Phase transition for random walks on graphs with added weighted random matching

TL;DR

This paper investigates the cutoff phenomenon for weighted random walks on graphs modified by adding a weighted perfect matching, establishing conditions under which cutoff occurs depending on graph growth and structure.

Contribution

It provides new criteria for cutoff in weighted random walks on graphs with added matchings, especially for polynomial growth, vertex-transitive, and expander graphs.

Findings

Log(1/εₙ) ≪ log|Vₙ| is sufficient for cutoff in polynomial growth graphs.

Necessary condition for cutoff in vertex-transitive graphs is log(1/εₙ) ≪ log|Vₙ|.

Complete characterization of cutoff regimes for expander graphs.

Abstract

For a finite graph $G=(V,E)$ let $G^*$ be obtained by considering a random perfect matching of $V$ and adding the corresponding edges to $G$ with weight $\varepsilon$, while assigning weight 1 to the original edges of $G$. We consider whether for a sequence $(G_n)$ of graphs with bounded degrees and corresponding weights $(\varepsilon_n)$, the (weighted) random walk on $(G_n^*)$ has cutoff. For graphs with polynomial growth we show that $\log\left(\frac{1}{\varepsilon_n}\right)\ll\log|V_n|$ is a sufficient condition for cutoff. Under the additional assumption of vertex-transitivity we establish that this condition is also necessary. For graphs where the entropy of the simple random walk grows linearly up to some time of order $\log|V_n|$ we show that $\frac{1}{\varepsilon_n}\ll\log|V_n|$ is sufficient for cutoff. In case of expander graphs we also provide a complete picture for the complementary regime $\frac{1}{\varepsilon_n}\gtrsim\log|V_n|$.