Random-walk renormalisation and the spectral dimension of harmonic models on a class of hierarchical lattice

Abstract

Random-walk recursion relations are developed on a simple example of hierarchical lattice and are found to be the generating functions for first-passage walks. With exact renormalisation of a general harmonic model the spectral dimension, F, for the bond hierarchy class is shown to be F=2ln(g)/(ln(g)+ln( lambda r)), where g and lambda r are respectively the aggregation number and resistance eigenvalue. The behaviour of F across families of hierarchies is discussed. It is noted that the renormalisation group has limitations in its abilities to give results for the above models on hierarchies and that F so found may not maintain on hierarchies its usual significance for random-walk statistics.