Complexity and Ultradiffusion

. We present the exact solution to the problem of diffusion in an arbitrary hierarchical space. We derive rigorous upper and lower bounds for the dynamic exponent describing the decay of the autocorrelation function. We show that the upper bound is saturated by both uniformly and randomly multifurcating hierarchical trees, and identify a class of highly unbalanced trees that saturate the lower bound. We conclude that the speed of relaxation is a measure of the --complexity, or lack of self-similarity of the underlyng tree. We point out that -complexity may be revealed by the temperature dependence of the dynamic exponent, and in particular by the nature of the transition from exponential to power-lay decay. Submitted to Journal of Physics A -*Work supported in part by the Department of Energy, contract DE-ACOS76SF00515 and by ONR contract NOOOl-14-82-0699. _.f Present Address: Centre de Physique Theorique, Ecole Polytechnique, 91128 Palaiseau, France