Moments of Moments of the Characteristic Polynomials of Random Orthogonal and Symplectic Matrices

TL;DR

This paper derives asymptotic formulas for the moments of the moments of characteristic polynomials of large random orthogonal and symplectic matrices, revealing symmetry-dependent phase transitions.

Contribution

It extends existing results for unitary matrices to orthogonal and symplectic groups using Toeplitz+Hankel determinant asymptotics, highlighting symmetry effects.

Findings

Asymptotic formulas for moments of moments for orthogonal and symplectic matrices

Identification of symmetry-dependent phase transitions

Extension of Fahs' results from unitary to other classical groups

Abstract

Using asymptotics of Toeplitz+Hankel determinants, we establish formulae for the asymptotics of the moments of the moments of the characteristic polynomials of random orthogonal and symplectic matrices, as the matrix-size tends to infinity. Our results are analogous to those that Fahs obtained for random unitary matrices in [14]. A key feature of the formulae we derive is that the phase transitions in the moments of moments are seen to depend on the symmetry group in question in a significant way.