Real-space renormalization of bond-disordered conductance lattices

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, ${ρ}({σ})=p{δ}({σ}{-}{{σ}}₁)+(1{-}p){δ}({σ}{-}{{σ}}₂)$. 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 ${{σ}}(p)$ which is superior to the effective-medium approximation for all values of ${{{σ}}₂}{{{σ}}₁}$ and $p$. In particular, the slopes of ${{σ}}(p)$ 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 (${{{σ}}₂}{{{σ}}₁}=0$) 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 $s$.