Random walks on fractals and stretched exponential relaxation

Stretched exponential relaxation ($\exp {-(t\mathrm{/}\tau )}^{{\beta }_K}$) is observed in a large variety of systems but has not been explained so far. Studying random walks on percolation clusters in curved spaces whose dimensions range from 2 to 7, we show that the relaxation is accurately a stretched exponential and is directly connected to the fractal nature of these clusters. Thus we find that in each dimension the decay exponent ${\beta }_K$ is related to well-known exponents of the percolation theory in the corresponding flat space. We suggest that the stretched exponential behavior observed in many complex systems (polymers, colloids, glasses…) is due to the fractal character of their configuration space.