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

TL;DR

This paper demonstrates that the sublinear variance property of passage times in first-passage percolation extends to models with dependent edge weights, including those related to Ising landscapes, broadening the scope of prior independent models.

Contribution

It extends the sublinearity of variance results from independent to dependent edge weight models in first-passage percolation, including Ising landscape cases.

Findings

Variance of passage time is sublinear in dependent models.

Includes models with dependent edge weights like Ising landscapes.

Generalizes previous results from independent to dependent settings.

Abstract

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\"{\i}m 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."