On the meaning of Parisi's functional order parameter

The author has recently proved that a famous formula discovered by G. Parisi gives at any temperature the correct value for the limiting free energy of a large class of mean field models for spin glasses (a class which contains in particular the Sherrington–Kirkpatrick model). Here we prove rigorously that (generically) the “functional order parameter” occuring in this formula can be interpreted as predicted by Parisi, namely as representing the limiting distribution of the overlap of two independent configurations. To cite this article: M. Talagrand, C. R. Acad. Sci. Paris, Ser. I 337 (2003).