Exact relations between critical exponents for elastic stiffness and electrical conductivity of two-dimensional percolating networks.

Abstract

It has long been known that the critical exponent T of the elastic stiffness C(e)proportional, Deltap(T) of a d-dimensional percolating network (Deltap identical with p - p(c)>0 measures the closeness of the network to its percolation threshold p(c)) satisfies the following inequalities: 1+dnu < or = T < or = t+2nu, where t is the critical exponent of the electrical conductivity sigma(e) proportional, Deltap(t) of the same network and nu is the critical exponent of the percolation correlation length xi proportional, Deltap(-nu). Similarly, the critical exponents that characterize the divergences C(e)proportional, /Deltap/(-S), sigma(e) proportional to /Deltap/(-s) of a rigid or normal and a superconducting or normal random mixture (Deltap identical with p-(c)<0 now measures the closeness of the rigid or superconducting constituent to its percolation threshold p(c)) have long been known to satisfy S < or = s. We now show that, when d=2, T is in fact exactly equal to t+2nu and S is exactly equal to s.