Electrical properties of percolation clusters: exact results on a deterministic fractal

Publisher Website
Google Scholar
Abstract
A recursively built deterministic fractal lattice, generalising a model proposed by Kirkpatrick (1979), is used to examine general electrical properties of percolation clusters in arbitrary dimensions. Most physical quantities are exactly analytically tractable, since the model admits an exact renormalisation group transformation, associated with a rational mapping T(x) of one complex variable. The model has one free parameter f, the fraction of conducting material, which is put equal to the self-dual point f=2/3 in two dimensions, and chosen to reproduce the numerical values of the exponent ratio t/ nu in higher dimensions. Results concern in particular the frequency dependence of the impedance and the loss angle, where the authors prove the existence of scaling laws at low and high frequency. They also determine exactly the transient response to an arbitrary input signal, and relate the distribution of relaxation times of the infinite lattice to well known mathematical objects associated to the rational transform T, namely its Julia set and its invariant measure. The critical amplification of flicker (or 1/f) resistor noise is also considered. It is shown to obey scaling laws with its own critical exponents, in agreement with other recent works. The relationships to other theoretical models and to experiments on metal-insulator mixtures are discussed.

This publication has 23 references indexed in Scilit: