Determination of the percolation critical exponent as measure of dimension

Abstract

The critical exponent β related to the percolation probability describes the dimensionality of a specific lattice topology. The accurate determination of β will aid in distinguishing different universality classes. This work describes the fine tuning of the Sigma method developed previously by the authors in obtaining β. The approach uses the log-log plot of the percolation probability versus the difference of the occupation probability and the percolation threshold. Considering a symmetric distribution of points about the midpoint of this plot yields the β with (1) the lowest standard deviation and (2) the appropriate asymptotic behavior for both one and two-dimensional lattices.