Past exam of the random matrix theory course of the University of Oxford / 2025 / 2 / c / i

The correlation functions of a point process use factorial normalization:Start with ${\rho }_3=D_3/6$. By Gaudin's lemma, integrating $D_3$ once gives $D_2$, and integrating $D_2$ once gives $2D_1$. The factorial factors cancel, so the eigenvalues form a determinantal point process withandThe last equality uses the real symmetry of this kernel. In particular, $\int R_3^{(1)}=3$ and $\iint R_3^{(2)}=6$, as required for three points and six ordered pairs of distinct points.