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On the number of zeros of linear combinations of independent characteristic polynomials of random unitary matrices


Barhoumi-Andréani, Yacine; Hughes, Christopher; Najnudel, Joseph; Nikeghbali, Ashkan (2015). On the number of zeros of linear combinations of independent characteristic polynomials of random unitary matrices. International Mathematics Research Notices, 2015(23):12366-12404.

Abstract

We show that for any linear combination of characteristic polynomials of independent random unitary matrices with the same determinant, the expected proportion of zeros lying on the unit circle tends to 1 as the dimension of the matrices tends to infinity. This result is the random matrix analog of an earlier result by Bombieri and Hejhal on the distribution of zeros of linear combinations of L-functions, and thus is consistent with the conjectured links between the value distribution of the characteristic polynomial of random unitary matrices and the value distribution of L-functions on the critical line.

Abstract

We show that for any linear combination of characteristic polynomials of independent random unitary matrices with the same determinant, the expected proportion of zeros lying on the unit circle tends to 1 as the dimension of the matrices tends to infinity. This result is the random matrix analog of an earlier result by Bombieri and Hejhal on the distribution of zeros of linear combinations of L-functions, and thus is consistent with the conjectured links between the value distribution of the characteristic polynomial of random unitary matrices and the value distribution of L-functions on the critical line.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:10 March 2015
Deposited On:14 Jan 2016 12:05
Last Modified:15 Feb 2018 06:15
Publisher:Oxford University Press
ISSN:1073-7928
OA Status:Green
Publisher DOI:https://doi.org/10.1093/imrn/rnv060

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