Advances in Applied Probability

The two-type continuum Richardson model: nondependence of the survival of both types on the initial configuration

Sebastian Carstens and Thomas Richthammer

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We consider the model of Deijfen, Häggström and Bagley (2004) for competing growth of two infection types in Rd, based on the Richardson model on Zd. Stochastic ball-shaped infection outbursts transmit the infection type of the center to all points of the ball that are not yet infected. Relevant parameters of the model are the initial infection configuration, the (type-dependent) growth rates, and the radius distribution of the infection outbursts. The main question is that of coexistence: Which values of the parameters allow the unbounded growth of both types with positive probability? Deijfen, Häggström and Bagley (2004) conjectured that the initial configuration is basically irrelevant for this question, and gave a proof for this under strong assumptions on the radius distribution, which, e.g. do not include the case of a deterministic radius. Here we give a proof that does not rely on these assumptions. One of the tools to be used is a slight generalization of the model with immune regions and delayed initial infection configurations.

Article information

Adv. in Appl. Probab., Volume 43, Number 3 (2011), 597-615.

First available in Project Euclid: 23 September 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Continuum growth model Richardson's model competing growth initial configuration shape theorem


Carstens, Sebastian; Richthammer, Thomas. The two-type continuum Richardson model: nondependence of the survival of both types on the initial configuration. Adv. in Appl. Probab. 43 (2011), no. 3, 597--615. doi:10.1239/aap/1316792661.

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  • Deijfen, M. (2003). Asymptotic shape in a continuum growth model. Adv. Appl. Prob. 35, 303–318.
  • Deijfen, M. and Häggström, O. (2004). Coexistence in a two-type continuum growth model. Adv. Appl. Prob. 36, 973–980.
  • Deijfen, M. and Häggström, O. (2006). The initial configuration is irrelevant for the possibility of mutual unbounded growth in the two-type Richardson model. Combinatorics Prob. Comput. 15, 345–353.
  • Deijfen, M., Häggström, O. and Bagley, J. (2004). A stochastic model for competing growth on $\R^d$. Markov Process. Relat. Fields 10, 217–248.
  • Häggström, O. and Pemantle, R. (1998). First passage percolation and a model for competing spatial growth. J. Appl. Prob. 35, 683–692.
  • Richardson, D. (1973). Random growth in a tessellation. Proc. Camb. Phil. Soc. 74, 515–528.