Open Access
August 2002 Strict inequalities for the time constant in first passage percolation
R. Marchand
Ann. Appl. Probab. 12(3): 1001-1038 (August 2002). DOI: 10.1214/aoap/1031863179


In this work we are interested in the variations of the asymptotic shape in first passage percolation on $\mathbb{Z}^2$ according to the passage time distribution. Our main theorem extends a result proved by van den Berg and Kesten, which says that the time constant strictly decreases when the distribution of the passage time is modified in a certain manner (according to a convex order extending stochastic comparison). Van den Berg and Kesten's result requires, when the minimum $r$ of the support of the passage time distribution is strictly positive, that the mass given to $r$ is less than the critical threshold of an embedded oriented percolation model. We get rid of this assumption in the two-dimensional case, and to achieve this goal, we entirely determine the flat edge occurring when the mass given to $r$ is greater than the critical threshold, as a functional of the asymptotic speed of the supercritical embedded oriented percolation process, and we give a related upper bound for the time constant.


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R. Marchand. "Strict inequalities for the time constant in first passage percolation." Ann. Appl. Probab. 12 (3) 1001 - 1038, August 2002.


Published: August 2002
First available in Project Euclid: 12 September 2002

zbMATH: 1062.60100
MathSciNet: MR1925450
Digital Object Identifier: 10.1214/aoap/1031863179

Primary: 60K35
Secondary: 82B43

Keywords: asymptotic shape , first passage percolation , flat edge , Time constant

Rights: Copyright © 2002 Institute of Mathematical Statistics

Vol.12 • No. 3 • August 2002
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