Densities in Hermitian Matrix Models

Orthogonal polynomials are traditionally studied as special functions in mathematical theories such as in the Hilbert space theory, differential equations and asymptotics. In this chapter, a new purpose of the generalized Hermite polynomials will be discussed in detail. The Lax pair obtained from the generalized Hermite polynomials can be applied to formulate the eigenvalue densities in the Hermitian matrix models with a general potential. The Lax pair method then solves the eigenvalue density problems on multiple disjoint intervals, which are associated with scalar Riemann-Hilbert problems for multi-cuts. The string equation can be applied to derive the nonlinear algebraic relations between the parameters in the density models by reformulating the potential function in terms of the trace function of the coefficient matrix obtained from the Lax pair and using the Cayley-Hamilton theorem in linear algebra. The Lax pair method improves the traditional methods for solving the eigenvalue densities by reducing the complexities in finding the nonlinear relations, and the parameters are then well organized for further analyzing the free energy function to discuss the phase transition problems.