Bifurcation Transitions and Expansions

  • C. B. Wang


It is believed in matrix model theory that when the eigenvalue density on one interval is split to a new density on two disjoint intervals, a phase transition occurs. The complexity for the mathematical details of this physical phenomenon comes not only from the elliptic integral calculations, but also from the organization of the parameters in the model. Generally, the elliptic integrals do not have simple analytic formulations for discussing the transition. The string equations can be applied to find the critical point for the transition from the parameter bifurcation, and the bifurcation clearly separates the different phases for analyzing the free energy. Based on the expansion method for elliptic integrals, the third-order bifurcation transition for the Hermitian matrix model with a general quartic potential is discussed in this chapter by applying the nonlinear relations obtained from the string equations. The density on multiple disjoint intervals for higher degree potential and the corresponding free energy are discussed in association with the Seiberg-Witten differential. In the symmetric cases for the quartic potential, the third-order phase transitions are explained with explicit formulations of the free energy function.


Expansion Free energy Merged and split densities Third-order phase transition Toda lattice Seiberg-Witten theory 


  1. 1.
    Bleher, P., Eynard, B.: Double scaling limit in random matrix models and a nonlinear hierarchy of differential equations. J. Phys. A 36, 3085–3106 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Braden, H.W., Krichever, I.M. (eds.): Integrability: The Seiberg-Witten and Whitham Equations. Gordon & Breach, Amsterdam (2000) Google Scholar
  3. 3.
    Brézin, E., Itzykson, C., Parisi, G., Zuber, J.B.: Planar diagrams. Commun. Math. Phys. 59, 35–51 (1978) CrossRefzbMATHGoogle Scholar
  4. 4.
    Chekhov, L., Mironov, A.: Matrix models vs. Seiberg–Witten/Whitham theories. Phys. Lett. B 552, 293–302 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chekhov, L., Marshakov, A., Mironov, A., Vasiliev, D.: Complex geometry of matrix models. Proc. Steklov Inst. Math. 251, 254–292 (2005) Google Scholar
  6. 6.
    Dijkgraaf, R., Vafa, C.: Matrix models, topological strings, and supersymmetric gauge theories. Nucl. Phys. B 644, 3–20 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dijkgraaf, R., Moore, G.W., Plesser, R.: The partition function of 2-D string theory. Nucl. Phys. B 394, 356–382 (1993) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Douglas, M.R.: Report on the status of the Yang-Mills millenium prize problem (2004).
  9. 9.
    Fuji, H., Mizoguchi, S.: Remarks on phase transitions in matrix models and N=1 supersymmetric gauge theory. Phys. Lett. B 578, 432–442 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gorsky, A., Krichever, I., Marshakov, A., Mironov, A., Morozov, A.: Integrability and Seiberg-Witten exact solution. Phys. Lett. B 355, 466–477 (1995) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jaffe, A., Witten, E.: Quantum Yang-Mills theory. In: Carlson, J., Jaffe, A., Wiles, A. (eds.) The Millennium Prize Problems, pp. 129–152. AMS, Providence (2006) Google Scholar
  12. 12.
    Jurkiewicz, J.: Regularization of one-matrix models. Phys. Lett. B 235, 178–184 (1990) MathSciNetGoogle Scholar
  13. 13.
    Korepin, V.E., Bogoliubov, N.M., Izergin, A.G.: Quantum Inverse Scattering Method and Correlation Functions. Cambridge University Press, Cambridge (1993) CrossRefzbMATHGoogle Scholar
  14. 14.
    Marshakov, A., Nekrasov, N.: Extended Seiberg-Witten theory and integrable hierarchy. J. High Energy Phys. 0701, 104 (2007) MathSciNetCrossRefGoogle Scholar
  15. 15.
    McLeod, J.B., Wang, C.B.: Eigenvalue density in Hermitian matrix models by the Lax pair method. J. Phys. A, Math. Theor. 42, 205205 (2009) MathSciNetCrossRefGoogle Scholar
  16. 16.
    Mehta, M.L.: Random Matrices, 3rd edn. Academic Press, New York (2004) zbMATHGoogle Scholar
  17. 17.
    Nakatsu, T., Takasaki, K.: Whitham-Toda hierarchy and N=2 supersymmetric Yang-Mills theory. Mod. Phys. Lett. A 11, 157–168 (1996) MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Nekrasov, N.: Seiberg-Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7, 831–864 (2004) MathSciNetGoogle Scholar
  19. 19.
    Seiberg, N., Witten, E.: Electric-magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory. Nucl. Phys. B 426, 19–52 (1994). Erratum, ibid. B 430, 485–486 (1994) MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Shimamune, Y.: On the phase structure of large N matrix models and gauge models. Phys. Lett. B 108, 407–410 (1982) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • C. B. Wang
    • 1
  1. 1.Institute of AnalysisTroyUSA

Personalised recommendations