×

zbMATH — the first resource for mathematics

Local percolative properties of the vacant set of random interlacements with small intensity. (English. French summary) Zbl 1319.60180
Summary: Random interlacements at level \(u\) is a one parameter family of connected random subsets of \(\mathbb{Z}^{d}\), \(d\geq3\) [A.-S. Sznitman, Ann. Math. (2) 171, No. 3, 2039–2087 (2010; Zbl 1202.60160)]. Its complement, the vacant set at level \(u\), exhibits a non-trivial percolation phase transition in \(u\) (see [loc. cit.; V. Sidoravicius and A.-S. Sznitman, Commun. Pure Appl. Math. 62, No. 6, 831–858 (2009; Zbl 1168.60036)]), and the infinite connected component, when it exists, is almost surely unique [A. Teixeira, Ann. Appl. Probab. 19, No. 1, 454–466 (2009; Zbl 1158.60046)].
In this paper we study local percolative properties of the vacant set of random interlacements at level \(u\) for all dimensions \(d\geq 3\) and small intensity parameter \(u>0\). We give a stretched exponential bound on the probability that a large (hyper)cube contains two distinct macroscopic components of the vacant set at level \(u\). In particular, this implies that finite connected components of the vacant set at level \(u\) are unlikely to be large. These results are new for \(d\in \{3,4\}\). The case of \(d\geq 5\) was treated in [A. Teixeira, Probab. Theory Relat. Fields 150, No. 3–4, 529–574 (2011; Zbl 1231.60117)] by a method that crucially relies on a certain “sausage decomposition” of the trace of a high-dimensional bi-infinite random walk. Our approach is independent from that of Teixeira [Zbl 1231.60117]. It only exploits basic properties of random walks, such as Green function estimates and Markov property, and, as a result, applies also to the more challenging low-dimensional cases. One of the main ingredients in the proof is a certain conditional independence property of the random interlacements, which is interesting in its own right.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G50 Sums of independent random variables; random walks
82B43 Percolation
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] I. Benjamini and A.-S. Sznitman. Giant component and vacant set for random walk on a discrete torus. J. Eur. Math. Soc. 10 (2008) 133-172. · Zbl 1141.60057 · doi:10.4171/JEMS/106
[2] J. T. Chayes, L. Chayes and C. M. Newman. Bernoulli percolation above threshold: An invasion percolation analysis. Ann. Probab. 15 (1987) 1272-1287. · Zbl 0627.60099 · doi:10.1214/aop/1176991976
[3] A. Drewitz, B. Ráth and A. Sapozhnikov. On chemical distances and shape theorems in percolation models with long-range correlations. Preprint. Available at . 1212.2885 · Zbl 1301.82027 · arxiv.org
[4] G. R. Grimmett. Percolation , 2nd edition. Springer-Verlag, Berlin, 1999. · Zbl 0926.60004
[5] H. Kesten. Aspects of first-passage percolation. In École d’été de Probabilité de Saint-Flour XIV 125-264. Lecture Notes in Math. 1180 . Springer-Verlag, Berlin, 1986. · Zbl 0602.60098
[6] H. Kesten and Y. Zhang. The probability of a large finite cluster in supercritical Bernoulli percolation. Ann. Probab. 18 (1990) 537-555. · Zbl 0705.60092 · doi:10.1214/aop/1176990844
[7] G. Lawler. A self-avoiding random walk. Duke Math. J. 47 (1980) 655-693. · Zbl 0445.60058 · doi:10.1215/S0012-7094-80-04741-9
[8] G. Lawler. Intersections of Random Walks . Birkhäuser, Basel, 1991. · Zbl 1228.60004
[9] B. Ráth and A. Sapozhnikov. The effect of small quenched noise on connectivity properties of random interlacements. Electron. J. Probab. 18 (2013) Article 4 1-20. · Zbl 1347.60132
[10] V. Sidoravicius and A.-S. Sznitman. Percolation for the vacant set of random interlacements. Comm. Pure Appl. Math. 62 (2009) 831-858. · Zbl 1168.60036 · doi:10.1002/cpa.20267
[11] A.-S. Sznitman. Vacant set of random interlacements and percolation. Ann. Math. 171 (2010) 2039-2087. · Zbl 1202.60160 · doi:10.4007/annals.2010.171.2039 · annals.princeton.edu
[12] A.-S. Sznitman. Decoupling inequalities and interlacement percolation on \(G\times\mathbb{Z}\). Invent. Math. 187 (2012) 645-706. · Zbl 1277.60183 · doi:10.1007/s00222-011-0340-9
[13] A. Teixeira. On the uniqueness of the infinite cluster of the vacant set of random interlacements. Ann. Appl. Probab. 19 (2009) 454-466. · Zbl 1158.60046 · doi:10.1214/08-AAP547
[14] A. Teixeira. On the size of a finite vacant cluster of random interlacements with small intensity. Probab. Theory Related Fields 150 (2011) 529-574. · Zbl 1231.60117 · doi:10.1007/s00440-010-0283-x
[15] A. Teixeira and D. Windisch. On the fragmentation of a torus by random walk. Comm. Pure Appl. Math. 64 (2011) 1599-1646. · Zbl 1235.60143 · doi:10.1002/cpa.20382
[16] Á. Tímár. Boundary-connectivity via graph theory. Proc. Amer. Math. Soc. 141 (2013) 475-480.
[17] D. Windisch. Random walk on a discrete torus and random interlacements. Electron. Commun. Probab. 13 (2008) 140-150. · Zbl 1187.60089 · doi:10.1214/ECP.v13-1359 · emis:journals/EJP-ECP/_ejpecp/ECP/viewarticlee04e.html · eudml:231549
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.