The Annals of Probability

Second order asymptotics for matrix models

Alice Guionnet and Edouard Maurel-Segala

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Abstract

We study several-matrix models and show that when the potential is convex and a small perturbation of the Gaussian potential, the first order correction to the free energy can be expressed as a generating function for the enumeration of maps of genus one. In order to do that, we prove a central limit theorem for traces of words of the weakly interacting random matrices defined by these matrix models and show that the variance is a generating function for the number of planar maps with two vertices with prescribed colored edges.

Article information

Source
Ann. Probab., Volume 35, Number 6 (2007), 2160-2212.

Dates
First available in Project Euclid: 8 October 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1191860419

Digital Object Identifier
doi:10.1214/009117907000000141

Mathematical Reviews number (MathSciNet)
MR2353386

Zentralblatt MATH identifier
1129.15020

Subjects
Primary: 15A52 05C30: Enumeration in graph theory

Keywords
Random matrices map enumeration

Citation

Guionnet, Alice; Maurel-Segala, Edouard. Second order asymptotics for matrix models. Ann. Probab. 35 (2007), no. 6, 2160--2212. doi:10.1214/009117907000000141. https://projecteuclid.org/euclid.aop/1191860419


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References

  • Albeverio, S., Pastur, L. and Shcherbina, M. (2001). On the $1/n$ expansion for some unitary invariant ensembles of random matrices. Comm. Math. Phys. 224 271--305.
  • Ambjørn, J., Chekhov, L., Kristjansen, C. F. and Makeenko, Y. (1993). Matrix model calculations beyond the spherical limit. Nuclear Phys. B 404 127--172.
  • Anderson, G. W. and Zeitouni, O. (2006). A CLT for a band matrix model. Probab. Theory Related Fields 134 283--338.
  • Ané, C., Blachère, S., Chafaï, D., Fougères, P., Gentil, I., Malrieu, F., Roberto, C. and Scheffer, G. (2000). Sur les inégalités de Sobolev logarithmiques. Société Mathématique de France, Paris.
  • Ben Arous, G., Dembo, A. and Guionnet, A. (2001). Aging of spherical spin glasses. Probab. Theory Related Fields 120 1--67.
  • Bertola, M., Eynard, B. and Harnad, J. (2002). Duality, biorthogonal polynomials and multi-matrix models. Comm. Math. Phys. 229 73--120.
  • Bessis, D., Itzykson, C. and Zuber, J. B. (1980). Quantum field theory techniques in graphical enumeration. Adv. in Appl. Math. 1 109--157.
  • Cabanal-Duvillard, T. (2001). Fluctuations de la loi empirique de grandes matrices aléatoires. Ann. Inst. H. Poincaré Probab. Statist. 37 373--402.
  • Di Francesco, P., Ginsparg, P. and Zinn-Justin, J. (1995). 2D gravity and random matrices. Phys. Rep. 254 133.
  • Ercolani, N. M. and McLaughlin, K. D. T.-R. (2003). Asymptotics of the partition function for random matrices via Riemann--Hilbert techniques and applications to graphical enumeration. Int. Math. Res. Not. 14 755--820.
  • Eynard, B., Kokotov, A. and Korotkin, D. (2005). $1/N\sp2$-correction to free energy in Hermitian two-matrix model. Lett. Math. Phys. 71 199--207.
  • Gross, D. J., Piran, T. and Weinberg, S., eds. (1992). Two-Dimensional Quantum Gravity and Random Surfaces. World Scientific, River Edge, NJ.
  • Guionnet, A. (2002). Large deviations upper bounds and central limit theorems for non-commutative functionals of Gaussian large random matrices. Ann. Inst. H. Poincaré Probab. Statist. 38 341--384.
  • Guionnet, A. and Maurel-Segala, E. (2006). Combinatorial aspects of matrix models. ALEA Lat. Am. J. Probab. Math. Stat. 1 241--279 (electronic).
  • Guionnet, A. and Zeitouni, O. (2002). Large deviations asymptotics for spherical integrals. J. Funct. Anal. 188 461--515.
  • Harer, J. and Zagier, D. (1986). The Euler characteristic of the moduli space of curves. Invent. Math. 85 457--485.
  • Hargé, G. (2004). A convex/log-concave correlation inequality for Gaussian measure and an application to abstract Wiener spaces. Probab. Theory Related Fields 130 415--440.
  • Johansson, K. (1998). On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91 151--204.
  • Maurel-Segala, E. (2005). High order expansion of matrix models and enumeration of maps. Available at http://front.math.ucdavis.edu/math.PR/0608192.
  • Mingo, J. and Speicher, R. (2004). Second order freeness and fluctuations of random matrices: I. Gaussian and Wishart matrices and cyclic Fock spaces. Available at http://front.math.ucdavis.edu/math.OA/0405191.
  • Pisier, G. and Xu, Q. (2003). Non-commutative $L\sp p$-spaces. In Handbook of the Geometry of Banach Spaces 2 1459--1517. North-Holland, Amsterdam.
  • Soshnikov, A. (1999). Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. 207 697--733.
  • Voiculescu, D. (1991). Limit laws for random matrices and free products. Invent. Math. 104 201--220.
  • Voiculescu, D. (2000). Lectures on free probability theory. Lectures on Probability Theory and Statistics (Saint-Flour, 1998). Lecture Notes in Math. 1738 279--349. Springer, Berlin.
  • Wigner, E. P. (1958). On the distribution of the roots of certain symmetric matrices. Ann. of Math. (2) 67 325--327.