Advertisement

Percolation of the Vacant Set

  • Alexander Drewitz
  • Balázs Ráth
  • Artëm Sapozhnikov
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

In this chapter we discuss basic geometric properties of the vacant set \({\mathcal{V}}^{u}\) defined in (5.2.4). We view this set as a subgraph of \({\mathbb{Z}}^{d}\) with edges drawn between any pair of vertices \(x,y \in {\mathcal{V}}^{u}\) with | xy |1 = 1.

References

  1. [15]
    Kesten, H.: Aspects of first passage percolation. In: École d’été de probabilités de Saint-Flour, XIV—1984. Lecture Notes in Mathematics, vol. 1180, pp. 125–264. Springer, Berlin (1986). DOI 10.1007/BFb0074919. URL http://dx.doi.org/10.1007/BFb0074919
  2. [16]
    Kingman, J.F.C.: Poisson processes. In: Oxford Studies in Probability, vol. 3. The Clarendon Press Oxford University Press, New York (1993). Oxford Science PublicationsGoogle Scholar
  3. [24]
    Montroll, E.W.: Random walks in multidimensional spaces, especially on periodic lattices. J. Soc. Indust. Appl. Math. 4, 241–260 (1956)CrossRefMathSciNetGoogle Scholar
  4. [25]
    Peierls, R.: On Ising’s model of ferromagnetism. Math. Proc. Camb. Philos. Soc. 32, 477–481 (1936). DOI 10.1017/S0305004100019174. URL http://journals.cambridge.org/article_S0305004100019174
  5. [41]
    Sznitman, A.S.: Vacant set of random interlacements and percolation. Ann. Math. 171(3), 2039–2087 (2010). DOI 10.4007/annals.2010.171.2039. URL http://dx.doi.org/10.4007/annals.2010.171.2039

Copyright information

© The Author(s) 2014

Authors and Affiliations

  • Alexander Drewitz
    • 1
  • Balázs Ráth
    • 2
  • Artëm Sapozhnikov
    • 3
  1. 1.Department of MathematicsColumbia UniversityNew York CityUSA
  2. 2.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  3. 3.Max-Planck Institute of Mathematics in the SciencesLeipzigGermany

Personalised recommendations