报告题目 (Title)：On the Pearcey Determinant: Differential Equations and Asymptotics（关于皮尔斯行列式:微分方程与渐近）

报告人 (Speaker)： 张仑 教授（复旦大学）

报告时间 (Time)：2022年7月13日 (周三) 15:00

报告地点 (Place)：腾讯会议（会议号：800-551-381）

邀请人(Inviter)：何卓衡

主办部门：理学院数学系

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.