Asymptotics of Fredholm Determinant Associated with the Pearcey Kernel

The Pearcey kernel is a classical and universal arising from random matrix theory, which describes the local statistics of eigenvalues when limiting mean eigenvalue density exhibits cusp-like singularity. It appears in variety statistical physics models beyond as well. We consider Fredholm determinant trace class operator acting on $L^2\left(-s, s\right)$ with kernel. Based steepest descent analysis for $3\times 3$ matrix-valued Riemann-Hilbert problem, we obtain asymptotics $s\to +\infty$, also interpreted large gap context theory.