Numerical methods for the determination of the properties and critical behaviour of percolation and the Ising model (PhD thesis)

For this thesis, numerical methods have been developed, based on Monte Carlo methods, which allow for investigating percolation and the Ising model with high precision. Emphasis is on methods to use modern parallel computers with high efficiency. Two basic approaches for parallelization were chosen: replication and domain decomposition, in conjunction with suitable algorithms.

For percolation, the Hoshen-Kopelman algorithm for cluster counting was adapted to different needs. For studying fluctuations of cluster numbers, its traditional version (i. e., which is already published in literature) was used with simple replication. For simulating huge lattices, the Hoshen-Kopelman algorithm was adapted to domain decomposition, by dividing the hyperplane of investigation into strips that were assigned to different processors. By using this way of domain decomposition, it is viable to simulate huge lattices (with world record sizes) even for dimensions d > 2 on massively-parallel computers with distributed memory and message passing. For studying properties of percolation in dependence of system size, the Hoshen-Kopelman algorithm was modified to work on changing domains, i. e., growing lattices. By using this method, it is possible to simulate a lattice of linear size L_max and investigate lattices of size L_i < L_max for free. Here again, replication is a viable parallelization strategy.

For the Ising model, the standard Monte Carlo method of importance sampling with Glauber kinetics and multi-spin coding is adapted to parallel computers by domain decomposition of the lattice into strips. Using this parallelization method, it is possible to use massively-parallel computers with distributed memory and message passing in order to study huge lattices (again world record sizes) over many Monte Carlo steps, in order to investigate the dynamical critical behaviour in two dimensions.