Links Between Quantum Chaos and Counting Problems

Yurievich, Orlov Alexander

doi:10.1007/978-3-030-01156-7_37

Cited by 6 publications

(2 citation statements)

References 43 publications

(62 reference statements)

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“…see [10][11][12]. Similar products in which complex matrices are replaced by unitary and certain generalizations have also been considered [16,21,22,[33][34][35].…”

Section: Preliminary On the Products Of Random Matricesmentioning

confidence: 99%

On Products of Random Matrices

Amburg,

Orlov,

Vasiliev

2022

Preprint

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We introduce a family of models, which we name matrix models associated with children's drawings-the so-called dessin d'enfant. Dessins d'enfant are graphs of a special kind drawn on a closed connected orientable surface (in the sky). The vertices of such a graph are small disks that we call stars. We attach random matrices to the edges of the graph and get multimatrix models. Additionally, to the stars we attach source matrices. They play the role of free parameters or model coupling constants. The answers for our integrals are expressed through quantities that we call the "spectrum of stars." The answers may also include some combinatorial numbers, such as Hurwitz numbers or characters from group representation theory.

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“…see [10][11][12]. Similar products in which complex matrices are replaced by unitary and certain generalizations have also been considered [16,21,22,[33][34][35].…”

Section: Preliminary On the Products Of Random Matricesmentioning

confidence: 99%

On Products of Random Matrices

Amburg,

Orlov,

Vasiliev

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Similar products in which complex matrices are replaced by unitary and certain generalizations have also been considered [ 16 , 21 , 22 , 33 , 34 , 35 ].…”

Section: Our Models Products Of Random Matrices We Choosementioning

confidence: 99%

On Products of Random Matrices

Amburg

Orlov

Vasiliev

2020

Entropy

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We introduce a family of models, which we name matrix models associated with children’s drawings—the so-called dessin d’enfant. Dessins d’enfant are graphs of a special kind drawn on a closed connected orientable surface (in the sky). The vertices of such a graph are small disks that we call stars. We attach random matrices to the edges of the graph and get multimatrix models. Additionally, to the stars we attach source matrices. They play the role of free parameters or model coupling constants. The answers for our integrals are expressed through quantities that we call the "spectrum of stars." The answers may also include some combinatorial numbers, such as Hurwitz numbers or characters from group representation theory.

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Weighted Hurwitz Numbers and Topological Recursion

Alexandrov

Chapuy

Eynard

et al. 2020

Commun. Math. Phys.

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The KP and 2D Toda τ -functions of hypergeometric type that serve as generating functions for weighted single and double Hurwitz numbers are related to the topological recursion programme. A graphical representation of such weighted Hurwitz numbers is given in terms of weighted constellations. The associated classical and quantum spectral curves are derived, and these are interpreted combinatorially in terms of the graphical model. The pair correlators are given a finite Christoffel-Darboux representation and determinantal expressions are obtained for the multipair correlators. The genus expansion of the multicurrent correlators is shown to provide generating series for weighted Hurwitz numbers of fixed ramification profile lengths. The WKB series for the Baker function is derived and used to deduce the loop equations and the topological recursion relations. * graphs such as maps, dessins d'enfants, or more generally, constellations [54]. There has also been important progress in relating Hurwitz numbers to other classes of enumerative geometric invariants [5,19,23,30,37,51,61,64,65,68] and matrix models [1,2,4,9,10,15,19,22,57,60,64,65].A key development was the identification by Pandharipande [68] and Okounkov [61] that certain special τ -functions for integrable hierarchies of the KP and 2D Toda type may serve as generating functions for simple (single and double) Hurwitz numbers (i.e., those for which all branch points, with the possible exception of one, or two, have simple ramification profiles). It was shown subsequently [9,10,38,39,41,44] that all weighted (single and double) Hurwitz numbers have KP or 2D Toda τ -functions of the special hypergeometric type [63, 66, 67] as generating functions.An alternative approach, particularly useful for studying genus dependence and recursive relations between the invariants involved [37], consists of using multicurrent correlators as generating functions for weighted Hurwitz numbers having a fixed ramification profile length n. These may be defined in a number of equivalent ways: either as the coefficients in multivariable Taylor series expansions of the τ -function about suitably defined n-parameter families of evaluation points, in terms of pair correlators, or as fermionic expectation values of products of current operators evaluated at n points [7].A very efficient way of computing Hurwitz numbers, which provides strong results about their structure, follows from the method of Topological Recursion (TR), introduced by Eynard and Orantin in [27]. This approach, originally inspired by results arising naturally in random matrix theory [24], has been shown applicable to many enumerative geometry problems, such as the counting of maps [25] or computation of Gromov-Witten invariants [18]. It has received a great deal of attention in recent years and found to have many far-reaching implications. The fact that simple Hurwitz numbers satisfy the TR relations was conjectured by Bouchard and Mariño [19], and proved in [15,26]. This provides an algorithm that allows them to...

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Links Between Quantum Chaos and Counting Problems

Cited by 6 publications

References 43 publications

On Products of Random Matrices

On Products of Random Matrices

On Products of Random Matrices

Weighted Hurwitz Numbers and Topological Recursion

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