Some exact results on the ultrametric overlap distribution in mean field spin glass models (I)

The mean field spin glass model is analyzed by a combination of exact metho ds 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 carri es the whole physical content of these models. A functional order parameter of Parisi type is introduced by rigorous methods, according to previous wo rks 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 d istribution. The physical interpretation of the functional order parameter is obtained, and ultrametricity of overlaps is derived as a natural consequ ence of a branching diffusion process. It is shown by explicit construction that ultrametricity of the 3-replicas overlap distribution together with t he Ghirlanda-Guerra relations determines the distribution of overlaps among s replicas, for any s, in terms of the one-overlap distribution.