In Chapter 3 of the textbook: [An Introduction to Random Matrices](https://www.google.com.hk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&cad=rja&uact=8&ved=2ahUKEwiC8aL504P7AhW_k2oFHTv9BK8QFnoECBUQAQ&url=http%3A%2F%2Fwww.wisdom.weizmann.ac.il%2F%7Ezeitouni%2Fcupbook.pdf&usg=AOvVaw0wbMPwbj788zuq0QFfT4Cp&cshid=1666985383466164), 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.



Do we have the universal property of the edge of the spectrum for the Wigner matrix?
There is an extensive literature on the universality of the Tracy-Widom distribution. Here are some pointers:


* [Quantitative Tracy-Widom
laws for the largest eigenvalue of generalized Wigner matrices](https://arxiv.org/abs/2207.00546)
studies the general case of a Wigner matrix.
* [A necessary and
sufficient condition for edge universality at the largest singular
values of covariance matrices](https://www.jstor.org/stable/26542349%20) considers Wishart matrices, of the
form $XX^\ast$.
* [Local law and Tracy–Widom
limit for sparse random matrices](https://arxiv.org/abs/1605.08767) considers sparse random
matrices.
* [Tracy-Widom distribution
for the edge eigenvalues of Gram type random matrices](https://arxiv.org/abs/2008.04166) considers
Gram matrices.


