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Journal of Theoretical Probability

, Volume 24, Issue 4, pp 1087–1096 | Cite as

A Note About Critical Percolation on Finite Graphs

  • Gady Kozma
  • Asaf Nachmias
Article

Abstract

In this note we study the geometry of the largest component \(\mathcal {C}_{1}\) of 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 n 2/3, and here we show that its diameter is n 1/3 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 \(\mathbb{Z}_{n}^{d}\) (with d large and n→∞) and the Hamming cube {0,1} n .

Keywords

Critical percolation Triangle condition Critical exponents Intrinsic metric 

Mathematics Subject Classification (2000)

60K35 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.The Weizmann Institute of ScienceRehovot POBIsrael
  2. 2.Microsoft ResearchRedmondUSA

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