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We show that the fluctuations of the largest eigenvalue of any generalizedWigner matrix $H$ converge to the Tracy-Widom laws at a rate nearly$O(N^{-1/3})$, as the matrix dimension $N$ tends to infinity. We allow thevariances of the entries of $H$ to have distinct values but of comparable sizessuch that $\sum_{i} \mathbb{E}|h_{ij}|^2=1$. Our result improves the previousrate $O(N^{-2/9})$ by Bourgade [8] and the proof relies on the first long-timeGreen function comparison theorem near the edges without the second momentmatching restriction.

Quantitative Tracy-Widom laws for the largest eigenvalue of generalized Wigner matrices