报告主题：酉随机矩阵系综在临界点附近的大间隙渐近性及其与可积偏微分方程的关系

主 讲 人：代丹教授（香港城市大学）

主 持 人：姚鹿鸣

时 间：2025年12月9日（二）10：00

地 点：致知楼701

嘉宾简介：

代丹，香港城市大学数学系教授，博士生导师。他的研究兴趣主要包括渐近分析，特殊函数，可积系统和随机矩阵等学术领域，在Advances in Mathematics，Communications in Mathematical Physics，SIAM Journal on Mathematical Analysis等国际重要期刊发表多篇论文，并主持多项香港研究资助局研究项目。

报告摘要：

In this talk, we are concerned with higher-order analogues of the Tracy-Widom distribution, which describe the eigenvalue distributions in unitary random matrix models near critical edge points. The associated kernels are constructed by functions related to the even members of the Painleve I hierarchy $\mathrm{P_{I}^{2k}}, k\in\mathbb{N}^{+}$, and are regarded as higher-order analogues of the Airy kernel. By performing a uniform asymptotic analysis on the associated Riemann-Hilbert problem, we derive the multiplicative constant in the large gap asymptotics of the distribution, resolving an open problem in the work of Clayes, Its and Krasovsky. In addition, we find the relation with integrable PDEs and derive the small and large time asymptotics of the solutions.

This is a joint work with Wen-Gao Long, Shuai-Xia Xu, Lu-Ming Yao and Lun Zhang.