Branching, self-annihilating random walk

The Model

A branching random walker explores a 2D lattice by either moving randomly from site to site or by spontaneously branching off an offspring, which in this model is placed at a randomly chosen neighbouring site on the lattice. The ratio with which branching occurs in relation to the frequency of random walking is σ. So far there is no interaction --- random walkers and their offspring are just all over the place.

Now the following constraint is added: Walkers leave a trace behind, whenever they leave a site. If a walker runs into (or is born into) a site that contains such a trace, it is annihilated instantly. There seems to be a phase transition (possibly/probably a well-known one): Below a certain value of σ an initial seed does not grow beyond a small, local neighbourhood (how does that size change with σ?). Above a certain value of σ the initial seed eventually covers the entire plane --- while the traces are apparently not sparse, the density of walkers probably displays a second order phase transition.

The aim of this project is to characterise the phase transition. The first step is to find a (possible/likely) link to the literature, in particular so-called absorbing state phase transitions. The project would then either focuses on the numerical characterisation of the phase transition using Monte-Carlo, or on field-theoretical calculations, using the renormalisation group.

The movies below show the trace left behind by the walkers in blue as a function of σ.

Literature:
John Cardy's lecture notes
Haye Hinrichsen's review

Transition

Far below: σ=0.66
Just below: σ=0.660877358
Just above: σ=0.660877359
Far above: σ=0.72

All movies