Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Connectivity bounds for the vacant set of random interlacements

Vladas Sidoravicius and Alain-Sol Sznitman

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Abstract

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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 46, Number 4 (2010), 976-990.

Dates
First available in Project Euclid: 4 November 2010

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1288878333

Digital Object Identifier
doi:10.1214/09-AIHP335

Mathematical Reviews number (MathSciNet)
MR2744881

Zentralblatt MATH identifier
1210.60107

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60G50: Sums of independent random variables; random walks 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

Keywords
Connectivity function Random interlacements Percolation

Citation

Sidoravicius, Vladas; Sznitman, Alain-Sol. Connectivity bounds for the vacant set of random interlacements. Ann. Inst. H. Poincaré Probab. Statist. 46 (2010), no. 4, 976--990. doi:10.1214/09-AIHP335. https://projecteuclid.org/euclid.aihp/1288878333


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References

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