Moments of a single entry of circular orthogonal ensembles and Weingarten calculus

Abstract: Consider a symmetric unitary random matrix $V=(v_{ij}){1 \le i,j \le N}$ 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.