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      Asymptotic behavior of the Eden model with positively homogeneous edge weights

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          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 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 substantial list of open problems and several computer simulations.

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          Author and article information

          Journal
          2015-08-20
          2015-08-25
          Article
          1508.05140
          c1effe2c-a9d2-4d2d-bb4e-91e1fa201519

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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          45 pages, 12 figures; version 2 has corrected metadata
          math.PR

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