Title: Cluster expansion approach to the mean field spin glass theory.
Abstract: Cluster expansion, as a powerful combinatorial scheme, has been widely used in many statistical mechanics problems. However, most of these focused on the non-disordered spin system. In disordered system, such as spin glasses, Aizenman-Lebowitz-Ruelle first utilized the cluster expansion idea to obtain some rigorous results in the Sherrington-Kirkpatrick (SK) model at zero external field. Nevertheless, since then, it was believed in the literature that this approach does not work once the external is present. In this talk, we will show that under some “weak” external field, the cluster expansion still works, but a new cluster appears. This implies some transition results for the free energy in SK model. Time permits, we will also discuss the extension of this framework to a very general setting, the mixed $p$-spin models, where a new multiple transition phenomenon was shown along with the discovery of a collection of fruitful cluster structures. This is based on joint works with Partha S. Dey.
