Weingarten Calculus

TL;DR

This paper develops a unified approach to Weingarten calculus, enabling explicit computation of integrals over the unitary group and generalizing moments of Haar random matrices.

Contribution

It provides a unified framework connecting classical results and deriving explicit formulas for unitary integrals and moments of Haar matrices.

Findings

Explicit formula for integrals over the unitary group.

Generalization of moments of Haar random matrices.

Connection of various existing results through a unified approach.

Abstract

We consider the problem of computing the integral $$ \int_{\mathcal{U}(d)} u_{i_1j_1}\cdots u_{i_nj_n} \bar{u}_{i'_1j'_1} \cdots \bar{u}_{i'_{n'}j'_{n'}} dU, $$ where the integration takes place with respect to the probability Haar measure on the unitary group $\mathcal{U}(d)$, and the $u_{ij}$ denotes the $ij$-th entry of a unitary matrix $U$. We present a unified approach connecting classical results, the explicit formula for the integral given by B. Collins and P. Sniady and subsequent works of various authors providing different points of view. Finally we are able to provide an explicit formula for the $2n$-th moment of the trace of a unitary Haar random matrix, generalizing a result of P. Diaconis.