The goal of this minicourse will be to outline some recent progress in the analysis of the Sherrington Kirkpatrick mean field, spin glass model. This is primarily work of
1. Guerra and Toninelli,
2. Guerra himself, and
3. Aizenman, Sims and myself.
In the first two weeks we will introduce the problem. We will describe the physical motivation of the SK model, some initial failed attempts by physicists to analyze the model, and Parisi's amazing, but highly heuristic exact solution by the replica symmetry breaking. We will also compare to the deterministic analogue, which is the Curie-Weiss, mean field spin system. The CW model really is exactly solvable as can be proved using a large deviation principle, and is a typical first example
in a basic statistical mechanics class. The SK model is harder.
In the second portion of class, we will describe the work of Guerra and Toninelli to prove the existence of a thermodynamic limit at all temperatures; the work of Guerra to prove that the Parisi-ansatz free energy is a rigorous lower bound; and the work of Aizenman, Sims and myself to obtain an extended variational principle for the free energy density in the thermodynamic limit. These works all use one trick, which is "quadratic interpolation".
We note that the problem of proving Parisi's ansatz is not solved by these techniques. Only certain parts of the problem.
In the last part of class we will describe Ruelle's random probability cascades, and show how they can be used to derive Parisi's ansatz as a variational bound, starting from the extended variational principle (thus rederiving 2). A less detailed and more heuristic version of this second derivation is included in the nice book of Mezard, Parisi and Viraso, "Spin Glass Theory and Beyond". But the mathematically rigorous version does not seem to be well known. It remains a mystery (to me) to understand how Parisi came up with his first derivation. I believe there must be a good answer and it probably leads to new math!