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 $\delta=\min\_{1\le i,j \le n}\{|\lambda\_i-\lambda\_j|\}$?



Asymptotic results for smallest gap of Gaussian random matrix
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](https://arxiv.org/abs/1111.2743) 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](https://arxiv.org/abs/1901.01567) (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.


