Competing Growth, Interfaces and Geodesics in First-Passage Percolation Models on Voronoi Tilings
First-passage percolation | competing growth | geodesics | competition interface | Voronoi tilings
This work deal with two-dimensional stochastic growth processes that are defined in terms of first-passage percolation models on Voronoi tilings. We begin by introducing the single spatial growth process based on the notion of first-passage time and answering some questions that are known for first-passage percolation models on some periodic (non random) graphs structures but were unanswered for the random graph case. These questions are related to the compactness of the limit set and also to the rate of convergence. After that we introduce a different type of spatial growth model by permitting more than one cluster to develop over the same environment. We focus on the long time behavior of these growing clusters that are competing for space and derive some limit theorems related to the morphology of the ''competition interface'' between them. To study the structure of this interface we use the notion of geodesic in first-passage percolation.