On the characteristic polynomial of a random unitary matrix

HP Laboratories Technical Report

Volumn BRIMS, Issue 17, 2000, Pages

  Hughes, C P ^a,b   Keating, J P ^a,b   O'Connell, Neil ^a  

^a HEWLETT PACKARD LABORATORIES   (United Kingdom)

^b UNIVERSITY OF BRISTOL   (United Kingdom)

Author keywords

Eigenvalue counting function; Fluctuations; Large deviations

Indexed keywords

COVARIANCE; HIGH-ORDER SCALINGS; UNIT CIRCLES; VARIABLES;

EIGENVALUES AND EIGENFUNCTIONS; PHASE TRANSITIONS; POLYNOMIALS; RANDOM PROCESSES; THEOREM PROVING;

MATRIX ALGEBRA;

EID: 84862448506     PISSN: None     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (33)

1. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover, 1965)
2. E.W. Barnes, "The Theory of the G-function", Quart. J. Pure Appl. Math. 31 (1899) 264-314
3. E. Basor, "Asymptotic formulas for Toeplitz determinants", Trans. Amer. Math. Soc. 239 (1978) 33-65
4. G. Ben Arous and A. Guionnet, "Large deviations for Wigner's law and Voiculescu's non-commutative entropy", Probab. Theor. Rel. Fields 108 (1997), no. 4, 517-542
5. G. Ben Arous and O. Zeitouni, "Large deviations from the circular law", ESAIM: Probability and Statistics 2 (1998) 123-134
6. A. Böttcher and B. Silbermann, Introduction to Large Truncated Toeplitz Matrices (Springer-Verlag, 1999)
7. J. Bucklew, Large Deviation Techniques in Decision, Simulation, and Estimation (Wiley Interscience, 1990)
8. M. Coram and P. Diaconis. "New tests of the correspondence between unitary eigenvalues and the zeros of Riemann's zeta function", preprint (2000)
9. O. Costin and J.L. Lebowitz, "Gaussian fluctuation in random matrices", Phys. Rev. Lett. 75 (1995) 69-72
10. A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, 2nd Ed. (Springer-Verlag, 1998)
11. P. Diaconis and S.N. Evans, "Linear functionals of eigenvalues of random matrices", preprint (2000)
12. P. Diaconis and M. Shashahani, "On the eigenvalues of random matrices", J. Appl. Probab. 31A (1994) 49-62
13. A. Ghosh, "On the Riemann zeta function - mean value theorems and the distribution of |S(T)|", J. Number Theory 17 (1983) 93-102
14. A. Guiounnet, "Fluctuations for strongly interacting random variables and Wigner's law", preprint (1999)
15. A. Guiounnet and O. Zeitouni, "Concentration of the spectral measure for large matrices", preprint (2000)
16. F. Hiai and D. Petz, "A large deviation theorem for the empirical eigenvalue distribution of random unitary matrices", preprint (2000)
17. T. Hida, Brownian Motion (Springer 1980)
18. K. Johansson, "On random matrices from the classical compact groups", Ann. Math. 145 (1997) 519-545
19. K. Johansson, "On fluctuations of eigenvalues of random Hermitian matrices", Duke J. Math. 91 (1998) 151-204
20. J.P. Keating and N.C. Snaith, "Random Matrix Theory and ς(1/2+it)", Commun. Math. Phys. (in the press) (1999)
21. A. Laurinčikas, Limit Theorems for the Riemann Zeta Function (Kluwer Academic Publishers, 1996)
22. M.L. Mehta, Random Matrices (Academic Press, 1991)
23. 20-th zero of the Riemann zeta function and 175 million of its neighbors", unpublished
24. E. Rains, "High powers of random elements of compact Lie groups", Prob. Th. Rel. Fields 107 (1997) 219-241
25. D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, (Springer-Verlag, 1990).
26. A.B. Shoshnikov, "Gaussian fluctuation for the number of particles in Airy, Bessel, Sine, and other determinantal random point fields", preprint (1999)
27. G. Szegö, Orthogonal Polynomials (AMS Colloqium Publications XXII, 1939)
28. E.G. Titchmarsh, The Theory of the Riemann Zeta-Function (Oxford Science Publications, 1986)
29. A. Voros, "Spectral functions, special functions and the Selberg zeta function", Comm. Math. Phys. 110 (1987) 439-465
30. H. Weyl, Classical Groups, (Princeton University Press, 1946)
31. Harold Widom, private communication.
32. K. Wieand, Eigenvalue Distributions of Random Matrices in the Permutation Group and Compact Lie Groups, PhD Thesis, (Harvard University, 1998)
33. K. Wieand, "Eigenvalue distributions of random unitary matrices", preprint (1999)