Phase transitions in multi-cut matrix models and matched solutions of Whitham hierarchies

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We present a method for studying phase transitions in the large N limit of matrix models using matched solutions of Whitham hierarchies. The endpoints of the eigenvalue spectrum as functions of the temperature are characterized both as solutions of hodograph equations and as solutions of a system of ordinary differential equations. In particular we show that the free energy of the matrix model is the quasiclassical tau-function of the associated hierarchy, and that critical processes in which the number of cuts changes in one unit are third-order phase transitions described by C(1) matched solutions of Whitham hierarchies. The method is illustrated with the Bleher-Eynard model for the merging of two cuts. We show that this model involves also the birth of a cut.
�© 2010 IOP Publishing Ltd and SISSA. The financial support of the Universidad Complutense under project GR58/08-910556, that of the Comision Interministerial de Ciencia y Tecnologia under projects FIS2008-00200 and FIS2008-00209, and that of the ESF programme MISGAM are gratefully acknowledged.
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[1] Metha, M. L., 1991, Random Matrices, (New York: Academic). [2] Di Francesco, P., Ginsparg, P. and Zinn-Justin, Z., 1995, Phys. Rep. , 254, p 1. [3] Deift, P., Kriecherbauer, T. and McLaughlin, K., New results for the asymptotic of orthogonal polynomials and related problems via the Lax–Levermore method, 1998, Recent Advances in Partial Differential Equations, Venice 1996, Proc. Symp. Appl. Math., vol 54, p 87. [4] Deift, P., Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach, Courant Lecture Notes in Mathematics, 1999, vol 3 (Providence, RI: American Mathematical Society). [5] Bonnet, G., David, F. and Eybard, B., J. Phys. A: Math. Gen., 2000, 33, p 6739. [6] Bleher, P. and Its, A., Commun. Pure Appl. Math., 2003, 56, p 433. [7] Bleher, P. and Eynard, B., J. Phys. A: Math. Gen., 2003, 36, p 3085. [8] Bleher, P., Lectures on Random Matrix Models. The Riemann–Hilbert Approach, 2008, The CRM Volume on Random Matrices (Berlin: Springer) at press arXiv:0801.1858 [math-ph]. [9] Eynard, B., J. Stat. Mech., 2006, p 07005. [10] Flume, R. and Klitz, A., J. Stat. Mech., 2008, N10001. [11] Krichever, I. M., Commun. Pure Appl. Math., 1992, 47, p 437. [12] Krichever, I. M., Funct. Anal. Appl., 1988, 22, p 200 [13] Grava, T., J. Phys. A: Math. Gen., 2006, 39, p 8905. [14] Flaschka, H., Forest, M. G. and MacLaughlin, D. W., Commun. Pure Appl. Math., 1980, 33, p 739. [15] Lax, P. D. and Levermore, C. D., Commun. Pure Appl. Math., 1983, 36, p 253, p 571, p 809. [16] Venakides, S., Commun. Pure Appl. Math., 1985, 38, p 125. [17] Tian, F. R., Commun. Pure Appl. Math., 1993, 46, p 1093. [18] Tian, F. R., Commun. Math. Phys., 1994, 166, p 79. [19] Tian, F. R., Duke Math. J., 1994, 74, p 203. [20] Grava, T., Math. Phys. Anal. Geom., 2001, 4, p 65. [21] Grava, T., Commun. Pure Appl. Math., 2002, 55, p 395. [22] Grava, T. and Tian, F. R., Commun. Pure Appl. Math., 2002, 55, p 1569. [23] Gross, D. and Witten, E., Phys. Rev. D, 1980, 21, p 446. [24] Springer, G., Introduction to the Riemann Surfaces, 1987, (Reading, MA: Addison-Wesley). [25] Farkas, H. M. and Kra, I., Riemann Surfaces, 1980, (New York: Springer). [26] Tsarev, S. P., Sov. Math. Dokl., 1985, 31, p 488. [27] [28] [29]