Renormalization group approach for percolation conductivity

A recently developed renormalization group approach to two-dimensional bond percolation problems is generalized to deal with the percolation conductivity sigma of dilute resistor networks. The critical exponent t, defined by sigma (p) varies as (p-p c ) t , is related to an eigenvalue of the renormalization group transformation of conductance probability distributions. The transformation and eigenvalue are obtained for various real-space scalings of the square and triangular lattices, resulting in t=1.13+or-0.09. The Last-Kirkpatrick scaling relationship is used to obtain the critical exponent for spin-wave stiffness in the dilute ferromagnet.