## Electronic Journal of Probability

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

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

#### 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

**Permanent link to this document**

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

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 60G50: Sums of independent random variables; random walks 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50] 05C80: Random graphs [See also 60B20]

**Keywords**

Giant component vacant set random walk discrete torus

**Rights**

This work is licensed under aCreative Commons Attribution 3.0 License.

#### Citation

Windisch, David. Logarithmic Components of the Vacant Set for Random Walk on a Discrete Torus. Electron. J. Probab. 13 (2008), paper no. 28, 880--897. doi:10.1214/EJP.v13-506. https://projecteuclid.org/euclid.ejp/1464819101