Integration over matrix spaces with unique invariant measures

TL;DR

This paper introduces a systematic method for calculating integrals over matrix spaces with invariant measures, providing exact results for low-order monomials and controlled approximations for higher orders, useful for symbolic computation.

Contribution

The paper develops a novel approach using Wick contractions to evaluate integrals over matrix spaces, with error bounds and systematic improvements for higher-order monomials.

Findings

Exact results for low-order monomials in matrix integrals.

Error bounds of order 1/N^alpha for higher-order monomials.

Explicit calculations for O(N), U(N), and circular orthogonal ensemble.

Abstract

We present a method to calculate integrals over monomials of matrix elements with invariant measures in terms of Wick contractions. The method gives exact results for monomials of low order. For higher--order monomials, it leads to an error of order 1/N^alpha where N is the dimension of the matrix and where alpha is independent of the degree of the monomial. We give a lower bound on the integer alpha and show how alpha can be increased systematically. The method is particularly suited for symbolic computer calculation. Explicit results are given for O(N), U(N) and for the circular orthogonal ensemble.