Real-space renormalization of bond-disordered conductance lattices

Abstract

We propose a new real-space renormalization approach for the conductivity of bond-disordered conductance lattices, and investigate two-dimensional square and three-dimensional simple cubic lattices with a binary distribution of conductances, $\rho \left(\sigma \right)=p\delta (\sigma -{\sigma }_1)+(1-p)\mathrm{}\delta (\sigma -{\sigma }_2)$. It is shown that our transformations not only give a good description of the percolation conductivity near the critical point, but lead to an approximation for the lattice conductivity $\overline{\sigma }\mathrm{}\left(p\right)\mathrm{}$ which is superior to the effective-medium approximation for all values of $\frac{{\sigma }_2}{{\sigma }_1}$ and $p$. In particular, the slopes of $\overline{\sigma }\mathrm{}\left(p\right)\mathrm{}$ at $p=0\mathrm{}$ and $p=1\mathrm{}$ are reproduced exactly, and in two dimensions the transformations satisfy the selfdual symmetry of the square lattice. For percolation conduction problems ($\frac{{\sigma }_2}{{\sigma }_1}=0\mathrm{}$) we determine the conductivity exponents $t$ and $s$, and compare our results with alternative estimates. We also present a simple approximate solution to the renormalization relations which is very accurate for all values of $p$ and produces reasonable rough estimates of $t$ and $s$.