Let E be the set of edges of the d-dimensional cubic lattice ℤd, 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 infπ : 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  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 .
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 , where the passage time corresponds with the minimal number of sign changes in a subcritical “Ising landscape.”
"Sublinearity of the travel-time variance for dependent first-passage percolation." Ann. Probab. 40 (2) 743 - 764, March 2012. https://doi.org/10.1214/10-AOP631