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79
On the characteristic polynomial of a random unitary matrix
 Comm. Math. Phys
, 2001
"... Abstract: We present a range of fluctuation and large deviations results for the logarithm of the characteristic√polynomial Z of a random N × N unitary matrix, as N →∞. First 12 we show that ln Z / ln N, evaluated at a finite set of distinct points, is asymptotically a collection of i.i.d. complex n ..."
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Cited by 43 (11 self)
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Abstract: We present a range of fluctuation and large deviations results for the logarithm of the characteristic√polynomial Z of a random N × N unitary matrix, as N →∞. First 12 we show that ln Z / ln N, evaluated at a finite set of distinct points, is asymptotically a collection of i.i.d. complex normal random variables. This leads to a refinement of a recent central limit theorem due to Keating and Snaith, and also explains the covariance structure of the eigenvalue counting function. Next we obtain a central limit theorem for ln Z in a Sobolev space of generalised functions on the unit circle. In this limiting regime, lowerorder terms which reflect the global covariance structure are no longer negligible and feature in the covariance structure of the limiting Gaussian measure. Large deviations results for ln Z/A, evaluated at a finite set of distinct points, can be obtained for √ ln N ≪ A ≪ ln N. For higherorder scalings we obtain large deviations results for ln Z/A evaluated at a single point. There is a phase transition at A = ln N (which only applies to negative deviations of the real part) reflecting a switch from global to local conspiracy.
On the frequency of vanishing of quadratic twists of modular Lfunctions
 in Number theory for the millennium, I (Urbana, IL, 2000), 301–315, A K Peters
, 2002
"... Abstract. We present theoretical and numerical evidence for a random matrix theoretical approach to a conjecture about vanishings of quadratic twists of certain Lfunctions. In this paper we 1 present some evidence that methods from random matrix theory can give insight into the frequency of vanishi ..."
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Cited by 38 (15 self)
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Abstract. We present theoretical and numerical evidence for a random matrix theoretical approach to a conjecture about vanishings of quadratic twists of certain Lfunctions. In this paper we 1 present some evidence that methods from random matrix theory can give insight into the frequency of vanishing for quadratic twists of modular Lfunctions. The central question is the following: given a holomorphic newform f with integral coefficients and associated Lfunction Lf(s), for how many fundamental discriminants d with d  ≤ x, does Lf(s, χd), the Lfunction twisted by the real, primitive, Dirichlet character associated with the discriminant d, vanish at the center of the critical strip to order at least 2? This question is of particular interest in the case that the Lfunction is associated with an elliptic curve, in light of the conjecture of Birch and SwinnertonDyer. This case corresponds to weight k = 2. We will focus on this case for most of the paper, though we do make some remarks about higher weights (see (26) and below). Suppose that E/Q is an elliptic curve with associated Lfunction (1) LE(s) = for ℜs> 1. Then, as a consequence of the TaniyamaShimura conjecture, recently solved by Wiles, Taylor, ([W], [TW]), and Breuil, Conrad, and Diamond, LE is entire and satisfies a functional equation n=1 a ∗ n n s
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 matrix theory over finite fields
 Bull. Amer. Math. Soc. (N.S
"... Abstract. The first part of this paper surveys generating functions methods in the study of random matrices over finite fields, explaining how they arose from theoretical need. Then we describe a probabilistic picture of conjugacy classes of the finite classical groups. Connections are made with sym ..."
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Cited by 22 (6 self)
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Abstract. The first part of this paper surveys generating functions methods in the study of random matrices over finite fields, explaining how they arose from theoretical need. Then we describe a probabilistic picture of conjugacy classes of the finite classical groups. Connections are made with symmetric function theory, Markov chains, RogersRamanujan type identities, potential theory, and various measures on partitions.
Universality for mathematical and physical systems
, 2006
"... Abstract. All physical systems in equilibrium obey the laws of thermodynamics. In other words, whatever the precise nature of the interaction between the atoms and molecules at the microscopic level, at the macroscopic level, physical systems exhibit universal behavior in the sense that they are all ..."
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Cited by 21 (0 self)
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Abstract. All physical systems in equilibrium obey the laws of thermodynamics. In other words, whatever the precise nature of the interaction between the atoms and molecules at the microscopic level, at the macroscopic level, physical systems exhibit universal behavior in the sense that they are all governed by the same laws and formulae of thermodynamics. In this paper we describe some recent history of universality ideas in physics starting with Wigner’s model for the scattering of neutrons off large nuclei and show how these ideas have led mathematicians to investigate universal behavior for a variety of mathematical systems. This is true not only for systems which have a physical origin, but also for systems which arise in a purely mathematical context such as the Riemann hypothesis, and a version of the card game solitaire called patience sorting. 1.
Howe pairs, supersymmetry, and ratios of random characteristic polynomials for the unitary groups UN
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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 17 (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
A.: On the average of characteristic polynomials from classical groups
 Comm. Math. Phys
"... Abstract. 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. 1. ..."
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Cited by 17 (1 self)
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Abstract. 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. 1.
The importance of Selberg integral
 Bull. Amer. Math. Soc
"... Abstract. It has been remarked that a fair measure of the impact of Atle Selberg’s work is the number of mathematical terms that bear his name. One of these is the Selberg integral, an ndimensional generalization of the Euler beta integral. We trace its sudden rise to prominence, initiated by a que ..."
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Cited by 17 (4 self)
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Abstract. It has been remarked that a fair measure of the impact of Atle Selberg’s work is the number of mathematical terms that bear his name. One of these is the Selberg integral, an ndimensional generalization of the Euler beta integral. We trace its sudden rise to prominence, initiated by a question to Selberg from Enrico Bombieri, more than thirty years after its initial publication. In quick succession the Selberg integral was used to prove an outstanding conjecture in random matrix theory and cases of the Macdonald conjectures. It further initiated the study of qanalogues, which in turn enriched the Macdonald conjectures. We review these developments and proceed to exhibit the sustained prominence of the Selberg integral as evidenced by its central role in random matrix theory, Calogero–Sutherland quantum manybody systems, Knizhnik–Zamolodchikov equations, and multivariable orthogonal polynomial
Averages of ratios of characteristic polynomials for the compact classical groups, preprint
 International Mathematics Research Notices 2005 (2005
, 2005
"... Averages of ratios of characteristic polynomials for the compact classical groups are evaluated in terms of determinants whose dimensions are independent of the matrix rank. These formulas are shown to be equivalent to expressions for the same averages obtained in a previous study, which was motivat ..."
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Cited by 15 (5 self)
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Averages of ratios of characteristic polynomials for the compact classical groups are evaluated in terms of determinants whose dimensions are independent of the matrix rank. These formulas are shown to be equivalent to expressions for the same averages obtained in a previous study, which was motivated by applications to analytic number theory. Our approach uses classical methods of random matrix theory, in particular determinants and orthogonal polynomials, and can be considered more elementary than the method of Howe pairs used in the previous study. 1