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Subcritical multiplicative chaos for regularized counting statistics from random matrix theory
Version 2 2023-06-12, 08:49
Version 1 2023-06-09, 11:57
journal contribution
posted on 2023-06-12, 08:49 authored by Gaultier Lambert, Dmitry Ostrovsky, Nicholas SimmNicholas SimmFor 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 ?_N>0. We prove that the renormalized exponential of this field converges as N ? 8 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.
Funding
Mesoscopic statistics of random matrices and the Gaussian free field; Leverhulme Trust; ECF-2014-309
History
Publication status
- Published
File Version
- Published version
Journal
Communications in Mathematical PhysicsISSN
0010-3616Publisher
Springer VerlagExternal DOI
Issue
1Volume
360Page range
1-54Department affiliated with
- Mathematics Publications
Research groups affiliated with
- Probability and Statistics Research Group Publications
Full text available
- Yes
Peer reviewed?
- Yes
Legacy Posted Date
2018-02-07First Open Access (FOA) Date
2018-05-16First Compliant Deposit (FCD) Date
2018-02-07Usage metrics
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