Percolation and Conductivity in Nearly-1D Systems

Abstract

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