1,323 research outputs found
The two-type Richardson model with unbounded initial configurations
The two-type Richardson model describes the growth of two competing
infections on and the main question is whether both infection
types can simultaneously grow to occupy infinite parts of . For
bounded initial configurations, this has been thoroughly studied. In this
paper, an unbounded initial configuration consisting of points
in the hyperplane is
considered. It is shown that, starting from a configuration where all points in
\mathcal{H} {\mathbf{0}\} are type 1 infected and the origin is
type 2 infected, there is a positive probability for the type 2 infection to
grow unboundedly if and only if it has a strictly larger intensity than the
type 1 infection. If, instead, the initial type 1 infection is restricted to
the negative -axis, it is shown that the type 2 infection at the origin
can also grow unboundedly when the infection types have the same intensity.Comment: Published in at http://dx.doi.org/10.1214/07-AAP440 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Competition between growths governed by Bernoulli Percolation
We study a competition model on where the two infections are
driven by supercritical Bernoulli percolations with distinct parameters and
. We prove that, for any , there exist at most countably many values of
such that coexistence can occur.Comment: 30 pages with figure
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