Universality of cutoff for graphs with an added random matching

TL;DR

This paper proves that adding a random perfect matching to certain bounded-degree graphs causes the simple random walk to exhibit cutoff, demonstrating a universal phenomenon in graph randomization.

Contribution

It establishes the universality of cutoff for random walks on graphs with added random perfect matchings, under specific conditions.

Findings

Random perfect matchings induce cutoff in simple random walks.

Universality holds for graphs with bounded degree and large size.

Cutoff occurs with high probability under the given conditions.

Abstract

We establish universality of cutoff for simple random walk on a class of random graphs defined as follows. Given a finite graph $G=(V,E)$ with $|V|$ even we define a random graph $ G^*=(V,E \cup E')$ obtained by picking $E'$ to be the (unordered) pairs of a random perfect matching of $V$. We show that for a sequence of such graphs $G_n$ of diverging sizes and of uniformly bounded degree, if the minimal size of a connected component of $G_n$ is at least 3 for all $n$, then the random walk on $G_n^*$ exhibits cutoff w.h.p. This provides a simple generic operation of adding some randomness to a given graph, which results in cutoff.