FRACTAL DIMENSIONS FOR PERCOLATION AND PHASE-TRANSITION MODELS - THEORY AND NEW MONTE-CARLO CALCULATION METHOD

The phase transitions of d-dimensional Ising-type spin models and hard-core particle models may be related to percolation transitions of corresponding correlated percolation models such that the order parameter exponent beta and the correlation length exponent nu of the former may be related to the fractal dimensions D of the latter by D = d - beta/nu as in the case of random percolation models. Based on such connections, we present a histogram Monte Carlo method to calculate the percolation probability and other physical quantities for such models as continuous functions of the system parameter. We then use physical quantities thus obtained in a percolation renormalization group method to calculate critical point, fractal dimensions, critical exponents, and thermodynamic order parameter. Our method is very simple, but it may give quite accurate results for many percolation and phase transition models.