Limit theorems and soft edge of freezing random matrix models via dual orthogonal polynomials

英文摘要：

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