An edge CLT for the log determinant of Wigner ensembles

We derive a Central Limit Theorem (CLT) for $\log \left\vert\det \left( W_{N}-E_{N}\right)\right\vert,$ where $W_{N}$ is a Wigner matrix, and $E_{N}$ is local to the edge of the semi-circle law. Precisely, $E_N=2+N^{-2/3}\sigma_N$ with $\sigma_N$ being either a constant (possibly negative), or a sequence of positive real numbers, slowly diverging to infinity so that $\sigma_N \ll \log^{2} N$. We also extend our CLT to cover spiked Wigner matrices. Our interest in the CLT is motivated by its applications to statistical testing in critically spiked models and to the fluctuations of the free energy in the spherical Sherrington-Kirkpatrick model of statistical physics.