## Duke Mathematical Journal

### The characteristic polynomial of a random unitary matrix: A probabilistic approach

#### Abstract

In this article, we propose a probabilistic approach to the study of the characteristic polynomial of a random unitary matrix. We recover the Mellin-Fourier transform of such a random polynomial, first obtained by Keating and Snaith in [8] using a simple recursion formula, and from there we are able to obtain the joint law of its radial and angular parts in the complex plane. In particular, we show that the real and imaginary parts of the logarithm of the characteristic polynomial of a random unitary matrix can be represented in law as the sum of independent random variables. From such representations, the celebrated limit theorem obtained by Keating and Snaith in [8] is now obtained from the classical central limit theorems of probability theory, as well as some new estimates for the rate of convergence and law of the iterated logarithm-type results

#### Article information

Source
Duke Math. J., Volume 145, Number 1 (2008), 45-69.

Dates
First available in Project Euclid: 17 September 2008

https://projecteuclid.org/euclid.dmj/1221656862

Digital Object Identifier
doi:10.1215/00127094-2008-046

Mathematical Reviews number (MathSciNet)
MR2451289

Zentralblatt MATH identifier
1155.15025

Subjects
Primary: 15A52
Secondary: 60F05: Central limit and other weak theorems 60F15: Strong theorems

#### Citation

Bourgade, P.; Hughes, C. P.; Nikeghbali, A.; Yor, M. The characteristic polynomial of a random unitary matrix: A probabilistic approach. Duke Math. J. 145 (2008), no. 1, 45--69. doi:10.1215/00127094-2008-046. https://projecteuclid.org/euclid.dmj/1221656862

#### References

• G. E. Andrews, R. Askey, and R. Roy, Special Functions, Encyclopedia Math. Appl. 71, Cambridge Univ. Press, Cambridge, 1999.
• P. Bourgade, A. Nikeghbali, and A. Rouault, Hua-Pickrell measures on general compact groups, preprint,\arxiv0712.0848v1[math.PR]
• P. Carmona, F. Petit, and M. Yor, On the distribution and asymptotic results for exponential functionals of Lévy processes'' in Exponential Functionals and Principal Values Related to Brownian Motion, Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoamericana, Madrid, 1997, 73--130.
• L. Chaumont and M. Yor, Exercises in Probability: A Guided Tour from Measure Theory to Random Processes, via Conditioning, Camb. Ser. Stat. Probab. Math. 13, Cambridge Univ. Press, Cambridge, 2003.
• P. Diaconis and M. Shahshahani, The subgroup algorithm for generating uniform random variables, Probab. Engrg. Inform. Sci. 1 (1987), 15--32.
• C. P. Hughes, J. P. Keating, and N. O'Connell, On the characteristic polynomial of a random unitary matrix, Comm. Math. Phys. 220 (2001), 429--451.
• J. Jacod and P. Protter, Probability Essentials, 2nd ed., Universitext, Springer, Berlin, 2003.
• J. P. Keating and N. C. Snaith, Random matrix theory and $\zeta(1/2+it)$, Comm. Math. Phys. 214 (2000), 57--89.
• —, Random matrix theory and $L$-functions at $s=1/2$, Comm. Math. Phys. 214 (2000), 91--110.
• M. L. Mehta, Random Matrices, 2nd ed., Academic Press, Boston, 1991.
• F. Mezzadri, How to generate random matrices from the classical compact groups, Notices Amer. Math. Soc. 54 (2007), 592--604.
• F. Mezzadri and N. C. Snaith, eds., Recent Perspectives in Random Matrix Theory and Number Theory, London Math. Soc. Lecture Note Ser. 322, Cambridge Univ. Press, Cambridge, 2005.
• A. Nikeghbali and M. Yor, The Barnes G-function and its relations with sums and products of generalized gamma convolution variables, preprint,\arxiv0707.3187v1[math.PR]
• B. Odgers, Random matrix theory and L-functions: Transitions between ensembles, Ph.D. dissertation, University of Bristol, Bristol, United Kingdom, 2006.
• V. V. Petrov, On a relation between an estimate of the remainder in the central limit theorem and the law of iterated logarithm, Theor. Probability Appl. 11 (1966), 454--458.
• —, Limit Theorems of Probability Theory: Sequences of Independent Random Variables, Oxford Stud. Probab. 4, Oxford Univ. Press, New York, 1995.
• —, On the law of the iterated logarithm for a sequence of independent random variables, Theory Probab. Appl. 46 (2003), 542--544.
• E. Royer, Fonction $\zeta$ et matrices aléatoires'' in Physics and Number Theory, IRMA Lect. Math. Theor. Phys. 10, Eur. Math. Soc., Zürich, 2006, 165--224.
• A. Selberg, Old and new conjectures and results about a class of Dirichlet series'' in Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, Italy, 1989), Univ. Salerno, Salerno, Italy, 1992, 367--385.
• D. N. Shanbhag and M. Sreehari, On certain self-decomposable distributions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 38 (1977), 217--222.
• D. W. Stroock, Probability Theory: An Analytic View, Cambridge Univ. Press, Cambridge, 1993.