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35
Introduction to the random matrix theory: Gaussian unitary ensemble and beyond
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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.
Universal behavior for averages of characteristic polynomials at the origin of the spectrum
 Commun.Math.Phys
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On absolute moments of characteristic polynomials of a certain class of complex random matrices
, 2006
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Average characteristic polynomials for multiple orthogonal polynomial ensembles
, 2009
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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
A universality theorem for ratios of random characteristic polynomials
 J. Approx. Theory
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Multivariable Christoffel–Darboux kernels and characteristic polynomials of random hermitian matrices
"... Abstract. We study multivariable Christoffel–Darboux kernels, which may be viewed as reproducing kernels for antisymmetric orthogonal polynomials, and also as correlation functions for products of characteristic polynomials of random hermitian matrices. Using their interpretation as reproducing kern ..."
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Abstract. We study multivariable Christoffel–Darboux kernels, which may be viewed as reproducing kernels for antisymmetric orthogonal polynomials, and also as correlation functions for products of characteristic polynomials of random hermitian matrices. Using their interpretation as reproducing kernels, we obtain simple proofs of pfaffian and determinant formulas, as well as Schur polynomial expansions, for such kernels. In subsequent work, these results are applied in combinatorics (enumeration of marked shifted tableaux) and number theory (representation of integers as sums of squares). 1.