Conductivity fluctuations in polymer's networks

Abstract

Polymer's network is treated as an anisotropic fractal with fractional dimensionality D = 1 + \epsilon close to one. Percolation model on such a fractal is studied. Using the real space renormalization group approach of Migdal and Kadanoff we find threshold value and all the critical exponents to be strongly nonanalytic functions of \epsilon, e.g. the critical exponent of the conductivity was obtained to be \epsilon^{-2}\exp(-1-1/\epsilon). The main part of the finite size conductivities distribution function at the threshold was found to be universal if expressed in terms of the fluctuating variable, which is proportional to the large power of the conductivity, but with dimensionally-dependent low-conductivity cut-off. Its reduced central momenta are of the order of \exp(-1/\epsilon) up to the very high order.