Densities in Unitary Matrix Models
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.
KeywordsDiscrete AKNS-ZS system Lax pair Orthogonal polynomials on unit circle String equation Strong and weak couplings
- 10.Simon, B.: Orthogonal Polynomials on the Unit Circle, Vol. 2: Spectral Theory. AMS Colloquium Series. AMS, Providence (2005) Google Scholar