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 ℤ 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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  1. [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. Zbl1141.60057MR2349899
  2. [2] J. T. Chayes, L. Chayes and C. M. Newman. Bernoulli percolation above threshold: An invasion percolation analysis. Ann. Probab.15 (1987) 1272–1287. Zbl0627.60099MR905331
  3. [3] A. Drewitz, B. Ráth and A. Sapozhnikov. On chemical distances and shape theorems in percolation models with long-range correlations. Preprint. Available at arXiv:1212.2885. Zbl1301.82027
  4. [4] G. R. Grimmett. Percolation, 2nd edition. Springer-Verlag, Berlin, 1999. MR1707339
  5. [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. Zbl0602.60098MR876084
  6. [6] H. Kesten and Y. Zhang. The probability of a large finite cluster in supercritical Bernoulli percolation. Ann. Probab.18 (1990) 537–555. Zbl0705.60092MR1055419
  7. [7] G. Lawler. A self-avoiding random walk. Duke Math. J.47 (1980) 655–693. Zbl0445.60058MR587173
  8. [8] G. Lawler. Intersections of Random Walks. Birkhäuser, Basel, 1991. Zbl1228.60004MR1117680
  9. [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. Zbl06247173MR3024098
  10. [10] V. Sidoravicius and A.-S. Sznitman. Percolation for the vacant set of random interlacements. Comm. Pure Appl. Math.62 (2009) 831–858. Zbl1168.60036MR2512613
  11. [11] A.-S. Sznitman. Vacant set of random interlacements and percolation. Ann. Math.171 (2010) 2039–2087. Zbl1202.60160MR2680403
  12. [12] A.-S. Sznitman. Decoupling inequalities and interlacement percolation on G × ℤ . Invent. Math.187 (2012) 645–706. Zbl1277.60183MR2891880
  13. [13] A. Teixeira. On the uniqueness of the infinite cluster of the vacant set of random interlacements. Ann. Appl. Probab.19 (2009) 454–466. Zbl1158.60046MR2498684
  14. [14] A. Teixeira. On the size of a finite vacant cluster of random interlacements with small intensity. Probab. Theory Related Fields150 (2011) 529–574. Zbl1231.60117MR2824866
  15. [15] A. Teixeira and D. Windisch. On the fragmentation of a torus by random walk. Comm. Pure Appl. Math.64 (2011) 1599–1646. Zbl1235.60143MR2838338
  16. [16] Á. Tímár. Boundary-connectivity via graph theory. Proc. Amer. Math. Soc.141 (2013) 475–480. Zbl1259.05049MR2996951
  17. [17] D. Windisch. Random walk on a discrete torus and random interlacements. Electron. Commun. Probab.13 (2008) 140–150. Zbl1187.60089MR2386070

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