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122
Universality at the edge of the spectrum in Wigner random matrices
, 1999
"... We prove universality at the edge for rescaled correlation functions of Wigner random matrices in the limit n → +∞. As a corollary, we show that, after proper rescaling, the 1st, 2nd, 3rd, etc. eigenvalues of Wigner random hermitian (resp. real symmetric) matrix weakly converge to the distributions ..."
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Cited by 101 (7 self)
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We prove universality at the edge for rescaled correlation functions of Wigner random matrices in the limit n → +∞. As a corollary, we show that, after proper rescaling, the 1st, 2nd, 3rd, etc. eigenvalues of Wigner random hermitian (resp. real symmetric) matrix weakly converge to the distributions established by Tracy and Widom in G.U.E. (G.O.E.) cases.
The Riemann Zeros and Eigenvalue Asymptotics
 SIAM Rev
, 1999
"... Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many feat ..."
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Cited by 40 (5 self)
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Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many features of H cl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the t n have a similar structure to those of the semiclassical En ; in particular, they display randommatrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the t n can be computed accurately from formulae with quantum analogues. The RiemannSiegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian H cl = XP. Key words. spectral asymptotics, number theory AMS subject classifications. 11M26, 11M06, 35P20, 35Q40, 41A60, 81Q10, 81Q50 PII. S0036144598347497 1.
The distribution of the largest eigenvalue in the Gaussian ensembles: β
 CalogeroMoserSutherland models (Montréal, QC, 1997), CRM Ser. Math. Phys
, 2000
"... The focus of this survey paper is on the distribution function FNβ(t) for the largest eigenvalue in the finite N Gaussian Orthogonal Ensemble (GOE, β = 1), the Gaussian Unitary Ensemble (GUE, β = 2), and the Gaussian Symplectic Ensemble (GSE, β = 4) in the edge scaling limit of N → ∞. These limiting ..."
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Cited by 30 (2 self)
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The focus of this survey paper is on the distribution function FNβ(t) for the largest eigenvalue in the finite N Gaussian Orthogonal Ensemble (GOE, β = 1), the Gaussian Unitary Ensemble (GUE, β = 2), and the Gaussian Symplectic Ensemble (GSE, β = 4) in the edge scaling limit of N → ∞. These limiting distribution functions are expressible in terms of a particular Painlevé II function. Comparisons are made with finite N simulations as well as a discussion of the universality of these distribution functions. 1.
Universality of the distribution functions of random matrix theory, preprint
"... Dedicated to James B. McGuire on the occasion of his sixtyfifth birthday. ..."
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Cited by 26 (5 self)
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Dedicated to James B. McGuire on the occasion of his sixtyfifth birthday.
Introduction to random matrices
 the proceedings of the 8 th Scheveningen Conference, Springer Lecture Notes in Physics
, 1993
"... ..."
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.
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 18 (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
Semiclassical trace formulae and eigenvalue statistics in quantum chaos Open Sys
 Information Dyn
, 1999
"... A detailed discussion of semiclassical trace formulae is presented and it is demonstrated how a regularized trace formula can be derived while dealing only with finite and convergent expressions. Furthermore, several applications of trace formula techniques to quantum chaos are reviewed. Then local ..."
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Cited by 17 (4 self)
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A detailed discussion of semiclassical trace formulae is presented and it is demonstrated how a regularized trace formula can be derived while dealing only with finite and convergent expressions. Furthermore, several applications of trace formula techniques to quantum chaos are reviewed. Then local spectral statistics, measuring correlations among finitely many eigenvalues, are reviewed and a detailed semiclassical analysis of the number variance is given. Thereafter the transition to global spectral statistics, taking correlations among infinitely many quantum energies into account, is discussed. It is emphasized that the resulting limit distributions depend on the way one passes to the global scale. A conjecture on the distribution of the fluctuations of the spectral staircase is explained in this general context and evidence supporting the conjecture is discussed. 1 Lectures held at the 3rd International Summer School/Conference Let’s face chaos through nonlinear dynamics at
Global Level Spacings Distribution for Large Random Matrices from Classical Compact Groups: Gaussian Fluctuations
 Ann. of Math
, 1997
"... We study the levelspacings distribution for eigenvalues of large N \Theta N matrices from the Classical Compact Groups in the scaling limit when the mean distance between nearest eigenvalues equals 1. Defining by jN (s) the number of nearest neighbors spacings, greater than s ? 0 (smaller than s ? ..."
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Cited by 16 (6 self)
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We study the levelspacings distribution for eigenvalues of large N \Theta N matrices from the Classical Compact Groups in the scaling limit when the mean distance between nearest eigenvalues equals 1. Defining by jN (s) the number of nearest neighbors spacings, greater than s ? 0 (smaller than s ? o ) we prove functional limit theorem for the process (j N (s) \Gamma IEj N (s))=N 1=2 , giving weak convergence of this distribution to some Gaussian random process on [0; 1) . The limiting Gaussian random process is universal for all Classical Compact Groups. It is Holder continuous with any exponent less than 1=2 : Numerical results suggest it not to be a standard Brownian bridge. Our methods can be also applied to study nlevel spacings distribution. AMS Subject classification : Probability theory and stochastic processes 1 Introduction and Formulation of Main Results The idea that statistical behavior of eigenvalues of large random matrices would give an information about spectra...
Statistics of energy levels and eigenfunctions in disordered systems,” Phys
 Rep
"... The article reviews recent developments in the theory of fluctuations and correlations of energy levels and eigenfunction amplitudes in diffusive mesoscopic samples. Various spatial geometries are considered, with emphasis on lowdimensional (quasi1D and 2D) systems. Calculations are based on the s ..."
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Cited by 15 (0 self)
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The article reviews recent developments in the theory of fluctuations and correlations of energy levels and eigenfunction amplitudes in diffusive mesoscopic samples. Various spatial geometries are considered, with emphasis on lowdimensional (quasi1D and 2D) systems. Calculations are based on the supermatrix σmodel approach. The method reproduces, in socalled zeromode approximation, the universal random matrix theory (RMT) results for the energylevel and eigenfunction fluctuations. Going beyond this approximation allows us to study systemspecific deviations from universality, which are determined by the diffusive classical dynamics in the system. These deviations are especially strong in the far “tails ” of the distribution function of the eigenfunction amplitudes (as well as of some related quantities, such as local density of states, relaxation time, etc.). These asymptotic “tails ” are governed by anomalously localized states which are formed in rare realizations of the random potential. The deviations of the level and eigenfunction statistics from their RMT form strengthen with increasing disorder and become especially pronounced at the Anderson metalinsulator