In the first part of the talk, we will quantify the universal Tracy-Widom laws for (generalized) Wigner matrices and derive an optimal Berry-Esseen type theorem for the fluctuations of the largest eigenvalue. The proof relies on the first long-time Green function comparison method near the edges without the second moment matching restriction. In the second part of the talk, we will discuss non-Hermitian matrices with i.i.d. entries and establish precise and universal three-term asymptotic expansions for the rightmost eigenvalue and spectral radius with optimal error estimates.

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