Percolation in half-spaces

equality of critical densities and continuity of the percolation probability

David J. Barsky, Geoffrey R. Grimmett, Charles M. Newman

Research output: Contribution to journalArticle

Abstract

Renormalization arguments are developed and applied to independent nearest-neighbor percolation on various subsets ℕ of ℤd, d≧2, yielding:Equality of the critical densities, pc(ℕ), for ℕ a half-space, quarter-space, etc., and (for d>2) equality with the limit of slab critical densities. Continuity of the phase transition for the half-space, quarter-space, etc.; i.e., vanishing of the percolation probability, θ(p), at p=pc(ℕ). Corollaries of these results include uniqueness of the infinite cluster for such ℕ's and sufficiency of the following for proving continuity of the full-space phase transition: showing that percolation in the full-space at density p implies percolation in the half-space at the same density.

Original languageEnglish (US)
Pages (from-to)111-148
Number of pages38
JournalProbability Theory and Related Fields
Volume90
Issue number1
DOIs
StatePublished - Mar 1991

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Half-space
Equality
Phase Transition
Sufficiency
Renormalization
Nearest Neighbor
Corollary
Uniqueness
Imply
Subset

ASJC Scopus subject areas

  • Statistics and Probability
  • Analysis
  • Mathematics(all)

Cite this

Percolation in half-spaces : equality of critical densities and continuity of the percolation probability. / Barsky, David J.; Grimmett, Geoffrey R.; Newman, Charles M.

In: Probability Theory and Related Fields, Vol. 90, No. 1, 03.1991, p. 111-148.

Research output: Contribution to journalArticle

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