A Generalisation of Dyson's Integration Theorem for Determinants

TL;DR

This paper generalizes Dyson's integration theorem for determinants, enabling the reduction of complex integrals involving arbitrary functions in random matrix theory, thus broadening its applicability beyond orthogonal functions.

Contribution

The authors derive a new formula that reduces (n-k)-fold integrals of determinants of arbitrary functions to smaller determinants, extending Dyson's theorem to non-orthogonal function sets.

Findings

Reduces complex integrals to smaller determinants

Applicable to non-orthogonal and non-bi-orthogonal functions

Recovers Dyson's original theorem in special cases

Abstract

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 beta = 2. We derive a formula reducing the (n-k)-fold integral of an n x n determinant of a kernel of two sets of arbitrary functions to a determinant of size k x 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.