Gaussian orthogonal ensemble ($\mathrm{GOE}(n)$)

In the unscaled variance-one convention, a Gaussian orthogonal ensemble matrix $M$ is a real symmetric matrix whose upper-triangular entries are independent, with $M_{ij}\sim N(0,1)$ for $i<j$ and $M_{ii}\sim N(0,2)$. Its joint probability density function relative to the upper-triangular coordinates is proportional to $\exp [-\mathrm{tr}(M^2)/4]$. The scaled convention uses $M/\sqrt{n}$; the scaling must be stated. If $X$ has independent standard normal entries, $(X+X^T)/\sqrt{2}$ realizes the unscaled convention. Expanding in the entries of $X$ gives the covariance identity $\mathbb{E}[M_{ij}{M}_{kl}]={\delta }_{ik}{\delta }_{jl}+{\delta }_{il}{\delta }_{jk}$, with Kronecker deltas.