An efficient method for computing genus expansions and counting numbers in the Hermitian matrix model
| dc.contributor.author | Álvarez Galindo, Gabriel | |
| dc.contributor.author | Martínez Alonso, Luis | |
| dc.contributor.author | Medina Reus, Elena | |
| dc.date.accessioned | 2023-06-20T03:31:29Z | |
| dc.date.available | 2023-06-20T03:31:29Z | |
| dc.date.issued | 2011-07-11 | |
| dc.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. | |
| dc.description.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. | |
| dc.description.department | Depto. de Física Teórica | |
| dc.description.faculty | Fac. de Ciencias Físicas | |
| dc.description.refereed | TRUE | |
| dc.description.sponsorship | Universidad Complutense | |
| dc.description.sponsorship | Comision Interministerial de Ciencia y Tecnologia | |
| dc.description.status | pub | |
| dc.eprint.id | https://eprints.ucm.es/id/eprint/20467 | |
| dc.identifier.doi | 10.1016/j.nuclphysb.2011.02.019 | |
| dc.identifier.issn | 0550-3213 | |
| dc.identifier.officialurl | http://pdn.sciencedirect.com/science?_ob=MiamiImageURL&_cid=271560&_user=144492&_pii=S0550321311001246&_check=y&_origin=article&_zone=toolbar&_coverDate=11-Jul-2011&view=c&originContentFamily=serial&wchp=dGLbVlV-zSkWz&md5=e155f623a2d9c04bb489d6b9a45028c8& | |
| dc.identifier.relatedurl | http://arxiv.org/pdf/1101.2727.pdf | |
| dc.identifier.relatedurl | http://pdn.sciencedirect.com | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14352/43708 | |
| dc.issue.number | 2 | |
| dc.journal.title | Nuclear Physics B | |
| dc.language.iso | eng | |
| dc.page.final | 429 | |
| dc.page.initial | 398 | |
| dc.publisher | Elsevier Science BV | |
| dc.relation.projectID | GR58/08-910556 | |
| dc.relation.projectID | FIS2008-00200 | |
| dc.relation.projectID | FIS2008-00209 | |
| dc.rights.accessRights | open access | |
| dc.subject.cdu | 51-73 | |
| dc.subject.keyword | Graphical Enumeration | |
| dc.subject.keyword | Partition-Function | |
| dc.subject.keyword | Asymptotics | |
| dc.subject.keyword | Universality | |
| dc.subject.keyword | Behavior | |
| dc.subject.keyword | Gravity | |
| dc.subject.keyword | Polynomials | |
| dc.subject.keyword | Limit | |
| dc.subject.keyword | Hermitian Matrix Model | |
| dc.subject.keyword | Genus Expansion | |
| dc.subject.keyword | Counting Maps | |
| dc.subject.ucm | Física-Modelos matemáticos | |
| dc.title | An efficient method for computing genus expansions and counting numbers in the Hermitian matrix model | |
| dc.type | journal article | |
| dc.volume.number | 848 | |
| dcterms.references | [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. | |
| dspace.entity.type | Publication | |
| relation.isAuthorOfPublication | 93e2c5ce-9576-43ad-99af-1f18cb650636 | |
| relation.isAuthorOfPublication | 896aafc0-9740-4609-bc38-829f249a0d2b | |
| relation.isAuthorOfPublication.latestForDiscovery | 93e2c5ce-9576-43ad-99af-1f18cb650636 |
