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