Electronic Journal of Probability

Greedy polyominoes and first-passage times on random Voronoi tilings

Raphaël Rossignol and Leandro Pimentel

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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.

Article information

Source
Electron. J. Probab., Volume 17 (2012), paper no. 12, 31 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465062334

Digital Object Identifier
doi:10.1214/EJP.v17-1788

Mathematical Reviews number (MathSciNet)
MR2878791

Zentralblatt MATH identifier
1246.60120

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Keywords
Random Voronoi tiling Delaunay graph First passage percolation connective constant greedy animal random walk

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Rossignol, Raphaël; Pimentel, Leandro. Greedy polyominoes and first-passage times on random Voronoi tilings. Electron. J. Probab. 17 (2012), paper no. 12, 31 pp. doi:10.1214/EJP.v17-1788. https://projecteuclid.org/euclid.ejp/1465062334


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