Abstract
For an {N times N} Haar distributed random unitary matrix UN, we consider the random field defined by counting the number of eigenvalues of UN in a mesoscopic arc centered at the point u on the unit circle. We prove that after regularizing at a small scale {epsilon_{N} > 0}, the renormalized exponential of this field converges as {N to infty} 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 L1-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.
Highlights
IntroductionApart from the weight e−N V (x)/2, fV is exactly the first column of the solution YN of the orthogonal polynomial Riemann–Hilbert problem, whose solution is derived in great detail in [19]
The study of Gaussian fields with logarithmic correlations has seen many recent developments in the last few years. One of those concerns a relation to the eigenvalues of random matrices, which can be traced back to a work of Hughes, Keating and O’Connell [37]. They studied the characteristic polynomial of random N × N matrices UN sampled from the unitary group according to the Haar measure, known as the Circular Unitary Ensemble (CUE)
We provide a detailed discussion of Gaussian multiplicative chaos (GMC) in Appendix B, but the basic idea can be summarised as follows
Summary
Apart from the weight e−N V (x)/2, fV is exactly the first column of the solution YN of the orthogonal polynomial Riemann–Hilbert problem, whose solution is derived in great detail in [19] In particular from their results, one can extract the universal oscillatory behavior of the functions fV and gV in the bulk, which indicates strongly that the approach we present for the sine process could be generalized and would provide a way to show that the limiting chaos measure has the same law for a large class of potentials V. Turning this heuristic into a rigorous computation is rather technical and we leave it as an open problem for future work
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