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General properties of overlap probability distributions in disordered spin systems. Towards Parisi ultrametricity. (English) Zbl 0953.82037

Summary: 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 \(q_{a,s+1}\) 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 \(q_{ab}\), with \(b\) running among the first \(s\) replicas, excluding \(a\). Each of these cases has equal probability \(1/s\).

MSC:

82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
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