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, $\ensuremath{\rho}(\ensuremath{\sigma})=p\ensuremath{\delta}(\ensuremath{\sigma}\ensuremath{-}{\ensuremath{\sigma}}_{1})+(1\ensuremath{-}p)\ensuremath{\delta}(\ensuremath{\sigma}\ensuremath{-}{\ensuremath{\sigma}}_{2})$. 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 $\overline{\ensuremath{\sigma}}(p)$ which is superior to the effective-medium approximation for all values of $\frac{{\ensuremath{\sigma}}_{2}}{{\ensuremath{\sigma}}_{1}}$ and $p$. In particular, the slopes of $\overline{\ensuremath{\sigma}}(p)$ at $p=0$ and $p=1$ are reproduced exactly, and in two dimensions the transformations satisfy the selfdual symmetry of the square lattice. For percolation conduction problems ($\frac{{\ensuremath{\sigma}}_{2}}{{\ensuremath{\sigma}}_{1}}=0$) 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 $s$.