Giant component and vacant set for random walk on a discrete torus

Benjamini, Itai, and Sznitman, Alain-Sol. "Giant component and vacant set for random walk on a discrete torus." Journal of the European Mathematical Society 010.1 (2008): 133-172. <http://eudml.org/doc/277578>.

@article{Benjamini2008,
abstract = {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 $uN^d$. 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 $\log N$. Moreover, this connected component occupies a non-degenerate fraction of the total number of sites $N^d$ of $E$, and any point of $E$ lies within distance $N^\beta $ of this component, with $\beta $ an arbitrary positive number.},
author = {Benjamini, Itai, Sznitman, Alain-Sol},
journal = {Journal of the European Mathematical Society},
keywords = {probability theory; random walk; percolation; stochastic processes; coupon collector; probability theory; random walk; percolation; stochastic processes; coupon collector},
language = {eng},
number = {1},
pages = {133-172},
publisher = {European Mathematical Society Publishing House},
title = {Giant component and vacant set for random walk on a discrete torus},
url = {http://eudml.org/doc/277578},
volume = {010},
year = {2008},
}

TY - JOUR
AU - Benjamini, Itai
AU - Sznitman, Alain-Sol
TI - Giant component and vacant set for random walk on a discrete torus
JO - Journal of the European Mathematical Society
PY - 2008
PB - European Mathematical Society Publishing House
VL - 010
IS - 1
SP - 133
EP - 172
AB - 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 $uN^d$. 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 $\log N$. Moreover, this connected component occupies a non-degenerate fraction of the total number of sites $N^d$ of $E$, and any point of $E$ lies within distance $N^\beta $ of this component, with $\beta $ an arbitrary positive number.
LA - eng
KW - probability theory; random walk; percolation; stochastic processes; coupon collector; probability theory; random walk; percolation; stochastic processes; coupon collector
UR - http://eudml.org/doc/277578
ER -