Percolation in contact processes: sharpness, robustness, and applications to vegetation patterns
The contact process, introduced by Harris in the 1970's,
provides a natural framework
to model processes such as the spread of disease
or vegetation. It
possesses a phase transition: depending on the parameters, the upper
invariant measure (limit distribution obtained when initially all
`sites' are `infected') is trivial or nontrivial.
We study versions of the
2-dimensional contact process with 2 or 3
states, and with spontaneous infections occurring as a
function of the particle density.
Motivated by a model for vegetation
patterns in arid landscapes,
we focus on percolation under
invariant measures of such processes.
For the 2-state process we prove that the percolation
transition is sharp and the same holds for the
3-state process under a reasonable assumption.
This is shown to contradict a form of
`robust critical behaviour' suggested in
a recent paper in Nature.
This is joint work with Rob van den Berg
(CWI, Amsterdam) and Markus Heydenreich
(Leiden).