Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-06-06T10:49:54.119Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Special processes Equilibrium statistical mechanics

Published online by Cambridge University Press:  01 July 2016

Sebastian Carstens  and
Thomas Richthammer
Sebastian Carstens*
Affiliation:
Ludwig-Maximilians-Universität Müchen
Thomas Richthammer*
Affiliation:
University of California, Los Angeles
*
∗ Postal address: Mathematisches Institut, Ludwig-Maximilians-Universität Müchen, Theresienstr. 39, D-80333 München, Germany.
∗ Postal address: Mathematisches Institut, Ludwig-Maximilians-Universität Müchen, Theresienstr. 39, D-80333 München, Germany.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

Continuum growth modelRichardson's modelcompeting growthinitial configurationshape theorem

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 82B43: Percolation
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 43 , Issue 3 , September 2011 , pp. 597 - 615
Copyright
Copyright © Applied Probability Trust 2011 

Footnotes

Research supported in part by NSF grant DMS-0300672.

References

Deijfen, M. (2003). Asymptotic shape in a continuum growth model. Adv. Appl. Prob. 35, 303–318.Google Scholar
Deijfen, M. and Häggström, O. (2004). Coexistence in a two-type continuum growth model. Adv. Appl. Prob. 36, 973–980.CrossRefGoogle Scholar
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.Google Scholar
Deijfen, M., Häggström, O. and Bagley, J. (2004). A stochastic model for competing growth on R d . Markov Process. Relat. Fields 10, 217–248.Google Scholar
Häggström, O. and Pemantle, R. (1998). First passage percolation and a model for competing spatial growth. J. Appl. Prob. 35, 683–692.CrossRefGoogle Scholar
Richardson, D. (1973). Random growth in a tessellation. Proc. Camb. Phil. Soc. 74, 515–528.Google Scholar