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

TL;DR

This paper proves that the inversion walk on tournaments exhibits a sharp cutoff at time n, with precise bounds on the total variation distance, and characterizes the state space of the k-restricted inversion walk as a coset of a subgroup.

Contribution

It establishes the cutoff phenomenon for the inversion walk on tournaments and describes the algebraic structure of the state space for the k-restricted inversion walk.

Findings

Inversion walk undergoes total-variation cutoff at time n.

Upper tail decays exponentially within O(1) above n.

Lower tail threshold is within O(√n) below n.

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} undergoes 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 $s\in\{0,1,\dots,n\}$, $d_n(n-s)\ge 1-2^{\,n+s\logtwo n-\binom{s}{2}}$. In particular, the lower tail threshold lies within $O(\sqrt{n})$ below $n$, while the upper tail decays within $O(1)$ above $n$. As a second result, we characterise the state space of the \emph{$k$-restricted inversion walk}, which inverts a uniformly random $k$-subset at each step. For $n\ge 4$ and $2\le k\le n-2$, the reachable states form a coset of a subgroup $H_k\le\F_2^{\binom{n}{2}}$ whose codimension is determined solely by $k\bmod 4$.