The Annals of Probability

Sublinearity of the travel-time variance for dependent first-passage percolation

Jacob van den Berg and Demeter Kiss

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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.”

Article information

Ann. Probab., Volume 40, Number 2 (2012), 743-764.

First available in Project Euclid: 26 March 2012

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Zentralblatt MATH identifier

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

First-passage percolation influence results greedy lattice animals Ising model


van den Berg, Jacob; Kiss, Demeter. Sublinearity of the travel-time variance for dependent first-passage percolation. Ann. Probab. 40 (2012), no. 2, 743--764. doi:10.1214/10-AOP631.

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