The Annals of Applied Probability

The two-type Richardson model with unbounded initial configurations

Maria Deijfen and Olle Häggström

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Abstract

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

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

Dates
First available in Project Euclid: 17 December 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1197909502

Digital Object Identifier
doi:10.1214/07-AAP440

Mathematical Reviews number (MathSciNet)
MR2358637

Zentralblatt MATH identifier
1146.60075

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

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

Citation

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. https://projecteuclid.org/euclid.aoap/1197909502


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