Relaxational dynamics for a class of disordered ultrametric models

Abstract

The relaxational dynamics for a class of disordered, ultrametric spaces with arbitrary and irregular branchings $K$ is considered. It is shown that, if the transfer rates between sites depend upon their ultrametric distance and obey a certain constraint, the relaxational eigenvalues and eigenmodes can be obtained exactly for an arbitrary tree. The plausibility of the constraint is given on physical grounds. Approximate ensemble averagings are performed for the decay law leading to a power law similar to that found in regular models. This generalizes the exactly soluble regular $K=2\mathrm{}$ model of Ogielski and Stein, and makes closer contact with real disordered systems.