Matrix Beta-integrals: An Overview

Neretin, Yury A.

doi:10.1007/978-3-319-18212-4_20

Cited by 8 publications

(6 citation statements)

References 32 publications

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“…N on H(N) which are also known in random matrix theory as the Cauchy ensemble and under the Cayley transform as the circular Jacobi ensemble on U(N). We refer the reader to the extensive literature for properties of these [40], [50], [9], [56], [17], [12], [11] and closely related measures [54], [14], [15], [16], [4], and connections ranging from Painlevé equations [31] to stochastic processes [2], [3].…”

Section: Acting On the Space Of Infinitementioning

confidence: 99%

Moments of moments of characteristic polynomials of random unitary matrices and lattice point counts

Assiotis

Keating

2020

Random Matrices: Theory Appl.

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The aim of this note is to give a combinatorial and non-computational proof of the asymptotics of the integer moments of the moments of the characteristic polynomials of Haar distributed unitary matrices as the size of the matrix goes to infinity. This is achieved by relating these quantities to a lattice point count problem. As a byproduct, we obtain a new expression for the leading order coefficient in the asymptotic as a volume of a certain region involving continuous Gelfand-Tsetlin patterns with constraints.

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Section: Acting On the Space Of Infinitementioning

confidence: 99%

Moments of moments of characteristic polynomials of random unitary matrices and lattice point counts

Assiotis

Keating

2020

Random Matrices: Theory Appl.

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“…Similar to an argument in the appendix of [33], this observation together with Theorem 3(3) and identity (27) leads for d = 2 to an alternative proof that for µ > µ 0 + kq + 1 and indices ν ∈ [0, ∞[ which do not belong to the Wallach set, the distribution β µ,ν cannot be a positive measure. In fact, otherwise identity (27) would imply that β µ,ν+l is a negative measure for l = 1 or l = 2, because the product on the right side of formula (27) will be negative for either ν or ν + 1.…”

Section: Beta Distributions and Extension Of The Sonine Formulamentioning

confidence: 56%

“…Let us mention at this point that there is a broad literature on beta probability distributions on matrix cones and their relevance in statistics, in particular in relation with Wishart distributions, see [6,13,21,25,28] well as the survey [8]. For some applications in mathematical physics and representation theory, see for example the survey [27] and references therein. To our knowledge, beta distributions have so far only rarely been considered for indices beyond the critical value µ 0 .…”

Section: Introductionmentioning

confidence: 99%

Beta Distributions and Sonine Integrals for Bessel Functions on Symmetric Cones

Rösler

Voit

2018

Stud Appl Math

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There exist several multivariate extensions of the classical Sonine integral representation for Bessel functions of some index μ+ν with respect to such functions of lower index μ. For Bessel functions on matrix cones, Sonine formulas involve beta densities βμ,ν on the cone and trace already back to Herz. The Sonine representations known so far on symmetric cones are restricted to continuous ranges ℜμ,ℜν>μ0, where the involved beta densities are probability measures and the limiting index μ0≥0 depends on the rank of the cone. It is zero only in the one‐dimensional case, but larger than zero in all multivariate cases. In this paper, we study the extension of Sonine formulas for Bessel functions on symmetric cones to values of ν below the critical limit μ0. This is achieved by an analytic extension of the involved beta measures as tempered distributions. Following recent ideas by A. Sokal for Riesz distributions on symmetric cones, we analyze for which indices the obtained beta distributions are still measures. At the same time, we characterize the indices for which a Sonine formula between the related Bessel functions exists. As for Riesz distributions, there occur gaps in the admissible range of indices, which are determined by the so‐called Wallach set.

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“…10 We include only those references that would help to understand the main points in the subject, a more complete list of references can be found in the review paper [11]. 11 There are many types of beta integrals, the interested reader is referred to [42][43][44] and to the review on the role of such integrals in supersymmetric gauge theories [28]. 12 The first example of the elliptic hypergeometric series was discovered about 25 years ago by Frenkel and Turaev [46] in the context of elliptic 6j-symbol [47].…”

Section: Beta Integrals and Gauge/ybe Correspondencementioning

confidence: 99%

Integrability from supersymmetric duality: a short review

Gahramanov¹

2022

Preprint

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Integrable models of statistical mechanics play a prominent role in modern mathematical physics, especially in conformal field theory, knot theory, combinatorics, topology, etc. In this brief review, we discuss a program of constructing integrable lattice spin models with the nearest neighbor interaction using methods inspired by the supersymmetric gauge theory computations, called gauge/YBE correspondence. After a brief introduction to the topic, we review some recent examples of this correspondence and the role of special functions and symmetries. Finally, we discuss future directions of research.

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Matrix Beta-integrals: An Overview

Cited by 8 publications

References 32 publications

Moments of moments of characteristic polynomials of random unitary matrices and lattice point counts

Moments of moments of characteristic polynomials of random unitary matrices and lattice point counts

Beta Distributions and Sonine Integrals for Bessel Functions on Symmetric Cones

Integrability from supersymmetric duality: a short review

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