A note on the vacant set of random walks on the hypercube and other regular graphs of high degree

TL;DR

This paper analyzes the component structure of the unvisited vertices in a random walk on high-degree regular graphs, especially hypercubes, revealing a phase transition in the size of the largest component around a specific time.

Contribution

It establishes the phase transition behavior of the vacant set in high-degree regular graphs, extending understanding beyond the hypercube to other similar graphs.

Findings

Largest component size remains large before the critical time

Largest component shrinks to logarithmic size after the critical time

Phase transition occurs around t* = n log d

Abstract

We consider a random walk on a $d$-regular graph $G$ where $d\to\infty$ and $G$ satisfies certain conditions. Our prime example is the $d$-dimensional hypercube, which has $n=2^d$ vertices. We explore the likely component structure of the vacant set, i.e. the set of unvisited vertices. Let $\Lambda(t)$ be the subgraph induced by the vacant set of the walk at step $t$. We show that if certain conditions are satisfied then the graph $\Lambda(t)$ undergoes a phase transition at around $t^*=n\log_ed$. Our results are that if $t\leq(1-\epsilon)t^*$ then w.h.p. as the number vertices $n\to\infty$, the size $L_1(t)$ of the largest component satisfies $L_1\gg\log n$ whereas if $t\geq(1+\e)t^*$ then $L_1(t)=o(\log n)$.