The Aizenman-Sims-Starr scheme and Parisi formula for mixed $p$-spin spherical models

Abstract

The Parisi formula for the free energy in the spherical models with mixed even $p$-spin interactions was proven in Michel Talagrand. In this paper we study the general mixed $p$-spin spherical models including $p$-spin interactions for odd $p$. We establish the Aizenman Sims-Starr scheme and from this together with many well-known results and Dmitry Panchenko's recent proof on the Parisi ultrametricity conjecture, we prove the Parisi formula.