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 $t^{\text{SC}}(a)=(d-2)\nu +\max [\varphi ,(1-a{)}^{-1}]$, where $d$ is the spatial dimension and $\nu$ and $\varphi$ 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.

arXiv comment14 pages, 1 figure, revised title + minor changes

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