Renormalization Multigrid (RMG): Statistically Optimal Renormalization Group Flow and Coarse-to-Fine Monte Carlo Acceleration

Achi Brandt and Dorit Ron
Department of Applied Math
The Weizmann Inst. of Science
Rehovot, 76100, Israel

Abstract

New renormalization-group algorithms are developed with adaptive representations of the renormalized action which automatically express only significant interactions. As the amount of statistics grows, more interactions enter, thereby systematically reducing the truncation error. This allows statistically optimal calculation of thermodynamic limits, in the sense that it achieves accuracy e in just O(1/(e*e)) random number generations. There are practically no finite-size effects and the renormalization transformation can be repeated arbitrarily many times. Consequently, the desired fixed point is obtained and the correlation-length critical exponent $\nu$ is extracted.In addition, we introduce a new multiscale coarse-to-fine acceleration method, based on a multigrid-like approach. This general (non-cluster) algorithm generates independent equilibrium configurations without slow down. A particularly simple version of it can be used at criticality. The methods are of great generality; here they are demonstrated on the 2D Ising model.

Key words. Ising model, Renormalization Multigrid, P+ probabilities, neighborhoods, criticalization, coarse-to-fine Monte Carlo acceleration, Compatible Monte Carlo, Post Relaxation.

Contributed July 15, 1999.