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13
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 25 (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.
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
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
Singularity dominated strong fluctuations for some random matrix averages
 Comm. Math. Phys
"... The circular and Jacobi ensembles of random matrices have their eigenvalue support on the unit circle of the complex plane and the interval (0, 1) of the real line respectively. The averaged value of the modulus of the corresponding characteristic polynomial raised to the power 2µ diverges, for 2µ ≤ ..."
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Cited by 10 (0 self)
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The circular and Jacobi ensembles of random matrices have their eigenvalue support on the unit circle of the complex plane and the interval (0, 1) of the real line respectively. The averaged value of the modulus of the corresponding characteristic polynomial raised to the power 2µ diverges, for 2µ ≤ −1, at points approaching the eigenvalue support. Using the theory of generalized hypergeometric functions based on Jack polynomials, the functional form of the leading asymptotic behaviour is established rigorously. In the circular ensemble case this confirms a conjecture of Berry and Keating. 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.
Triple correlation of the Riemann zeros
"... We use the conjecture of Conrey, Farmer and Zirnbauer for averages of ratios of the Riemann zeta function [11] to calculate all the lower order terms of the triple correlation function of the Riemann zeros. A previous approach was suggested by Bogomolny and Keating [6] taking inspiration from semi ..."
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Cited by 4 (2 self)
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We use the conjecture of Conrey, Farmer and Zirnbauer for averages of ratios of the Riemann zeta function [11] to calculate all the lower order terms of the triple correlation function of the Riemann zeros. A previous approach was suggested by Bogomolny and Keating [6] taking inspiration from semiclassical methods. At that point they did not write out the answer explicitly, so we do that here, illustrating that by our method all the lower order terms down to the constant can be calculated rigourously if one assumes the ratios conjecture of Conrey, Farmer and Zirnbauer. Bogomolny and Keating [4] returned to their previous results simultaneously with this current work, and have written out the full expression. The result presented in this paper agrees precisely with their formula, as well as with our numerical computations, which we include here. We also include an alternate proof of the triple correlation of eigenvalues from random U(N) matrices which follows a nearly identical method to that for the Riemann zeros, but is based on
The derivative of SO(2N + 1) characteristic polynomials and rank 3 elliptic curves
 In this volume
"... Here we calculate the value distribution of the first derivative of characteristic polynomials of matrices from SO(2N + 1) at the point 1, the symmetry point on the unit circle of the eigenvalues of these matrices. The connection between the values of random matrix characteristic polynomials and val ..."
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Cited by 3 (1 self)
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Here we calculate the value distribution of the first derivative of characteristic polynomials of matrices from SO(2N + 1) at the point 1, the symmetry point on the unit circle of the eigenvalues of these matrices. The connection between the values of random matrix characteristic polynomials and values of the Lfunctions of families of elliptic curves implies that this calculation in random matrix theory is relevant to the problem of predicting the frequency of rank three curves within these families, since the Birch and SwinnertonDyer conjecture relates the value of an Lfunction and its derivatives to the rank of the associated elliptic curve. This article is based on a talk given at the Isaac Newton Institute for Mathematical Sciences during the “Clay Mathematics
Random Matrix Theory Predictions for the Asymptotics of the Moments of the Riemann Zeta Function and Numerical Tests of the Predictions
, 2006
"... In 1972, H.L. Montgomery and F. Dyson uncovered a surprising connection between the Theory of the Riemann Zeta function and Random Matrix Theory. For the next few decades, the major developments in the area were the numerical calculations of Odlyzko and conjectures for the moments of the Riemann Zet ..."
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In 1972, H.L. Montgomery and F. Dyson uncovered a surprising connection between the Theory of the Riemann Zeta function and Random Matrix Theory. For the next few decades, the major developments in the area were the numerical calculations of Odlyzko and conjectures for the moments of the Riemann Zeta function (and other Lfunctions) found by Conrey, Ghosh, Gonek, HeathBrown, Hejhal and Sarnak. Recently, there have been two important advances. First, Keating and Snaith, in a 2000 paper, conjectured connections between the moments of the characteristic polynomials of random matrices and the moments of the Riemann Zeta function. Second, Katz and Sarnak proposed connections between certain families of Lfunctions and other matrix groups. Our goal in this paper is twofold. First, we discuss links between Random Matrix Theory and the Zeta function. Then, we describe our numerical calculation of the moments of the Zeta function and compare initial results with Random Matrix Theory predictions. 1
Let D = 1 orD be a fundamental discriminant [1]. The KroneckerJacobi
, 2005
"... symbol (D/n) is a completely multiplicative function on the positive ..."