Gaussian unitary ensemble ($\mathrm{GUE}(n)$)

In the unscaled convention with diagonal variance one, a Gaussian unitary ensemble matrix $H$ is a Hermitian matrix with independent real diagonal entries $N(0,1)$ and upper off-diagonal entries $(A_{ij}+i{B}_{ij})/\sqrt{2}$, where all the $A_{ij},B_{ij}$ and diagonal entries are independent and have the standard normal distribution. Its density on the real coordinates of Hermitian matrices is proportional to $\exp [-\mathrm{tr}(H^2)/2]$. Its exchangeably labelled real eigenvalues have symmetric joint probability density function proportional to ${\prod }_{i<j}({\lambda }_i-{\lambda }_j{)}^2{e}^{-{\sum }_i{\lambda }_i^2/2}$. The squared Vandermonde determinant suppresses coincident eigenvalues; orthogonal polynomial projection kernels express the correlation functions of a point process as determinants.