## The Annals of Applied Probability

### 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