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30
Autocorrelation of random matrix polynomials
 COMMUN. MATH. PHYS
, 2003
"... We calculate the autocorrelation functions (or shifted moments) of the characteristic polynomials of matrices drawn uniformly with respect to Haar measure from the groups U(N), O(2N) and USp(2N). In each case the result can be expressed in three equivalent forms: as a determinant sum (and hence in t ..."
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Cited by 32 (17 self)
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We calculate the autocorrelation functions (or shifted moments) of the characteristic polynomials of matrices drawn uniformly with respect to Haar measure from the groups U(N), O(2N) and USp(2N). In each case the result can be expressed in three equivalent forms: as a determinant sum (and hence in terms of symmetric polynomials), as a combinatorial sum, and as a multiple contour integral. These formulae are analogous to those previously obtained for the Gaussian ensembles of Random Matrix Theory, but in this case are identities for any size of matrix, rather than largematrix asymptotic approximations. They also mirror exactly the autocorrelation formulae conjectured to hold for Lfunctions in a companion paper. This then provides further evidence in support of the connection between Random Matrix Theory and the theory of Lfunctions.
Universal Results for Correlations of Characteristic Polynomials
 RiemannHilbert Approach. Commun. Math. Phys
, 2003
"... Abstract We prove that general correlation functions of both ratios and products of characteristic polynomials of Hermitian random matrices are governed by integrable kernels of three different types: a) those constructed from orthogonal polynomials; b) constructed from Cauchy transforms of the same ..."
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Cited by 26 (6 self)
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Abstract We prove that general correlation functions of both ratios and products of characteristic polynomials of Hermitian random matrices are governed by integrable kernels of three different types: a) those constructed from orthogonal polynomials; b) constructed from Cauchy transforms of the same orthogonal polynomials and finally c) those constructed from both orthogonal polynomials and their Cauchy transforms. These kernels are related with the RiemannHilbert problem for orthogonal polynomials. For the correlation functions we obtain exact expressions in the form of determinants of these kernels. Derived representations enable us to study asymptotics of correlation functions of characteristic polynomials via DeiftZhou steepestdescent/stationary phase method for RiemannHilbert problems, and in particular to find negative moments of characteristic polynomials. This reveals the universal parts of the correlation functions and moments of characteristic polynomials for arbitrary invariant ensemble of β = 2 symmetry class. 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 20 (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.
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
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 18 (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
Random matrix theory and discrete moments of the Riemann zeta function
 J. Phys. AMath. Gen
"... Abstract. We calculate the discrete moments of the characteristic polynomial of a random unitary matrix, evaluated a small distance away from an eigenangle. Such results allow us to make conjectures about similar moments for the Riemann zeta function, and provide a uniform approach to understanding ..."
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Cited by 15 (0 self)
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Abstract. We calculate the discrete moments of the characteristic polynomial of a random unitary matrix, evaluated a small distance away from an eigenangle. Such results allow us to make conjectures about similar moments for the Riemann zeta function, and provide a uniform approach to understanding moments of the zeta function and its derivative. 1.
APPLICATIONS OF THE LFUNCTIONS RATIOS CONJECTURES
 PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY
, 2006
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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
Characteristic polynomials of random Hermitian matrices and DuistermaatHeckman localisation on noncompact Kahler manifolds
 Nucl. Phys. B
"... We reconsider the problem of calculating a general spectral correlation function containing an arbitrary number of products and ratios of characteristic polynomials for a N × N random matrix taken from the Gaussian Unitary Ensemble (GUE). Deviating from the standard ”supersymmetry ” approach, we int ..."
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Cited by 10 (5 self)
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We reconsider the problem of calculating a general spectral correlation function containing an arbitrary number of products and ratios of characteristic polynomials for a N × N random matrix taken from the Gaussian Unitary Ensemble (GUE). Deviating from the standard ”supersymmetry ” approach, we integrate out Grassmann variables at the early stage and circumvent the use of the HubbardStratonovich transformation in the ”bosonic ” sector. The method, suggested recently by one of us [19], is shown to be capable of calculation when reinforced with a generalization of the ItzyksonZuber integral to a noncompact integration manifold. We arrive to such a generalisation by discussing the DuistermaatHeckman localization principle for integrals over noncompact homogeneous Kähler manifolds. In the limit of large N the asymptotic expression for the correlation function reproduces the result outlined earlier by Andreev and Simons [14]. 1
A hybrid EulerHadamard product formula for the Riemann zeta function
, 2005
"... We use a smoothed version of the explicit formula to find an approximation to the Riemann zeta function as a product over its nontrivial zeros multiplied by a product over the primes. We model the first product by characteristic polynomials of random matrices. This provides a statistical model of ..."
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Cited by 9 (2 self)
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We use a smoothed version of the explicit formula to find an approximation to the Riemann zeta function as a product over its nontrivial zeros multiplied by a product over the primes. We model the first product by characteristic polynomials of random matrices. This provides a statistical model of the zeta function that involves the primes in a natural way. We then employ the model in a heuristic calculation of the moments of the modulus of the zeta function on the critical line. This calculation illuminates recent conjectures for these moments based on connections with random matrix theory.