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

doi:10.1007/s10473-023-0418-7

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

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

Liangzhen LEI¹, Yutao MA^2,†

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 MA^2,†

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/Abstract

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

Liangzhen LEI, Yutao MA. LARGE DEVIATIONS FOR TOP EIGENVALUES OF ß -JACOBI ENSEMBLES AT SCALING TEMPERATURES^∗[J]. 数学物理学报(英文版), 2023, 43(4): 1767-1780.

Liangzhen LEI, Yutao MA. LARGE DEVIATIONS FOR TOP EIGENVALUES OF ß -JACOBI ENSEMBLES AT SCALING TEMPERATURES^∗[J]. Acta mathematica scientia,Series B, 2023, 43(4): 1767-1780.

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

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

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