## The Annals of Applied Probability

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

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

#### 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