Narrow Results

A novel approach to parallelize the well-known Hoshen–Kopelman algorithm has been chosen, suitable for simulating huge lattices in high dimensions on massively-parallel computers with distributed memory and message passing. This method consists of domain decomposition of the simulated lattice into strips perpendicular to the hyperplane of investigation that is used in the Hoshen–Kopelman algorithm. Systems of world record sizes, up to L = 4 000 256 in two dimensions, L = 20 224 in three, and L = 1036 in four, gave precise estimates for the Fisher exponent τ, the corrections to scaling Δ₁, and for the critical number density n_c.

A new and efficient algorithm is presented for the calculation of the partition function in the S=±1 Ising model. As an example, we use the algorithm to obtain the thermal-dependence of the magnetic spin susceptibility of an Ising antiferromagnet for a 8×8 square lattice with open boundary conditions. The results agree qualitatively with the prediction of the Monte Carlo simulations and the experimental data, and they are better than the mean field approach results. For the 8×8 lattice, the algorithm reduces the computation time by nine orders of magnitude.

Monte Carlo simulations with up to 176⁵ spins show no significant deviations between four different random number generators in multispin coding, as opposed to multiplication with 16807 for one-word-per-spin coding for 32⁵. Also world record sizes 1000192², 9984³, 880⁴ and 48⁶ spins were simulated.