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

Itai Benjamini; Alain-Sol Sznitman

Journal of the European Mathematical Society (2008)

- Volume: 010, Issue: 1, page 133-172
- ISSN: 1435-9855

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topBenjamini, 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 -

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