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October 1997 Percolation and contact processes with low-dimensional inhomogeneity
Charles M. Newman, C. Chris Wu
Ann. Probab. 25(4): 1832-1845 (October 1997). DOI: 10.1214/aop/1023481113

Abstract

We consider inhomogeneous nearest neighbor Bernoulli bond percolation on $mathbb{Z}^d$ where the bonds in a fixed $s$-dimensional hyperplane $1\leq s\leq d-1)$ have density $p_1$ and all other bonds have fixed density, $p_c(\mathbb{Z}^d)$, the homogeneous percolation critical value. For $s\leq 2$, it is natural to conjecture that there is a new critical value, $p_c^s(\mathbb{Z}^d)$ for $p_1$, strictly between $p_c(\mathbb{Z}^d)$ and $p_c(\mathbb{Z}^s)$ ; we prove this for large $d$ and $2 \leq s \leq d-3$. For $s=1$, it is natural to conjecture that $p_c^1(\mathbb{Z}^d) =1$, as shown for $d =2$ by Zhang; we prove this for large $d$. Related results for the contact process are also presented.

Citation

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Charles M. Newman. C. Chris Wu. "Percolation and contact processes with low-dimensional inhomogeneity." Ann. Probab. 25 (4) 1832 - 1845, October 1997. https://doi.org/10.1214/aop/1023481113

Information

Published: October 1997
First available in Project Euclid: 7 June 2002

zbMATH: 0901.60074
MathSciNet: MR1487438
Digital Object Identifier: 10.1214/aop/1023481113

Subjects:
Primary: 60K35, 82B43

Rights: Copyright © 1997 Institute of Mathematical Statistics

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Vol.25 • No. 4 • October 1997
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