Dynamics on Ultrametric Spaces

Abstract

We present an exact solution to the problem of a random walk on an ultrametric space with arbitrary transition probabilities. The solution provides a description of fluctuation dynamics of systems with many degenerate states separated by a hierarchy of energy barriers. Dynamical behavior is found to depend on the scaling of barriers with ultrametric distance. We find a range of interesting behavior, from temperature-dependent power-law decay to the Kohlrausch law.