A Note About Critical Percolation on Finite Graphs

Gady Kozma; Asaf Nachmias

doi:10.1007/s10959-010-0283-x

A Note About Critical Percolation on Finite Graphs

Gady Kozma, Asaf Nachmias

Department of Mathematics

Weizmann Institute of Science

Research output: Contribution to journal › Article › peer-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 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 ℤ^d_n(with d large and n→∞) and the Hamming cube {0,1}ⁿ.

Original language	English
Pages (from-to)	1087-1096
Number of pages	10
Journal	Journal of Theoretical Probability
Volume	24
Issue number	4
DOIs	https://doi.org/10.1007/s10959-010-0283-x
State	Published - Dec 2011

Keywords

Critical exponents
Critical percolation
Intrinsic metric
Triangle condition

All Science Journal Classification (ASJC) codes

Statistics and Probability
General Mathematics
Statistics, Probability and Uncertainty

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10.1007/s10959-010-0283-x

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title = "A Note About Critical Percolation on Finite Graphs",

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.",

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AU - Nachmias, Asaf

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N2 - 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.

AB - 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.

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KW - Intrinsic metric

KW - Triangle condition

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DO - 10.1007/s10959-010-0283-x

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JO - Journal of Theoretical Probability

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ER -

A Note About Critical Percolation on Finite Graphs

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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