Convergence of moments of twisted COE matrices

TL;DR

This paper studies the eigenvalue moments of perturbed Circular Orthogonal Ensemble matrices, showing universal behavior and fast convergence to Circular Unitary Ensemble moments when the permutation has long cycles.

Contribution

It establishes the universality and convergence rate of eigenvalue moments for COE matrices perturbed by permutations with long cycles, using Weingarten calculus.

Findings

Eigenvalue moments converge to CUE moments with rate 1/N^3.

Universality holds for permutations with only long cycles.

Cancellations eliminate lower-order error terms.

Abstract

We investigate eigenvalue moments of matrices from Circular Orthogonal Ensemble multiplicatively perturbed by a permutation matrix. More precisely we investigate variance of the sum of the eigenvalues raised to power $k$, for arbitrary but fixed $k$ and in the limit of large matrix size. We find that when the permutation defining the perturbed ensemble has only long cycles, the answer is universal and approaches the corresponding moment of the Circular Unitary Ensemble with a particularly fast rate: the error is of order $1/N^3$ and the terms of orders $1/N$ and $1/N^2$ disappear due to cancellations. We prove this rate of convergence using Weingarten calculus and classifying the contributing Weingarten functions first in terms of a graph model and then algebraically.