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

Let \(E\) be the set of edges of the \(d\)-dimensional cubic lattice \(\mathbb{Z}^d\), with \(d\geq2\), and let \(t(e),e\in E\), be nonnegative values. The passage time from a vertex \(v\) to a vertex \(w\) is defined as \(\inf_{\pi:v\rightarrow w}\sum_{e\in\pi}t(e)\), where the infimum is over all paths \(\pi\) from \(v\) to \(w\), and the sum is over all edges \(e\) of \(\pi\). 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 Bena\"ım 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."