## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 3, Number 1 (1993), 56-80.

### Inequalities for the Time Constant in First-Passage Percolation

#### Abstract

Consider first-passage percolation on $\mathbb{Z}^d$. A classical result says, roughly speaking, that the shortest travel time from $(0, 0,\ldots, 0)$ to $(n, 0, \ldots, 0)$ is asymptotically equal to $n \mu$, for some constant $\mu$, which is called the time constant, and which depends on the distribution of the time coordinates. Except for very special cases, the value of $\mu$ is not known. We show that certain changes of the time coordinate distribution lead to a decrease of $\mu$; usually $\mu$ will strictly decrease. Two examples of our results are: (i) If $F$ and $G$ are distribution functions with $F \leq G, F \not\equiv G$, then, under mild conditions, the time constant for $G$ is strictly smaller than that for $F$. (ii) For $0 < \varepsilon_1 < \varepsilon_2 \leq a < b$, the time constant for the uniform distribution on $\lbrack a - \varepsilon_2, b + \varepsilon_1 \rbrack$ is strictly smaller than for the uniform distribution on $\lbrack a, b\rbrack$. We assume throughout that all our distributions have finite first moments.

#### Article information

**Source**

Ann. Appl. Probab., Volume 3, Number 1 (1993), 56-80.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1177005507

**Digital Object Identifier**

doi:10.1214/aoap/1177005507

**Mathematical Reviews number (MathSciNet)**

MR1202515

**Zentralblatt MATH identifier**

0771.60092

**JSTOR**

links.jstor.org

**Subjects**

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

Secondary: 82A42

**Keywords**

First-passage percolation time constant inequality

#### Citation

van den Berg, J.; Kesten, H. Inequalities for the Time Constant in First-Passage Percolation. Ann. Appl. Probab. 3 (1993), no. 1, 56--80. doi:10.1214/aoap/1177005507. https://projecteuclid.org/euclid.aoap/1177005507