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

Drewitz, Alexander; Ráth, Balázs; Sapozhnikov, Artëm

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 4, pp. 1165-1197.

Un entrelac aléatoire au niveau $u$ est une famille à un paramètre de sous-ensembles connexes aléatoires de ${\mathbb{Z}}^d$, $d\ge 3$, 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 $d\ge 3$ 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 $d\ge 5$. Notre approche est différente (de celle de (Probab. Theory Related Fields 150 (2011) 529-574)) et est valide pour $d\ge 3$. L’un des ingrédients principaux de la preuve est une certaine propriété d’indépendence conditionelle des entrelacs aléatoires, qui est intéressante en elle-même.

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 $u>0$. 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 $d\ge 5$ 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.

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