Weekly Bulletin

In this talk I will address the question of bonding the operator norm of the covariance matrix of a disordered spin system. A first motivation stems from random matrix theory, where one would like to develop techniques for understanding matrices with subtle dependencies between the entries. A second motivation is about establishing functional inequalities and rapid mixing of Markov chains for these systems, where it has become recently apparent that bounding the norm of certain covariance matrices is a crucial step. I will explain the above connections and present new results in the case of the Sherrington-Kirkpatrick model and the random field Ising model. I will then pose open problems and challenges arising when trying to extend these techniques to other settings of interest. Relevant readings: https://arxiv.org/abs/2212.02445 (with Gaitonde) https://arxiv.org/abs/2311.06171 (with Eldan, Gheissari and Piana) https://arxiv.org/abs/2212.14476 (Brennecke, Schertzer, Xu, Yau)