数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (4): 1767-1780.doi: 10.1007/s10473-023-0418-7

• • 上一篇    下一篇

LARGE DEVIATIONS FOR TOP EIGENVALUES OF ß -JACOBI ENSEMBLES AT SCALING TEMPERATURES

Liangzhen LEI1, Yutao MA2,†   

  1. 1. School of Mathematical Science, Capital Normal University, Beijing 100048, China;
    2. School of Mathematical Sciences & Laboratory of Mathematics and Complex Systems of Ministry of Education, Beijing Normal University, Beijing 100875, China
  • 收稿日期:2022-04-02 修回日期:2022-10-22 发布日期:2023-08-08
  • 通讯作者: †Yutao MA, E-mail: mayt@bnu.edu.cn
  • 作者简介:Liangzhen, LEI E-mail: leiliangzhen@cnu.edu.cn
  • 基金资助:
    *Ma's research was supported by the NSFC (12171038, 11871008) and 985 Projects.

LARGE DEVIATIONS FOR TOP EIGENVALUES OF ß -JACOBI ENSEMBLES AT SCALING TEMPERATURES

Liangzhen LEI1, Yutao MA2,†   

  1. 1. School of Mathematical Science, Capital Normal University, Beijing 100048, China;
    2. School of Mathematical Sciences & Laboratory of Mathematics and Complex Systems of Ministry of Education, Beijing Normal University, Beijing 100875, China
  • Received:2022-04-02 Revised:2022-10-22 Published:2023-08-08
  • Contact: †Yutao MA, E-mail: mayt@bnu.edu.cn
  • About author:Liangzhen, LEI E-mail: leiliangzhen@cnu.edu.cn
  • Supported by:
    *Ma's research was supported by the NSFC (12171038, 11871008) and 985 Projects.

摘要: Let $\lambda=(\lambda_1, \cdots, \lambda_n)$ be $\beta$-Jacobi ensembles with parameters $p_1, p_2, n$ and $\beta,$ with $\beta$ varying with $n.$ Set $\gamma=\lim\limits_{n\rightarrow\infty}\frac{n}{p_1}$ and $\sigma=\lim\limits_{n\rightarrow\infty}\frac{p_1}{p_2}.$ Suppose that $\lim\limits_{n\to\infty}\frac{\log n}{\beta n}=0$ and $0\le \sigma\gamma< 1.$ We offer the large deviation for $\frac{p_1+p_2}{p_1}\max\limits_{1\le i\le n}\lambda_{i}$ when $\gamma>0$ via the large deviation of the corresponding empirical measure and via a direct estimate, respectively, when $\gamma=0.$

关键词: $\beta$-Jacobi ensemble, large deviation, Wachter law, extremal eigenvalue

Abstract: Let $\lambda=(\lambda_1, \cdots, \lambda_n)$ be $\beta$-Jacobi ensembles with parameters $p_1, p_2, n$ and $\beta,$ with $\beta$ varying with $n.$ Set $\gamma=\lim\limits_{n\rightarrow\infty}\frac{n}{p_1}$ and $\sigma=\lim\limits_{n\rightarrow\infty}\frac{p_1}{p_2}.$ Suppose that $\lim\limits_{n\to\infty}\frac{\log n}{\beta n}=0$ and $0\le \sigma\gamma< 1.$ We offer the large deviation for $\frac{p_1+p_2}{p_1}\max\limits_{1\le i\le n}\lambda_{i}$ when $\gamma>0$ via the large deviation of the corresponding empirical measure and via a direct estimate, respectively, when $\gamma=0.$

Key words: $\beta$-Jacobi ensemble, large deviation, Wachter law, extremal eigenvalue