Asymptotics for averages over classical orthogonal ensembles

TL;DR

This paper derives new asymptotic formulas for eigenvalue statistics in orthogonal ensembles, especially near singularities and gaps, using orthogonal polynomial techniques and extending known results from unitary groups.

Contribution

It provides novel asymptotic results for Toeplitz+Hankel determinants with Fisher-Hartwig singularities, including merging singularities and gaps, in orthogonal ensembles.

Findings

Asymptotics for gap probabilities in COE and CSE

Upper bounds for eigenvalue rigidity in orthogonal ensembles

Extension of Fisher-Hartwig asymptotics to merging singularities

Abstract

We study averages of multiplicative eigenvalue statistics in ensembles of orthogonal Haar distributed matrices, which can alternatively be written as Toeplitz+Hankel determinants. We obtain new asymptotics for symbols with Fisher-Hartwig singularities in cases where some of the singularities merge together, and for symbols with a gap or an emerging gap. We obtain these asymptotics by relying on known analogous results in the unitary group and on asymptotics for associated orthogonal polynomials on the unit circle. As consequences of our results, we derive asymptotics for gap probabilities in the Circular Orthogonal and Symplectic Ensembles, and an upper bound for the global eigenvalue rigidity in the orthogonal ensembles.