The Classical Compact Groups and Gaussian Multiplicative Chaos

TL;DR

This paper demonstrates that certain random measures derived from powers of characteristic polynomials of Haar-distributed orthogonal and symplectic matrices converge to Gaussian multiplicative chaos measures, extending previous results from unitary matrices.

Contribution

It establishes the convergence of these measures for classical compact groups, completing the link between these groups and Gaussian multiplicative chaos, and extends the results to the entire unit circle.

Findings

Convergence of measures to Gaussian multiplicative chaos for orthogonal and symplectic groups.

Extension of results to the whole unit circle using recent asymptotic formulas.

Completion of the connection between classical compact groups and Gaussian multiplicative chaos.

Abstract

We consider powers of the absolute value of the characteristic polynomial of Haar distributed random orthogonal or symplectic matrices, as well as powers of the exponential of its argument, as a random measure on the unit circle minus small neighborhoods around $\pm 1$. We show that for small enough powers and under suitable normalization, as the matrix size goes to infinity, these random measures converge in distribution to a Gaussian multiplicative chaos measure. Our result is analogous to one on unitary matrices previously established by Christian Webb in [31]. We thus complete the connection between the classical compact groups and Gaussian multiplicative chaos. To prove this we establish appropriate asymptotic formulae for Toeplitz and Toeplitz+Hankel determinants with merging singularities. Using a recent formula communicated to us by Claeys et al., we are able to extend our result to the whole of the unit circle.