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First passage percolation and a model for competing spatial growth

Part of: Special processes Equilibrium statistical mechanics

Published online by Cambridge University Press:  14 July 2016

Olle Häggström  and
Robin Pemantle
Olle Häggström*
Affiliation:
Chalmers University of Technology
Robin Pemantle*
Affiliation:
University of Wisconsin
*
∗Postal address: Department of Mathematics, Chalmers University of Technology, S-41296 Goteborg, Sweden. Email address: [email protected]
∗∗Postal address: Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA.
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Abstract

An interacting particle system modelling competing growth on the ℤ2 lattice is defined as follows. Each x ∈ ℤ2 is in one of the states {0,1,2}. 1's and 2's remain in their states for ever, while a 0 flips to a 1 (a 2) at a rate equal to the number of its neighbours which are in state 1 (2). This is a generalization of the well-known Richardson model. 1's and 2's may be thought of as two types of infection, and 0's as uninfected sites. We prove that if we start with a single site in state 1 and a single site in state 2, then there is positive probability for the event that both types of infection reach infinitely many sites. This result implies that the spanning tree of time-minimizing paths from the origin in first passage percolation with exponential passage times has at least two topological ends with positive probability.

Keywords

First passage percolationRichardson's modeltree of infectioncompeting growth

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 82B43: Percolation
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 3 , September 1998 , pp. 683 - 692
Copyright
Copyright © Applied Probability Trust 1998 

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Footnotes

Research supported by grants from the Swedish Natural Science Council and the Royal Swedish Academy of Sciences.

Research supported in part by a grant from the Alfred P. Sloan Foundation, by a Presidential Faculty Fellowship and by NSF grant # DMS9300191.

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