Translator Disclaimer
2012 Greedy polyominoes and first-passage times on random Voronoi tilings
Raphaël Rossignol, Leandro Pimentel
Author Affiliations +
Electron. J. Probab. 17(none): 1-31 (2012). DOI: 10.1214/EJP.v17-1788


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.


Download Citation

Raphaël Rossignol. Leandro Pimentel. "Greedy polyominoes and first-passage times on random Voronoi tilings." Electron. J. Probab. 17 1 - 31, 2012.


Accepted: 1 February 2012; Published: 2012
First available in Project Euclid: 4 June 2016

zbMATH: 1246.60120
MathSciNet: MR2878791
Digital Object Identifier: 10.1214/EJP.v17-1788

Primary: 60K35
Secondary: 60D05


Vol.17 • 2012
Back to Top