Workshop: Using RMT to understand phase transitions in spherical spin glass

Speaker
Collins-Woodfin, McGill University

Abstract
One of the fascinating phenomena of spin glasses is the dramatic change in behavior that occurs between the high and low temperature regimes.  The free energy of the spherical Sherrington-Kirkpatrick (SSK) model, for example, has Gaussian fluctuations at high temperature, but Tracy-Widom fluctuations at low temperature.  A similar phenomenon holds for the bipartite SSK model, and we show that, when the temperature is within a small window around the critical temperature, the free energy fluctuations converge to an independent sum of Gaussian and Tracy-Widom random variables (joint work with Han Le).  Our work follows two recent papers that proved similar results for the SSK model (by Landon and by Johnstone, Klochkov, Onatski, Pavlyshyn).  Analyzing bipartite SSK at critical temperature requires a variety of tools including classical random matrix results, contour integral techniques, and a CLT for the log-characteristic polynomial of Laguerre (Wishart) random matrices evaluated near the spectral edge.  This last ingredient was not present in the literature when we began our project, so I will discuss our proof of this CLT, which has other applications separate from bipartite spin glasses.