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Large Deviation Theorem for Zeros of Polynomials and Hermitian Random Matrices

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Abstract

We give abstract versions of the large deviation theorem for the distribution of zeros of polynomials and apply them to the characteristic polynomials of Hermitian random matrices. We obtain new estimates related to the local semi-circular law for the empirical spectral distribution of these matrices when the 4th moments of their entries are controlled.

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Acknowledgements

This work was supported by the Start-Up Grant R-146-000-204-133 from National University of Singapore. It was partially written during my visit at Paris 11 University. I would like to thank Viet-Anh Nguyen and Nessim Sibony for their hospitality and help. I also would like to thank the referee for his remarks which allow me to improve the presentation of this article.

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  1. Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore, 119076, Singapore

    Tien-Cuong Dinh

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  1. Tien-Cuong Dinh
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Correspondence to Tien-Cuong Dinh.

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In memory of Professor Gennadi Henkin.

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Dinh, TC. Large Deviation Theorem for Zeros of Polynomials and Hermitian Random Matrices. J Geom Anal 30, 2558–2580 (2020). https://doi.org/10.1007/s12220-017-9951-8

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