Asymptotic Behavior of the Eden Model with Positively Homogeneous Edge Weights

Опубликовано 7 июля 2016, 22:41

Asymptotic Behavior of the Eden Model with Positively Homogeneous Edge Weights (a.k.a. The Polya Aggregate Model)
The Eden model is a random family of growing clusters in the d-dimensional integer lattice which can be defined inductively as follows. Start with the single point 0, then iteratively sample an edge uniformly from the set of all edges on the boundary of the current cluster, and add an endpoint of this edge to the cluster. We consider a generalization of the Eden model where edges are sampled with weights proportional to the values of a function f which is positively homogeneous of some real degree α, rather than uniformly. This model can be viewed as a d-dimensional generalization of the Polya urn model, so is called the Polya aggregate model. We prove that the Polya aggregate model exhibits a phase transition at α = 1. In particular, if α 1, then there exists an α-positively homogenous function f (which we can take to be the α-th power of a norm) such that for this choice of f, the Polya aggregate clusters are a.s. contained in a Euclidean cone of opening angle 1. We also list some open problems. This is based on a joint work with S. Bubeck.