Vacant set of random interlacements and percolation

Abstract

We introduce a model of random interlacements made of a countable collection of doubly infinite paths on $\mathbb{Z}^d$, $d \ge 3$. A nonnegative 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 $(\mathbb{Z}/N\mathbb{Z})^{d-1} \times \mathbb{Z}$ by simple random walk, or the set of points visited by simple random walk on the discrete torus $(\mathbb{Z}/N \mathbb{Z})^d$ at times of order $u N^d$. 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 $\mathbb{Z}^d$. 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 $d \ge 3$ and strictly positive when $d \ge 7$.

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Authors

Alain-Sol Sznitman

ETH
Departement Mathematik
HG G 36.2
Rämistrasse 101
8092 Zürich
Switzerland