The 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. We also consider the case where these measures are restricted to the unit circle minus small neighborhoods around ±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 (GMC) measure. Our result is analogous to one relating to unitary matrices previously established by Christian Webb (2015 Electron. J. Probab. 20). We thus complete the connection between the classical compact groups and GMC. To prove this convergence when excluding small neighborhoods around ±1 we establish appropriate asymptotic formulae for Toeplitz and Toeplitz + Hankel determinants with merging singularities. Using a recent formula due to Claeys et al (2021 Int. Math. Res. Not. rnaa354), we are able to prove convergence on the whole of the unit circle.