The Annals of Applied Probability

The two-type Richardson model with unbounded initial configurations

Maria Deijfen and Olle Häggström

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The two-type Richardson model describes the growth of two competing infections on ℤd and the main question is whether both infection types can simultaneously grow to occupy infinite parts of ℤd. For bounded initial configurations, this has been thoroughly studied. In this paper, an unbounded initial configuration consisting of points x=(x1, …, xd) in the hyperplane $\mathcal{H}=\{x\in\mathbb{Z}^{d}:x_{1}=0\}$ is considered. It is shown that, starting from a configuration where all points in $\mathcal{H}\backslash\{\mathbf{0}\}$ are type 1 infected and the origin 0 is type 2 infected, there is a positive probability for the type 2 infection to grow unboundedly if and only if it has a strictly larger intensity than the type 1 infection. If, instead, the initial type 1 infection is restricted to the negative x1-axis, it is shown that the type 2 infection at the origin can also grow unboundedly when the infection types have the same intensity.

Article information

Ann. Appl. Probab., Volume 17, Number 5/6 (2007), 1639-1656.

First available in Project Euclid: 17 December 2007

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35]

Richardson’s model first-passage percolation competing growth initial configuration coexistence


Deijfen, Maria; Häggström, Olle. The two-type Richardson model with unbounded initial configurations. Ann. Appl. Probab. 17 (2007), no. 5/6, 1639--1656. doi:10.1214/07-AAP440.

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