An edge CLT for the log determinant of Laguerre beta ensembles

TL;DR

This paper establishes a central limit theorem for the log determinant of scaled Laguerre beta ensembles near the spectral edge, with implications for statistical testing and spin glass models.

Contribution

It extends CLT results to Laguerre beta ensembles at the spectral edge, including special cases like LUE and LOE, with new conditions on the scaling parameter.

Findings

CLT holds for $ ext{log}| ext{det}(M_n - s_n)|$ near the spectral edge.

Results apply to LUE and LOE with constant order $\sigma_n$.

Implications for statistical testing and spin glass models.

Abstract

We obtain a CLT for $\log|\det(M_n-s_n)|$ where $M_n$ is a scaled Laguerre $\beta$ ensemble and $s_n=d_++\sigma_n n^{-2/3}$ with $d_+$ denoting the upper edge of the limiting spectrum of $M_n$ and $\sigma_n$ a slowly growing function ($\log\log^2 n\ll\sigma_n\ll\log^2 n$). In the special cases of LUE and LOE, we prove that the CLT also holds for $\sigma_n$ of constant order. A similar result was proved for Wigner matrices by Johnstone, Klochkov, Onatski, and Pavlyshyn. Obtaining this type of CLT of Laguerre matrices is of interest for statistical testing of critically spiked sample covariance matrices as well as free energy of bipartite spherical spin glasses at critical temperature.