Limit theorems and soft edge of freezing random matrix models via dual orthogonal polynomials
Abstract
N-dimensional Bessel and Jacobi processes describe interacting particle systems with N particles and are related to β-Hermite, β-Laguerre, and β-Jacobi ensembles. For fixed N there exist associated weak limit theorems (WLTs) in the freezing regime β∞ in the β-Hermite and β-Laguerre case by Dumitriu and Edelman (2005) with explicit formulas for the covariance matrices N in terms of the zeros of associated orthogonal polynomials. Recently, the authors derived these WLTs in a different way and computed N-1 with formulas for the eigenvalues and eigenvectors of N-1 and thus of N. In the present paper we use these data and the theory of finite dual orthogonal polynomials of de Boor and Saff to derive formulas for N from N-1 where, for β-Hermite and β-Laguerre ensembles, our formulas are simpler than those of Dumitriu and Edelman. We use these polynomials to derive asymptotic results for the soft edge in the freezing regime for N∞ in terms of the Airy function. For β-Hermite ensembles, our limit expressions are different from those of Dumitriu and Edelman.