The Calogero-Sutherland Model and Generalized Classical Polynomials

Baker, T. H.; Forrester, Peter J.

doi:10.1007/s002200050161

Cited by 220 publications

(413 citation statements)

References 27 publications

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“…In the study of these ground states, identities of a different type to Theorem 2 relating β to 4/β (β even) have previously been encountered. These are so called duality relations, an example being [2] …”

Section: Significance Of the β Ensemblesmentioning

confidence: 99%

A Random Matrix Decimation Procedure Relating β = 2/(r + 1) to β = 2(r + 1)

Forrester

2008

Commun. Math. Phys.

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A random matrix decimation procedure relating β = 2/(r + 1) to β = 2(r + 1)Peter J. Forrester † Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia AbstractClassical random matrix ensembles with orthogonal symmetry have the property that the joint distribution of every second eigenvalue is equal to that of a classical random matrix ensemble with symplectic symmetry. These results are shown to be the case r = 1 of a family of interrelations between eigenvalue probability density functions for generalizations of the classical random matrix ensembles referred to as β-ensembles. The inter-relations give that the joint distribution of every (r + 1)-st eigenvalue in certain β-ensembles with β = 2/(r + 1) is equal to that of another β-ensemble with β = 2(r + 1). The proof requires generalizing a conditional probability density function due to Dixon and Anderson.

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Section: Significance Of the β Ensemblesmentioning

confidence: 99%

A Random Matrix Decimation Procedure Relating β = 2/(r + 1) to β = 2(r + 1)

Forrester

2008

Commun. Math. Phys.

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“…9 The β-ensemble generalized this "three-fold way" by constructing a β dependent ensemble of tridiagonal Hermitian matrices in which β is a parameter that can assume any positive real value. 10,11 For the integer 1, 2, 4 values of β, the statistical properties of the Gaussian ensemble 12 are reproduced.…”

Section: Introductionmentioning

confidence: 99%

Non-Hermitian β-ensemble with real eigenvalues

Bohigas

Pato

2013

AIP Advances

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“…(26)) and may also have important implications for lattice gas theory (27). Furthermore, it is a long-standing observation that nuclear systems with two-body interactions display an average density of states whose profile is much closer to a Gaussian distribution (28) (29) than to a semicircle.…”

Section: β-Ensembles Of Dumitriu-edelmanmentioning

confidence: 99%

“…In particular, the apparent divergences of the Gamma functions in (27) cancel out and the final density reads:…”

Section: Marginal Distribution Of Entriesmentioning

confidence: 99%

On invariant 2×2 β -ensembles of random matrices

Vivo

Majumdar

2008

Physica A: Statistical Mechanics and its Applications

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We introduce and solve exactly a family of invariant 2×2 random matrices, depending on one parameter η, and we show that rotational invariance and real Dyson index β are not incompatible properties. The probability density for the entries contains a weight function and a multiple trace-trace interaction term, which corresponds to the representation of the Vandermonde-squared coupling on the basis of power sums. As a result, the effective Dyson index β eff of the ensemble can take any real value in an interval. Two weight functions (Gaussian and non-Gaussian) are explored in detail and the connections with β-ensembles of Dumitriu-Edelman and the so-called Poisson-Wigner crossover for the level spacing are respectively highlighted. A curious spectral twinning between ensembles of different symmetry classes is unveiled: as a consequence, the identification between symmetry group (orthogonal, unitary or symplectic) and the exponent of the Vandermonde (β = 1, 2, 4) is shown to be potentially deceptive. The proposed technical tool more generically allows for designing actual matrix models which i) are rotationally invariant; ii) have a real Dyson index β eff ; iii) have a pre-assigned confining potential or alternatively level-spacing profile. The analytical results have been checked through numerical simulations with Preprint submitted to Elsevier 8 March 2018an excellent agreement. Eventually, we discuss possible generalizations and further directions of research.

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The Calogero-Sutherland Model and Generalized Classical Polynomials

Cited by 220 publications

References 27 publications

A Random Matrix Decimation Procedure Relating β = 2/(r + 1) to β = 2(r + 1)

A Random Matrix Decimation Procedure Relating β = 2/(r + 1) to β = 2(r + 1)

Non-Hermitian β-ensemble with real eigenvalues

On invariant 2×2 β -ensembles of random matrices

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