matrices at the cusps [@EKS], where the requirement on the identical distribution in Wigner matrices is dropped. It is worthwhile to mention that, for Wigner-type matrices, the density of states exhibits only square root or cubic root cusp singularities; see the classification theorem in [@AjEK2017; @AlEK2018]. On the other hand, one also encounters the Pearcey kernel beyond matrix models, as can be seen from its connection with non-intersecting Brownian motions at cusps [@AOV; @AM; @BK3] and a combinatorial model on random partitions [@OR].

Let $K^{\mathrm{Pe}}_{s,\rho}$ be the trace class operator acting on $L^2\left(-s, s\right)$ with the Pearcey kernel , it is well-known that the associated Fredholm determinant $\det\left(I-K^{\mathrm{Pe}}_{s,\rho}\right)$ gives us the probability of finding no particles (also known as the gap probability) on the interval $(-s,s)$ in a determinantal point process on the real line characterized by the Pearcey kernel. Moreover, it is shown in [@AM; @BC1; @BH; @TW] that the gap probability satisfies some nonlinear differential equations under more general settings. Since one cannot evaluate the Fredholm determinant explicitly for any fixed $s$, a natural and fundamental question is then to ask for its large $s$ asymptotics, which will be the aim of the present work. Denote by $$\label{def:Fnotation}
