Year: 2021
Type: article
Abstract: Consider Jacobi random matrix ensembles with the distributions $$c_{k_1,k_2,k_3}\prod_{1\leq i< j \leq N}\left(x_j-x_i\right)^{k_3}\prod_{i=1}^N \left(1-x_i\right)^{\frac{k_1+k_2}{2}-\frac{1}{2}}\left(1+x_i\right)^{\frac{k_2}{2}-\frac{1}{2}} dx$$ of the eigenvalues on the alcoves $$A:=\{x\in\mathbb R^N| \> -1\leq x_1\le ...\le x_N\leq 1\}.$$ For $(k_1,k_2,k_3)=κ\cdot (a,b,1)$ with $a,b>0$ fixed, we derive a central limit theorem for the distributions above for $κ\to\infty$. The drift and the inverse of the limit covariance matrix are expressed in terms of the zeros of classical Jacobi polynomials. We also rewrite the CLT in trigonometric form and determine the eigenvalues and eigenvectors of the limit covariance matrices. These results are related to corresponding limits for $β$-Hermite and $β$-Laguerre ensembles for $β\to\infty$ by Dumitriu and Edelman and by Voit.
Source: Tunisian Journal of Mathematics
Authors: Kilian Hermann, Michael Voit
Institution: TU Dortmund University
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2945445666