MARK KAC SEMINAR

The Sherrington-Kirkpatrick model is the archetype of mean field spin glasses, a particular type of disordered systems introduced in the 70’s to model certain magnetic alloys with remarkable features. A breakthrough in the analysis of these systems was achieved by the theoretical physicist and recent Nobel laureate Giorgio Parisi in 1978, when he devised is celebrated “replica symmetry breaking” mechanism. In the first hour of these lecture series, I will recall some basic from statistical mechanics through the paradigmatic Curie-Weiss model, then move to its disordered counterpart, the SK-model, and conclude with a sketchy rendition of Parisi's mysteriously powerful computations.

In the second hour of these lectures, I will recall what is now known rigorously about the SK-model. I will, in particular, give an overview of the groundbreaking work by Talagrand, Guerra, and Panchenko. Keywords will be: Talagrand’s cavity approach for the high temperature regime, Guerra’s interpolations ( as well as their re-formulation by Aizenman, Sims, Starr ), Ghirlanda-Guerra identities, and finally Panchenko’s proof of the ultrametricity conjecture.

In this third hour of the lecture series, we will go back to theoretical physics: we will recall the main ideas from an alternative treatment to the SK-model (which, in fact, pre-dates the Parisi replica computations) devised by Thouless, Anderson and Palmer (TAP) in the late 70’s, and later improved by Plefka. The TAP-Plefka analysis amounts to non-rigorous expansions of the Gibbs potential, the critical points of which are the celebrated TAP equations for the spin magnetisations under the quenched Gibbs measure. We will discuss, in a sketchy way, these diagrammatic expansions, as well as the unsettled conceptual enigmas of the framework.

We propose a new iterative construction of solutions of the classical TAP equations for the Sherrington-Kirkpatrick model, i.e. with finite-size Onsager correction. The algorithm can be started in an arbitrary point, and converges up to the AT line. The analysis relies on a novel treatment of mean field algorithms through Stein's method. As such, the approach also yields weak convergence of the effective fields at all temperatures towards Gaussians, and can be applied, upon proper alterations, to all models where TAP-like equations and a Stein-operator are available. Joint work with Adrien Schertzer, and Stephan Gufler.