Abstract
In this paper we study the asymptotic behavior of mesoscopic fluctuations for the thinned Circular Unitary Ensemble. The effect of thinning is that the eigenvalues start to decorrelate. The decorrelation is stronger on the larger scales than on the smaller scales. We investigate this behavior by studying mesoscopic linear statistics. There are two regimes depending on the scale parameter and the thinning parameter. In one regime we obtain a CLT of a classical type and in the other regime we retrieve the CLT for CUE. The two regimes are separated by a critical line. On the critical line the limiting fluctuations are no longer Gaussian, but described by infinitely divisible laws. We argue that this transition phenomenon is universal by showing that the same transition and their laws appear for fluctuations of the thinned sine process in a growing box. The proofs are based on a Riemann-Hilbert problem for integrable operators.
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Acknowledgements
We thank Kurt Johansson, Gaultier Lambert and Boualem Djehiche for fruitful and inspiring discussions. We also thank the anonymous referees for many valuable comments that helped improving the presentation of the paper.
Tomas Berggren was supported by the grant KAW 2010.0063 from the Knut and Alice Wallenberg Foundation. Maurice Duits was supported by the Swedish Research Council (VR) Grant no. 2012-3128.
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Berggren, T., Duits, M. Mesoscopic Fluctuations for the Thinned Circular Unitary Ensemble. Math Phys Anal Geom 20, 19 (2017). https://doi.org/10.1007/s11040-017-9250-4
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DOI: https://doi.org/10.1007/s11040-017-9250-4