### 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

rw_retrace_0.66.mp4

rw_retrace_0.6605.mp4

rw_retrace_0.6608.mp4

rw_retrace_0.66085.mp4

rw_retrace_0.66086.mp4

rw_retrace_0.66087.mp4

rw_retrace_0.660875.mp4

rw_retrace_0.660877.mp4

rw_retrace_0.6608772.mp4

rw_retrace_0.6608773.mp4

rw_retrace_0.66087735.mp4

rw_retrace_0.660877355.mp4

rw_retrace_0.660877357.mp4

rw_retrace_0.660877358.mp4

rw_retrace_0.660877359.mp4

rw_retrace_0.66087736.mp4

rw_retrace_0.66087737.mp4

rw_retrace_0.6608774.mp4

rw_retrace_0.6608775.mp4

rw_retrace_0.660878.mp4

rw_retrace_0.66088.mp4

rw_retrace_0.6609.mp4

rw_retrace_0.661.mp4

rw_retrace_0.6610.mp4

rw_retrace_0.6615.mp4

rw_retrace_0.6625.mp4

rw_retrace_0.665.mp4

rw_retrace_0.67.mp4

rw_retrace_0.68.mp4

rw_retrace_0.69.mp4

rw_retrace_0.7.mp4

rw_retrace_0.70.mp4

rw_retrace_0.71.mp4

rw_retrace_0.72.mp4