  The mean field spin glass model is analyzed by a combination of
mathematically rigororous methods and a powerful 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.
