We consider random walk on a discrete torus of side-length , in sufficiently high dimension . 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 . We show that when is chosen small, as tends to infinity, there is with overwhelming probability a unique connected component in the vacant set which contains segments of length const . Moreover, this connected component occupies a non-degenerate fraction of the total number of sites of , and any point of lies within distance of this component, with an arbitrary positive number.
Cite this article
Alain-Sol Sznitman, Itai Benjamini, Giant component and vacant set for random walk on a discrete torus. J. Eur. Math. Soc. 10 (2008), no. 1, pp. 133–172DOI 10.4171/JEMS/106