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2006 The time constant and critical probabilities in percolation models
Leandro Pimentel
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Electron. Commun. Probab. 11: 160-167 (2006). DOI: 10.1214/ECP.v11-1210


We consider a first-passage percolation (FPP) model on a Delaunay triangulation $\mathcal{D}$ of the plane. In this model each edge $\mathbf{e}$ of $\mathcal{D}$ is independently equipped with a nonnegative random variable $\tau_\mathbf{e}$, with distribution function $\mathbb{F}$, which is interpreted as the time it takes to traverse the edge. Vahidi-Asl and Wierman [9] have shown that, under a suitable moment condition on $\mathbb{F}$, the minimum time taken to reach a point $\mathbf{x}$ from the origin $\mathbf{0}$ is asymptotically $\mu(\mathbb{F})|\mathbf{x}|$, where $\mu(\mathbb{F})$ is a nonnegative finite constant. However the exact value of the time constant $\mu(\mathbb{F})$ still a fundamental problem in percolation theory. Here we prove that if $\mathbb{F}(0)<1-p_c^*$ then $\mu(\mathbb{F})>0$, where $p_c^*$ is a critical probability for bond percolation on the dual graph $\mathcal{D}^*$.


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Leandro Pimentel. "The time constant and critical probabilities in percolation models." Electron. Commun. Probab. 11 160 - 167, 2006.


Accepted: 7 August 2006; Published: 2006
First available in Project Euclid: 4 June 2016

zbMATH: 1112.60082
MathSciNet: MR2240709
Digital Object Identifier: 10.1214/ECP.v11-1210

Primary: 60K35
Secondary: 82D30


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