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 [Ann. Inst. Henri Poincare, Sect. B 41, 1083 (2005)] with explicit formulas for the covariance matrices ΣN in terms of the zeros of associated orthogonal polynomials. Recently, the authors derived these WLTs in a different way and computed ΣN−1 with formulas for the eigenvalues and eigenvectors of ΣN−1 and thus of ΣN. 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 ΣN from ΣN−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.