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February, 1993 Inequalities for the Time Constant in First-Passage Percolation
J. van den Berg, H. Kesten
Ann. Appl. Probab. 3(1): 56-80 (February, 1993). DOI: 10.1214/aoap/1177005507


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.


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J. van den Berg. H. Kesten. "Inequalities for the Time Constant in First-Passage Percolation." Ann. Appl. Probab. 3 (1) 56 - 80, February, 1993.


Published: February, 1993
First available in Project Euclid: 19 April 2007

zbMATH: 0771.60092
MathSciNet: MR1202515
Digital Object Identifier: 10.1214/aoap/1177005507

Primary: 60K35
Secondary: 82A42

Keywords: First-passage percolation , inequality , Time constant

Rights: Copyright © 1993 Institute of Mathematical Statistics


Vol.3 • No. 1 • February, 1993
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