Moments of Random Matrices and Weingarten Functions

Let G be the unitary group, orthogonal group, or (compact) symplectic group, equipped with its Haar probability measure, and suppose that G is realized as a matrix group. Consider a random matrix X = (xi,j)1≤i,j≤N picked up from G. We would like to know how to compute the moments E [ xi1j1 · · ·xinjnxi′1j′ 1 · · ·xi′nj′ n ] or E [xi1j1 · · ·xi2nj2n ] . In this report, we focus on the unitary group UN and use the methods established in [5] and [9] which express the moments as sums in terms of Weingarten functions. The function Wg(·, N), called the unitary Weingarten function, has rich combinatorial structures involving Jucys-Murphy elements. We discuss and prove some of its properties. Finally, we consider some applications of the formula for integration with respect to the Haar measure over the unitary group UN . We compute matrix-valued expectations with the goal of having a better understanding of the operator-valued Cauchy transform.