Abstract
The unitary matrix model is another important topic in quantum chromodynamics (QCD) and lattice gauge theory. The Gross-Witten weak and strong coupling densities are the most popular density models in QCD for studying the third-order phase transition problems, which are related to asymptotic freedom and confinement. For the Gross-Witten weak and strong coupling densities and the generalizations to be discussed in this chapter, it should be noted that the densities are defined on the complement of the cuts in the unit circle, and there are two essential singularities, which are different from the Hermitian models. The orthogonal polynomials on the unit circle are applied to study these problems by using the string equation. The recursion formula now becomes the discrete AKNS-ZS system, and the reduction of the eigenvalue density is now based on new linear systems of equations satisfied by the orthogonal polynomials on the unit circle. The integrable systems and string equation discussed in this chapter provide a structure for finding the generalized density models and parameter relations that will be used as the mathematical foundation to investigate the transition problems discussed in next chapter.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Cresswell, C., Joshi, N.: The discrete first, second and thirty-fourth Painlevé hierarchies. J. Phys. A 32, 655–669 (1999)
Grammaticos, B., Nijhoff, F.W., Papageorgiou, V., Ramani, A., Satsuma, J.: Linearization and solutions of the discrete Painlevé III equation. Phys. Lett. A 185, 446–452 (1994)
Grammaticos, B., Ohta, Y., Ramani, A., Sakai, H.: Degenerate through coalescence of the q-Painlevé VI equation. J. Phys. A 31, 3545–3558 (1998)
Gross, D.J., Witten, E.: Possible third-order phase transition in the large-N lattice gauge theory. Phys. Rev. D 21, 446–453 (1980)
McLeod, J.B., Wang, C.B.: Discrete integrable systems associated with the unitary matrix model. Anal. Appl. 2, 101–127 (2004)
Periwal, V., Shevitz, D.: Unitary-matrix models as exactly solvable string theories. Phys. Rev. Lett. 64, 1326–1329 (1990)
Periwal, V., Shevitz, D.: Exactly solvable unitary matrix models: multicritical potentials and correlations. Nucl. Phys. B 344, 731–746 (1990)
Rossi, P., Campostrini, M., Vicari, E.: The large-N expansion of unitary-matrix models. Phys. Rep. 302, 143–209 (1998)
Simon, B.: Orthogonal Polynomials on the Unit Circle, Vol. 1: Classical Theory. AMS Colloquium Series. AMS, Providence (2005)
Simon, B.: Orthogonal Polynomials on the Unit Circle, Vol. 2: Spectral Theory. AMS Colloquium Series. AMS, Providence (2005)
Szegö, G.: Orthogonal Polynomials, 4th edn. American Mathematical Society Colloquium Publications, vol. 23. AMS, Providence (1975)
Wang, C.B.: Orthonormal polynomials on the unit circle and spatially discrete Painlevé II equation. J. Phys. A 32, 7207–7217 (1999)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Wang, C.B. (2013). Densities in Unitary Matrix Models. In: Application of Integrable Systems to Phase Transitions. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38565-0_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-38565-0_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-38564-3
Online ISBN: 978-3-642-38565-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)