Diffusion noise of fractal networks and percolation clusters

Diffusion noise and Nyquist noise on fractal lattices and percolation clusters are discussed in the high- and low-frequency limits. Diffusion noise can also be considered as a sort of ``1/f noise.'' Even for system sizes much larger than the correlation length, the fractal structure of the percolation clusters reveals itself at sufficiently high frequencies through the anomalous frequency dependence f^-(3+theta)/(2+theta), where theta is the anomalous diffusion exponent. This law takes the 1/f form in the large-theta limit and reduces to the universal Lax-Mengert law f^-3/2 in the case of ordinary diffusion (theta=0). The new inequality -βL2 for the localization {β} function of certain fractals is also derived. The corresponding inequality for percolation clusters is t/{ν}d where t is the conductivity exponent and {ν} the correlation length exponent.