Asymptotics of Fredholm Determinant Associated with the Pearcey Kernel

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. We consider the Fredholm determinant of a trace class operator acting on L²(-s, s) with the Pearcey kernel. Based on a steepest descent analysis for a 3 × 3 matrix-valued Riemann-Hilbert problem, we obtain asymptotics of the Fredholm determinant as s→ + ∞, which is also interpreted as large gap asymptotics in the context of random matrix theory.