Cutoff for rewiring dynamics on perfect matchings

Abstract

We establish cutoff for a natural random walk (RW) on the set of perfect matchings (PMs). An $n$-PM is a pairing of $2n$ objects. The $k$-PM RW selects $k$ pairs uniformly at random, disassociates the corresponding $2k$ objects, then chooses a new pairing on these $2k$ objects uniformly at random. The equilibrium distribution is uniform over the set of all $n$-PM. We establish cutoff for the $k$-PM RW whenever $2 \le k \ll n$. If $k \gg 1$, then the mixing time is $\tfrac nk \log n$ to leading order. The case $k = 2$ was established by Diaconis and Holmes (2002) by relating the $2$-PM RW to the random transpositions card shuffle and also by Ceccherini-Silberstein, Scarabotti and Tolli (2007, 2008) using representation theory. We are the first to handle $k>2$. Our argument builds on previous work of Berestycki, Schramm, \c{S}eng\"ul and Zeitouni (2005, 2011, 2019) regarding conjugacy-invariant RWs on the permutation group.