A Brief Introduction to Mean Field Spin Glass Models
Historically, mean field spin glass models come from the study of statistical physics and have served as prototypical examples of complex energy landscapes. To tackle these questions statistical physicists developed a new class of tools, such as the cavity method and the replica symmetry breaking. Since their introduction, these methods have been applied to a wide variety of problems from statistical physics, to combinatorics, to data science.
This course will serve as a high-level introduction to the basics of mean field spin glasses and is intended to introduce the students to the basic notions that will arise in other courses during the Seminaire. On the first day, we plan to cover the random energy model, the ultrametric decomposition of Gibbs measures in p-spin glass models and the connection to Poisson-Dirichlet statistics. On the second day, if there is time, we will also introduce notions of free energy barriers and overlap gaps and their connection to spectral gap inequalities and algorithmic hardness results.
- Measure theoretic probability;
- Point processes and their definition as random probability measures;
- Basic notions from Gaussian analysis (concentration of measure, Slepian's interpolation inequality)