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

Abstract

Let E be the set of edges of the d-dimensional cubic lattice ℤ^d, with d ≥ 2, and let t(e), e ∈ E, be nonnegative values. The passage time from a vertex v to a vertex w is defined as inf_{π : v→w} ∑_e∈π t(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.”