Connectivity bounds for the vacant set of random interlacements
Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 4, pp. 976-990.

The model of random interlacements on ℤd, d≥3, was recently introduced in [Vacant set of random interlacements and percolation. Available at http://www.math.ethz.ch/u/sznitman/preprints]. A non-negative parameter u parametrizes the density of random interlacements on ℤd. In the present note we investigate connectivity properties of the vacant set left by random interlacements at level u, in the non-percolative regime u>u∗, with u∗ the non-degenerate critical parameter for the percolation of the vacant set, see [Vacant set of random interlacements and percolation. Available at http://www.math.ethz.ch/u/sznitman/preprints], [Comm. Pure Appl. Math. 62 (2009) 831-858]. We prove a stretched exponential decay of the connectivity function for the vacant set at level u, when u>u∗∗, where u∗∗ is another critical parameter introduced in [Ann. Probab. 37 (2009) 1715-1746]. It is presently an open problem whether u∗∗ actually coincides with u∗.

Le modèle des entrelacs aléatoires sur ℤd, d≥3, a été récemment introduit dans [Vacant set of random interlacements and percolation. Available at http://www.math.ethz.ch/u/sznitman/preprints]. Un nombre positif ou nul u contrôle la densité des entrelacs aléatoires sur ℤd. Dans la note présente, nous étudions les propriétés de connectivité du complémentaire de l'entrelac au niveau u, dans le régime non percolatif u>u∗, avec u∗ le nombre positif qui est le paramètre critique de la percolation du complémentaire des entrelacs, voir [Vacant set of random interlacements and percolation. Available at http://www.math.ethz.ch/u/sznitman/preprints], [Comm. Pure Appl. Math. 62 (2009) 831-858]. Nous montrons une propriété de décroissance sous-exponentielle de la fonction de connectivité au niveau u, lorsque u>u∗∗, où u∗∗ est un autre paramètre critique introduit dans [Ann. Probab. 37 (2009) 1715-1746]. La question de savoir si u∗ et u∗∗ sont égaux est pour le moment ouverte.

DOI: 10.1214/09-AIHP335
Classification: 60K35,  60G50,  82C41
Keywords: connectivity function, random interlacements, percolation
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Sidoravicius, Vladas; Sznitman, Alain-Sol. Connectivity bounds for the vacant set of random interlacements. Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 4, pp. 976-990. doi : 10.1214/09-AIHP335. http://www.numdam.org/articles/10.1214/09-AIHP335/

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