โ Scribed by J.-P. Hovi; Amnon Aharony
No coin nor oath required. For personal study only.
The spanning probability function for percolation is discussed using ideas from the renormalization group theory. We find that, apart from a few scale factors, the scaling functions are determined by the fixed point, and therefore are universal for every system with the same dimensionality, spanning rule, aspect ratio and boundary conditions, being independent of lattice structure and (finite) interaction length. This yields general results concerning the finite-size corrections and other corrections to scaling in general dimensions. For the special case of the square lattice with free boundaries, this theory, combined with duality arguments, give strong relations among different derivatives of the spanning function with respect to the scaling variables, thus yielding several new universal amplitude ratios and allowing systematic study of the corrections to scaling. The theoretical predictions are numerically confirmed with excellent accuracy.
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