Thermodynamic relations of the Hermitian matrix ensembles

Applying the Coulomb fluid approach to the Hermitian random matrix ensembles, universal derivatives of the free energy for a system of N logarithmically repelling classical particles under the influence of an external confining potential are derived. It is shown that the elements of the Jacobi matrix associated with the three-term recurrence relation for a system of orthogonal polynomials can be expressed in terms of these derivatives and therefore give an interpretation of the recurrence coefficients as thermodynamic susceptibilities. This provides an algorithm for the computation of the asymptotic recurrence coefficients for a given weight function. We also show that a pair of quasilinear partial differential equations, obtained in the continuum limit of the Toda lattice, can be integrated exactly in terms of certain auxilliary functions related to the initial data, and in our formulation in terms of integrals of the logarithm of the weight function. To demonstrate this procedure we give some examples where the initial data increases along the half line. Combining identities of the theory of orthogonal polynomials and certain Coulomb fluid relations, a second-order ordinary differential equation (with coefficients determined by the Coulomb fluid density) satisfied by the polynomials is derived. We use this to prove some conjectures put forward in previous papers. We show that, if the confining potential is convex, then near the edges of the spectrum of the Jabcobi matrix, orthogonal polynomials of large degree is uniformly asymptotic to Airy function.