Small eigenvalues of large Hankel matrices

In this paper we investigate the smallest eigenvalue, denoted as lambda(N), of a (N+1) X (N+1) Hankel or moments matrix, associated with the weight, w(x) = exp(-x(beta)), x > 0, beta > 0, in the large N limit. Using a previous result, the asymptotics for the polynomials, P-n(z), z is not an element of [0, infinity), orthonormal with respect to w, which are required in the determination of lambda(N) are found. Adopting an argument of Szego the asymptotic behaviour of lambda(N), for beta > 1/2 where the related moment problem is determinate, is derived. This generalizes the result given by Szego for beta = 1. It is shown that for beta > 1/2 the smallest eigenvalue of the infinite Hankel matrix is zero, while for 0 < beta < 1/2 it is greater then a positive constant. This shows a phase transition in the corresponding identified as the critical Hermitian random matrix model as the parameter beta varies with beta = 1/2 point. The smallest eigenvalue at this point is conjectured.