The Annals of Applied Probability

Strict inequalities for the time constant in first passage percolation

R. Marchand

Full-text: Open access

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.

Article information

Source
Ann. Appl. Probab., Volume 12, Number 3 (2002), 1001-1038.

Dates
First available in Project Euclid: 12 September 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1031863179

Digital Object Identifier
doi:10.1214/aoap/1031863179

Mathematical Reviews number (MathSciNet)
MR1925450

Zentralblatt MATH identifier
1062.60100

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B43: Percolation [See also 60K35]

Keywords
First passage percolation time constant asymptotic shape flat edge

Citation

Marchand, R. Strict inequalities for the time constant in first passage percolation. Ann. Appl. Probab. 12 (2002), no. 3, 1001--1038. doi:10.1214/aoap/1031863179. https://projecteuclid.org/euclid.aoap/1031863179


Export citation