Stochastic Growth Models

Gregor Šega

Abstract

Our main interest in this article will be the limit shape in a two-dimensional stochastic growth model. This model can represent spreading of fire (or virus, information, etc.) on a fixed lattice. Limit shape depends on various parameters: the shape and size of chosen neighbourhood and the excitability of the media. First we define what a growth model is, then we focus on one kind of such a model. We derive a certain equation in one dimension which tells us the speed of a spreading. Then we use Wulff transformation to determine the limit shape in two dimensions. This method is used only for square neighbourhood, but minor modifications make it work for other shapes. However, technical problems do occur. At the end we consider possible interesting modifications to the system.