91社区

罢颈迟濒别:听Free energy fluctuations of spherical spin glasses near the critical temperature threshold.

Abstract:

One of the fascinating phenomena of spin glasses is the dramatic change in behavior that occurs between the high and low temperature regimes. In addition to its physical meaning, this phase transition corresponds to a detection threshold with respect to the signal-to-noise ratio in a spiked matrix model. The free energy of the spherical Sherrington-Kirkpatrick (SSK) model 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). From a statistical perspective, the free energy of SSK and bipartite SSK correspond to log-likelihood ratios for spiked Wigner and spiked Wishart matrices respectively. 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 Wishart random matrices evaluated near the spectral edge.

Speaker

Elizabeth Collins-Woodfin is a postdoc in the probability group at 91社区. Her research involves high-dimensional probability in the context of spin glasses, random matrix theory, and stochastic gradient descent. She received her PhD from the University of Michigan in 2022, advised by Jinho Baik.