## Electronic Journal of Probability

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

David Windisch

#### 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.

#### Article information

Source
Electron. J. Probab., Volume 13 (2008), paper no. 28, 880-897.

Dates
Accepted: 9 May 2008
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464819101

Digital Object Identifier
doi:10.1214/EJP.v13-506

Mathematical Reviews number (MathSciNet)
MR2413287

Zentralblatt MATH identifier
1191.60118

Rights