In this talk we will discuss two results on symmetric random matrices.
The first deals with a cross-over regime for the largest eigenvalue: when matrix tails are at a transition point from Tracy-Widom to Poisson law for the top eigenvalue, we can uncover a new deformed Poisson point process structure and eigenvector localization beyond spectral edge. This applies to Wigner, Wishart and other ensembles. In the second part the author discusses joint small ball estimates on multiple smallest singular values at different locations in the spectrum, under a natural entry density assumption, highlighting a quantitative version of independence.
