A Generalisation of Dyson’s Integration Theorem for Determinants

Dyson’s integration theorem is widely used in the computation of eigenvalue correlation functions in Random Matrix Theory. Here we focus on the variant of the theorem for determinants, relevant for the unitary ensembles with Dyson index β = 2. We derive a formula reducing the (n − k)-fold integral of an n × n determinant of a kernel of two sets of arbitrary functions to a determinant of size k× k. Our generalisation allows for sets of functions that are not orthogonal or bi-orthogonal with respect to the integration measure. In the special case of orthogonal functions Dyson’s theorem is recovered. 1 Motivation Random Matrix Theory (RMT) has many applications in all areas of Physics and beyond (see e.g. the introduction of [1]). For the class of invariant RMT Dyson’s integration theorem is at the heart of the method of orthogonal polynomials when computing all eigenvalue correlation functions exactly, for finite n × n matrices. The resulting expressions are then amenable to the large-n limit, in which universal RMT predictions follow. In the following we restrict ourselves to the integration theorem for determinants. Before presenting our generalisation thereof we briefly recall how it reveals all eigenvalue correlations in the unitary ensembles. We start by stating Dyson’s integration theorem, as cited in [1] (Theorem 5.1.4). Given K(x, y) is a real valued function satisfying the following self-contraction property: