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On the overlap in the multiple spherical SK models. (English) Zbl 1128.60086
Summary: In order to study certain questions concerning the distribution of the overlap in Sherrington-Kirkpatrick type models, such as the chaos and ultrametricity problems, it seems natural to study the free energy of multiple systems with constrained overlaps. One can write analogues of Guerra’s replica symmetry breaking bound for such systems but it is not at all obvious how to choose informative functional order parameters in these bounds. We were able to make some progress for spherical pure \(p\)-spin SK models where many computations can be made explicitly. For the pure 2-spin model, we prove ultrametricity and chaos in an external field. For the pure \(p\)-spin model for even \(p>4\) without an external field, we describe two possible values of the overlap of two systems at different temperatures. We also prove a somewhat unexpected result which shows that in the 2-spin model the support of the joint overlap distribution is not always witnessed at the level of the free energy and, for example, ultrametricity holds only in a weak sense.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
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