On the Complex Asymptotics of the HCIZ and BGW Integrals

Novak, Jonathan

doi:10.48550/arxiv.2006.04304

Cited by 8 publications

(11 citation statements)

References 66 publications

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“…When G = U(N) one should take the only term with q = 0 in expression (2). It can be proven that the resulting formula agrees with the U(N) result [18,19]. Furthermore, the corresponding representation for U(N) integral has recently been used to rigorously prove the large N asymptotic expansion in the strong coupling region (small h region) [19].…”

Section: Introductionmentioning

confidence: 87%

“…It can be proven that the resulting formula agrees with the U(N) result [18,19]. Furthermore, the corresponding representation for U(N) integral has recently been used to rigorously prove the large N asymptotic expansion in the strong coupling region (small h region) [19]. Our method relies on calculating large N limit of sums over all partitions λ.…”

Section: Introductionmentioning

confidence: 91%

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The large N limit of SU(N) integrals in lattice models

Borisenko,

Chelnokov,

Voloshyn

2020

Preprint

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The standard U (N ) and SU (N ) integrals are calculated in the large N limit. Our main finding is that for an important class of integrals this limit is different for two groups. We describe the critical behaviour of SU (N ) models and discuss implications of our results for the large N behaviour of SU (N ) lattice gauge theories at finite temperatures and non-zero baryon chemical potential. The key ingredients of our approach are 1) expansion of the integrals into a sum over irreducible representations and 2) calculation of sums over partitions of r of products of dimensions of two different representations of a symmetric group S r .

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Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 91%

The large N limit of SU(N) integrals in lattice models

Borisenko,

Chelnokov,

Voloshyn

2020

Preprint

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“…When G = U(N ), the transform L λ (x) is known as the Harish-Chandra-Itzykson-Zuber (HCIZ) integral; it has been widely studied in theoretical physics and random matrix theory since the 1980's [19]. More recently, the HCIZ integral has become an important object in combinatorics [11,28] and probability [3,27]. For further background on these functions and their diverse applications, we refer the reader to [23,24].…”

Section: Lie Algebrasmentioning

confidence: 99%

“…Up to a normalizing factor, the Heckman-Opdam polynomials turn out to be specializations of the Heckman-Opdam hypergeometric function. In particular, for λ ∈ V , define (28) c(λ, k)…”

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confidence: 99%

Majorization and Spherical Functions

McSwiggen¹,

Novak²

2020

Preprint

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Majorization is a partial order on real vectors which plays an important role in a variety of subjects, ranging from algebra and combinatorics to probability and statistics. A basic goal in the study of majorization is to construct functions which characterize this order. In this paper, we introduce a generalized notion of majorization associated to an arbitrary root system Φ, and show that it admits a natural characterization in terms of the values of spherical functions on any Riemannian symmetric space with restricted root system Φ.

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“…where dU is the Haar measure on either the orthogonal or the unitary group. Introduced by Harish-Chandra in [7], and studied then by Itzykson and Zuber in [9] who gave an explicit formula in the complex case, this integral has proven to be an interesting object in a wide variety of fields, from algebraic geometry [3,11] to physics [2]. In random matrix theory, it is a powerful tool when one looks at the large deviations of the spectrum.…”

Section: Introductionmentioning

confidence: 99%