A Note About Critical Percolation on Finite Graphs

Gady Kozma () and Asaf Nachmias ()
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Gady Kozma: The Weizmann Institute of Science
Asaf Nachmias: Microsoft Research

Journal of Theoretical Probability, 2011, vol. 24, issue 4, 1087-1096

Abstract: 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; 60K35 (search for similar items in EconPapers)
Date: 2011
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DOI: 10.1007/s10959-010-0283-x

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