Full large deviation principles for the largest eigenvalue of sub-Gaussian Wigner matrices

We establish precise estimates for the probability of rare events of the largest eigenvalue of Wigner matrices with sub-Gaussian entries. In contrast to the case of Wigner matrices with heavier tails, where deviations are governed by the appearance of a few large entries, and the sharp sub-Gaussian case that is governed by the collective deviation of entries in a delocalized rank-one pattern, we show that in the general sub-Gaussian case that deviations can be caused by a mixture of localized and delocalized changes in the entries. Our key result is a finite-$N$ approximation for the probability of rare events by an optimization problem involving \emph{restricted annealed free energies} for a spherical spin glass model. This allows us to derive full large deviation principles for the largest eigenvalue in several cases, including when the law of the matrix entries is compactly supported and symmetric, as well as the case of randomly sparsified GOE matrices.