A Note About Critical Percolation on Finite Graphs

Gady Kozma, Asaf Nachmias

Research output: Contribution to journalArticlepeer-review

Abstract

In this note we study the geometry of the largest component C1of critical percolation on a finite graph G which satisfies the finite triangle condition, defined by Borgs et al. (Random Struct. Algorithms 27:137-184, 2005). There it is shown that this component is of size n2/3, and here we show that its diameter is n1/3 and that the simple random walk takes n steps to mix on it. By Borgs et al. (Ann. Probab. 33:1886-1944, 2005), our results apply to critical percolation on several high-dimensional finite graphs such as the finite torus ℤdn(with d large and n→∞) and the Hamming cube {0,1}n.

Original languageEnglish
Pages (from-to)1087-1096
Number of pages10
JournalJournal of Theoretical Probability
Volume24
Issue number4
DOIs
StatePublished - Dec 2011
Externally publishedYes

Keywords

  • Critical exponents
  • Critical percolation
  • Intrinsic metric
  • Triangle condition

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