Phase transitions for a planar quadratic contact process
Masha Bessonov and Rick Durrett
Abstract.
We study a two dimensional version of Neuhauser's long range sexual reproduction model and prove results
that give bounds on the critical values λf for that the process to survive from a finite set
and λf for the existence of a nontrivial stationary distribution. Our first result comes from a standard
block construction, while the second involves a comparison with the ``generic population model'' of Bramson
and Gray. An interesting new feature of our work is the suggestion that, as in the one dimensional contact
process, edge speeds characterize critical values. We are able to prove the following for our quadratic contact process
when the range is large but suspect they are true for two dimensional finite range attractive particle systems that are symmetric with respect
to reflection in each axis. There is a speed c(θ) for the expansion of the process in each direction.
If c(θ) > 0 in all directions, then λ > λf, while if at least one speed is positive, then
λ > λe. It is a challenging open problem to show that
if some speeds is negative, then the system dies out from any finite set.
Final version to applear in Advances in Applied Math
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