### 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.

### 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