Cutoff for rewiring dynamics on perfect matchings

Abstract

We establish cutoff for a natural random walk (RW) on the set of perfect matchings (PMs), based on “rewiring”. 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 all n-PMs. The 2-PM RW was first introduced by Diaconis and Holmes (Proc. Natl. Acad. Sci. USA 95 (1998) 14600–14602; Electron. J. Probab. 7 (2002) no. 6), seen as a RW on phylogenetic trees. They established cutoff in this case. We establish cutoff for the k-PM RW whenever 2≤k≪n. If k≫1, then the mixing time is nklogn to leading order. Diaconis and Holmes (Electron. J. Probab. 7 (2002) no. 6) relate the 2-PM RW to the random transpositions card shuffle. Ceccherini-Silberstein, Scarabotti and Tolli (J. Math. Sci. 141 (2007) 1182–1229; Harmonic Analysis on Finite Groups: Representation Theory, Gelfand Pairs and Markov Chains (2008) Cambridge Univ. Press) establish the same result using representation theory. We are the first to handle k>2. We relate the PM RW to conjugacy-invariant RWs on the permutation group by introducing a “cycle structure” for PMs, then build on work of Berestycki, Schramm, Şengül and Zeitouni (Israel J. Math. 147 (2005) 221–243; Ann. Probab. 39 (2011) 1815–1843; Probab. Theory Related Fields 173 (2019) 1197–1241) on such RWs.