Greedy polyominoes and first-passage times on random Voronoi tilings

Abstract

Let $\mathcal{N}$ be distributed as a Poisson random set on $\mathbb{R}^d$, $d\geq 2$, with intensity comparable to the Lebesgue measure. Consider the Voronoi tiling of $\mathbb{R}^d$, $\{C_v\}_{v\in \mathcal{N}}$, where $C_v$ is composed of points $\mathbf{x}\in\mathbb{R}^d$ that are closer to $v\in\mathcal{N}$ than to any other $v'\in\mathcal{N}$.  A polyomino $\mathcal{P}$ of size $n$ is a connected union (in the usual $\mathbb{R}^d$ topological sense) of $n$ tiles, and we denote by $\Pi_n$ the collection of all polyominos $\mathcal{P}$ of size $n$ containing the origin. Assume that the weight of a Voronoi tile $C_v$ is given by $F(C_v)$, where $F$ is a nonnegative functional on Voronoi tiles. In this paper we investigate for some functionals $F$, mainly when $F(C_v)$ is a polynomial function of the number of faces of $C_v$,  the tail behavior of the maximal weight among polyominoes in $\Pi_n$: $F_n=F_n(\mathcal{N}):=\max_{\mathcal{P}\in\Pi_n} \sum_{v\in \mathcal{P}} F(C_v)$. Next we apply our results to study self-avoiding paths, first-passage percolation models and the stabbing number on the dual graph, named the Delaunay triangulation. As the main application we show that first passage percolation has at most linear variance.