Small Eigenvalues of Large Hankel Matrices
arXiv:math/0009238 · doi:10.1088/0305-4470/32/42/306
Abstract
In this paper we investigate the smallest eigenvalue, denoted as $\la_N,$ of a Hankel or moments matrix, associated with the weight, $w(x)=\exp(-x^{\bt}),x>0,\bt>0$, in the large limit. Using a previous result, the asymptotics for the polynomials, , orthonormal with respect to which are required in the determination of $\la_N$ are found. Adopting an argument of Szegö the asymptotic behaviour of $\la_N$, for $\bt>1/2$ where the related moment problem is determinate, is derived. This generalises the result given by Szegö for $\bt=1$. It is shown that for $\bt>1/2$ the smallest eigenvalue of the infinite Hankel matrix is zero, while for $0<\bt<1/2$ it is greater then a positive constant. This shows a phase transition in the corresponding Hermitian random matrix model as the parameter $\bt$ varies with $\bt=1/2$ identified as the critical point. The smallest eigenvalue at this point is conjectured.
15 pages,1 figure