Integer moments of complex Wishart matrices and Hurwitz numbers

Cunden, Fabio Deelan; Dahlqvist, Antoine; O’Connell, Neil

doi:10.4171/aihpd/103

Cited by 22 publications

(43 citation statements)

References 60 publications

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“…Therefore the results of the present work about the JUE directly imply analogous results for the LUE; these results are already known from [17,31]. See also Remark 2.9.…”

Section: Example 19supporting

confidence: 85%

“…The Hurwitz numbers corresponding to these limit functions can be identified as in Proposition 2.8 in terms of double strictly (+) or weakly (−) Hurwitz numbers, respectively. Thus, bearing in mind the scaling limit for β → ∞ of JUE correlators to the correlators of the Laguerre Unitary Ensemble of Remark 1.10, the Theorem 1.3 recovers the results of [17].…”

Section: Remark 29supporting

confidence: 73%

“…There is a similar result for the Laguerre Unitary Ensemble (LUE) [17] which is recovered by the limit explained in Remark 1.10. However, the proof presented in this paper uses substantially different methods than those employed in [17]; in particular our proof is completely self-contained and uses the notion of multiparametric weighted Hurwitz numbers, see, e.g. [36] and Sect.…”

Section: Jue Correlators and Hurwitz Numberssupporting

confidence: 66%

“…This provides an analogue of the classical result of Bessis, Itzykson and Zuber [14] expressing correlators of the Gaussian Unitary Ensemble as a generating function counting ribbon graphs weighted according to their genus, see also [25]. At the same time, it is more similar in spirit (and actually a generalization, see Remark 1.10) of the analogous result for the Laguerre Unitary Ensemble, whose correlators are expressed in terms of double monotone Hurwitz numbers [17], and (for a specific value of the parameter) in terms of Hodge integrals [20,22,31]; in particular in [31] we provide an ELSV-type formula [24] for weighted double monotone Hurwitz numbers in terms of Hodge integrals.…”

Section: Jue Correlators and Hurwitz Numberssupporting

confidence: 52%

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Jacobi Ensemble, Hurwitz Numbers and Wilson Polynomials

2021

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We express the topological expansion of the Jacobi Unitary Ensemble in terms of triple monotone Hurwitz numbers. This completes the combinatorial interpretation of the topological expansion of the classical unitary invariant matrix ensembles. We also provide effective formulæ for generating functions of multipoint correlators of the Jacobi Unitary Ensemble in terms of Wilson polynomials, generalizing the known relations between one point correlators and Wilson polynomials.

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“…Therefore the results of the present work about the JUE directly imply analogous results for the LUE; these results are already known from [17,31]. See also Remark 2.9.…”

Section: Example 19supporting

confidence: 85%

Section: Remark 29supporting

confidence: 73%

Section: Jue Correlators and Hurwitz Numberssupporting

confidence: 66%

Section: Jue Correlators and Hurwitz Numberssupporting

confidence: 52%

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Jacobi Ensemble, Hurwitz Numbers and Wilson Polynomials

2021

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“…which is the k-th Catalan number. An analogous expansion to (1.9) holds for the LUE with Laguerre parameter a = αN [13,15]. The latter of these references involves so-called double monotone Hurwitz numbers, which have both combinatorial and topological significance.…”

mentioning

confidence: 97%

$q$-Pearson pair and moments in $q$-deformed ensembles

Forrester¹,

Li²,

Shen³

et al. 2021

Preprint

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The generalisation of continuous orthogonal polynomial ensembles from random matrix theory to the q-lattice setting is considered. We take up the task of initiating a systematic study of the corresponding moments of the density from two complementary viewpoints. The first requires knowledge of the ensemble average with respect to a general Schur polynomial, from which the spectral moments follow as a corollary. In the case of little q-Laguerre weight, a particular 3 φ 2 basic hypergeometric polynomial is used to express density moments. The second approach is to study the q-Laplace transform of the un-normalised measure. Using integrability properties associated with the q-Pearson equation for the q-classical weights, a fourth order qdifference equation is obtained, generalising a result of Ledoux in the continuous classical cases.

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$$\beta $$-Ensembles and higher genera Catalan numbers

Cassia,

Posch,

Zabzine

2024

Lett Math Phys

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We propose formulas for the large N expansion of the generating function of connected correlators of the $$\beta $$ β -deformed Gaussian and Wishart–Laguerre matrix models. We show that our proposal satisfies the known transformation properties under the exchange of $$\beta $$ β with $$1/\beta $$ 1 / β and, using Virasoro constraints, we derive a recursion formula for the coefficients of the expansion. In the undeformed limit $$\beta =1$$ β = 1 , these coefficients are integers and they have the combinatorial interpretation of generalized Catalan numbers. For generic $$\beta $$ β , we define the higher genus Catalan polynomials $$C_{g,\nu }(\beta )$$ C g , ν ( β ) whose coefficients are integer numbers.

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Integer moments of complex Wishart matrices and Hurwitz numbers

Abstract: Neil (2021) Integer moments of complex Wishart matrices and Hurwitz numbers. Annales De L'institut Henri Poincaré D.

Cited by 22 publications

References 60 publications

Jacobi Ensemble, Hurwitz Numbers and Wilson Polynomials

Jacobi Ensemble, Hurwitz Numbers and Wilson Polynomials

$q$-Pearson pair and moments in $q$-deformed ensembles

$$\beta $$-Ensembles and higher genera Catalan numbers

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