On the Pearcey Determinant: Differential Equations and Asymptotics
报告题目(Title)：On the Pearcey Determinant: Differential Equations and Asymptotics
报告人(Speaker)：张仑 教授（复旦大学数学科学学院）
地点(Place)：腾讯会议：453 850 0147
时间(Time)：11月10日(周三)，下午13:30-14:30
邀请人(Inviter)：王灯山
报告摘要
The Pearcey kernel is a classical and universal kernel arising from random matrix theory, which describes the local statistics of eigenvalues when the limiting mean eigenvalue density exhibits a cusp-like singularity. It appears in a variety of statistical physics models beyond matrix models as well. In this talk, we are concerned with the Fredholm determinant $\det\left(I-\gamma K^{\mathrm{Pe}}_{s,\rho}\right)$, where $0 \leq \gamma \leq 1$ and $K^{\mathrm{Pe}}_{s,\rho}$ stands for the trace class operator acting on $L^2\left(-s, s\right)$ with the Pearcey kernel. We establish an integral representation of the Pearcey determinant involving the Hamiltonian associated with a family of special solutions to a system of nonlinear differential equations and obtain asymptotics of this determinant as $s\to +\infty$, which is also interpreted as large gap asymptotics in the context of random matrix theory. It comes out that the Pearcey determinant exhibits a significantly different asymptotic behavior for $\gamma=1$ and $0<\gamma<1$, which suggests a transition will occur as the parameter $\gamma$ varies. Based on joint works with Dan Dai and Shuai-Xia Xu.
主讲人简介
张仑，复旦大学数学科学学院教授，博士生导师。研究方向为随机矩阵理论，Riemann-Hilbert方法与渐进分析，特殊函数与正交多项式等。迄今已在Comm. Pure Appl. Math.，Comm. Math. Phys.，Numer. Math.，SIAM J. Math. Anal.等国际重要学术期刊发表学术论文30余篇。主持包括优秀青年基金在内的多项国家自然科学基金，并入选上海高校特聘教授（东方学者）及其跟踪计划。
