Logarithmic Components of the Vacant Set for Random Walk on a Discrete Torus

David Windisch

doi:10.1214/ejp.v13-506

Abstract

This work continues the investigation, initiated in a recent work by Benjamini and Sznitman, of percolative properties of the set of points not visited by a random walk on the discrete torus $({\mathbb Z}/N{\mathbb Z})^d$ up to time $uN^d$ in high dimension $d$. If $u>0$ is chosen sufficiently small it has been shown that with overwhelming probability this vacant set contains a unique giant component containing segments of length $c_0 \log N$ for some constant $c_0 > 0$, and this component occupies a non-degenerate fraction of the total volume as $N$ tends to infinity. Within the same setup, we investigate here the complement of the giant component in the vacant set and show that some components consist of segments of logarithmic size. In particular, this shows that the choice of a sufficiently large constant $c_0>0$ is crucial in the definition of the giant component.

Highlights

In a recent work by Benjamini and Sznitman [1], the authors consider a simple random walk on the d-dimensional integer torus E = (Z/N Z)d for a sufficiently large dimension d and investigate properties of the set of points in the torus not visited by the walk after [uN d] steps for a sufficiently small parameter u > 0 and large N
Among other properties of this so-called vacant set, the authors of [1] find that for a suitably defined dimension-dependent constant c0 > 0, there is a unique component of the vacant set containing segments of length at least [c0 log N ] with probability tending to 1 as N tends to infinity, provided u > 0 is chosen small enough
This component is referred to as the giant component. It is shown in [1] that with overwhelming probability, the giant component is at |.|∞-distance of at most N β from any point and occupies at least a constant fraction γ of the total volume of the torus for arbitrary β, γ ∈ (0, 1), when u > 0 is chosen sufficiently small

Summary

Introduction

Among other properties of this so-called vacant set, the authors of [1] find that for a suitably defined dimension-dependent constant c0 > 0, there is a unique component of the vacant set containing segments of length at least [c0 log N ] with probability tending to 1 as N tends to infinity, provided u > 0 is chosen small enough This component is referred to as the giant component. The proof of Lemma 3.3 uses a result on the ubiquity of segments of logarithmic size in the vacant set from [1] From this ubiquity result, we know that for any β > 0, with overwhelming probability, there is a segment of length l in the vacant set left until the beginning of every considered subinterval ( even until [uN d] for small u > 0) in the N β-neighborhood of any point. The numbered constants c0, c1, c2, c3, c4 are fixed and refer to their first place of appearance in the text

Some definitions and useful results

Profusion of logarithmic components until time a1

Nc log

Survival of a logarithmic segment

Proof of the main result