Abstract
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.
Citation
R. Marchand. "Strict inequalities for the time constant in first passage percolation." Ann. Appl. Probab. 12 (3) 1001 - 1038, August 2002. https://doi.org/10.1214/aoap/1031863179
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