The Annals of Probability

Critical probabilities for site and bond percolation models

G. R. Grimmett and A. M. Stacey

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.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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Ann. Probab., Volume 26, Number 4 (1998), 1788-1812.

First available in Project Euclid: 31 May 2002

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35]

Percolation enhancement critical probability


Grimmett, G. R.; Stacey, A. M. Critical probabilities for site and bond percolation models. Ann. Probab. 26 (1998), no. 4, 1788--1812. doi:10.1214/aop/1022855883.

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