AC response near the percolation threshold: Transfer-matrix results in two and three dimensions

The complex admittivities of two- and three-dimensional networks of resistors and capacitors are calculated at pc using a transfer matrix. As predicted by scaling theories, the loss tangent becomes frequency independent in a critical range of frequencies. In three dimensions, for the usual model of unit conductors and capacitors at pc for the conductors, the dynamical exponents t and s are found to have values consistent with previous dc numerical studies. Similarly, the two-dimensional square site lattice and a bond lattice of superconductors and normal capacitors are both found to display dynamical scaling with universal (two-dimensional) exponents. Finally, two-dimensional networks with unit capacitors, but a power-law distribution of conductors, p()-± for ±<0 (and zero otherwise), are studied for the two cases ±=0 and 0.6. Dynamical scaling behavior is also seen on these lattices. In both cases, the bulk conductivities inferred from the dynamical scaling display nonuniversal values of the exponent t, while the exponent s remains at its universal value. © 1987 The American Physical Society.