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
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
The movies below show the trace left behind by the walkers in blue as a
function of σ.
Haye Hinrichsen's review
Far below: σ=0.66
Just below: σ=0.660877358
Just above: σ=0.660877359
Far above: σ=0.72