Avalanches and $\frac{1}{f}$ Noise in Evolution and Growth Models

We formally establish the relationship between spatial fractal behavior and long-range temporal correlations for a broad range of self-organized (and not self-organized) critical phenomena including directed percolation, interface depinning, and a simple evolution model. The recurrent activity at any particular site forms a fractal in time, with a power spectrum $S\left(f\right)\mathrm{}\sim \frac{1}{f^{\tilde{d}}}$. The exponent $\tilde{d}=\frac{\mathrm{}(D-d)\mathrm{}}{z}$, where $d$ is the spatial dimension, $D$ is the avalanche dimension, and $z$ is the usual dynamical exponent. Theoretical results agree with numerical simulations.