论文标题
限制冻结随机矩阵模型的定理和软边
Limit theorems and soft edge of freezing random matrix models via dual orthogonal polynomials
论文作者
论文摘要
$ n $维贝塞尔和雅各比过程描述了与$ n $颗粒的交互粒子系统，并且与$β$ -HERMITE，$β$ -Laguerre和$β$ -Jacobi ensembles有关。 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.最近，作者以不同的方式得出了这些wlts，并以$σ_n^{ - 1} $的eigenValues和eigenVectors计算$σ_n^{ - 1} $，以及$σ_n$的特征。在本文中，我们使用这些数据和de boor和saff有限的双重正交多项式的理论来得出$σ_n$的公式，从$σ_n^{ - 1} $中，对于$β$ -Hermite和$β$ -Hermite和$β$-β$ -laguerre and-laguerre ememembles，我们的公式是简单的，而不是这些公式。我们使用这些多项式来得出在通风函数方面以$ n \ to \ infty $的冻结状态中软边的渐近结果。对于$β$ -HERMITE的合奏，我们的极限表达式与Dumitriu和Edelman的表达式不同。
$N$-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.
