Percolation for the vacant set of random interlacements

✍ Scribed by Vladas Sidoravicius; Alain-Sol Sznitman

• Publisher: John Wiley and Sons
• Year: 2009
• Tongue: English
• Weight: 262 KB
• Volume: 62
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/cpa.20267

✦ Synopsis

Abstract

We investigate random interlacements on ℤ^d^, d ≥ 3. This model, recently introduced in [8], corresponds to a Poisson cloud on the space of doubly infinite trajectories modulo time shift tending to infinity at positive and negative infinite times. A nonnegative parameter u measures how many trajectories enter the picture. Our main interest lies in the percolative properties of the vacant set left by random interlacements at level u. We show that for all d ≥ 3 the vacant set at level u percolates when u is small. This solves an open problem of [8], where this fact has only been established when d ≥ 7. It also completes the proof of the nondegeneracy in all dimensions d ≥ 3 of the critical parameter u~*~ of [8]. © 2008 Wiley Periodicals, Inc.