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 Electronic Communications in Probability > Vol. 11 (2006) > Paper 16 open journal systems 


The time constant and critical probabilities in percolation models

Leandro Pimentel, Ecole Polytechnique Federale de Lausanne


Abstract
We consider a first-passage percolation (FPP) model on a Delaunay triangulation D of the plane. In this model each edge e of D is independently equipped with a nonnegative random variable, with distribution function F, which is interpreted as the time it takes to traverse the edge. Vahidi-Asl and Wierman (1990) have shown that, under a suitable moment condition on F, the minimum time taken to reach a point at distance n from the origin is asymptotically m(F)n, where m(F) is a nonnegative finite constant (the time constant). However, its exact value still a fundamental problem in percolation theory. Here we prove that if F(0) < 1-p'c then m(F)>0, where p'c is a critical probability for bond percolation on the dual graph D'.


Full text: PDF

Pages: 160-167

Published on: August 7, 2006
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Electronic Communications in Probability. ISSN: 1083-589X