General properties of overlap probability distributions in disordered spin systems. Towards Parisi ultrametricity

For a very general class of probability distributions in disordered Ising spin systems, in the thermodynamical limit, we prove the following property for overlaps among real replicas. Consider the overlaps among s replicas. Add one replica s + 1. Then, the overlap between one of the first s replicas, let us say a , and the added s + 1 is either independent of the former ones, or it is identical to one of the overlaps , with b running among the first s replicas, excluding a . Each of these cases has equal probability .