Hopping conduction in thed-dimensional lattice bond-percolation problem

Hopping conduction of a bond-percolation model in $d$-dimensional lattices is studied by making use of the coherent-medium approximation. The dc conductivity vanishes when $p<\mathrm{}{p}_c \left(\equiv \frac{2}{z}\right)$ and is proportional to $p-p_c$ when $p\ge p_c$, where $p$ is the probability that a given bond is not broken and $z$ is the coordination number of the lattice. In the low frequency region, the leading term of the imaginary and real parts of the ac part of the conductivity are written as $A\mathrm{}(z,d,p)f\left(\omega \right)$ and $B\mathrm{}(z,d,p)g\left(\omega \right)$, respectively. It is shown that when $p<\mathrm{}{p}_c$, $f\left(\omega \right)=\omega$ and $g\left(\omega \right)={\omega }^2$, regardless of the dimensionality; at $p=p_c$, $f\left(\omega \right)=g\left(\omega \right)={\omega }^{\frac{d}{\mathrm{}(d+2\mathrm{})\mathrm{}}}$ if $1<d<\mathrm{}2\mathrm{}$, $f\left(\omega \right)=g\left(\omega \right)={(-\omega \ln \omega )}^{\frac{1}{2}}$ if $d=2\mathrm{}$, and $f\left(\omega \right)=g\left(\omega \right)={\omega }^{\frac{1}{2}}$ if $d>\mathrm{}2\mathrm{}$; when $p>\mathrm{}{p}_c$, $f\left(\omega \right)=g\left(\omega \right)={\omega }^{\frac{d}{2}}$ if $1<d<\mathrm{}2\mathrm{}$, $f\left(\omega \right)=-\omega \ln \omega$ and $g\left(\omega \right)=\omega$ if $d=2\mathrm{}$, $f\left(\omega \right)=\omega$ and $g\left(\omega \right)={\omega }^{\frac{d}{2}}$ if $2<d<\mathrm{}4\mathrm{}$, $f\left(\omega \right)=\omega$ and