Nonmonotonic Coexistence Regions for the Two-Type Richardson Model on Graphs
Maria Deijfen, Stockholm University, Sweden
Olle Haggstrom, Chalmers University of Technology, Sweden
Abstract
In
the two-type Richardson model on a graph
G=(V,E), each vertex is at a given time in state 0,1 or 2.
A 0 flips to a 1 (resp. 2) at rate λ1 (λ2)
times the number of neighboring 1's (2's), while
1's and 2's never flip. When G is infinite, the main
question is whether, starting from a single 1 and a single 2,
with positive probability we will see both types of infection
reach infinitely many sites. This has previously been studied on
the d-dimensional cubic lattice Z2, d≥2, where the
conjecture (on which a good deal of progress has been made) is
that such coexistence has positive probability if and only if
λ1 =λ2. In the present paper examples are given of
other graphs where the set of points in the parameter space which
admit such coexistence has a more surprising form. In particular,
there exist graphs exhibiting coexistence at some value of
(λ1 /λ2) and non-coexistence when this
ratio is brought closer to 1.
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