Time-Dependent Orthogonal Polynomials and Theory of Soliton –Applications to Matrix Model, Vertex Model and Level Statistics

By introducing a time variable to the theory of orthogonal polynomials, partition functions of both the matrix model and the vertex model are shown to be described by the Toda molecule equation. In short, the soliton is at the center of orthogonal polynomials, the matrix model and the vertex model . The differential-difference Painlevé equation appears in the case of pure gravity of the matrix model. An application to the random matrix theory of level statistics is also shown.