We derive a Central Limit Theorem (CLT) for log | det (WN − EN )|, where WN is a Wigner matrix, and EN is local to the edge of the semi-circle law. Precisely, EN = 2 + N−2/3 σN with σN being either a constant (possibly negative), or a sequence of positive real numbers, slowly diverging to infinity so that σN ≪ log2 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.
Johnstone et al. (Sat,) studied this question.