Local percolative properties of the vacant set of random interlacements with small intensity

Alexander Drewitz; Balázs Ráth; Artëm Sapozhnikov

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 4, page 1165-1197
  • ISSN: 0246-0203

Abstract

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Random interlacements at level u is a one parameter family of connected random subsets of d , d 3 (Ann. Math.171(2010) 2039–2087). Its complement, the vacant set at level u , exhibits a non-trivial percolation phase transition in u (Comm. Pure Appl. Math.62 (2009) 831–858; Ann. Math.171 (2010) 2039–2087), and the infinite connected component, when it exists, is almost surely unique (Ann. Appl. Probab.19(2009) 454–466). In this paper we study local percolative properties of the vacant set of random interlacements at level u for all dimensions d 3 and small intensity parameter u g t ; 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 { 3 , 4 } . The case of d 5 was treated in (Probab. Theory Related Fields150 (2011) 529–574) 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 (Probab. Theory Related Fields150 (2011) 529–574). 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.

How to cite

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Drewitz, Alexander, Ráth, Balázs, and Sapozhnikov, Artëm. "Local percolative properties of the vacant set of random interlacements with small intensity." Annales de l'I.H.P. Probabilités et statistiques 50.4 (2014): 1165-1197. <http://eudml.org/doc/271952>.

@article{Drewitz2014,
abstract = {Random interlacements at level $u$ is a one parameter family of connected random subsets of $\mathbb \{Z\}^\{d\}$, $d\ge 3$ (Ann. Math.171(2010) 2039–2087). Its complement, the vacant set at level $u$, exhibits a non-trivial percolation phase transition in $u$ (Comm. Pure Appl. Math.62 (2009) 831–858; Ann. Math.171 (2010) 2039–2087), and the infinite connected component, when it exists, is almost surely unique (Ann. Appl. Probab.19(2009) 454–466). In this paper we study local percolative properties of the vacant set of random interlacements at level $u$ for all dimensions $d\ge 3$ and small intensity parameter $u&gt;0$. 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 \lbrace 3,4\rbrace $. The case of $d\ge 5$ was treated in (Probab. Theory Related Fields150 (2011) 529–574) 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 (Probab. Theory Related Fields150 (2011) 529–574). 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.},
author = {Drewitz, Alexander, Ráth, Balázs, Sapozhnikov, Artëm},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {random interlacement; random walk; large finite cluster; supercriticality; conditional independence; random interlacements; local percolative properties},
language = {eng},
number = {4},
pages = {1165-1197},
publisher = {Gauthier-Villars},
title = {Local percolative properties of the vacant set of random interlacements with small intensity},
url = {http://eudml.org/doc/271952},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Drewitz, Alexander
AU - Ráth, Balázs
AU - Sapozhnikov, Artëm
TI - Local percolative properties of the vacant set of random interlacements with small intensity
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 4
SP - 1165
EP - 1197
AB - Random interlacements at level $u$ is a one parameter family of connected random subsets of $\mathbb {Z}^{d}$, $d\ge 3$ (Ann. Math.171(2010) 2039–2087). Its complement, the vacant set at level $u$, exhibits a non-trivial percolation phase transition in $u$ (Comm. Pure Appl. Math.62 (2009) 831–858; Ann. Math.171 (2010) 2039–2087), and the infinite connected component, when it exists, is almost surely unique (Ann. Appl. Probab.19(2009) 454–466). In this paper we study local percolative properties of the vacant set of random interlacements at level $u$ for all dimensions $d\ge 3$ and small intensity parameter $u&gt;0$. 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 \lbrace 3,4\rbrace $. The case of $d\ge 5$ was treated in (Probab. Theory Related Fields150 (2011) 529–574) 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 (Probab. Theory Related Fields150 (2011) 529–574). 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.
LA - eng
KW - random interlacement; random walk; large finite cluster; supercriticality; conditional independence; random interlacements; local percolative properties
UR - http://eudml.org/doc/271952
ER -

References

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