Electronic Communications in Probability

The time constant and critical probabilities in percolation models

Leandro Pimentel

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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 $\mathcal{D}^*$.

Article information

Electron. Commun. Probab., Volume 11 (2006), paper no. 16, 160-167.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82D30: Random media, disordered materials (including liquid crystals and spin glasses)

Percolation time constant critical probabilities Delaunay triangulations

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Pimentel, Leandro. The time constant and critical probabilities in percolation models. Electron. Commun. Probab. 11 (2006), paper no. 16, 160--167. doi:10.1214/ECP.v11-1210. https://projecteuclid.org/euclid.ecp/1465058859

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