Geodesic
Giant component and vacant set for random walk on a discrete torus
Journal of the European Mathematical Society, Tome 10 (2008) no. 1, pp. 133-172.

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We consider random walk on a discrete torus E of side-length N, in sufficiently high dimension d. We investigate the percolative properties of the vacant set corresponding to the collection of sites which have not been visited by the walk up to time uNd. We show that when u is chosen small, as N tends to infinity, there is with overwhelming probability a unique connected component in the vacant set which contains segments of length const logN. Moreover, this connected component occupies a non-degenerate fraction of the total number of sites Nd of E, and any point of E lies within distance Nβ of this component, with β an arbitrary positive number.
DOI : 10.4171/jems/106
Classification : 60-XX, 00-XX 
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     title = {Giant component and vacant set for random walk on a discrete torus},
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Itai Benjamini; Alain-Sol Sznitman. Giant component and vacant set for random walk on a discrete torus. Journal of the European Mathematical Society, Tome 10 (2008) no. 1, pp. 133-172. doi : 10.4171/jems/106. http://geodesic.mathdoc.fr/articles/10.4171/jems/106/

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