Semiclassical expansions in the Toda hierarchy and the Hermitian matrix model

Alonso, Luis Martı́nez; Медина, Елена

doi:10.1088/1751-8113/40/47/013

Cited by 10 publications

(16 citation statements)

References 46 publications

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“…Our analysis extends previous work pertaining to matrix models with polynomial potentials [25,32,33,34]. The main idea is to combine the standard string equations with certain identities for the resolvent (L − z) −1 .…”

Section: Introductionmentioning

confidence: 61%

Partition functions and the continuum limit in Penner matrix models

Álvarez¹,

Alonso²,

Медина³

2014

J. Phys. A: Math. Theor.

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Abstract. We present an implementation of the method of orthogonal polynomials which is particularly suitable to study the partition functions of Penner random matrix models, to obtain their explicit forms in the exactly solvable cases, and to determine the coefficients of their perturbative expansions in the continuum limit. The method relies on identities satisfied by the resolvent of the Jacobi matrix in the three-term recursion relation of the associated families of orthogonal polynomials. These identities lead to a convenient formulation of the string equations. As an application, we show that in the continuum limit the free energy of certain exactly solvable models like the linear and double Penner models can be written as a sum of gaussian contributions plus linear terms. To illustrate the one-cut case we discuss the linear, double and cubic Penner models, and for the two-cut case we discuss theoretically and numerically the existence of a double-branch structure of the free energy for the gaussian Penner model.

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Section: Introductionmentioning

confidence: 61%

Partition functions and the continuum limit in Penner matrix models

Álvarez¹,

Alonso²,

Медина³

2014

J. Phys. A: Math. Theor.

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“…Although in this paper we restricted our analysis to the genus expansions of one-cut even models, since both the Bleher-Its representation of the freeenergy [2] and the method for solving the string equation using the resolvent of the Lax operator [30] can be used for models associated with general (not necessarily even) potentials, there is no obstacle to apply our analysis to these problems too. Furthermore, Bleher and Its [2] applied the integral representation (10) to determine a three-term large N asymptotic expansion for the free energy for the quartic model in the neighborhood of a critical point at the boundary between the phases G 1 and G 2 .…”

Section: Discussionmentioning

confidence: 99%

An efficient method for computing genus expansions and counting numbers in the Hermitian matrix model

2011

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We present a method to compute the genus expansion of the free energy of Hermitian matrix models from the large N expansion of the recurrence coefficients of the associated family of orthogonal polynomials. The method is based on the Bleher-Its deformation of the model, on its associated integral representation of the free energy, and on a method for solving the string equation which uses the resolvent of the Lax operator of the underlying Toda hierarchy. As a byproduct we obtain an efficient algorithm to compute generating functions for the enumeration of labeled k-maps which does not require the explicit expressions of the coefficients of the topological expansion. Finally we discuss the regularization of singular one-cut models within this approach.

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“…As our first application of these results consider the quartic potential (23) in the regular one-cut region. Equation (62) gives…”

Section: The Quartic Potential In the Regular One-cut Casementioning

confidence: 99%

“…Our analysis is based on a method for solving continuum limits of the discrete string equation [7]. This method uses the resolvent of the Lax operator of the underlying Toda hierarchy and can be applied to obtain the large N expansions both in regular and in critical models [23,24]. The asymptotic behavior of r n,N for regular models is given by power series in N −2 whose coefficients can be determined recursively.…”

Section: Introductionmentioning

confidence: 99%

LargeNexpansions and Painlevé hierarchies in the Hermitian matrix model

Álvarez¹,

Alonso²,

Медина³

2011

J. Phys. A: Math. Theor.

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Abstract. We present a method to characterize and compute the large N formal asymptotics of regular and critical Hermitian matrix models with general even potentials in the one-cut and two-cut cases. Our analysis is based on a method to solve continuum limits of the discrete string equation which uses the resolvent of the Lax operator of the underlying Toda hierarchy. This method also leads to an explicit formulation, in terms of coupling constants and critical parameters, of the members of the Painlevé I and Painlevé II hierarchies associated with one-cut and two-cut critical models respectively.

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Semiclassical expansions in the Toda hierarchy and the Hermitian matrix model

Cited by 10 publications

References 46 publications

Partition functions and the continuum limit in Penner matrix models

Partition functions and the continuum limit in Penner matrix models

An efficient method for computing genus expansions and counting numbers in the Hermitian matrix model

LargeNexpansions and Painlevé hierarchies in the Hermitian matrix model

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