Monotone Hurwitz Numbers and the HCIZ Integral

Goulden, I. P.; Guay-Paquet, Mathieu; Novak, Jonathan

doi:10.5802/ambp.336

Cited by 95 publications

(158 citation statements)

References 37 publications

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“…Theorem 1, when combined with the results of [5] and [11], establishes the following topological expansion of the HCIZ free energy…”

Section: Collins Guionnet and Maurel-segalamentioning

confidence: 76%

“…For more on monotone double Hurwitz numbers, see [11]. Theorem 5 is the perturbative version of an asymptotic expansion of the HCIZ free energy conjectured to hold by Matytsin in [17].…”

Section: Collins Guionnet and Maurel-segalamentioning

confidence: 98%

“…We then establish the asymptotic expansion of the free energy of µ V N . Finally, we combine Theorem 1 with results from [11] to obtain a proof of Theorem 5.…”

Section: Collins Guionnet and Maurel-segalamentioning

confidence: 99%

“…Decomposing all relevant functions in Maclaurin series, as N → ∞, for each fixed d ≥ 1. The asymptotics of the Maclaurin coefficients of F N (t) were obtained by a different method in [11], where a representation-theoretic argument was used to show that…”

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confidence: 99%

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Asymptotics of unitary multimatrix models: The Schwinger–Dyson lattice and topological recursion

Guionnet

Novak

2015

Journal of Functional Analysis

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Abstract. We prove the existence of a 1/N expansion in unitary multimatrix models which are Gibbs perturbations of the Haar measure, and express the expansion coefficients recursively in terms of the unique solution of a noncommutative initial value problem. The recursion obtained is closely related to the "topological recursion" which underlies the asymptotics of many random matrix ensembles and appears in diverse enumerative geometry problems. Our approach consists of two main ingredients: an asymptotic study of the Schwinger-Dyson lattice over noncommutative Laurent polynomials, and uniform control on the cumulants of Gibbs measures on product unitary groups. The required cumulant bounds are obtained by concentration of measure arguments and change of variables techniques.

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“…Theorem 1, when combined with the results of [5] and [11], establishes the following topological expansion of the HCIZ free energy…”

Section: Collins Guionnet and Maurel-segalamentioning

confidence: 76%

“…For more on monotone double Hurwitz numbers, see [11]. Theorem 5 is the perturbative version of an asymptotic expansion of the HCIZ free energy conjectured to hold by Matytsin in [17].…”

Section: Collins Guionnet and Maurel-segalamentioning

confidence: 98%

“…We then establish the asymptotic expansion of the free energy of µ V N . Finally, we combine Theorem 1 with results from [11] to obtain a proof of Theorem 5.…”

Section: Collins Guionnet and Maurel-segalamentioning

confidence: 99%

mentioning

confidence: 99%

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Asymptotics of unitary multimatrix models: The Schwinger–Dyson lattice and topological recursion

Guionnet

Novak

2015

Journal of Functional Analysis

Self Cite

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“…The former problem has been solved from our workings in Sections 1 and 2 for the initial condiitions (1.6) and (2.25), while the evaluation of Sτ in the first of these is given in [10]. For a discussion of analyticity properties of (6.4) see [19].…”

Section: (62)mentioning

confidence: 99%

Hydrodynamical spectral evolution for random matrices

Forrester

Grela

2016

J. Phys. A: Math. Theor.

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Abstract. The eigenvalues of the matrix structure X + X (0) , where X is a random Gaussian Hermitian matrix and X (0) is non-random or random independent of X, are closely related to Dyson Brownian motion. Previous works have shown how an infinite hierarchy of equations satisfied by the dynamical correlations become triangular in the infinite density limit, and give rise to the complex Burgers equation for the Green's function of the corresponding one-point density function. We show how this and analogous partial differential equations, for chiral, circular and Jacobi versions of Dyson Brownian motion follow from a macroscopic hydrodynamical description involving the current density and continuity equation. The method of characteristics gives a systematic approach to solving the PDEs, and in the chiral case we show how this efficiently reclaims the characterisation of the global eigenvalue density for non-central Wishart matrices due to Dozier and Silverstein. Collective variables provide another approach to deriving the complex Burgers equation in the Gaussian case, and we show that this approach applies equally as well to chiral matrices. We relate both the Gaussian and chiral cases to the asymptotics of matrix integrals.

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