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

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