Time-Dependent Orthogonal Polynomials and Theory of Soliton (統計物理学の展開と応用--多様性の中の類似性(研究会報告))

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 Painleve 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.