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April, 1981 The Shape of the Limit Set in Richardson's Growth Model
Richard Durrett, Thomas M. Liggett
Ann. Probab. 9(2): 186-193 (April, 1981). DOI: 10.1214/aop/1176994460

Abstract

Let $C_p$ be the limiting shape of Richardson's growth model with parameter $p \in (0, 1\rbrack$. Our main result is that if $p$ is sufficiently close to one, then $C_p$ has a flat edge. This means that $\partial C_p \cap \{x \in R^2:x_1 + x_2 = 1\}$ is a nondegenerate interval. The value of $p$ at which this first occurs is shown to be equal to the critical probability for a related contact process. For $p < 1$, we show that $C_p$ is not the full diamond $\{x \in R^2:\|x\| = |x_1| + |x_2| \leq 1\}$. We also show that $C_p$ is a continuous function of $p$, and that when properly rescaled, $C_p$ converges as $p \rightarrow 0$ to the limiting shape for exponential site percolation.

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Richard Durrett. Thomas M. Liggett. "The Shape of the Limit Set in Richardson's Growth Model." Ann. Probab. 9 (2) 186 - 193, April, 1981. https://doi.org/10.1214/aop/1176994460

Information

Published: April, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0457.60083
MathSciNet: MR606981
Digital Object Identifier: 10.1214/aop/1176994460

Subjects:
Primary: 60K35
Secondary: 60J80 , 60K99

Keywords: branching random walks , contact processes , percolation processes , Richardson's model

Rights: Copyright © 1981 Institute of Mathematical Statistics

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Vol.9 • No. 2 • April, 1981
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