Asymptotic behavior of the Eden model with positively homogeneous edge weights

Abstract

Let $d\in\mathbb N$, $α\in\mathbb R$, and let $f :\mathbb R^d\setminus \{0\} \rightarrow (0,\infty)$ be locally Lipschitz and positively homogeneous of degree $α$ (e.g. $f$ could be the $α$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 $α< 1$, 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 $α>1$, then there is a norm $ν$ on $\mathbb R^d$ (depending on $α$) such that if we set $f(z) = ν(z)^{ α}$, then the $f$-weighted Eden model clusters are a.s.\ contained in a Euclidean cone with opening angle $<π$ 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 $α>1$; and that there is no choice of function $f$ for which the above statement holds with $α=1$. 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.