In the Thouless-Anderson-Palmer approach to mean-field spin glasses, the free energy is presented as the infimum of a functional which TAP defined over the space of all possible magnetization vectors, subject to a convergence condition. Its self-averaging over exponentially many solutions at low temperature seems to be taken for granted, though that topic has not been fully addressed.

The main focus of this talk is a related but different free energy functional, defined over magnetization vectors, with more explicit relation to the Gibbs measure. The functional was introduced in the context of the spherical mixed p-spin models (arXiv:1804.10576) and then analyzed for Ising spin models in a joint work with Wei-Kuo Chen and Dmitry Panchenko (arXiv:1812.05066).  I will explain how this free energy functional self-averages uniformly for all magnetization vectors.  As a consequence of that, we obtain a characterization for all approximately pure or ancestral states of the Gibbs measure, and a family of representations for the free energy which generalize the TAP representation. Time permitting, I will discuss recent results about the free energy for the pure spherical models and their multi-species version.