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2008 Logarithmic Components of the Vacant Set for Random Walk on a Discrete Torus
David Windisch
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Electron. J. Probab. 13: 880-897 (2008). DOI: 10.1214/EJP.v13-506


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.


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David Windisch. "Logarithmic Components of the Vacant Set for Random Walk on a Discrete Torus." Electron. J. Probab. 13 880 - 897, 2008.


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

zbMATH: 1191.60118
MathSciNet: MR2413287
Digital Object Identifier: 10.1214/EJP.v13-506

Primary: 60K35
Secondary: 05C80 , 60G50 , 82C41

Keywords: discrete torus , Giant component , Random walk , Vacant set

Vol.13 • 2008
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