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March 2012 Sublinearity of the travel-time variance for dependent first-passage percolation
Jacob van den Berg, Demeter Kiss
Ann. Probab. 40(2): 743-764 (March 2012). DOI: 10.1214/10-AOP631


Let E be the set of edges of the d-dimensional cubic lattice ℤd, with d ≥ 2, and let t(e), eE, be nonnegative values. The passage time from a vertex v to a vertex w is defined as infπ : vw ∑eπt(e), where the infimum is over all paths π from v to w, and the sum is over all edges e of π.

Benjamini, Kalai and Schramm [2] proved that if the t(e)’s are i.i.d. two-valued positive random variables, the variance of the passage time from the vertex 0 to a vertex v is sublinear in the distance from 0 to v. This result was extended to a large class of independent, continuously distributed t-variables by Benaïm and Rossignol [1].

We extend the result by Benjamini, Kalai and Schramm in a very different direction, namely to a large class of models where the t(e)’s are dependent. This class includes, among other interesting cases, a model studied by Higuchi and Zhang [9], where the passage time corresponds with the minimal number of sign changes in a subcritical “Ising landscape.”


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Jacob van den Berg. Demeter Kiss. "Sublinearity of the travel-time variance for dependent first-passage percolation." Ann. Probab. 40 (2) 743 - 764, March 2012.


Published: March 2012
First available in Project Euclid: 26 March 2012

zbMATH: 1252.60096
MathSciNet: MR2952090
Digital Object Identifier: 10.1214/10-AOP631

Primary: 60K35
Secondary: 82B43

Keywords: First-passage percolation , greedy lattice animals , influence results , Ising model

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 2 • March 2012
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