Phase transition for the vacant set left by random walk on the giant component of a random graph

TL;DR

This paper investigates the phase transition in the structure of the vacant set left by a simple random walk on the giant component of a supercritical Erdős-Rényi graph, identifying a critical parameter where the component structure changes dramatically.

Contribution

It establishes the existence of a phase transition in the vacant set's component structure and links the critical parameter to that of random interlacements on a Poisson-Galton-Watson tree.

Findings

For u < u_* the vacant set has a unique giant component of order n.

For u > u_* all components of the vacant set are small.

The critical parameter u_* matches that of random interlacements on a Poisson-Galton-Watson tree.

Abstract

We study the simple random walk on the giant component of a supercritical Erd\H{o}s-R\'enyi random graph on $n$ vertices, in particular the so-called vacant set at level $u$, the complement of the trajectory of the random walk run up to a time proportional to $u$ and $n$. We show that the component structure of the vacant set exhibits a phase transition at a critical parameter $u_{\star}$: For $u<u_{\star}$ the vacant set has with high probability a unique giant component of order $n$ and all other components small, of order at most $\log^{7}n$, whereas for $u>u_{\star}$ it has with high probability all components small. Moreover, we show that $u_{\star}$ coincides with the critical parameter of random interlacements on a Poisson-Galton-Watson tree, which was identified in [Tas10].