Connectivity Bounds for the Vacant Set of Random Interlacements

2009/08/17 by Vladas Sidoravicius, Alain-Sol Sznitman
Mathematics · Physics and Astronomy · #math.PR #math-ph #math.MP #msc:60K35 #msc:60G50 #msc:82C41

paper · pdf · doi:10.1214/09-aihp335

published as Ann. Inst. H. Poincare Probab. Statist., 2010, Vol. 46, No. 4, 976-990 · 16 pages, 1 figure, accepted for publication in Ann. Inst. H. Poincare

arxiv created 2009/08/17 · arxiv updated 2015/05/13

Abstract

The model of random interlacements on Z^d, d bigger or equal to 3, was recently introduced in arXiv:0704.2560. A non-negative parameter u parametrizes the density of random interlacements on Z^d. In the present note we investigate the connectivity properties of the vacant set left by random interlacements at level u, in the non-percolative regime, where u is bigger than the non-degenerate critical parameter for percolation of the vacant set, see arXiv:0704.2560, arXiv:0808.3344. We prove a stretched exponential decay of the connectivity function for the vacant set at level u, when u is bigger than an other critical parameter. It is presently an open problem whether these two critical parameters actually coincide.

Citations

• Vacant Set of Random Interlacements and Percolation
• Percolation for the Vacant Set of Random Interlacements