A Classification of Critical Phenomena on Quasi-Crystals and Other Aperiodic Structures

We investigate statistical-mechanical models on aperiodic structures ( e.g. quasi-periodic, random, self-similar). We derive a condition for the relevance of any kind of aperiodicity near a continuous phase transition, in the spirit of the Harris criterion. This explains why crystals and quasi-crystals exhibit the same critical behaviour. Novel universality classes of critical phenomena are predicted on structures with unbounded geometrical fluctuations, whenever their wandering exponent exceeds a model-dependent threshold value.