Percolation and contact processes with low-dimensional inhomogeneity

Charles Newman, C. Chris Wu

Research output: Contribution to journalArticle

Abstract

We consider inhomogeneous nearest neighbor Bernoulli bond percolation on ℤd where the bonds in a fixed s-dimensional hyperplane (1 ≤ s ≤ d - 1) have density p1 and all other bonds have fixed density, pc(ℤd), the homogeneous percolation critical value. For s ≥ 2, it is natural to conjecture that there is a new critical value, psc(ℤd), for p1, strictly between pc(ℤd) and pc(ℤs); we prove this for large d and 2 ≤ s ≤ d - 3. For s = 1, it is natural to conjecture that p1c(ℤd) = 1, as shown for d = 2 by Zhang; we prove this for large d. Related results for the contact process are also presented.

Original languageEnglish (US)
Pages (from-to)1832-1845
Number of pages14
JournalAnnals of Probability
Volume25
Issue number4
StatePublished - Oct 1997

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Contact Process
Inhomogeneity
Critical value
Bernoulli
Hyperplane
Nearest Neighbor
Strictly
Nearest neighbor

Keywords

  • Contact process
  • Inhomogeneity
  • Percolation

ASJC Scopus subject areas

  • Mathematics(all)
  • Statistics and Probability

Cite this

Percolation and contact processes with low-dimensional inhomogeneity. / Newman, Charles; Wu, C. Chris.

In: Annals of Probability, Vol. 25, No. 4, 10.1997, p. 1832-1845.

Research output: Contribution to journalArticle

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AB - We consider inhomogeneous nearest neighbor Bernoulli bond percolation on ℤd where the bonds in a fixed s-dimensional hyperplane (1 ≤ s ≤ d - 1) have density p1 and all other bonds have fixed density, pc(ℤd), the homogeneous percolation critical value. For s ≥ 2, it is natural to conjecture that there is a new critical value, psc(ℤd), for p1, strictly between pc(ℤd) and pc(ℤs); we prove this for large d and 2 ≤ s ≤ d - 3. For s = 1, it is natural to conjecture that p1c(ℤd) = 1, as shown for d = 2 by Zhang; we prove this for large d. Related results for the contact process are also presented.

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