Speaker
Description
In their seminal `77 paper, Thouless, Anderson and Palmer (TAP) proposed their famous free energy for the SK model. The main focus of these talks is a related but different free energy, whose definition is guided in a natural way by properties of the Gibbs measure. It was introduced for spherical mixed p-spin models (arXiv:1804.10576) and in a joint work with Wei-Kuo Chen and Dmitry Panchenko (arXiv:1812.05066) for Ising models, and I will discuss both types of models in a unified way.
Like the TAP free energy, this free energy is a function defined over the convex hull of the configuration space, which is equal to the energy plus a deterministic function. Under some conditions it coincides with the TAP free energy, but in general it only lower bounds it. It satisfies two principles: 1. its maximum over certain radii is equal to the free energy of the model; 2. its maximizers are exactly the barycenters of pure/ancestral states, or subsets that are similar to them in an appropriate sense. The first principle yields a different representation for the free energy for any overlap in the support of the limiting overlap distribution (the order parameter), that generalizes the TAP representation which only corresponds to the largest overlap in the support. If time permits, I will also discuss important properties of this free energy like uniform concentration (self-averaging), its value on the vertices of the ultrametric tree, and how it can be used to compute the free energy of the spherical pure p-spin models and their multi-species version.
Based in part on a joint work with Wei-Kuo Chen and Dmitry Panchenko.
