a regular point inside the bulk. Thus, we expect that $F(s;\rho)$ might be related to the determinant of (generalized) sine kernel under certain scaling limits when $\rho  \to -\infty$, from which the constant term can be derived. We will leave this issue to a future publication.

Finally, we note that the asymptotics of Fredholm determinant associated with the Pearcey kernel is also investigated from the viewpoint of phase transition in [@ACV; @BC1], i.e., to show how the Pearcey process becomes an Airy process by sending both $s$ and the parameter $\rho$ to positive infinity. We emphasize the asymptotic results therein are essentially different from ours.

The rest of this paper is devoted to the proof of Theorem \[main-thm\]. We mainly follow the general strategy established in [@Bor:Dei2002; @DIZ97]. In Section \[sec:DiffIdentity\], we relate the partial derivatives of $F(s;\rho)$ to a $3 \times 3$ RH problem with constant jumps, which is essential in the proof. After introducing some auxiliary functions defined on a Riemann surface with a specified sheet structure in Section \[sec:auxiliary function\], we then perform a Deift-Zhou steepest descent analysis [@DZ93] on this RH problem for large positive $s$ in Section \[sec:asymanalyX\]. This asymptotic outcome, together with the differential 