Subcritical Multiplicative Chaos for Regularized Counting Statistics from Random Matrix Theory

For an Haar distributed random unitary matrix U (N) , we consider the random field defined by counting the number of eigenvalues of U (N) in a mesoscopic arc centered at the point u on the unit circle. We prove that after regularizing at a small scale , the renormalized exponential of this field converges as to a Gaussian multiplicative chaos measure in the whole subcritical phase. We discuss implications of this result for obtaining a lower bound on the maximum of the field. We also show that the moments of the total mass converge to a Selberg-like integral and by taking a further limit as the size of the arc diverges, we establish part of the conjectures in Ostrovsky (Nonlinearity 29(2):426-464, 2016). By an analogous construction, we prove that the multiplicative chaos measure coming from the sine process has the same distribution, which strongly suggests that this limiting object should be universal. Our approach to the L (1)-phase is based on a generalization of the construction in Berestycki (Electron Commun Probab 22(27):12, 2017) to random fields which are only asymptotically Gaussian. In particular, our method could have applications to other random fields coming from either random matrix theory or a different context.