## Electronic Communications in Probability

### The time constant and critical probabilities in percolation models

Leandro Pimentel

#### 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

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

Rights

#### 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

#### References

• B. Bollobas and O. Riordan. The critical probability for random Voronoi percolation in the plane is 1/2. Preprint available from arXiv.org:math/0410336
• B. Bollobas and O. Riordan. Sharp thresholds and percolation in the plane. Preprint available from arXiv.org:math/0412510
• G. Grimmett. Percolation (second edition), Springer (1999).
• J.M. Hammersley and D.J.A. Welsh. First-passage percolation, sub-additive process, stochastic network and generalized renewal theory. Springer-Verlag (1965), 61-110.
• H. Kesten. Aspects of first-passage percolation. Lectures Notes in Math. 1180, Springer-Verlag (1986), 125-264.
• T.M. Ligget, R.H. Schonmann and A.M. Stacey. Domination by product measures. Ann. Probab. 25 (1997), 71-95.
• J. Moller. Lectures on random Voronoi tessellations. Lectures Notes in Stat. 87, Springer-Verlag (1991).
• L.P.R. Pimentel. Competing growth, interfaces and geodesics in first-passage percolation on Voronoi tilings. Phd Thesis, IMPA, Rio de Janeiro (2004).
• M.Q. Vahidi-Asl and J.C. Wierman. First-passage percolation on the Voronoi tessellation and Delaunay triangulation. Random Graphs 87 (M. Karonske, J. Jaworski and A. Rucinski, eds.) Wiley (1990), 341-359.
• M.C. Vahidi-Asl and J.C. Wierman. A shape result for first-passage percolation on the Voronoi tessellation and Delaunay triangulation. Random Graphs 89 (A. Frieze and T. Luczak, eds.), Wiley (1992), 247-262.
• A. Zvavitch. The critical probability for Voronoi percolation. MSc. thesis, Weizmann Institute of Science (1996).