Higher Order Spacing Distributions for a Class of Unitary Ensembles

We consider the $n\mathrm{th}$-order spacing distribution, $P^n\mathrm{}\left(s\right)\mathrm{}$, in the statistical theory of energy levels of complex systems. Each $P^n$ is written as a sum of multiple integrals over correlation functions. This procedure is used to establish the identity of the spacing distributions for all members of a class of Hamiltonian unitary ensembles. A power-series expansion of $P^n\mathrm{}\left(s\right)\mathrm{}$, valid for all $n$, is developed.