Local percolative properties of the vacant set of random interlacements with small intensity
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 4, p. 1165-1197

Random interlacements at level u is a one parameter family of connected random subsets of d , d3 (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 d3 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{3,4}. The case of d5 was treated in (Probab. Theory Related Fields 150 (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 Fields 150 (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.

Un entrelac aléatoire au niveau u est une famille à un paramètre de sous-ensembles connexes aléatoires de d , d3, introduit dans (Ann. Math. 171 (2010) 2039-2087). Son complémentaire, l’ensemble vacant au niveau u, possède une transition de percolation non triviale en u, comme il a été montré dans (Comm. Pure Appl. Math. 62 (2009) 831-858) et (Ann. Math. 171 (2010) 2039-2087). La composante connexe infinie, lorsqu'elle existe, est presque sûrement unique, voir (Ann. Appl. Probab. 19 (2009) 454-466). Dans ce papier, nous étudions les propriétés percolatives locales de l’ensemble vacant au niveau u en toutes dimensions d3 et pour un petit paramètre d’intensité u. Nous donnons une borne exponentielle tendue sur la probabilité qu’un grand (hyper)cube contienne deux composantes macroscopiques distinctes de l’ensemble vacant au niveau u. Nos résultats impliquent qu’il est peu probable que les composantes connexes finies de l’ensemble vacant au niveau u soient grandes. Ces résultats ont été prouvés dans (Probab. Theory Related Fields 150 (2011) 529-574) pour d5. Notre approche est différente (de celle de (Probab. Theory Related Fields 150 (2011) 529-574)) et est valide pour d3. L’un des ingrédients principaux de la preuve est une certaine propriété d’indépendence conditionelle des entrelacs aléatoires, qui est intéressante en elle-même.

DOI : https://doi.org/10.1214/13-AIHP540
Classification:  60K35,  82B43
Keywords: random interlacement, random walk, large finite cluster, supercriticality, conditional independence
@article{AIHPB_2014__50_4_1165_0,
     author = {Drewitz, Alexander and R\'ath, Bal\'azs and Sapozhnikov, Art\"em},
     title = {Local percolative properties of the vacant set of random interlacements with small intensity},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {4},
     year = {2014},
     pages = {1165-1197},
     doi = {10.1214/13-AIHP540},
     zbl = {06377550},
     mrnumber = {3269990},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2014__50_4_1165_0}
}
Drewitz, Alexander; Ráth, Balázs; Sapozhnikov, Artëm. Local percolative properties of the vacant set of random interlacements with small intensity. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 4, pp. 1165-1197. doi : 10.1214/13-AIHP540. http://www.numdam.org/item/AIHPB_2014__50_4_1165_0/

[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. | MR 2349899 | Zbl 1141.60057

[2] J. T. Chayes, L. Chayes and C. M. Newman. Bernoulli percolation above threshold: An invasion percolation analysis. Ann. Probab. 15 (1987) 1272-1287. | MR 905331 | Zbl 0627.60099

[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. | Zbl 1301.82027

[4] G. R. Grimmett. Percolation, 2nd edition. Springer-Verlag, Berlin, 1999. | MR 1707339

[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. | MR 876084 | 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. | MR 1055419 | Zbl 0705.60092

[7] G. Lawler. A self-avoiding random walk. Duke Math. J. 47 (1980) 655-693. | MR 587173 | Zbl 0445.60058

[8] G. Lawler. Intersections of Random Walks. Birkhäuser, Basel, 1991. | MR 1117680 | 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. | MR 3024098 | Zbl pre06247173

[10] V. Sidoravicius and A.-S. Sznitman. Percolation for the vacant set of random interlacements. Comm. Pure Appl. Math. 62 (2009) 831-858. | MR 2512613 | Zbl 1168.60036

[11] A.-S. Sznitman. Vacant set of random interlacements and percolation. Ann. Math. 171 (2010) 2039-2087. | MR 2680403 | Zbl 1202.60160

[12] A.-S. Sznitman. Decoupling inequalities and interlacement percolation on G×. Invent. Math. 187 (2012) 645-706. | MR 2891880 | Zbl 1277.60183

[13] A. Teixeira. On the uniqueness of the infinite cluster of the vacant set of random interlacements. Ann. Appl. Probab. 19 (2009) 454-466. | MR 2498684 | Zbl 1158.60046

[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. | MR 2824866 | Zbl 1231.60117

[15] A. Teixeira and D. Windisch. On the fragmentation of a torus by random walk. Comm. Pure Appl. Math. 64 (2011) 1599-1646. | MR 2838338 | Zbl 1235.60143

[16] Á. Tímár. Boundary-connectivity via graph theory. Proc. Amer. Math. Soc. 141 (2013) 475-480. | MR 2996951 | Zbl 1259.05049

[17] D. Windisch. Random walk on a discrete torus and random interlacements. Electron. Commun. Probab. 13 (2008) 140-150. | MR 2386070 | Zbl 1187.60089