Topological regulation of activation barriers on fractal substrates

We study the phase-ordering dynamics of a ferromagnetic system with a scalar order-parameter on fractal graphs. We propose a scaling approach, inspired by renormalization-group ideas, where a crossover between distinct dynamical behaviors is induced by the presence of a length λ associated with the topological properties of the graph. The transition between the early and the asymptotic stages is observed when the typical size L(t) of the growing ordered domains reaches the crossover length λ. We consider two classes of inhomogeneous substrates with different activated processes, where the effects of the free-energy barriers can be analytically controlled during the evolution. On finitely ramified graphs, the free-energy barriers encountered by domains walls grow logarithmically with L(t ), whereas they increase as a power law on all other structures. This produces different asymptotic growth laws (power laws vs logarithmic) and a different dependence of the crossover length λ on the model parameters. Our theoretical picture agrees very well with extensive numerical simulations.