A classic problem in this connection is to derive GOE or GUE statistics for the spectrum of a random self-adjoint operator of the form $H=-\nabla^2 + V(\vec{r})$, in some bounded domain of $\mathbb{R}^3$. The measure is the Gaussian measure for $V(\vec{r})$, of zero mean and given two-point correlation function. Since this is a real operator, one would expect GOE statistics, to obtain GUE statistics one would replace $\nabla\mapsto \nabla+i\vec{B}\times\vec{r}$ for some given vector $\vec{B}$.

This problem was solved by Konstantin Efetov in 1982, as described in much detail in his book on <A HREF="https://books.google.nl/books?id=E4qpedgudPEC">Supersymmetry in Disorder and Chaos.</A> The GOE or GUE statistics is found to hold only over a limited range $E_{0}$ (the so-called <A HREF="https://en.wikipedia.org/wiki/Thouless_energy">Thouless energy</A>): eigenvalues that differ by more than $E_0$ become uncorrelated, reverting to Poisson statistics.