An edge CLT for the log determinant of Laguerre beta ensembles

We obtain a CLT for $\log \mathrm{\mid }\det (M_n-s_n)\mathrm{\mid }$ where $M_n$ is a scaled Laguerre $\beta$ ensemble and $s_n=d_{+}+{\sigma }_n{n}^{-2\mathrm{/}3}$ 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.