Central limit theorems for percolation models

J. Theodore Cox, Geoffrey Grimmett

Research output: Contribution to journalArticle

6 Scopus citations

Abstract

Let p ≠ 1/2 be the open-bond probability in Broadbent and Hammersley's percolation model on the square lattice. Let Wxbe the cluster of sites connected to x by open paths, and let γ(n) be any sequence of circuits with interiors {Mathematical expression}. It is shown that for certain sequences of functions {fn}, {Mathematical expression} converges in distribution to the standard normal law when properly normalized. This result answers a problem posed by Kunz and Souillard, proving that the number Snof sites inside γ(n) which are connected by open paths to γ(n) is approximately normal for large circuits γ(n).

Original languageEnglish (US)
Pages (from-to)237-251
Number of pages15
JournalJournal of Statistical Physics
Volume25
Issue number2
DOIs
StatePublished - Jun 1 1981

Keywords

  • Percolation
  • asymptotic normality
  • circuits
  • semi-invariants

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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