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October 1998 Critical probabilities for site and bond percolation models
G. R. Grimmett, A. M. Stacey
Ann. Probab. 26(4): 1788-1812 (October 1998). DOI: 10.1214/aop/1022855883

Abstract

.Any infinite graph $G = (V, E)$ has a site percolation critical probability $p_c^{\rm site}$ and a bond percolation critical probability $p_c^{\rm bond}$. The well-known weak inequality $p_c^{\rm site} \geq p_c^{\rm bond}$ is strengthened to strict inequality for a c c broad category of graphs $G$, including all the usual finite-dimensional lattices in two and more dimensions. The complementary inequality $p_c^{\rm site} \leq 1 - (1 - p_c^{\rm bond})^{\Delta - 1}$ is proved also, where $\Delta$ denotes the supremum of the vertex degrees of $G$.

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G. R. Grimmett. A. M. Stacey. "Critical probabilities for site and bond percolation models." Ann. Probab. 26 (4) 1788 - 1812, October 1998. https://doi.org/10.1214/aop/1022855883

Information

Published: October 1998
First available in Project Euclid: 31 May 2002

zbMATH: 0935.60097
MathSciNet: MR1675079
Digital Object Identifier: 10.1214/aop/1022855883

Subjects:
Primary: 60K35 , 82B43

Keywords: critical probability , enhancement , percolation

Rights: Copyright © 1998 Institute of Mathematical Statistics

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Vol.26 • No. 4 • October 1998
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