Asymptotics of the partition function of a random matrix model
Annales de l'Institut Fourier, Volume 55 (2005) no. 6, pp. 1943-2000.

We prove a number of results concerning the large N asymptotics of the free energy of a random matrix model with a polynomial potential. Our approach is based on a deformation of potential and on the use of the underlying integrable structures of the matrix model. The main results include the existence of a full asymptotic expansion in even powers of N of the recurrence coefficients of the related orthogonal polynomials for a one-cut regular potential and the double scaling asymptotics of the free energy for a singular quartic potential. We also prove the analyticity of the coefficients of the asymptotic expansions of the recurrence coefficients and the free energy, with respect to the coefficients of the potential, and the one-sided analyticity of the recurrent coefficients and the free energy for a one-cut singular potential.

Nous prouvons de nombreux résultats concernant les comportements asymptotiques de l’énergie libre d’un modèle matriciel aléatoire à potentiel polynômial. Notre approche est fondée sur la déformation du potentiel et de l’utilisation de la structure intégrable sous-jacente du modèle. Les principaux résultats incluent l’existence du développement asymptotique en puissances de N impaires des coefficients de récurrence des polynômes orthogonaux d’un potentiel régulier à une coupe et de la double réduction asymptotique de l’énergie libre pour un potentiel quartique singulier. Nous prouvons aussi l’analyticité des coefficients du développement asymptotique des coefficients de récurrence et de l’énergie selon ceux du potentiel libre, ainsi que l’analyticité unilatérale des coefficients et de l’énergie libre d’un potentiel singulier à une coupe.

DOI: 10.5802/aif.2147
Classification: 42C05
Keywords: Matrix Models, orthogonal polynomials, partition function
Mot clés : modèles matriciels, polynômes orthogonaux, fonction de partition
M. Bleher, Pavel 1; Its, Alexander 

1 Indiana University-Purdue University Indianapolis, department of mathematical sciences, 402 N. Blackford Street, Indianapolis IN 46202 (USA)
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     title = {Asymptotics of the partition function of a random matrix model},
     journal = {Annales de l'Institut Fourier},
     pages = {1943--2000},
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M. Bleher, Pavel; Its, Alexander. Asymptotics of the partition function of a random matrix model. Annales de l'Institut Fourier, Volume 55 (2005) no. 6, pp. 1943-2000. doi : 10.5802/aif.2147.

[BDE] G. Bonnet; F. David; B. Eynard Breakdown of universality in multi-cut matrix models, J. Phys., Volume A33 (2000), pp. 6739-6768 | MR | Zbl

[BDJ] J. Baik; P. Deift; K. Johansson On the distribution of the length of the longest increasing subsequence of random permutations, J. Am. Math. Soc., Volume 12 (1999), pp. 1119-1178 | DOI | MR | Zbl

[BE] P.M. Bleher; B. Eynard Double scaling limit in random matrix models and a nonlinear hierarchy of differential equations, J. Phys. A: Math. Gen., Volume 36 (2003), pp. 3085-3105 | DOI | MR | Zbl

[BEH] M. Bertola; B. Eynard; J. Harnad Partition functions for matrix models and Isomonodromic Tau functions, J. Phys. A. Math, Gen., Volume 36 (2003), pp. 3067-3983 | DOI | MR | Zbl

[BI1] P.M. Bleher; A.R. Its Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model, Annals Math., Volume 150 (1999), pp. 185-266 | DOI | EuDML | MR | Zbl

[BI2] P.M. Bleher; A.R. Its Double scaling limit in the matrix model: the Riemann-Hilbert approach, Com. Pure Appl. Math., Volume 56 (2003), pp. 433-516 | DOI | MR | Zbl

[BIZ] D. Bessis; C. Itzykson; J.B. Zuber Quantum field theory techniques in graphical enumeration, Adv. in Appl. Math., Volume 1 (1980) no. 2, pp. 109-157 | DOI | MR | Zbl

[BPS] A. Boutet de Monvel; L. Pastur; M. Shcherbina On the statistical mechanics approach in the random matrix theory: integrated density of states, J. Statist. Phys., Volume 79 (1995), pp. 585-611 | DOI | MR | Zbl

[DGZ] Ph. Di Francesco; P. Ginsparg; J. Zinn-Justin 2D gravity and random matrices, Phys. Rep., Volume 254 (1995) no. 1-2, pp. 133 pp. | MR

[DKM] P. Deift; T. Kriecherbauer; K.D. T.-R. McLaughlin New results on the equilibrium measure for logarithmic potentials in the presence of an external field, J. Approx. Theory, Volume 95 (1998), pp. 388-475 | DOI | MR | Zbl

[DKMVZ] P. Deift; T. Kriecherbauer; K.D. T.-R. McLaughlin; S. Venakides; X. Zhou Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Com. Pure Appl. Math., Volume 52 (1999), pp. 1335-1425 | DOI | MR | Zbl

[EM] N.M. Ercolani; K.D. T.-R. McLaughlin Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques and applications to graphical enumeration., Int. Math. Res. Not., Volume 14 (2003), pp. 755-820 | MR | Zbl

[Ey] B. Eynard A concise expression for the ODE's of orthogonal polynomials (2001) (Preprint, arXiv:math-ph/0109018, | MR

[FIK] A.S. Fokas; A.R. Its; A.V. Kitaev The isomonodromy approach to matrix models in 2D quantum gravity, Com. Math. Phys., Volume 147 (1992), pp. 395-430 | DOI | MR | Zbl

[Fl] H. Flashka The Toda Lattice II. Inverse scattering solution, Prog. Theor. Phys., Volume 51 (1974) no. 3, pp. 703-716 | MR | Zbl

[HM] S.P. Hastings; J.B. McLeod A boundary value problem associated with the second Painlevé transcendent and the Korteweg de Vries equation, Arch. Rat. Mech. Anal., Volume 73 (1980), pp. 31-51 | DOI | MR | Zbl

[IKF] A.R. Its; A.V. Kitaev; A.S. Fokas Matrix models of two-dimensional quantum gravity and isomonodromy solutions of `discrete Painlevé equations' (Zap. Nauch. Sem. LOMI, 187 (1991), 3–30. Russian transl. in J. Math. Sci.), Volume 73/4 (1995), pp. 415-429 | Zbl

[KM] A.B.J. Kuijlaars; K.D. T.-R. McLaughlin Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields, Com. Pure Appl. Math., Volume 53 (2000), pp. 736-785 | DOI | MR | Zbl

[KvM] M. Kac; P. van Moerbeke On an explicitly soluble system of non-linear differential equations related to certain Toda lattices, Adv. in Math., Volume 16 (1975), pp. 160-164 | DOI | Zbl

[Ma] S.V. Manakov On complete integrability and stochastization in the discrete dynamical systems, Zh. Exp. Teor. Fiz., Volume 67 (1974) no. 2, pp. 543-555 | MR

[TW] C.A. Tracy; H. Widom Level-spacing distributions and the Airy kernel, Com. Math. Phys., Volume 159 (1994) no. 1, pp. 151-174 | DOI | MR | Zbl

[vM1] P. van Moerbeke Random matrices and permutations, matrix integrals and Integrable systems, Séminaire Bourbaki, 52e année, Volume 879 (1999-2000), pp. 1-21 | Numdam | Zbl

[vM2] P. van Moerbeke; P.M. Bleher and A.R. Its Integrable lattices: random matrices and random permutations, Random Matrices and Their Applications (Mathematical Sciences Research) (2001) | Zbl

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