Vacant set of random interlacements and percolation

Alain-Sol Sznitman¹

¹ ETH<br/>Departement Mathematik<br/>HG G 36.2<br/>Rämistrasse 101<br/>8092 Zürich<br/>Switzerland

Annals of mathematics, Tome 171 (2010) no. 3, pp. 2039-2087.

Résumé

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 $d \ge 7$.

Détail

DOI : 10.4007/annals.2010.171.2039