Introduction

Phase transition problems in the one-matrix models in QCD are discussed in this book by using string equations. The phase transition models are formulated by using eigenvalue density which represents the momentum operator described in quantum mechanics. By using orthogonal polynomials, the eigenvalue densities in various matrix models can be unified. The string equation establishes a connection between the position and momentum aspects described in the Heisenberg uncertainty principle, which is important and usually hard to find in other methods. The recursion formula of the orthogonal polynomials can be applied to introduce an index folding technique to reorganize the wave functions in order to achieve a renormalization in the momentum aspect. It will be discussed that the critical phenomenon associated with the Gross-Witten model can be found by using Toda lattice and double scaling. The hypergeometric-type differential equations improve on some shortages of integrable systems to work on physical problems, such as the fact that a soliton system does not have a differential equation along the spectrum direction, and illustrate a new background to study the singularities of physical quantities, such as mass.