Electronic Communications in Probability

The time constant and critical probabilities in percolation models

Leandro Pimentel

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465058859

Digital Object Identifier
doi:10.1214/ECP.v11-1210

Mathematical Reviews number (MathSciNet)
MR2240709

Zentralblatt MATH identifier
1112.60082

Subjects
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)

Keywords
Percolation time constant critical probabilities Delaunay triangulations

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

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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