This question is motivated by Guido Li's question Expected minimal distance of eigenvalues, which concerns the sum of a deterministic matrix and a Gaussian random matrix. The deformation of the Gaussian ensemble is a complication that I would like to remove:
Consider the Gaussian ensemble of $n\times n$ real symmetric matrices $H$, with probability distribution
$$dP(H)=C\exp(-n\,{\rm tr}\, H^2)\prod_{i\leq j}dH_{ij},$$
with $C$ a normalization constant.
Order the eigenvalues $\lambda_n$ from small to large and compute the set of spacings $s_n=\lambda_{n+1}-\lambda_n$, $n=1,2,\ldots n-1$. The average spacing is $1$ independent of $n$ for large $n$.
Q: Denote the smallest spacing for a given $H$ in the ensemble by $s_{\rm min}$. How does the expectation value $\mathbb{E}[s_{\rm min}]$, averaged over the ensemble, decay with increasing $n$?
If the $s_n$'s would be independent I would expect $\mathbb{E}[s_{\rm min}]$ to decay as $1/\sqrt n$ (based on the Wigner surmise for the spacing distribution). Is this the correct scaling?
