On Some fundamental aspects of Polyominoes on Random Voronoi Tilings

TL;DR

This paper investigates the probabilistic properties of Voronoi polyominoes on random tilings, providing tail bounds that are essential for analyzing various stochastic models on these structures.

Contribution

It introduces tail bounds for the intersection counts of Voronoi polyominoes with boxes, advancing the understanding of geometric and probabilistic aspects of Voronoi and Delaunay structures.

Findings

Established tail bounds for the number of boxes intersected by Voronoi polyominoes.

Provided bounds for the intersection of Voronoi tiles with polyominoes and vice versa.

Facilitated analysis of stochastic processes like first-passage percolation on Voronoi and Delaunay graphs.

Abstract

Consider a Voronoi tiling of the Euclidean space based on a realization of a inhomogeneous Poisson random set. A Voronoi polyomino is a finite and connected union of Voronoi tiles. In this paper we provide tail bounds for the number of boxes that are intersected by a Voronoi polyomino, and vice-versa. These results will be crucial to analyze self-avoiding paths, greedy polyominoes and first-passage percolation models on Voronoi tilings and on the dual graph, named the Delaunay triangulation.