# zbMATH — the first resource for mathematics

Transience of the vacant set for near-critical random interlacements in high dimensions. (English. French summary) Zbl 1333.60202
Summary: The model of random interlacements is a one-parameter family $$\mathcal{I}^{u}$$, $$u\geq0$$, of random subsets of $$\mathbb{Z}^{d}$$, which locally describes the trace of a simple random walk on a $$d$$-dimensional torus run up to time $$u$$ times its volume. Its complement, the so-called vacant set $$\mathcal{V}^{u}$$, has been shown to undergo a non-trivial percolation phase-transition in $$u$$, i.e., there exists $$u_{*}(d)\in(0,\infty)$$ such that for $$u\in[0,u_{*}(d))$$ the vacant set $$\mathcal{V}^{u}$$ contains a unique infinite connected component $$\mathcal{V}_{\infty}^{u}$$, while for $$u>u_{*}(d)$$ it consists of finite connected components. It is known [A.-S, Sznitman, Probab. Theory Relat. Fields 150, No. 3–4, 575–611 (2011; Zbl 1230.60103); Ann. Probab. 39, No. 1, 70–103 (2011; Zbl 1210.60047)] that $$u_{*}(d) \sim \log d$$, and in this article we show the existence of $$u(d)>0$$ with $$\frac{u(d)}{u_{*}(d)}\to1$$ as $$d\to\infty$$ such that $$\mathcal{V}_{\infty}^{u}$$ is transient for all $$u\in[0,u(d))$$.
##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 82B43 Percolation
Full Text:
##### References:
 [1] O. Angel, I. Benjamini, N. Berger and Y. Peres. Transience of percolation clusters on wedges. Electron. J. Probab. 11 (2006) 655-669. · Zbl 1109.60062 [2] R. M. Burton and M. Keane. Density and uniqueness in percolation. Comm. Math. Phys. 121 (3) (1989) 501-505. · Zbl 0662.60113 [3] J. Černý and S. Popov. On the internal distance in the interlacement set. Electron. J. Probab. 17 (2012) 29. · Zbl 1245.60090 [4] A. Drewitz, B. Ráth and A. Sapozhnikov. Local percolative properties of the vacant set of random interlacements with small intensity. Ann. Inst. Henri Poincaré Probab. Stat. 50 (4) (2014) 1164-1197. · Zbl 1201.60099 [5] A. Drewitz, B. Ráth and A. Sapozhnikov. On chemical distances and shape theorems in percolation models with long-range correlations. J. Math. Phys. 55 (2014) 083307. · Zbl 1301.82027 [6] H. Lacoin and J. Tykesson. On the easiest way to connect $$k$$ points in the random interlacements process. ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013) 505-524. · Zbl 1277.60177 [7] R. Lyons and Y. Peres. Probability on Trees and Networks . Cambridge Univ. Press, Cambridge. Current version available at . · Zbl 1376.05002 [8] Y. Peres. Probability on trees: An introductory climb. In Lectures on Probability Theory and Statistics 193-280. Lecture Notes in Math. 1717 . Springer, Berlin, 1999. · Zbl 0957.60001 [9] S. Popov and A. Teixeira. Soft local times and decoupling of random interlacements. J. Eur. Math. Soc. (JEMS) 17 (10) (2015) 2545-2593. · Zbl 1329.60342 [10] E. Procaccia, R. Rosenthal and A. Sapozhnikov. Quenched invariance principle for simple random walk on clusters in correlated percolation models. Probab. Theory Related Fields . To appear. Available at . arXiv:1310.4764 · Zbl 1353.60034 [11] E. Procaccia and J. Tykesson. Geometry of the random interlacement. Electron. Commun. Probab. 16 (2011) 528-544. · Zbl 1254.60018 [12] B. Ráth and A. Sapozhnikov. On the transience of random interlacements. Electron. Commun. Probab. 16 (2011) 379-391. · Zbl 1231.60115 [13] B. Ráth and A. Sapozhnikov. Connectivity properties of random interlacement and intersection of random walks. ALEA Lat. Am. J. Probab. Math. Stat. 9 (2012) 67-83. · Zbl 1277.60182 [14] V. Sidoravicius and A.-S. Sznitman. Percolation for the vacant set of random interlacements. Comm. Pure Appl. Math. 62 (6) (2009) 831-858. · Zbl 1168.60036 [15] A.-S. Sznitman. Vacant set of random interlacement and percolation. Ann. of Math. (2) 171 (2010) 2039-2087. · Zbl 1202.60160 [16] A.-S. Sznitman. A lower bound on the critical parameter of interlacement percolation in high dimension. Probab. Theory Related Fields 150 (2011) 575-611. · Zbl 1230.60103 [17] A.-S. Sznitman. On the critical parameter of interlacement percolation in high dimension. Ann. Probab. 39 (2011) 70-103. · Zbl 1210.60047 [18] A.-S. Sznitman. Decoupling inequalities and interlacement percolation on $$G\times{\mathbb {Z}}$$. Invent. Math. 187 (3) (2012) 645-706. · Zbl 1277.60183 [19] A. Teixeira. On the uniqueness of the infinite cluster of the vacant set of random interlacements. Ann. Probab. 19 (2009) 454-466. · Zbl 1158.60046 [20] A. Teixeira. On the size of a finite vacant cluster of random interlacements with small intensity. Probab. Theory Related Fields 150 (3-4) (2011) 529-574. · Zbl 1231.60117 [21] A. Teixeira and D. Windisch. On the fragmentation of a torus by random walk. Comm. Pure Appl. Math. 64 (12) (2011) 1599-1646. · Zbl 1235.60143 [22] A. Timár. Boundary-connectivity via graph theory. Proc. Amer. Math. Soc. 141 (2) (2013) 475-480. · Zbl 1259.05049 [23] D. Windisch. Random walk on a discrete torus and random interlacements. Electron. Commun. Probab. 13 (2008) 140-150. · Zbl 1187.60089
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.