Giant vacant component left by a random walk in a random d-regular graph

TL;DR

This paper analyzes the structure of the unvisited vertices in a random walk on large, regular, tree-like graphs, revealing a phase transition in the size of the largest unvisited component related to a critical threshold.

Contribution

It establishes a precise phase transition for the vacant set in random regular graphs, linking it to the random interlacement process and providing explicit threshold calculations.

Findings

Existence of a critical threshold u* for the size of the largest vacant component.

Below u*, the largest component is proportional to the total number of vertices.

Above u*, the largest component is logarithmic in size.

Abstract

We study the trajectory of a simple random walk on a d-regular graph with d>2 and locally tree-like structure as the number n of vertices grows. Examples of such graphs include random d-regular graphs and large girth expanders. For these graphs, we investigate percolative properties of the set of vertices not visited by the walk until time un, where u>0 is a fixed positive parameter. We show that this so-called vacant set exhibits a phase transition in u in the following sense: there exists an explicitly computable threshold u* such that, with high probability as n grows, if u<u*, then the largest component of the vacant set has a volume of order n, and if u>u*, then it has a volume of order log(n). The critical value u* coincides with the critical intensity of a random interlacement process (introduced by Sznitman [arXiv:0704.2560]) on a d-regular tree. We also show that the random interlacement model describes the structure of the vacant set in local neighbourhoods.