Construction of pure states in mean field models for spin glasses

If a mean field model for spin glasses is generic in the sense that it satisfies the extended Ghirlanda-Guerra identities, and if the law of the overlaps has a point mass at the largest point q* of its support, we prove that one can decompose the configuration space into a sequence of sets (A (k) ) such that, generically, the overlap of two configurations is equal to q* if and only if they belong to the same set A (k) . For the study of the overlaps each set A (k) can be replaced by a single point. Combining this with a recent result of Panchenko (A connection between Ghirlanda-Guerra identities and ultrametricity. Ann Probab (2008, to appear)) this proves that if the overlaps take only finitely many values, ultrametricity occurs. We give an elementary, self-contained proof of this result based on simple inequalities and an averaging argument.