In this paper we review and discuss some fundamental aspects of the random version of the Olami-Feder- Christensen model, and its relevance for the understanding of self-organized criticality (SOC). We review the universal character of the exponent <img width=32 height=32 id="_x0000_i1026" src="../../img/revistas/bjp/v33n3/a09img01.gif" align=absmiddle > or = 3/2, related to avalanche size distributions in random SOC models, and its connection to branching processes theory. We also generalize previous results, that had been obtained for the random OFC model with four neighbors, to any coordination number. Finally we present some connections between our generalization and recent discussions involving the branching rate approach to this model.

Recent studies of the effects of geometric fluctuations, associated with aperiodic exchange inter-actions, on the critical behavior of ferromagnetic Ising models on hierarchical lattices, were the motivation to investigate some properties of two-letter sequences generated by uniform substitutions. We rigorously identify the substitutions that generate either periodic or aperiodic (almost periodic or not) sequences. For instance, two known substitutions, the Thue-Morse and the period-doubling rules, generate aperiodic sequences of almost periodic type.

We review recent investigations of the critical behavior of ferromagnetic q-state Potts models on a class of hierarchical lattices, with exchange interactions according to some deterministic but aperiodic substitution rules. The problem is formulated in terms of exact recursion relations on a suitable parameter space. The analysis of the fixed points of these equations leads to a criterion to gauge the relevance of the aperiodic geometric fluctuations. For irrelevant fluctuations, the critical behavior remains unchanged with respect to the underlying uniform models. In the presence of relevant fluctuations, a non-trivial symmetric fixed point, associated with the critical behavior of the uniform model, becomes fully unstable, and there appears a two-cycle of the recursion relations. A scaling analysis, supported by direct numerical thermodynamical calculations, shows the existence of a novel critical universality class associated with relevant geometric fluctuations.

We write exact renormalization-group recursion relations for a ferromagnetic Ising model on the diamond hierarchical lattice with an aperiodic distribution of exchange interactions according to a class of generalized two-letter Fibonacci sequences. For small geometric fluctuations, the critical behavior is unchanged with respect to the uniform case. For large fluctuations, the uniform fixed point in the parameter space becomes fully unstable. We analyze some limiting cases, and propose a heuristic criterion to check the relevance of the fluctuations.