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 $f^{\mathrm{-}(3+\mathrm{theta})/(2+\mathrm{theta})}$, 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 $f^{\mathrm{-}3/2}$ in the case of ordinary diffusion (theta=0). The new inequality -${\beta }_L$<2 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.