Asymptotic results for smallest gap of Gaussian random matrix For a symmetric Gaussian random matrix $G=\{G\}_{1\le i,j \le n}$ with iid $E[G_{ij}]=0$ and $E[G_{ij}^2]=1/n$ (it is normalized), ordering its eigenvalues $\lambda_1\le \lambda_2\le\cdots \lambda_n$.
Is there any results about the asymptotic result for the smallest gap $\delta=\min_{1\le i,j \le n}\{|\lambda_i-\lambda_j|\}$?
 A: For a complex Hermitian matrix (GUE ensemble) the probability distribution of the smallest eigenvalue spacing $\delta_{\rm min}$ is such that the rescaled minimal spacing $x=\delta_{\rm min}n^{4/3}$ has for large $n$ the asymptotic distribution
$$P(x)=3x^2e^{-x^3},$$
see Closest Spacing of Eigenvalues by J.P. Vinson (2011).
This result has been generalized to real symmetric matrices (GOE ensemble) by Feng, Tian, and Wei, in Small gaps of GOE (2019). In  that case the rescaled minimal spacing $x=\delta_{\rm min}n^{3/2}$ has for large $n$ the limiting distribution
$$P(x)=2xe^{-x^2}.$$
Note that for independent eigenvalues (with a mean spacing that scales as $1/n$) the minimal spacing would scale as $n^{-2}$ (Poisson distribution). The $n^{-3/2}$ scaling for the GOE and the $n^{-4/3}$ scaling for the GUE are indicative of "level repulsion" (stronger for the GUE than for the GOE). More generally, for the three classical ensembles (GOE: $\beta=1$, GUE: $\beta=2$, GSE: $\beta=4$) the minimal spacing scales as $n^{-(\beta+2)/(\beta+1)}$, and the rescaled minimal spacing has limiting distribution $P(x)\propto x^\beta e^{-x^{\beta+1}}$.
 In the OP the scaling $\delta_{\rm min} n^{2/3}\log n$ is suggested, I don't know where that comes from, it does not agree with these results from the literature, the scaling should be $\delta_{\rm min} n^{3/2}$ in the GOE. Note also that, since the mean level spacing scales as $1/n$, the smallest spacing cannot have an exponent smaller than $1$ -- a scaling as $n^{-2/3}$ is not possible.
