Moments of a Single Entry of Circular Orthogonal Ensembles and Weingarten Calculus

Abstract

Consider a symmetric unitary random matrix V = (v ij )1 ≤ i, j ≤ N from a circular orthogonal ensemble. In this paper, we study moments of a single entry v ij . For a diagonal entry v ii , we give the explicit values of the moments, and for an off-diagonal entry v ij , we give leading and subleading terms in the asymptotic expansion with respect to a large matrix size N. Our technique is to apply the Weingarten calculus for a Haar-distributed unitary matrix.