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

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 $L_1(t)=o(\log n)$.