Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses

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Abstract

We extend the theorem of Burton and Keane on uniqueness of the infinite component in dependent percolation to cover random graphs on ℤd or ℤd × ℕ with long-range edges. We also study a short-range percolation model related to nearest-neighbor spin glasses on ℤd or on a slab ℤd × {0,... K} and prove both that percolation occurs and that the infinite component is unique for V=ℤ2×{0,1} or larger.

Original languageEnglish (US)
Pages (from-to)511-527
Number of pages17
JournalProbability Theory and Related Fields
Volume92
Issue number4
DOIs
StatePublished - Dec 1 1992

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ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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