a.c. response of fractal networks

Abstract

We calculate the a.c. frequency response of Sierpinski-gasket networks, in which the bonds consist of resistors R (or of impedances Z h) and all nodes are connected to the circuit ground by identical capacitors C (or by impedances Zv). The resulting complex, size-dependent admittance between any of the « principal » nodes and the circuit ground can be accurately described at all frequencies less than 1/RC by a finite-size scaling function whose exponents are combinations of the fractal dimension df and the spectral or « fracton » dimension ds of the Sierpinski gasket. The response function also bears a striking similarity to experimental observations of the a.c. response of a random mixture of conducting and insulating particles