}+\varepsilon)\sqrt{n}$ and $d_n(t)\to 0$ for $t\ge n+\omega(1)$ as $n\to\infty$. Thus $\{W_n\}$ exhibits total-variation cutoff at time $n$, with a pre-cutoff window of order at most $O(\sqrt{n})$.

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: Inversion distance
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Lemma 4.1: Rank tail for random symmetric matrices AlonPowierskiSaveryScottWilmer2024
  • ...and 11 more