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18
Howe pairs, supersymmetry, and ratios of random characteristic polynomials for the unitary groups UN
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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 28 (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.
Applications of the Lfunctions ratios conjectures
"... In upcoming papers by Conrey, Farmer and Zirnbauer there appear conjectural formulas for averages, over a family, of ratios of products of shifted Lfunctions. In this paper we will present various applications of these ratios conjectures to a wide variety of problems that are of interest in number ..."
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Cited by 22 (6 self)
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In upcoming papers by Conrey, Farmer and Zirnbauer there appear conjectural formulas for averages, over a family, of ratios of products of shifted Lfunctions. In this paper we will present various applications of these ratios conjectures to a wide variety of problems that are of interest in number theory, such as lower order terms in the zero statistics of Lfunctions, mollified moments of Lfunctions and discrete averages over zeros of the Riemann zeta function. In particular, using the ratios conjectures we easily derive the answers to a number of notoriously difficult
Autocorrelation of ratios of Lfunctions
 COMM. NUMBER THEORY AND PHYSICS
, 2007
"... We give a new heuristic for all of the main terms in the quotient of products of Lfunctions averaged over a family. These conjectures generalize the recent conjectures for mean values of Lfunctions. Comparison is made to the analogous quantities for the characteristic polynomials of matrices ave ..."
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Cited by 21 (3 self)
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We give a new heuristic for all of the main terms in the quotient of products of Lfunctions averaged over a family. These conjectures generalize the recent conjectures for mean values of Lfunctions. Comparison is made to the analogous quantities for the characteristic polynomials of matrices averaged over a classical compact group.
Giambelli compatible point processes
 ADV. IN APPL. MATH
, 2006
"... We distinguish a class of random point processes which we call Giambelli compatible point processes. Our definition was partly inspired by determinantal identities for averages of products and ratios of characteristic polynomials for random matrices found earlier by Fyodorov and Strahov. It is clos ..."
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Cited by 8 (4 self)
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We distinguish a class of random point processes which we call Giambelli compatible point processes. Our definition was partly inspired by determinantal identities for averages of products and ratios of characteristic polynomials for random matrices found earlier by Fyodorov and Strahov. It is closely related to the classical Giambelli formula for Schur symmetric functions. We show that orthogonal polynomial ensembles, zmeasures on partitions, and spectral measures of characters of generalized regular representations of the infinite symmetric group generate Giambelli compatible point processes. In particular, we prove determinantal identities for averages of analogs of characteristic polynomials for partitions. Our approach provides a direct derivation of determinantal formulas for correlation functions.
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 7 (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
On absolute moments of characteristic polynomials of a certain class of complex random matrices
, 2006
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Riemann zeros and random matrix theory
, 2009
"... In the past dozen years random matrix theory has become a useful tool for conjecturing answers to old and important questions in number theory. It was through the Riemann zeta function that the connection with random matrix theory was first made in the 1970s, and although there has also been much re ..."
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Cited by 4 (0 self)
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In the past dozen years random matrix theory has become a useful tool for conjecturing answers to old and important questions in number theory. It was through the Riemann zeta function that the connection with random matrix theory was first made in the 1970s, and although there has also been much recent work concerning other varieties of Lfunctions, this article will concentrate on the zeta function as the simplest example illustrating the role of random matrix theory. 1
The covariance of almostprimes in Fq[t
 Int. Math. Res. Not. IMRN
, 2014
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