Multiresolution analysis in statistical mechanics. II. The wavelet transform as a basis for Monte Carlo simulations on lattices

In this paper, we extend our analysis of lattice systems using the wavelettransform to systems for which exact enumeration is impractical. For suchsystems, we illustrate a wavelet-accelerated Monte Carlo (WAMC) algorithm,which hierarchically coarse-grains a lattice model by computing the probabilitydistribution for successively larger block spins. We demonstrate that althoughthe method perturbs the system by changing its Hamiltonian and by allowingblock spins to take on values not permitted for individual spins, the resultsobtained agree with the analytical results in the preceding paper, and``converge'' to exact results obtained in the absence of coarse-graining.Additionally, we show that the decorrelation time for the WAMC is no worse thanthat of Metropolis Monte Carlo (MMC), and that scaling laws can be constructedfrom data performed in several short simulations to estimate the results thatwould be obtained from the original simulation. Although the algorithm is notasymptotically faster than traditional MMC, because of its hierarchical design,the new algorithm executes several orders of magnitude faster than a fullsimulation of the original problem. Consequently, the new method allows forrapid analysis of a phase diagram, allowing computational time to be focused onregions near phase transitions.Comment: 11 pages plus 7 figures in PNG format (downloadable separately