Results 1  10
of
28
Howe pairs, supersymmetry, and ratios of random characteristic polynomials for the unitary groups UN
"... by ..."
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 ..."
Abstract

Cited by 22 (6 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
(Show Context)
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.
On absolute moments of characteristic polynomials of a certain class of complex random matrices
, 2006
"... ..."
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 ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
(Show Context)
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
qdeformations of twodimensional YangMills theory: Classification, categorification and refinement
 2013) [arXiv:1305.1580 [hepth]]. 16 GEORGIOS GIASEMIDIS, RICHARD J. SZABO, AND MIGUEL TIERZ
"... Abstract. We characterise the quantum group gauge symmetries underlying qdeformations of twodimensional YangMills theory by studying their relationships with the matrix models that appear in ChernSimons theory and sixdimensional N = 2 gauge theories, together with their refinements and supersym ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
(Show Context)
Abstract. We characterise the quantum group gauge symmetries underlying qdeformations of twodimensional YangMills theory by studying their relationships with the matrix models that appear in ChernSimons theory and sixdimensional N = 2 gauge theories, together with their refinements and supersymmetric extensions. We develop uniqueness results for quantum deformations and refinements of gauge theories in two dimensions, and describe several potential analytic and geometric realisations of them. We reconstruct standard qdeformed YangMills amplitudes via gluing rules in the representation category of the quantum group associated to the gauge group, whose numerical invariants are the usual characters in the Grothendieck group of the category. We apply this formalism to compute refinements of qdeformed amplitudes in terms of generalised characters, and relate them to refined ChernSimons matrix models and generalized unitary matrix integrals in the quantum βensemble which compute refined topological string amplitudes. We also describe applications of our results to gauge theories in five and seven dimensions, and to the dual superconformal field theories in four dimensions which descend from the N = (2, 0) sixdimensional superconformal theory.
method, heat kernel, and traces of powers of elements of compact Lie groups
, 2010
"... Abstract Combining Stein's method with heat kernel techniques, we show that the trace of the jth power of an element of U (n, C), U Sp(n, C) or SO(n, R) has a normal limit with error term C · j/n, with C an absolute constant. In contrast to previous works, here j may be growing with n. The tec ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
Abstract Combining Stein's method with heat kernel techniques, we show that the trace of the jth power of an element of U (n, C), U Sp(n, C) or SO(n, R) has a normal limit with error term C · j/n, with C an absolute constant. In contrast to previous works, here j may be growing with n. The technique might prove useful in the study of the value distribution of approximate eigenfunctions of Laplacians.
JOINT MOMENTS OF DERIVATIVES OF CHARACTERISTIC POLYNOMIALS
, 2007
"... We investigate the joint moments of the 2kth power of the characteristic polynomial of random unitary matrices with the 2hth power of the derivative of this same polynomial. We prove that for a fixed h, the moments are given by rational functions of k, up to a wellknown factor that already arise ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
We investigate the joint moments of the 2kth power of the characteristic polynomial of random unitary matrices with the 2hth power of the derivative of this same polynomial. We prove that for a fixed h, the moments are given by rational functions of k, up to a wellknown factor that already arises when h = 0. We fully describe the denominator in those rational functions (this had already been done by Hughes experimentally), and define the numerators through various formulas, mostly sums over partitions. We also use this to formulate conjectures on joint moments of the zeta function and its derivatives, or even the same questions for the Hardy function, if we use a “real ” version of characteristic polynomials. Our methods should easily be applicable to other similar problems, for instance with higher derivatives of characteristic polynomials. More data is available online, either on the author’s web site or attached