Conductivity of continuum percolating systems

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 $\varepsilon$ expansion that the critical conductivity exponent of the Swiss-cheese model is given by $t^{\mathrm{SC}}\mathrm{}\left(a\right)=\left(d-2\right)\nu +\max [\phi ,(1\mathrm{}-{a)}^{-1}],$ 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 that is based on the “nodes, links, and blobs” picture of percolation clusters.