Some critical exponent inequalities for percolation

Research output: Contribution to journalArticle

Abstract

For a large class of independent (site or bond, short- or long-range) percolation models, we show the following: (1) If the percolation density P(p) is discontinuous at pc, then the critical exponent γ (defined by the divergence of expected cluster size, ∑nPn(p) ∼ (Pc-P) as p ↑pc) must satisfy γ ≥ 2. (2)γ or γ′ (defined analogously to γ, but as p ↓pc) and δ [Pn(pc) ∼ (n-1-1/δ) as n → ∞ ] must satisfy γ, γ′ ≥ 2(1 - 1/δ). These inequalities for γ improve the previously known bound γ ≥ 1(Aizenman and Newman), since δ ≥ 2 (Aizenman and Barsky). Additionally, result 1 may be useful, in standard d-dimensional percolation, for proving rigorously (in d>2) that, as expected, Px has no discontinuity at pc.

Original languageEnglish (US)
Pages (from-to)359-368
Number of pages10
JournalJournal of Statistical Physics
Volume45
Issue number3-4
DOIs
StatePublished - Nov 1986

Keywords

  • Percolation
  • critical exponent inequalities
  • rigorous results

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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