Connectivity bounds for the vacant set of random interlacements

Vladas Sidoravicius; Alain-Sol Sznitman

Annales de l'I.H.P. Probabilités et statistiques (2010)

  • Volume: 46, Issue: 4, page 976-990
  • ISSN: 0246-0203

Abstract

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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∗.

How to cite

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Sidoravicius, Vladas, and Sznitman, Alain-Sol. "Connectivity bounds for the vacant set of random interlacements." Annales de l'I.H.P. Probabilités et statistiques 46.4 (2010): 976-990. <http://eudml.org/doc/241134>.

@article{Sidoravicius2010,
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&gt;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&gt;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∗.},
author = {Sidoravicius, Vladas, Sznitman, Alain-Sol},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {connectivity function; random interlacements; percolation},
language = {eng},
number = {4},
pages = {976-990},
publisher = {Gauthier-Villars},
title = {Connectivity bounds for the vacant set of random interlacements},
url = {http://eudml.org/doc/241134},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Sidoravicius, Vladas
AU - Sznitman, Alain-Sol
TI - Connectivity bounds for the vacant set of random interlacements
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2010
PB - Gauthier-Villars
VL - 46
IS - 4
SP - 976
EP - 990
AB - 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&gt;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&gt;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∗.
LA - eng
KW - connectivity function; random interlacements; percolation
UR - http://eudml.org/doc/241134
ER -

References

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  7. [7] 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
  8. [8] A. Teixeira. Interlacement percolation on transient weighted graphs. Electron. J. Probab. 14 (2009) 1604–1627. Zbl1192.60108MR2525105
  9. [9] A. Teixeira. On the size of a finite vacant cluster of random interlacements with small intensity. Preprint. Available at http://www.math.ethz.ch/~teixeira/. Zbl1231.60117
  10. [10] D. Windisch. Random walk on a discrete torus and random interlacements. Electron. Commun. Probab. 13 (2008) 140–150. Zbl1187.60089MR2386070
  11. [11] D. Windisch. Random walks on discrete cylinders with large bases and random interlacements. Ann. Probab. To appear. Available at arXiv:0907.1627. Zbl1191.60062MR2642893

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