We study the characteristic polynomials Z(U,#)of matrices U in the Circular Unitary Ensemble (CUE) of Random Matrix Theory. Exact expressions for any matrix size N are derived for the moments of and Z/Z # , and from these we obtain the asymptotics of the value distributions and cumulants of the real and imaginary parts of log Z as N ##. In the

Random matrix theory (RMT) is used to model the asymptotics of the discrete moments of the derivative of the Riemann zeta function, ? (s), evaluated at the complex zeros + iγn, using the methods introduced by Keating and Snaith in [14]. We also discuss the probability distribution of ln |? ´(1/2 + iγn)|, proving the central limit theorem for the corresponding random matrix distribution and analysing its large deviations.

ABSTRACT. The problem is: Let Hn = n∑ n ≥ 1, that with equality only for n = 1. j=1 1 j d ≤ Hn + exp(Hn)log(Hn),

In recent years there has been a growing interest in connections between the statistical properties of number theoretical L-functions and random matrix theory. We review the history of these connections, some of the major achievements, and a number of applications.

There has been much recent interest, initiated by work of the physicists Hatano and Nelson, in the eigenvalues of certain random non-hermitian periodic tridiagonal matrices and their bidiagonal limits. These eigenvalues cluster along a \bubble with wings" in the complex plane, and the corresponding eigenvectors are localized in the wings, delocalized in the bubble. Here, in addition to eigenvalues, pseudospectra are analyzed, making it possible to treat the non-periodic analogues of these random matrix problems. Inside the bubble, the resolvent norm grows exponentially with the dimension. Outside, it grows subexponentially in a bounded region that is the spectrum of the in nite-dimensional operator. Localization and delocalization correspond to resolvent matrices whose entries exponentially decrease or increase, respectively, with distance from the diagonal. This article presents theorems that characterize the spectra, pseudospectra, and numerical range for the four cases of nite bidiagonal matrices, in nite bidiagonal matrices (\stochastic Toeplitz operators"), nite periodic matrices, and doubly in nite bidiagonal matrices (\stochastic Laurent operators").

A d−dimensional rational polytope P is a polytope whose vertices are located at the nodes of Z d lattice. Consider the number ∣ ∣kP ∩ Z d ∣ ∣ of points inside the inflated P with coefficient of inflation k (k = 1, 2, 3,...). The Ehrhart polynomial of P counts the number of such lattice points inside the inflated P and (may be) at its faces (including vertices). In Part I [ JGP 55 (2005) 50] of our four parts work we noticed that Veneziano amplitude is just the Laplace transform of the generating function (considered as a partition function in the sence of statistical mechanics) for the Ehrhart polynomial for the regular inflated simplex obtained as deformation retract of the Fermat (hyper) surface living in the complex projective space. This observation is sufficient for development of new symplectic (this work) and supersymmetric (Part II) physical models reproducing the Veneziano (and Veneziano-like) amplitudes. General ideas (e.g. those related to the properties of Ehrhart polynomials) are illustrated by simple practical examples (e.g. use of mirror symmetry for explanation of available experimental data on ππ scattering, etc.) worked out in some detail. Obtained final results are in formal accord with those earlier obtained by Vergne [PNAS

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The aim of this short lecture course is to develop Selberg’s trace formula for a compact hyperbolic surface M, and discuss some of its applications. The main motivation for our studies is quantum chaos: the Laplace-Beltrami operator − ∆ on the surface M represents the quantum Hamiltonian of a particle, whose classical dynamics is governed by the (strongly chaotic) geodesic flow on the unit tangent bundle of M. The trace formula is currently the only available tool to analyze the fine structure of the spectrum of −∆; no individual formulas for its eigenvalues are known. In the case of more general quantum systems, the role of Selberg’s formula is taken over by the semiclassical Gutzwiller trace formula [10], [7]. We begin by reviewing the trace formulas for the simplest compact manifolds, the circle S 1 (Section 1) and the sphere S 2 (Section 2). In both cases, the corresponding geodesic flow is integrable, and the trace formula is a consequence of the Poisson summation formula. In the remaining sections we shall discuss the following topics: the Laplacian on the hyperbolic plane and isometries (Section 3); Green’s functions (Section 4); Selberg’s point pair invariants (Section 5); The ghost of the sphere (Section 6); Linear operators on hyperbolic surfaces (Section 7); A trace formula for hyperbolic cylinders and poles of the scattering matrix (Section 8); Back to general hyperbolic surfaces (Section 9); The spectrum of a compact surface, Selberg’s pre-trace and trace formulas (Section 10); Heat kernel and Weyl’s law (Section 11); Density of closed geodesics (Section 12); Trace of the resolvent (Section 13); Selberg’s zeta function (Section 14); Suggestions for exercises and further reading (Section 15). Our main references are Hejhal’s classic lecture notes [12, Chapters one and two], Balazs and Voros ’ excellent introduction [1], and Cartier and Voros’ nouvelle interprétation [6]. Section 15 comprises a list of references for further reading. These notes are based on lectures given at the International School Quantum

Abstract. We show that the Circular Orthogonal Ensemble of random matrices arises naturally from a family of random polynomials. This sheds light on the appearance of random matrix statistics in the zeros of the Riemann zeta-function. 1.