Do we have the universal property of the edge of the spectrum for the Wigner matrix?

Question

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.

Answer 1

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 studies the general case of a Wigner matrix.
• A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices considers Wishart matrices, of the form $XX^\ast$.
• Local law and Tracy–Widom limit for sparse random matrices considers sparse random matrices.
• Tracy-Widom distribution for the edge eigenvalues of Gram type random matrices considers Gram matrices.