Weighted Hurwitz Numbers, "Equation missing" -Functions, and Matrix Integrals

A direct relation between the enumeration of ordinary maps and that of fully simple maps first appeared in the work of the first and last authors. The relation is via monotone Hurwitz numbers and was originally proved using Weingarten calculus for matrix integrals. The goal of this paper is to present two independent proofs that are purely combinatorial and generalise in various directions, such as to the setting of stuffed maps and hypermaps. The main motivation to understand the relation between ordinary and fully simple maps is the fact that it could shed light on fundamental, yet still not well-understood, problems in free probability and topological recursion.

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Genus expansion of matrix models and ћ expansion of KP hierarchy

Journal of High Energy Physics ◽

10.1007/jhep12(2020)038 ◽

2020 ◽

Vol 2020 (12) ◽

Author(s):

A. Andreev ◽

A. Popolitov ◽

A. Sleptsov ◽

A. Zhabin

Keyword(s):

Matrix Models ◽

Matrix Model ◽

Special Interest ◽

Hermitian Matrix ◽

Geometric Meaning ◽

Kp Hierarchy ◽

Hurwitz Numbers ◽

Kp Equations ◽

Genus Expansion ◽

Hermitian Matrix Model

Abstract We study ћ expansion of the KP hierarchy following Takasaki-Takebe [1] considering several examples of matrix model τ-functions with natural genus expansion. Among the examples there are solutions of KP equations of special interest, such as generating function for simple Hurwitz numbers, Hermitian matrix model, Kontsevich model and Brezin-Gross-Witten model. We show that all these models with parameter ћ are τ-functions of the ћ-KP hierarchy and the expansion in ћ for the ћ-KP coincides with the genus expansion for these models. Furthermore, we show a connection of recent papers considering the ћ-formulation of the KP hierarchy [2, 3] with original Takasaki-Takebe approach. We find that in this approach the recovery of enumerative geometric meaning of τ-functions is straightforward and algorithmic.

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Generating functions for weighted Hurwitz numbers: enumeration of branched coverings

Tau Functions and their Applications ◽

10.1017/9781108610902.015 ◽

2021 ◽

pp. 369-387

Keyword(s):

Generating Functions ◽

Hurwitz Numbers ◽

Branched Coverings

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Jacobi Ensemble, Hurwitz Numbers and Wilson Polynomials

Letters in Mathematical Physics ◽

10.1007/s11005-021-01396-z ◽

2021 ◽

Vol 111 (3) ◽

Author(s):

Massimo Gisonni ◽

Tamara Grava ◽

Giulio Ruzza

Keyword(s):

Generating Functions ◽

Combinatorial Interpretation ◽

Hurwitz Numbers ◽

Jacobi Ensemble ◽

Matrix Ensembles ◽

Wilson Polynomials ◽

Unitary Ensemble ◽

Topological Expansion

AbstractWe 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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First Order Asymptotics of Matrix Integrals; A Rigorous Approach Towards the Understanding of Matrix Models

Communications in Mathematical Physics ◽

10.1007/s00220-003-0992-4 ◽

2004 ◽

Vol 244 (3) ◽

pp. 527-569 ◽

Cited By ~ 18

Author(s):

Alice Guionnet

Keyword(s):

Matrix Models ◽

First Order ◽

Rigorous Approach ◽

Matrix Integrals

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The replica limit of unitary matrix integrals

Nuclear Physics B ◽

10.1016/s0550-3213(00)00571-x ◽

2001 ◽

Vol 592 (3) ◽

pp. 419-444 ◽

Cited By ~ 27

Author(s):

D. Dalmazi ◽

J.J.M. Verbaarschot

Keyword(s):

Unitary Matrix ◽

Matrix Integrals

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Matrix integrals & finite holography

Journal of High Energy Physics ◽

10.1007/jhep06(2021)120 ◽

2021 ◽

Vol 2021 (6) ◽

Author(s):

Dionysios Anninos ◽

Beatrix Mühlmann

Keyword(s):

Critical Exponents ◽

Continuum Theory ◽

Point Of View ◽

Leading Order ◽

Minimal Models ◽

Point Evaluation ◽

Brst Cohomology ◽

Matrix Integrals ◽

The Matrix ◽

The Continuum

Abstract We explore the conjectured duality between a class of large N matrix integrals, known as multicritical matrix integrals (MMI), and the series (2m − 1, 2) of non-unitary minimal models on a fluctuating background. We match the critical exponents of the leading order planar expansion of MMI, to those of the continuum theory on an S2 topology. From the MMI perspective this is done both through a multi-vertex diagrammatic expansion, thereby revealing novel combinatorial expressions, as well as through a systematic saddle point evaluation of the matrix integral as a function of its parameters. From the continuum point of view the corresponding critical exponents are obtained upon computing the partition function in the presence of a given conformal primary. Further to this, we elaborate on a Hilbert space of the continuum theory, and the putative finiteness thereof, on both an S2 and a T2 topology using BRST cohomology considerations. Matrix integrals support this finiteness.

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