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 $\delta=\min_{1\le i,j \le n}\{|\lambda_i-\lambda_j|\}$?

In this paper arxiv.org/abs/1010.1294, On page 6, it shows that for any $k$ $$ n^{2/3}(\lambda_n-2,\lambda_{n-1}-2,\dots, \lambda_1-2)\to \mbox{ multivariate Tracy–Widom distribution} $$ weakly as $n\to \infty$.

It follows that $$ n^{2/3}(\lambda_1-2, \lambda_2-2)\to Y_2=(Y_2^1, Y_2^2) $$ for some $R^2$ valued random variable $Y_2$.

Hence, by the continuous mapping theorem $$ n^{2/3}(\lambda_2-\lambda_1)\to Y_2^2-Y_2^1 $$

Hence, $\lambda_2-\lambda_1=O_p(n^{-2/3})$.

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 $\delta=\min_{1\le i,j \le n}\{|\lambda_i-\lambda_j|\}$?

For a symmetric Gaussian random matrix $G=\{G\}_{1\le i,j \le n}$ with iid $E[G_{ij}]=0$ and $E[G_{ij}^2]=1/n$, ordering its eigenvalues $\lambda_1\le \lambda_2\le\cdots \lambda_n$.

Is there any results about the non-asymptotic result for the smallest gap $\delta=\min_{1\le i,j \le n}\{|\lambda_i-\lambda_j|\}$? $$ P(\delta n^{2/3}\log n\ge x)\ge \mbox{something large e.g. } ( 1-\alpha e^{-\beta x})... $$

Or the asymptotic result like as n large enough $$ \delta n^{2/3}\log n\to ? $$

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 $\delta=\min_{1\le i,j \le n}\{|\lambda_i-\lambda_j|\}$?

In this paper arxiv.org/abs/1010.1294, On page 6, it shows that for any $k$ $$ n^{2/3}(\lambda_n-2,\lambda_{n-1}-2,\dots, \lambda_1-2)\to \mbox{ multivariate Tracy–Widom distribution} $$ weakly as $n\to \infty$.

It follows that $$ n^{2/3}(\lambda_1-2, \lambda_2-2)\to Y_2=(Y_2^1, Y_2^2) $$ for some $R^2$ valued random variable $Y_2$.

Hence, by the continuous mapping theorem $$ n^{2/3}(\lambda_2-\lambda_1)\to Y_2^2-Y_2^1 $$

Hence, $\lambda_2-\lambda_1=O_p(n^{-2/3})$.