Random determinants, the elastic manifold, and landscape complexity beyond invariance

09/23/2022 11:00 am - 12:00 pm
Benjamin McKenna Fried Tong
CMSA Room G10
Address: CMSA, 20 Garden Street, Cambridge, MA 02138 USA

Member Seminar

Speaker: Ben McKenna

Title: Random determinants, the elastic manifold, and landscape complexity beyond invariance

Abstract: The Kac-Rice formula allows one to study the complexity of high-dimensional Gaussian random functions (meaning asymptotic counts of critical points) via the determinants of large random matrices. We present new results on determinant asymptotics for non-invariant random matrices, and use them to compute the (annealed) complexity for several types of landscapes. We focus especially on the elastic manifold, a classical disordered elastic system studied for example by Fisher (1986) in fixed dimension and by Mézard and Parisi (1992) in the high-dimensional limit. We confirm recent formulas of Fyodorov and Le Doussal (2020) on the model in the Mézard-Parisi setting, identifying the boundary between simple and glassy phases. Joint work with Gérard Ben Arous and Paul Bourgade.