A Monte Carlo renormalisation approach to fractal dimensions of infinite cluster, backbone and cutting bonds for percolation

Abstract

A Monte Carlo renormalisation group method is presented to study the cluster structure of an infinite cluster, at the percolation threshold. Applying this technique to bond percolation on the square lattice, fractal dimensions of an infinite cluster, its backbone and cutting bonds are calculated. This method converges very rapidly with an increase in the size of the cell. The author estimates that the fractal dimension of an infinite cluster D=1.88+or-0.02, the fractal dimension of its backbone Db=1.62+or-0.02 and the fractal dimension of its cutting bonds Dc=0.75+or-0.01, up to the scale factor b=10, in excellent agreement with the large-cell Monte Carlo simulation result.