Hostname: page-component-6bf8c574d5-w79xw Total loading time: 0 Render date: 2025-02-27T18:13:36.782Z Has data issue: false hasContentIssue false
  • English
  • Français

Coexistence in a two-type continuum growth model

Part of: Equilibrium statistical mechanics Special processes

Published online by Cambridge University Press:  01 July 2016

Maria Deijfen  and
Olle Häggström
Maria Deijfen*
Affiliation:
Stockholm University
Olle Häggström*
Affiliation:
Chalmers University of Technology
*
∗ Postal address: Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden. Email address: mia@matematik.su.se
∗∗ Postal address: Department of Mathematical Statistics, Chalmers University of Technology, SE-412 91 Göteborg, Sweden. Email address: olleh@math.chalmers.se
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a stochastic model describing the growth of two competing infections on ℝd. The growth takes place by way of spherical outbursts in the infected region, an outburst in the type-1 or -2 infected region causing all previously uninfected points within a stochastic distance from the outburst location to become type-1 or -2 infected, respectively. The main result is that, if the infection types have the same intensity, then there is a strictly positive probability that both infection types grow unboundedly.

Keywords

Richardson's modelcompeting growthcoexistence

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 36 , Issue 4 , December 2004 , pp. 973 - 980
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Deijfen, M. (2003). Asymptotic shape in a continuum growth model. Adv. Appl. Prob. 35, 303–318.Google Scholar
Deijfen, M. and Häggström, O. (2003). The initial configuration is irrelevant for the possibility of mutual unbounded growth in the two-type Richardson model. To appear in Comb. Prob. Comput. Google Scholar
Deijfen, M., Häggström, O. and Bagley, J. (2003). A stochastic model for competing growth on ℝd . Markov Proc. Relat. Fields 10, 217–248.Google Scholar
Garet, O. and Marchand, R. (2004). Coexistence in two-type first-passage percolation models. To appear in Ann. Appl. Prob. Google Scholar
Häggström, O. and Pemantle, R. (1998). First passage percolation and a model for competing spatial growth. J. Appl. Prob. 35, 683–692.Google Scholar
Häggström, O. and Pemantle, R. (2000). Absence of mutual unbounded growth for almost all parameter values in the two-type Richardson model. Stoch. Process. Appl. 90, 207–222.Google Scholar
Liggett, T. M. (1985). An improved subadditive ergodic theorem. Ann. Prob. 13, 1279–1285.Google Scholar
Richardson, D. (1973). Random growth in a tessellation. Proc. Camb. Phil. Soc. 74, 515–528.Google Scholar