Asymptotic behavior of the Eden model with positively homogeneous edge weights

TL;DR

This paper investigates the asymptotic shape and growth properties of a generalized Eden model with positively homogeneous edge weights, revealing phase transitions at critical homogeneity degrees and providing explicit limit shape characterizations.

Contribution

It introduces a weighted Eden model with homogeneous edge weights, establishes conditions for deterministic limit shapes, and analyzes the geometric containment for different homogeneity degrees.

Findings

For <1, the cluster has a deterministic limit shape explicitly related to the standard Eden model.

For >1, clusters are contained in a Euclidean cone with an angle less than .

No universal norm exists for all >1 or =1 cases.

Abstract

Let $d\in\mathbb N$, $\alpha\in\mathbb R$, and let $f :\mathbb R^d\setminus \{0\} \rightarrow (0,\infty)$ be locally Lipschitz and positively homogeneous of degree $\alpha$ (e.g. $f$ could be the $\alpha$th power of a norm on $\mathbb R^d$). We study a generalization of the Eden model on $\mathbb Z^d$ wherein the next edge added to the cluster is chosen from the set of all edges incident to the current cluster with probability proportional to the value of $f$ at the midpoint of this edge, rather than uniformly. This model is equivalent to a variant of first passage percolation where the edge passage times are independent exponential random variables with parameters given by the value of $f$ at the midpoint of the edge. We prove that the $f$-weighted Eden model clusters have an a.s. deterministic limit shape if $\alpha< 1$, which is an explicit functional of $f$ and the limit shape of the standard Eden model, and estimate the rate of convergence to this limit shape. We also prove that if $\alpha>1$, then there is a norm $\nu$ on $\mathbb R^d$ (depending on $\alpha$) such that if we set $f(z) = \nu(z)^{ \alpha}$, then the $f$-weighted Eden model clusters are a.s.\ contained in a Euclidean cone with opening angle $<\pi$ for all time. We further show that there does \emph{not} exist a norm on $\mathbb R^d$ for which this latter statement holds for all $\alpha>1$; and that there is no choice of function $f$ for which the above statement holds with $\alpha=1$. Our basic approach is to compare the local behavior of the $f$-weighted first passage percolation to that of unweighted first passage percolation with iid exponential edge weights (which is equivalent to the unweighted Eden model). We include a list of open problems and several computer simulations.