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 $\beta$-Hermite, $\beta$-Laguerre, and $\beta$-Jacobi ensembles. For fixed $N$ there exist associated weak limit theorems (WLTs) in the freezing regime $\beta \to \infty$ in the $\beta$-Hermite and $\beta$-Laguerre case by Dumitriu and Edelman (2005) with explicit formulas for the covariance matrices ${\Sigma }_N$ in terms of the zeros of associated orthogonal polynomials. Recently, the authors derived these WLTs in a different way and computed ${\Sigma }_N^{-1}$ with formulas for the eigenvalues and eigenvectors of ${\Sigma }_N^{-1}$ and thus of ${\Sigma }_N$. 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 ${\Sigma }_N$ from ${\Sigma }_N^{-1}$ where, for $\beta$-Hermite and $\beta$-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 $N\to \infty$ in terms of the Airy function. For $\beta$-Hermite ensembles, our limit expressions are different from those of Dumitriu and Edelman.

32 pages, made small improvements and added references

References in corpus (3)

• Central limit theorems for multivariate Bessel processes in the freezing regime II: the covariance matrices
• Functional central limit theorems for multivariate Bessel processes in the freezing regime
• Limit theorems for Jacobi ensembles with large parameters

Cited by in corpus (5)

• Limit theorems for Jacobi ensembles with large parameters
• Freezing Limits for Beta-Cauchy Ensembles
• Hua-Pickrell diffusions and differential equations related with pseudo-Jacobi polynomials
• The Tracy-Widom distribution at large Dyson index
• On the extremal eigenvalues of Jacobi ensembles at zero temperature