»ã±¨±êÌâ (Title)£ºOn the Pearcey Determinant: Differential Equations and Asymptotics£¨¹ØÓÚƤ¶û˹ÐÐÁÐʽ:΢·Ö·½³ÌÓë½¥½ü£©

»ã±¨ÈË (Speaker)£º ÕÅÂØ ½ÌÊÚ£¨¸´µ©´óѧ£©

»ã±¨¹¦·ò (Time)£º2022Äê7ÔÂ13ÈÕ (ÖÜÈý) 15:00

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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.