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October 2003 First passage percolation has sublinear distance variance
Itai Benjamini, Gil Kalai, Oded Schramm
Ann. Probab. 31(4): 1970-1978 (October 2003). DOI: 10.1214/aop/1068646373

Abstract

Let $0 < a < b < \infty$, and for each edge e of $\Z^d$ let $\omega_e=a$ or $\omega_e=b$, each with probability $1/2$, independently. This induces a random metric $\dist_\omega$ on the vertices of $\Z^d$, called first passage percolation. We prove that for $d>1$, the distance $\dist_\omega(0,v)$ from the origin to a vertex $v$, $|v|>2$, has variance bounded by $C|v|/\log|v|$, where $C=C(a,b,d)$ is a constant which may only depend on a, b and d. Some related variants are also discussed.

Citation

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Itai Benjamini. Gil Kalai. Oded Schramm. "First passage percolation has sublinear distance variance." Ann. Probab. 31 (4) 1970 - 1978, October 2003. https://doi.org/10.1214/aop/1068646373

Information

Published: October 2003
First available in Project Euclid: 12 November 2003

zbMATH: 1087.60070
MathSciNet: MR2016607
Digital Object Identifier: 10.1214/aop/1068646373

Subjects:
Primary: 60K35
Secondary: 28A35, 60B15, 60E15

Rights: Copyright © 2003 Institute of Mathematical Statistics

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Vol.31 • No. 4 • October 2003
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