An edge CLT for the log determinant of Laguerre ensembles

Elizabeth W Collins-Woodfin, Han Gia Le

Published 2022-09-07Version 1

We obtain a CLT for $\log|\det(M_n-s_n)|$ where $M_n$ is a 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 ($\log\log^2 n\ll\sigma_n\ll\log^2 n$). 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.