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

We show that the fluctuations of the largest eigenvalue of any generalized Wigner matrix $H$ converge to the Tracy-Widom laws at a rate nearly $O(N^{-1\mathrm{/}3})$, as the matrix dimension $N$ tends to infinity. We allow the variances of the entries of $H$ to have distinct values but of comparable sizes such that ${\sum }_i\mathbb{E}\mathrm{\mid }{h}_{ij}{\mathrm{\mid }}^2=1$. Our result improves the previous rate $O(N^{-2\mathrm{/}9})$ by Bourgade [8] and the proof relies on the first long-time Green function comparison theorem near the edges without the second moment matching restriction.