DMCA
Vacant set of random interlacements and percolation
| Citations: | 66 - 16 self |
BibTeX
@MISC{Sznitman_vacantset,
author = {Alain-sol Sznitman},
title = {Vacant set of random interlacements and percolation},
year = {}
}
Share
 |
 |
 |
 |
||
OpenURL
Abstract
We introduce a model of random interlacements made of a countable collection of doubly infinite paths on Z d, d ≥ 3. A non-negative parameter u measures how many trajectories enter the picture. This model describes in the large N limit the microscopic structure in the bulk, which arises when considering the disconnection time of a discrete cylinder (Z/NZ) d−1 × Z by simple random walk, or the set of points visited by simple random walk on the discrete torus (Z/NZ) d at times of order uN d. In particular we study the percolative properties of the vacant set left by the interlacement at level u, which is an infinite connected translation invariant random subset of Z d. We introduce a critical value u ∗ such that the vacant set percolates for u < u ∗ and does not percolate for u> u∗. Our main results show that u ∗ is finite when d ≥ 3 and strictly positive when d ≥ 7.
Keyphrases
random interlacements vacant set simple random walk infinite path order un countable collection discrete torus vacant set percolate microscopic structure many trajectory disconnection time main result non-negative parameter large limit discrete cylinder critical value percolative property
