THERMODYNAMIC RELATIONS OF THE HERMITIAN MATRIX-ENSEMBLES

Applying the Coulomb fluid approach to the Hermitian random matrix ens embles, universal derivatives of the free energy for a system of N log arithmically repelling classical particles under the influence of an e xternal confining potential are derived. It is shown that the elements of the Jacobi matrix associated with the three-term recurrence relati on for a system of orthogonal polynomials can be expressed in terms of these derivatives and therefore give an interpretation of the recurre nce coefficients as thermodynamic susceptibilities. This provides an a lgorithm 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 auxilliar y functions related to the initial data, and in our formulation in ter ms of integrals of the logarithm of the weight function. To demonstrat e this procedure we give some examples where the initial data increase s along the half line. Combining identities of the theory of orthogona l polynomials and certain Coulomb fluid relations, a second-order ordi nary differential equation (with coefficients determined by the Coulom b 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 spe ctrum of the Jabcobi matrix, orthogonal polynomials of large degree is uniformly asymptotic to Airy function.