## The Annals of Applied Probability

### Strict inequalities for the time constant in first passage percolation

R. Marchand

#### 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

https://projecteuclid.org/euclid.aoap/1031863179

Digital Object Identifier
doi:10.1214/aoap/1031863179

Mathematical Reviews number (MathSciNet)
MR1925450

Zentralblatt MATH identifier
1062.60100

#### 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