On the Characteristic Polynomial¶ of a Random Unitary Matrix

Hughes, C. P.; Keating, Jon P; O’Connell, Neil

doi:10.1007/s002200100453

Cited by 187 publications

(270 citation statements)

References 23 publications

Supporting

Mentioning

254

Contrasting

Unclassified

Order By: Relevance

“…Our methods also recover the results [HKO00] (continued from those in [KS00]) on the asymptotic normality of the suitably normalised logarithm of the characteristic polynomial of M n . As remarked in [HKO00], this result coupled with an application of the argument principle gives another "explanation" of the covariance structure of [Wie98].…”

Section: Introductionsupporting

confidence: 53%

“…As remarked in [HKO00], this result coupled with an application of the argument principle gives another "explanation" of the covariance structure of [Wie98].…”

Section: Introductionmentioning

confidence: 56%

“…We essentially follow the notation and development given in [HKOS00] (see also [HKO00]). For an n × n unitary matrix U , write…”

Section: Asymptotics For the Characteristic Polynomialmentioning

confidence: 99%

See 2 more Smart Citations

Linear functionals of eigenvalues of random matrices

Diaconis

Evans

2001

Trans. Amer. Math. Soc.

228

194

View full text Add to dashboard Cite

Abstract. Let Mn be a random n × n unitary matrix with distribution given by Haar measure on the unitary group. Using explicit moment calculations, a general criterion is given for linear combinations of traces of powers of Mn to converge to a Gaussian limit as n → ∞. By Fourier analysis, this result leads to central limit theorems for the measure on the circle that places a unit mass at each of the eigenvalues of Mn. For example, the integral of this measure against a function with suitably decaying Fourier coefficients converges to a Gaussian limit without any normalisation. Known central limit theorems for the number of eigenvalues in a circular arc and the logarithm of the characteristic polynomial of Mn are also derived from the criterion. Similar results are sketched for Haar distributed orthogonal and symplectic matrices.

show abstract

Section: Introductionsupporting

confidence: 53%

“…As remarked in [HKO00], this result coupled with an application of the argument principle gives another "explanation" of the covariance structure of [Wie98].…”

Section: Introductionmentioning

confidence: 56%

See 1 more Smart Citation

Linear functionals of eigenvalues of random matrices

Diaconis

Evans

2001

Trans. Amer. Math. Soc.

228

194

View full text Add to dashboard Cite

show abstract

“…Different scalings, characterizing the large deviations of log Λ A , were also computed in [44], and shown to agree with numerical calculations and other results known to hold for the zeta function.…”

Section: Modelling Zeta At Finite Height On the Critical Linesupporting

confidence: 73%

“…It is natural to ask how close this average is to an average with respect to θ when A is fixed; that is, about ergodicity. It was proved in [44] that indeed the average is ergodic, in the sense that in the limit as N → ∞, the average over θ equals that over A for all but a set of matrices of zero measure.…”

Section: Modelling Zeta At Finite Height On the Critical Linementioning

confidence: 99%

Riemann Zeros and Random Matrix Theory

Snaith

2010

Milan J. Math.

View full text Add to dashboard Cite

In the past dozen years random matrix theory has become a useful tool for conjecturing answers to old and important questions in number theory. It was through the Riemann zeta function that the connection with random matrix theory was first made in the 1970s, and although there has also been much recent work concerning other varieties of L-functions, this article will concentrate on the zeta function as the simplest example illustrating the role of random matrix theory.

show abstract

Averages of characteristic polynomials in random matrix theory

Borodin

Strahov

2005

Comm. Pure Appl. Math.

103

159

View full text Add to dashboard Cite

We compute averages of products and ratios of characteristic polynomials associated with orthogonal, unitary, and symplectic ensembles of random matrix theory. The Pfaffian/determinantal formulae for these averages are obtained, and the bulk scaling asymptotic limits are found for ensembles with Gaussian weights. Classical results for the correlation functions of the random matrix ensembles and their bulk scaling limits are deduced from these formulae by a simple computation. We employ a discrete approximation method: the problem is solved for discrete analogues of random matrix ensembles originating from representation theory, and then a limit transition is performed. Exact Pfaffian/determinantal formulae for the discrete averages are proven using standard tools of linear algebra; no application of orthogonal or skew‐orthogonal polynomials is needed. © 2005 Wiley Periodicals, Inc.

show abstract

On the Characteristic Polynomial¶ of a Random Unitary Matrix

Cited by 187 publications

References 23 publications

Linear functionals of eigenvalues of random matrices

Linear functionals of eigenvalues of random matrices

Riemann Zeros and Random Matrix Theory

Averages of characteristic polynomials in random matrix theory

Contact Info

Product

Resources

About