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.

On the moments of moments of random matrices and Ehrhart polynomials

“…In the symplectic case much less is known currently and as far as we are aware there is no precise conjecture for the asymptotics of the maximum of log |P SP(2N) θ; g |. Nevertheless, the asymptotics of MoM SP(2N) (k; q) for positive integer parameters k, q ∈ N were established in [6] and connections to GMC in [27], see also [37].…”

Asymptotics for averages over classical orthogonal ensembles

“…In the case m = 1 where we have only two singularities, this multiplicative constant can be evaluated explicitly in terms of quantities related to a solution of the fifth Painlevé equation [20]. Simultaneously with this work, Forkel and Keating [27] evaluated the e O (1) factor in (2.10) explicitly in terms of the same Painlevé V solution when m = 2, as long as the singularities e ±it1 , e ±it2 do not approach ±1. When there are more than two singularities approaching each other, one might expect a multiplicative constant connected to a generalization of the fifth Painlevé equation, but the problem of evaluating the constant remains open.…”

“…independent of N ) with Fisher-Hartwig singularities, see [21,Theorem 1.25] for the most complete result in this respect and [4,8,9,10] for earlier developments. However, the picture for averages in O ± N is incomplete because, as far as we know, asymptotics are not known for symbols vanishing on an arc, and no results are available about transition asymptotics in situations where either several singularities approach each other in the large N limit (except for the results from [27] obtained simultaneously with ours, see Remark 2.4 below), or parameters tune in such a way that a gap in the support emerges as N → ∞. The objective in this paper is to complete this task.…”

“…The question of sharpness of the upper bound is closely related to the theory of Gaussian multiplicative chaos, see e.g. [2,12,39] in general and [27] in this specific situation.…”

“…This work was supported by the Fonds de la Recherche Scientifique-FNRS under EOS project O013018F. We are grateful to Johannes Forkel and Jon Keating for sharing an early version of their work [27] and for useful comments on an early version of our manuscript.…”