The time constant and critical probabilities in percolation models. (English) Zbl 1112.60082

Pimentel, Leandro P. R.

Electron. Commun. Probab. 11, 160-167 (2006).

Summary: We consider a first-passage percolation (FPP) model on a Delaunay triangulation \({\mathcal D}\) of the plane. In this model each edge \(e\) of \({\mathcal D}\) is independently equipped with a nonnegative random variable \(\tau_e\), with distribution function \(\mathbb{F}\), which is interpreted as the time it takes to traverse the edge. M. Q. Vahidi-Asl and J. C. Wierman [in: Random graphs ’87, 341–359 (1990; Zbl 0760.05023)] have shown that, under a suitable moment condition on \(\mathbb{F}\), the minimum time taken to reach a point \(x\) from the origin 0 is asymptotically \(\mu (\mathbb{F})|x|\) where \(\mu(\mathbb{F})\) is a nonnegative finite constant. However the exact value of the time constant \(\mu(\mathbb{F})\) is 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 \({\mathcal D}^*\).

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
82C43	Time-dependent percolation in statistical mechanics
82D30	Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)

Citations:

Zbl 0760.05023