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124
Periodic Orbit Theory and Spectral Statistics for Quantum Graphs
 ANN. PHYS. (N.Y
, 2000
"... We quantize graphs (networks) which consist of a finite number of bonds and vertices. We show that the spectral statistics of fully connected graphs is well reproduced by random matrix theory. We also define a classical phase space for the graphs, where the dynamics is mixing and the periodic orb ..."
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Cited by 44 (5 self)
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We quantize graphs (networks) which consist of a finite number of bonds and vertices. We show that the spectral statistics of fully connected graphs is well reproduced by random matrix theory. We also define a classical phase space for the graphs, where the dynamics is mixing and the periodic orbits proliferate exponentially. An exact trace formula for the quantum spectrum is developed in terms of the same periodic orbits, and it is used to investigate the origin of the connection between random matrix theory and the underlying chaotic classical dynamics. Being an exact theory, and due to its relative simplicity, it offers new insights into this problem which is at the forefront of the research in Quantum Chaos and related fields.
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.
Random matrices: Universality of the local eigenvalue statistics, submitted
"... Abstract. This is a continuation of our earlier paper [25] on the universality of the eigenvalues of Wigner random matrices. The main new results of this paper are an extension of the results in [25] from the bulk of the spectrum up to the edge. In particular, we prove a variant of the universality ..."
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Cited by 40 (1 self)
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Abstract. This is a continuation of our earlier paper [25] on the universality of the eigenvalues of Wigner random matrices. The main new results of this paper are an extension of the results in [25] from the bulk of the spectrum up to the edge. In particular, we prove a variant of the universality results of Soshnikov [23] for the largest eigenvalues, assuming moment conditions rather than symmetry conditions. The main new technical observation is that there is a significant bias in the Cauchy interlacing law near the edge of the spectrum which allows one to continue ensuring the delocalization of eigenvectors. 1.
Advanced Determinant Calculus
, 1999
"... The purpose of this article is threefold. First, it provides the reader with a few useful and efficient tools which should enable her/him to evaluate nontrivial determinants for the case such a determinant should appear in her/his research. Second, it lists a number of such determinants that have ..."
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Cited by 37 (0 self)
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The purpose of this article is threefold. First, it provides the reader with a few useful and efficient tools which should enable her/him to evaluate nontrivial determinants for the case such a determinant should appear in her/his research. Second, it lists a number of such determinants that have been already evaluated, together with explanations which tell in which contexts they have appeared. Third, it points out references where further such determinant evaluations can be found.
Nonasymptotic theory of random matrices: extreme singular values
 PROCEEDINGS OF THE INTERNATIONAL CONGRESS OF MATHEMATICIANS
, 2010
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Markov processes related with Dunkl Operators
 Adv. Appl. Math
, 1998
"... Dunkl operators are parametrized differentialdifference operators on R N which are related to finite reflection groups. They can be regarded as a generalization of partial derivatives and play a major role in the study of CalogeroMoserSutherlandtype quantum manybody systems. Dunkl operators l ..."
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Cited by 28 (5 self)
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Dunkl operators are parametrized differentialdifference operators on R N which are related to finite reflection groups. They can be regarded as a generalization of partial derivatives and play a major role in the study of CalogeroMoserSutherlandtype quantum manybody systems. Dunkl operators lead to generalizations of various analytic structures, like the Laplace operator, the Fourier transform, Hermite polynomials and the heat semigroup. In this paper we investigate some probabilistic aspects of this theory in a systematic way. For this, we introduce a concept of homogeneity of Markov processes on R N which generalizes the classical notion of processes with independent, stationary increments to the Dunkl setting. This includes analogues of Brownian motions and Cauchy processes. The generalizations of Brownian motions have the c`adl`ag property and form, after symmetrization with respect to the underlying reflection groups, diffusions on the Weyl chambers. A major part of the p...
On quantum statistical inference
 J. Roy. Statist. Soc. B
, 2001
"... [Read before The Royal Statistical Society at a meeting organized by the Research Section ..."
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Cited by 24 (5 self)
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[Read before The Royal Statistical Society at a meeting organized by the Research Section
Communities of practice: performance and evolution
 Computational and Mathematical Organization Theory
, 1995
"... We present a detailed model of collaboration in communities of practice and we examine its dynamical consequences for the group as a whole. We establish the existence of a novel mechanism that allows the community to naturally adapt to growth, specialization, or changes in the environment without th ..."
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Cited by 24 (6 self)
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We present a detailed model of collaboration in communities of practice and we examine its dynamical consequences for the group as a whole. We establish the existence of a novel mechanism that allows the community to naturally adapt to growth, specialization, or changes in the environment without the need for central controls. This mechanism relies on the appearance of a dynamical instability that initates an exploration of novel interactions, eventually leading to higher performance for the community as a whole. 1
Random matrices: Universality of esds and the circular law
, 2009
"... Given an n × n complex matrix A, let ..."
Random matrices: The circular law
, 2008
"... Let x be a complex random variable with mean zero and bounded variance σ². Let Nn be a random matrix of order n with entries being i.i.d. 1 copies of x. Let λ1,..., λn be the eigenvalues of ..."
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Cited by 23 (10 self)
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Let x be a complex random variable with mean zero and bounded variance σ². Let Nn be a random matrix of order n with entries being i.i.d. 1 copies of x. Let λ1,..., λn be the eigenvalues of