Subcritical multiplicative chaos for regularized counting statistics from random matrix theory

Lambert, Gaultier, Ostrovsky, Dmitry and Simm, Nick (2017) Subcritical multiplicative chaos for regularized counting statistics from random matrix theory. Communications in Mathematical Physics. ISSN 0010-3616 (Accepted)

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Abstract

For an N×N random unitary matrix U_N, we consider the random field defined by counting the number of eigenvalues of U_N in a mesoscopic arc of the unit circle, regularized at an N-dependent scale Ɛ_N>0. We prove that the renormalized exponential of this field converges as N → ∞ to a Gaussian multiplicative chaos measure in the whole subcritical phase. In addition, we 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 [55]. 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. The proofs are based on the asymptotic analysis of certain Toeplitz or Fredholm determinants using the Borodin-Okounkov formula or a Riemann-Hilbert problem for integrable operators. Our approach to the L¹-phase is based on a generalization of the construction in Berestycki [5] 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.

Item Type: Article
Keywords: Random matrix theory, multiplicative chaos, multifractal random measures, Gaussian processes
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Research Centres and Groups: Probability and Statistics Research Group
Subjects: Q Science > QA Mathematics > QA0273 Probabilities. Mathematical statistics > QA0274 Stochastic processes
Q Science > QA Mathematics > QA0273 Probabilities. Mathematical statistics > QA0274 Stochastic processes > QA0274.4 Gaussian processes
Q Science > QA Mathematics > QA0273 Probabilities. Mathematical statistics
Depositing User: Nicholas Simm
Date Deposited: 07 Feb 2018 12:58
Last Modified: 13 Apr 2018 14:13
URI: http://sro.sussex.ac.uk/id/eprint/73392

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Project NameSussex Project NumberFunderFunder Ref
Mesoscopic statistics of random matrices and the Gaussian free fieldUnsetLeverhulme TrustECF-2014-309