Dynamics on scale-invariant structures
We investigate dynamical processes on random and regular fractals. The (static) problem of percolation in the semi-infinite plane introduces many pertinent ideas including real space renormalisation group (RSRG) fugacity transformations and scaling forms. We study the percolation probability to determine the surface critical behaviour and to establish exponent relations. The fugacity approach is generalised to study random walks on diffusion-limited aggregates (DLA). Using regular and random models, we calculate the walk dimensionality and demonstrate that it is consistent with a conjecture by Aharony and Stauffer. It is shown that the kinetically grown DLA is in a distinct dynamic universality class to lattice animals. Similarly, the speculation of Helman-Coniglio-Tsallis regarding diffusion on self-avoiding walks (SAWs) is shown to be incorrect. The results are corroborated by an exact enumeration analysis of the internal structure of SAWs. A 'spin' and field theoretic Hamiltonian formulation for the conformational and resistance properties of random walks is presented. We consider Gaussian random walks, SAWs, spiral SAWs and valence walks. We express resistive susceptibilities as correlation functions and hence e-expansions are calculated for the resistance exponents. For SAWs, the local crosslinks are shown to be irrelevant and we calculate corrections to scaling. A scaling description is introduced into an equation-of-motion method in order to study spin wave damping in d-dimensional isotropic Heisenberg ferro-, antiferro- and ferri- magnets near pc . Dynamic scaling is shown to be obeyed by the Lorentzian spin wave response function and lifetime. The ensemble of finite clusters and multicritical behaviour is also treated. In contrast, the relaxational dynamics of the dilute Anisotropic Heisenberg model is shown to violate conventional dynamic scaling near the percolation bicritical point but satisfies instead a singular scaling behaviour arising from activation of Bloch walls over percolation cluster energy barriers.