Coarsening in inhomogeneous systems

I will review the topic of coarsening phenomena occurring in ferromagnetic systems where quenched features - such as random field, varying coupling constants or lattice vacancies - spoil homogeneity. I will discuss the current understanding of the problem in systems with a non-conserved scalar order parameter by focusing primarily on the form of the growth-law of the ordered domains and on the scaling properties. A general picture emerges characterized by two possible growth-laws, either logarithmic or algebraic, connected by a crossover phenomenon. In systems where lack of homogeneity is due to dilution an interpretation of these two growth-forms can be given in terms of the topology of the underlying diluted network hosting the magnetic system. In particular, a conjecture is proposed where a logarithmic or a power-law growth are associated to the presence/absence of an equilibrium phase-transition. I will discuss how such a conjecture is supported by the results of numerical simulations of paradigmatic model systems such as the Ising model with either site or bond diluted, both in the case of a stochastic dilution and in that of deterministic fractal graphs. The relevance of this to the case of different systems with other sources of inhomogeneities will be also discussed.