Abstract

We consider random walks on two classes of random graphs and explore the likely structure of the the set of unvisited vertices (or vacant set). Let Γ(t) be the subgraph induced by the vacant set. We show that for random graphs Gn,p above the connectivity threshold, and for random regular graphs Gr, for constant r ≥ 3, there is a phase transition in the sense of the well-known Erdős-Renyi phase transition. Thus for t ≤ (1 − ∊)t* we have a unique giant plus components of size O(log n) and for t ≥ (1 + ∊)t* we have only components of size O(log n). In the case of Gr we describe the likely degree sequence and structure of the small (O(log n)) size components.

Highlights

  • The problem we consider can be described as follows

  • What is the likely component structure induced by the unvisited vertices of G at step t of the walk?

  • Equivalently add random edges, we find that the maximum component size increases from logarithmic size for p = c/n, c < 1 and after the phase transition for c > 1 there is jump in maximum component size to linear in n and all other components have logarithmic size

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Summary

Introduction

The problem we consider can be described as follows. We have a finite graph G = (V, E), and a simple random walk Wu on G, starting at u ∈ V. Paper [5] deals with random walks on Gr, and shows that whp Γ(t) is sub-critical for t ≥ (1 + )t∗ and there is a unique linear size component for t ≤ (1 − )t∗. Consider a random walk on Gr. Conditional on N = |R(t)| and R(t) having degree sequence d = dΓ(t)(v), v ∈ R(t), Γ(t) is distributed as GN,d, the random graph with vertex set [N ] and degree sequence d. The following lemma gives the probability that a walk, starting from near stationarity makes a first visit to vertex v at a given step. Let X be the walk on T , starting from v, and let Xt be the distance of X from the root vertex at step t.

The vertices outside
We write
Thus fT
Then r
Define α to be the smallest positive solution to r
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