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In Chapter 3 of the textbook: An Introduction to Random Matrices, we have that for normalized GUE/GOE/GSE and ordering its eigenvalues $\lambda_1\le \lambda_2\le \cdots \le \lambda_n$, we have that $$ \lim_{n\to \infty}P(n^{2/3}(\lambda_n-2)\le x)\to \mbox{Tracy-Widom law} $$

Thereby, we have $\lambda_2-\lambda_1\approx O_p(n^{-2/3})$.

Question: Can we weaken the conditions of "GOE" (Gaussian orthogonal ensemble)? For example for any symmetric Wigner matrix ensembles with some conditions on the moment?

Here symmetric Wigner matrix ensembles mean the upper triangular coefficients $\xi_{ij}, j\ge i$ are jointly independent and real with $\xi_{ij}=\xi_{ji}$, and the strictly upper triangular coefficients will be iid, as will the diagonal coefficients, but the diagonal classes' of coefficients may have a different distribution.

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There is an extensive literature on the universality of the Tracy-Widom distribution. Here are some pointers:

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  • $\begingroup$ Thank you very much! $\endgroup$
    – Hermi
    Oct 31, 2022 at 18:55

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