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

TL;DR

This paper analyzes the moments of individual entries in circular orthogonal ensemble matrices, providing explicit formulas for diagonal entries and asymptotic expansions for off-diagonal entries using Weingarten calculus.

Contribution

It offers explicit moment formulas for diagonal entries and asymptotic expansions for off-diagonal entries in circular orthogonal ensembles, utilizing Weingarten calculus.

Findings

Explicit moments for diagonal entries derived.

Asymptotic behavior of off-diagonal entries characterized.

Application of Weingarten calculus to ensemble moments.

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.