Conductivity of continuum percolating systems

Abstract

We study the conductivity of a class of disordered continuum systems represented by the Swiss-cheese model, where the conducting medium is the space between randomly placed spherical holes, near the percolation threshold. This model can be mapped onto a bond percolation model where the conductance $\sigma$ of randomly occupied bonds is drawn from a probability distribution of the form $\sigma^{-a}$. Employing the methods of renormalized field theory we show to arbitrary order in $\epsilon$-expansion that the critical conductivity exponent of the Swiss-cheese model is given by $t^{\text{SC}} (a) = (d-2)\nu + \max [\phi, (1-a)^{-1}]$, where $d$ is the spatial dimension and $\nu$ and $\phi$ denote the critical exponents for the percolation correlation length and resistance, respectively. Our result confirms a conjecture which is based on the 'nodes, links, and blobs' picture of percolation clusters.