Perco

Percolation, directed percolation, and reaction-diffusion systems

One reason to study such systems is that the PERM algorithm can be applied to a number of problems involving very small probabilities. These involve e.g. the problem of multiple spanning clusters on finite percolating lattices, percolation and directed percolation below the critical point, and lattice animals. The latter are efficiently simulated by growing percolation clusters, computing their weights in the animal ensemble, and using resampling to enrich the sample with clusters having high weight.

Similarly, we applied a variant of PERM to the Dosker-Varadhan trapping problem, both with isotropic and with biased diffusion. For the latter we studied in detail the (de-)localization transition, mapping it also onto the problem of a stretched collapsed polymer. Similarly, we studied the reaction-diffusion system A+B -> B in low dimensions, proving the existence of very large corrections to scaling.

Apart from applications of PERM to these systems, we study them also for other reasons. We developed a hashing method and an improved estimator which allowed us to study percolation in high dimensions (d=4 to 13) with unprecedented accuracy. The same methods are now also applied to directed percolation in d+1 dimension, with d=3 and d=4.

We developed a new fast algorithm for identifying backbones of percolation clusters, and applied it to check predictions of the renormalization group.

Finally, we showed that a widely studied forest fire model does not obey the expected scaling behaviour.