Abstract
We demonstrate the convergence of the characteristic polynomial of several random matrix ensembles to a limiting universal function, at the microscopic scale. The random matrix ensembles we treat are classical compact groups and the Gaussian Unitary Ensemble. In fact, the result is the by-product of a general limit theorem for the convergence of random entire functions whose zeros present a simple regularity property.
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Acknowledgements
We would like to thank Elizabeth Meckes for an informative response regarding some of the bounds proved in Sect. 3, Sasha Sodin likewise for a helpful discussion, and an anonymous referee for several useful comments and corrections. B.R. was partially supported during this research by the NSF grant DMS-1701577.
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Communicated by Vadim Gorin.
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Chhaibi, R., Hovhannisyan, E., Najnudel, J. et al. The Limiting Characteristic Polynomial of Classical Random Matrix Ensembles. Ann. Henri Poincaré 20, 1093–1119 (2019). https://doi.org/10.1007/s00023-019-00769-4
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DOI: https://doi.org/10.1007/s00023-019-00769-4