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38
Random matrix theory
, 2005
"... Random matrix theory is now a big subject with applications in many disciplines of science, engineering and finance. This article is a survey specifically oriented towards the needs and interests of a numerical analyst. This survey includes some original material not found anywhere else. We includ ..."
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Cited by 80 (4 self)
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Random matrix theory is now a big subject with applications in many disciplines of science, engineering and finance. This article is a survey specifically oriented towards the needs and interests of a numerical analyst. This survey includes some original material not found anywhere else. We include the important mathematics which is a very modern development, as well as the computational software that is transforming the theory into useful practice.
MOPS: Multivariate orthogonal polynomials (symbolically)
, 2007
"... In this paper we present a Maple library (MOPS) for computing Jack, Hermite, Laguerre, and Jacobi multivariate polynomials, as well as eigenvalue statistics for the Hermite, Laguerre, and Jacobi ensembles of random matrix theory. We also compute multivariate hypergeometric functions, and offer both ..."
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Cited by 23 (9 self)
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In this paper we present a Maple library (MOPS) for computing Jack, Hermite, Laguerre, and Jacobi multivariate polynomials, as well as eigenvalue statistics for the Hermite, Laguerre, and Jacobi ensembles of random matrix theory. We also compute multivariate hypergeometric functions, and offer both symbolic and numerical evaluations for all these quantities. We prove that all algorithms are welldefined, analyze their complexity, and illustrate their performance in practice. Finally, we present a few applications of this library.
Connection coefficients, matchings, maps and combinatorial conjectures for Jack symmetric functions
 TRANS. AMER. MATH. SOC
, 1996
"... A power series is introduced that is an extension to three sets of variables of the Cauchy sum for Jack symmetric functions in the Jack parameter α. We conjecture that the coefficients of this series with respect to the power sum basis are nonnegative integer polynomials in b, the Jack parameter sh ..."
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Cited by 16 (5 self)
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A power series is introduced that is an extension to three sets of variables of the Cauchy sum for Jack symmetric functions in the Jack parameter α. We conjecture that the coefficients of this series with respect to the power sum basis are nonnegative integer polynomials in b, the Jack parameter shifted by 1. More strongly, we make the MatchingsJack Conjecture, that the coefficients are counting series in b for matchings with respect to a parameter of nonbipartiteness. Evidence is presented for these conjectures and they are proved for two infinite families. The coefficients of a second series, essentially the logarithm of the first, specialize at values 1 and 2 of the Jack parameter to the numbers of hypermaps in orientable and locally orientable surfaces, respectively. We conjecture that these coefficients are also nonnegative integer polynomials in b, andwemake the HypermapJack Conjecture, that the coefficients are counting series in b for hypermaps in locally orientable surfaces with respect to a parameter of nonorientability.
Supersymmetric CalogeroMoserSutherland models: superintegrability structure and aigenfunctions, to appear
 in the proceedings of the Workshop on superintegrability in classical and quantum systems, ed. P Winternitz, CRM series
"... A new generalization of the Jack polynomials that incorporates fermionic variables is presented. These Jack superpolynomials are constructed as those eigenfunctions of the supersymmetric extension of the trigonometric CalogeroMoserSutherland (CMS) model that decomposes triangularly in terms of the ..."
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Cited by 14 (12 self)
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A new generalization of the Jack polynomials that incorporates fermionic variables is presented. These Jack superpolynomials are constructed as those eigenfunctions of the supersymmetric extension of the trigonometric CalogeroMoserSutherland (CMS) model that decomposes triangularly in terms of the symmetric monomial superfunctions. Many explicit examples are displayed. Furthermore, various new results have been obtained for the supersymmetric version of the CMS models: the Lax formulation, the construction of the Dunkl operators and the explicit expressions for the conserved charges. The reformulation of the models in terms of the exchangeoperator formalism is a crucial aspect of our analysis.
Quantum vs Classical Integrability in RuijsenaarsSchneider Systems
, 2003
"... The relationship (resemblance and/or contrast) between quantum and classical integrability in RuijsenaarsSchneider systems, which are one parameter deformation of CalogeroMoser systems, is addressed. Many remarkable properties of classical Calogero and Sutherland systems (based on any root system) ..."
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Cited by 12 (9 self)
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The relationship (resemblance and/or contrast) between quantum and classical integrability in RuijsenaarsSchneider systems, which are one parameter deformation of CalogeroMoser systems, is addressed. Many remarkable properties of classical Calogero and Sutherland systems (based on any root system) at equilibrium are reported in a previous paper (CorriganSasaki). For example, the minimum energies, frequencies of small oscillations and the eigenvalues of Lax pair matrices at equilibrium are all “integer valued”. In this paper we report that similar features and results hold for the RuijsenaarsSchneider type of integrable systems based on the classical root systems.
Hidden Algebras of the (super) Calogero and Sutherland models
, 1997
"... We propose to parametrize the configuration space of onedimensional quantum systems of N identical particles by the elementary symmetric polynomials of bosonic and fermionic coordinates. It is shown that in this parametrization the Hamiltonians of the AN, BCN, BN, CN and DN Calogero and Sutherland ..."
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Cited by 8 (5 self)
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We propose to parametrize the configuration space of onedimensional quantum systems of N identical particles by the elementary symmetric polynomials of bosonic and fermionic coordinates. It is shown that in this parametrization the Hamiltonians of the AN, BCN, BN, CN and DN Calogero and Sutherland models, as well as their supersymmetric generalizations, can be expressed — for arbitrary values of the coupling constants — as quadratic polynomials in the generators of a Borel subalgebra of the Lie algebra gl(N + 1) or the Lie superalgebra gl(N + 1N) for the supersymmetric case. These algebras are realized by first order differential operators. This fact establishes the exact solvability of the models according to the general definition given by one of the authors in [1], and implies that the Calogero and JackSutherland polynomials, as well as their supersymmetric generalizations, are related to finitedimensional irreducible representations of the Lie algebra gl(N+ 1) and the Lie superalgebra gl(N + 1N).
The random matrix technique of ghosts and shadows
 Markov Processes and Related Fields
"... We propose to abandon the notion that a random matrix has to be sampled for it to exist. Much of today's applied nite random matrix theory concerns real or complex random matrices (β = 1, 2). The threefold way so named by Dyson in 1962 [2] adds quaternions (β = 4). While it is true there are on ..."
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Cited by 7 (5 self)
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We propose to abandon the notion that a random matrix has to be sampled for it to exist. Much of today's applied nite random matrix theory concerns real or complex random matrices (β = 1, 2). The threefold way so named by Dyson in 1962 [2] adds quaternions (β = 4). While it is true there are only three real division algebras (β = dimension over the reals), this mathematical fact while critical in some ways, in other ways is irrelevant and perhaps has been over interpreted over the decades. We introduce the notion of a ghost random matrix quantity that exists for every beta, and a shadow quantity which may be real or complex which allows for computation. Any number of computations have successfully given reasonable answers to date though di culties remain in some cases. Though it may seem absurd to have a three and a quarter dimensional or pi dimensional algebra, that is exactly what we propose and what we compute with. In the end β becomes a noisiness parameter rather than a dimension. 1
Zonal polynomials via Stanley’s coordinates and free cumulants
 J. Algebra
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