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34
E.: Averages of characteristic polynomials in Random Matrix Theory
 Commun. Pure and Applied Math
, 2006
"... Abstract. We compute averages of products and ratios of characteristic polynomials associated with Orthogonal, Unitary, and Symplectic Ensembles of Random Matrix Theory. The pfaffian/determinantal formulas for these averages are obtained, and the bulk scaling asymptotic limits are found for ensemble ..."
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Cited by 24 (3 self)
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Abstract. We compute averages of products and ratios of characteristic polynomials associated with Orthogonal, Unitary, and Symplectic Ensembles of Random Matrix Theory. The pfaffian/determinantal formulas 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 formulas 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 formulas for the discrete averages are proved using standard tools of linear algebra; no application of orthogonal or skeworthogonal polynomials is needed. 1.
Random matrices and Lfunctions
 J. PHYS A MATH GEN
, 2003
"... In recent years there has been a growing interest in connections between the statistical properties of number theoretical Lfunctions and random matrix theory. We review the history of these connections, some of the major achievements and a number of applications. ..."
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Cited by 22 (7 self)
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In recent years there has been a growing interest in connections between the statistical properties of number theoretical Lfunctions and random matrix theory. We review the history of these connections, some of the major achievements and a number of applications.
Developments in random matrix theory
 J. Phys. A: Math. Gen
, 2000
"... In this preface to the Journal of Physics A, Special Edition on Random Matrix Theory, we give a review of the main historical developments of random matrix theory. A short summary of the papers that appear in this special edition is also given. 1 1 ..."
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Cited by 20 (0 self)
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In this preface to the Journal of Physics A, Special Edition on Random Matrix Theory, we give a review of the main historical developments of random matrix theory. A short summary of the papers that appear in this special edition is also given. 1 1
Howe pairs, supersymmetry, and ratios of random characteristic polynomials for the unitary groups UN
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Random matrices, magic squares and matching polynomials
 Research Paper
"... Characteristic polynomials of random unitary matrices have been intensively studied in recent years: by number theorists in connection with Riemann zetafunction, and by theoretical physicists in connection with Quantum Chaos. In particular, Haake and collaborators have computed the variance of the c ..."
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Cited by 19 (3 self)
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Characteristic polynomials of random unitary matrices have been intensively studied in recent years: by number theorists in connection with Riemann zetafunction, and by theoretical physicists in connection with Quantum Chaos. In particular, Haake and collaborators have computed the variance of the coefficients of these polynomials and raised the question of computing the higher moments. The answer turns out to be intimately related to counting integer stochastic matrices (magic squares). Similar results are obtained for the moments of secular coefficients of random matrices from orthogonal and symplectic groups. Combinatorial meaning of the moments of the secular coefficients of GUE matrices is also investigated and the connection with matching polynomials is discussed. 1
On the average of characteristic polynomials from classical groups
 COMM. MATH. PHYS
, 2005
"... We provide an elementary and selfcontained derivation of formulae for products and ratios of characteristic polynomials from classical groups using classical results due to Weyl and Littlewood. ..."
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Cited by 17 (1 self)
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We provide an elementary and selfcontained derivation of formulae for products and ratios of characteristic polynomials from classical groups using classical results due to Weyl and Littlewood.
INTEGRAL MOMENTS OF LFUNCTIONS
, 2005
"... We give a newheuristic for all of the main terms in the integral moments of various families of primitive Lfunctions. The results agree with previous conjectures for the leading order terms. Our conjectures also have an almost identical form to exact expressions for the corresponding moments of the ..."
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Cited by 15 (8 self)
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We give a newheuristic for all of the main terms in the integral moments of various families of primitive Lfunctions. The results agree with previous conjectures for the leading order terms. Our conjectures also have an almost identical form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary, orthogonal, or symplectic matrices, where the moments are de ned by the appropriate group averages. This lends support to the idea that arithmetical Lfunctions have a spectral interpretation, and that their value distributions can be modeled using Random Matrix Theory. Numerical examples show good agreement with our conjectures.
Derivatives of random matrix characteristic polynomials with applications to elliptic curves
 J. Phys. A
"... The value distribution of derivatives of characteristic polynomials of matrices from SO(N) is calculated at the point 1, the symmetry point on the unit circle of the eigenvalues of these matrices. We consider subsets of matrices from SO(N) that are constrained to have at least n eigenvalues equal to ..."
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Cited by 12 (3 self)
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The value distribution of derivatives of characteristic polynomials of matrices from SO(N) is calculated at the point 1, the symmetry point on the unit circle of the eigenvalues of these matrices. We consider subsets of matrices from SO(N) that are constrained to have at least n eigenvalues equal to 1, and investigate the first nonzero derivative of the characteristic polynomial at that point. The connection between the values of random matrix characteristic polynomials and values of Lfunctions in families has been wellestablished. The motivation for this work is the expectation that through this connection with Lfunctions derived from families of elliptic curves, and using the Birch and SwinnertonDyer conjecture to relate values of the Lfunctions to the rank of elliptic curves, random matrix theory will be useful in probing important questions concerning these ranks. 1
APPLICATIONS OF THE LFUNCTIONS RATIOS CONJECTURES
 PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY
, 2006
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Secondary terms in the number of vanishings of quadratic twists of elliptic curve Lfunctions (2005), to appear
 in Proceedings of the Issac Newton Institute Workshop on Elliptic Curve and Random Matrix Theory, arxiv.org/math/0509059
"... We examine the number of vanishings of quadratic twists of the Lfunction associated to an elliptic curve. Applying a conjecture for the full asymptotics of the moments of critical Lvalues we obtain a conjecture for the first two terms in the ratio of the number of vanishings of twists sorted accor ..."
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Cited by 12 (6 self)
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We examine the number of vanishings of quadratic twists of the Lfunction associated to an elliptic curve. Applying a conjecture for the full asymptotics of the moments of critical Lvalues we obtain a conjecture for the first two terms in the ratio of the number of vanishings of twists sorted according to arithmetic progressions. 1