Multifractality of entangled random walks and non-uniform hyperbolic spaces

Multifractal properties of the distribution of topological invariants for a model of trajectories randomly entangled with a non-symmetric lattice of obstacles are investigated. Using the equivalence of the model to random walks on a locally non-symmetric tree, statistical properties of topological invariants, such as drift and return probabilities, have been studied by means of a renormalization-group (RG) technique. The comparison of the analytical RG results with numerical simulations as well as with the rigorous results of Gerl and Woess demonstrates clearly the validity of our approach. It is shown explicitly, by direct counting for the discrete version of the model and by conformal methods for the continuous version, that multifractality occurs when local uniformity of the phase space (which has an exponentially large number of states) has been broken.