Gaussian Multiplicative Chaos for Gaussian Orthogonal and Symplectic Ensembles

TL;DR

This paper demonstrates that for Gaussian Orthogonal and Symplectic Ensembles, the normalized powers of characteristic polynomials converge to Gaussian multiplicative chaos measures, revealing new asymptotic relations among ensemble moments.

Contribution

It introduces a novel asymptotic relation between fractional moments of characteristic polynomials across Gaussian ensembles and establishes convergence to Gaussian multiplicative chaos.

Findings

Convergence of characteristic polynomial powers to chaos measures

New asymptotic relations between ensemble moments

Applicable to small real powers in the ensembles

Abstract

We study the characteristic polynomials of both the Gaussian Orthogonal and Symplectic Ensembles. We show that for both ensembles, powers of the absolute value of the characteristic polynomials converge in law to Gaussian multiplicative chaos measures after normalization for sufficiently small real powers. The main tool is a new asymptotic relation between the fractional moments of the absolute characteristic polynomials of Gaussian Orthogonal, Unitary, and Symplectic Ensembles.