Complexity and ultradiffusion

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Abstract
The authors present the exact solution to the problem of ultradiffusion in an arbitrary hierarchical space. They derive rigorous upper and lower bounds for the dynamic exponent describing the decay of the autocorrelation function. They 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. They conclude that the speed of relaxation is a measure of the complexity or lack of self-similarity of the underlying tree. They 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-law decay.

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