Publication: An efficient method for computing genus expansions and counting numbers in the Hermitian matrix model
Loading...
Full text at PDC
Publication Date
2011-07-11
Authors
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Elsevier Science BV
Abstract
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.
Description
© 2011 Elsevier B.V. The financial support of the Universidad Complutense under project GR58/08-910556 and the Comisión Interministerial de Ciencia y Tecnología under projects FIS2008-00200 and FIS2008-00209 are gratefully acknowledged.
UCM subjects
Unesco subjects
Keywords
Citation
[1] E. Brezin, C. Itzykson, G. Parisi, J.B. Zuber, Planar diagrams, Commun. Math. Phys. 59 (1978) 35.
[2] P. Bleher, N. Its, Asymptotics of the partition function of a random matrix model, Ann. Inst. Fourier (Grenoble) 55 (2005) 1943.
[3] P. Bleher, Lectures on random matrix models. The Riemann-Hilbert approach, North Holland, Amsterdam, 2008.
[4] P. Bleher, A. Deaño, Topological expansion in the cubic random matrix model, Arxiv:1011.6338 (2010).
[5] P. Di Francesco, P. Ginsparg, J. Zinn-Justion, 2D gravity and random matrices, Phys. Rep. 254 (1995) 1.
[6] P. Di Francesco, 2D quantum gravity, matrix models and graph combinatorics, in: Applications of random matrices in physics, Springer, Dordrecht, 2006, p. 33.
[7] N. M. Ercolani, K.D.T.-R. McLaughlin, Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques and applications to graphical numeration, Int. Math. Res. Not. 14 (2003)755.
[8] G. Bonnet, F. David, B. Eynard, Breakdown of universality in multi-cut matrix models, J. Phys. A: Math. Gen. 33 (2000) 6739.
[9] B. Eynard, Large N expansion of convergent matrix integrals, holomorphic anomalies, and background independence, J. High Energy Phys. 0903 (2009) 003.
[10] P. Deift, T. Kriecherbauer, K.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, Commun. Pure. Appl. Math. 52 (1999) 1335.
[11] E.B. Saff, V. Totik, Logaritmic potentials with external fields, Springer, Berlin, 1997.
[12] P. Deift, Orthogonal polynomials and random matrices: A Riemann–Hilbert approach, American Mathematical Society, Providence, 1999.
[13] J. Ambjørn, L. Chekhov, C.F. Kristjansen, Y. Makkeenko, N. Deo,Matrix model calculations beyond the spherical limit, Nuc. Phys. B 404 (1993) 127.
[14] B. Eynard, Topological expansion for the 1-hermitian matrix model correlation functions, J. High Energy Phys. 11 (2004) 031.
[15] I. Kostov, Matrix models as cft: genus expansion, Nuc. Phys. B 837 (2010) 221.
[16] D. Bessis, A new method in the combinatorics of the topological expansion, Commun. Math. Phys. 69 (1979) 147.
[17] D. Bessis, C. Itzykson, J. Zuber, Quantum field theory techniques in graphical enumeration, Adv. in Appl. Math. 1 (1980) 109.
[18] K. Demeterfi, N. Deo, S. Jain, C.-I. Tan, Multiband structure and critical behavior of matrix models, Phys. Rev. D 42 (1990) 4105.
[19] H. Shirokura, Exact solution of 1-matrix model, in: Frontiers in quantum field theory, World Scientific, River Edge, 1995, p. 136.
[20] H. Shirokura, Generating functions in two-dimensional quantum gravity, Nuc. Phys. B 462 (1996) 99.
[21] M. Mariño, R. Schiappa, M. Weiss, Nonperturbative effects and the large-order behavior of matrix models and topological strings, Commun. Number Theory Phys. 2 (2008) 349.
[22] N.M. Ercolani, K.D.T.-R. McLaughlin, V.U. Pierce, Random matrices, graphical enumeration and the continuum limit of Toda lattices, Commun. Math. Phys. 278 (2008) 31.
[23] C. Itzykson, J.B. Zuber, The planar approximation. II, J. Math. Phys. 21 (1980) 411.
[24] A. Gerasimov, A.Marshakov, A.Mironov, A.Morozov, A. Orlov, Matrix models of two-dimensional gravity and Toda theory, Nuc. Phys. B 357 (1991) 565.
[25] E. Brézin, E. Marinari, G. Parisi, A nonperturbative ambiguity free solution of a string model, Phys. Lett. B 242 (1990) 35.
[26] P. Bleher, A. Its, Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model, Ann. Math. 150 (1999) 185.
[27] A.B.J. Kuijlaars, K.D. McLaughlin, Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields, Commun. Pure Appl. Math. 53 (2000) 736.
[28] N.M. Ercolani, Caustics, counting maps and semiclassical asymptotics, arXiv:0912.1904v2 (2009).
[29] Y. Kodama, V.U. Pierce, Combinatorics of dispersionless integrable systems and universality in random matrix theory, Commun. Math. Phys. 292 (2009) 529.
[30] L. Martínez Alonso, E. Medina, Semiclassical expansions in the Toda hierarchy and the Hermitian matrix model, J. Phys. A: Math. And Theor. 40 (2007) 14223.
[31] I. M. Gel’fand, L.A. Dikii, Asymptotic behavior of the resolvent of Sturm-Liouville equations and the algebra of the Korteweg-de Vries equations, Uspekhi Mat. Nauk 30 (1975) 67.
[32] C.A. Tracy, H. Widom, Level spacing distributions and the Airy kernel, Commun. Math. Phys. 159 (1994) 151.
[33] M. Mariño, Nonperturbative effects and nonperturbative definitions in matrix models and topological strings, J. High Energy Phys. 12 (2008) 114.
[34] B. Eynard, Universal distribution of random matrix eigenvalues near the birth of a cut, J. Stat. Mech. Theory Exp. (2006) P07005.
[35] R. Flume, A. Klitz, A new type of critical behaviour in random matrix models, J. Stat. Mech. Theory Exp. (2008) 1742.
[36] G. Álvarez, L. Martínez Alonso, E. Medina, Phase transitions in multicut matrix models and matched solutions of Whitham hierarchies, J. Stat. Mech. Theory Exp. (2010) P03023.