Extreme values of CUE characteristic polynomials: a numerical study

Yan V. Fyodorov; Sven Gnutzmann; Jonathan P. Keating

doi:10.1088/1751-8121/aae65a

Extreme values of CUE characteristic polynomials: a numerical study

Yan V. Fyodorov, Sven Gnutzmann^*, Jonathan P. Keating

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

11 Citations (Scopus)

Abstract

We present the results of systematic numerical computations relating to the extreme value statistics of the characteristic polynomials of random unitary matrices drawn from the circular unitary ensemble (CUE) of random matrix theory. In particular, we investigate a range of recent conjectures and theoretical results inspired by analogies with the theory of logarithmically-correlated Gaussian random fields. These include phenomena related to the conjectured freezing transition. Our numerical results are consistent with, and therefore support, the previous conjectures and theory. We also go beyond previous investigations in several directions: we provide the first quantitative evidence in support of a correlation between extreme values of the characteristic polynomials and large gaps in the spectrum, we investigate the rate of convergence to the limiting formulae previously considered, and we extend the previous analysis of the CUE to the CβE which corresponds to allowing the degree of the eigenvalue repulsion to become a parameter.

Original language	English
Article number	464001
Journal	Journal of Physics A: Mathematical and Theoretical
Volume	51
Issue number	46
DOIs	https://doi.org/10.1088/1751-8121/aae65a
Publication status	Published - 22 Oct 2018

Keywords

freezing transition
log-correlated processes
random matrix theory

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10.1088/1751-8121/aae65a

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abstract = "We present the results of systematic numerical computations relating to the extreme value statistics of the characteristic polynomials of random unitary matrices drawn from the circular unitary ensemble (CUE) of random matrix theory. In particular, we investigate a range of recent conjectures and theoretical results inspired by analogies with the theory of logarithmically-correlated Gaussian random fields. These include phenomena related to the conjectured freezing transition. Our numerical results are consistent with, and therefore support, the previous conjectures and theory. We also go beyond previous investigations in several directions: we provide the first quantitative evidence in support of a correlation between extreme values of the characteristic polynomials and large gaps in the spectrum, we investigate the rate of convergence to the limiting formulae previously considered, and we extend the previous analysis of the CUE to the CβE which corresponds to allowing the degree of the eigenvalue repulsion to become a parameter.",

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T2 - a numerical study

AU - Fyodorov, Yan V.

AU - Gnutzmann, Sven

AU - Keating, Jonathan P.

PY - 2018/10/22

Y1 - 2018/10/22

N2 - We present the results of systematic numerical computations relating to the extreme value statistics of the characteristic polynomials of random unitary matrices drawn from the circular unitary ensemble (CUE) of random matrix theory. In particular, we investigate a range of recent conjectures and theoretical results inspired by analogies with the theory of logarithmically-correlated Gaussian random fields. These include phenomena related to the conjectured freezing transition. Our numerical results are consistent with, and therefore support, the previous conjectures and theory. We also go beyond previous investigations in several directions: we provide the first quantitative evidence in support of a correlation between extreme values of the characteristic polynomials and large gaps in the spectrum, we investigate the rate of convergence to the limiting formulae previously considered, and we extend the previous analysis of the CUE to the CβE which corresponds to allowing the degree of the eigenvalue repulsion to become a parameter.

AB - We present the results of systematic numerical computations relating to the extreme value statistics of the characteristic polynomials of random unitary matrices drawn from the circular unitary ensemble (CUE) of random matrix theory. In particular, we investigate a range of recent conjectures and theoretical results inspired by analogies with the theory of logarithmically-correlated Gaussian random fields. These include phenomena related to the conjectured freezing transition. Our numerical results are consistent with, and therefore support, the previous conjectures and theory. We also go beyond previous investigations in several directions: we provide the first quantitative evidence in support of a correlation between extreme values of the characteristic polynomials and large gaps in the spectrum, we investigate the rate of convergence to the limiting formulae previously considered, and we extend the previous analysis of the CUE to the CβE which corresponds to allowing the degree of the eigenvalue repulsion to become a parameter.

KW - freezing transition

KW - log-correlated processes

KW - random matrix theory

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JO - Journal of Physics A: Mathematical and Theoretical

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ER -

Extreme values of CUE characteristic polynomials: a numerical study

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