Dissertation Defense: Log-Determinant of the Laguerre Beta Ensembles and Free Energy of the Bipartite Spherical Sherrington–Kirkpatrick Model

Abstract:

This thesis consists of two parts, each addressing a distinct problem concerning the asymptotic behavior of large disordered systems. The first part studies random matrices from the Laguerre beta ensembles. We prove that when a Laguerre matrix is shifted by a scalar multiple of the identity—where the scalar is near the edge of the Mar\v{c}enko–Pastur law—the logarithm of its determinant exhibits Gaussian fluctuations. Additionally, we establish that this result extends to the Laguerre Orthogonal and Laguerre Unitary Ensembles under a relaxed assumption on the shift. This edge central limit theorem has applications in statistical testing of critically spiked sample covariance matrices, and in analyzing free energy fluctuations of bipartite spherical spin glasses at critical temperature. The second part examines the free energy fluctuations of the bipartite spherical Sherrington–Kirkpatrick model, an extension of the spherical Sherrington–Kirkpatrick model by incorporating heterogeneity. Previous work by Baik and Lee shows that at non-critical temperatures, the free energy exhibits Gaussian fluctuations at high temperatures and Tracy–Widom fluctuations at low temperatures. We focus on the critical temperature case and demonstrate that within a vanishing window around criticality, the fluctuations are given by the sum of independent Gaussian and Tracy–Widom random variables. A key component of our analysis is the central limit theorem for Laguerre matrices established in the first part of the thesis.