Some exact results on the ultrametric overlap distribution in mean field spin glass models (I)
Dipartimento di Matematica, Università di Roma "Tor Vergata", via della Ricerca Scientifica 1, 00133 Roma,
2 Dipartimento di Fisica, Università di Roma "Tor Vergata", via della Ricerca Scientifica 1, 00133 Roma, Italy
Published online: 15 October 2000
The mean field spin glass model is analyzed by a combination of exact methods and a simple Ansatz. The method exploited is general, and can be applied to others disordered mean field models such as, e.g., neural networks. It is well known that the probability measure of overlaps among replicas carries the whole physical content of these models. A functional order parameter of Parisi type is introduced by rigorous methods, according to previous works by F. Guerra. By the Ansatz that the functional order parameter is the correct order parameter of the model, we explicitly find the full overlap distribution. The physical interpretation of the functional order parameter is obtained, and ultrametricity of overlaps is derived as a natural consequence of a branching diffusion process. It is shown by explicit construction that ultrametricity of the 3-replicas overlap distribution together with the Ghirlanda-Guerra relations determines the distribution of overlaps among s replicas, for any s, in terms of the one-overlap distribution.
PACS: 05.20.-y – Classical statistical mechanics / 64.70.Pf – Glass transitions / 75.10.Nr – Spin-glass and other random models
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2000