Cumulant renormalisation group and its application to the incipient infinite cluster in percolation

Abstract

The authors introduce the cumulant method into the context of position-space renormalisation group. The advantage of this method over previous procedures is that one need not systematically evaluate all 2^N states of each Kadanoff cell, where N is the number of elements (sites or bonds) in the cell. They illustrate this method by calculating, for bond percolation with an eight-bond cell, the critical exponents characterizing the following quantities: (i) the mean number of backbone bonds, (ii) the mean number of bonds in the minimum and maximum paths along the backbone, (iii) the mean number of singly connected ('red') bonds, and (iv) the mean number of bonds in all the self-avoiding walks connecting the two extreme points. They derive previously obtained results of the conductivity and resistivity.