Preprint C35/2004
Competing Growth, Interfaces and Geodesics in First-Passage Percolation Models on Voronoi Tilings

Leandro Pimentel

**Keywords: **
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.