identities for $F(s;\rho)$, will finally lead to the proof of Theorem \[main-thm\], as presented in Section \[sec:proof\].

Differential identities for the Fredholm determinant {#sec:DiffIdentity}
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A Riemann-Hilbert characterization of the Pearcey kernel
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The starting point toward the proof of Theorem \[main-thm\] is an alternative representation of the Pearcey kernel $K^{\mathrm{Pe}}$ via a $3 \times 3$ RH problem, as shown in [@BK3] and stated next.

\[rhp: Pearcey\] We look for a $3 \times 3$ matrix-valued function $\Psi(z)=\Psi(z;\rho)$ satisfying

-   $\Psi(z)$ is defined and analytic in ${\mathbb{C}}\setminus
    \{\cup_{j=0}^5\Sigma_j \cup \{ 0 \} \}$, where $$\label{def:sigmai}
    \begin{aligned}
    &\Sigma_0=(0,+\infty), ~~ \Sigma_1=e^{\frac{\pi i}{4}}(0,+\infty), ~~\Sigma_2=e^{\frac{ 3 \pi i}{4}}(0,+\infty),
    \\
    &\Sigma_3=(-\infty, 0), ~~ \Sigma_4=e^{-\frac{3\pi i}{4}}(0,+\infty),~~ \Sigma_5=e^{-\frac{\pi i}{4}}(0,+\infty),
    \end{aligned}$$ with the orientations as shown in Figure \[fig:Pearcey\].

    (100,70)(-5,2) (40,40)[(-1,-1)[20]{}]{} (40,40)[(-1,1)[20]{}]{} (40,40)[(-1,0)[30]{}]{} (40,40)[(1,0)[30]{}]{} (40,40)[(1,1)[20]{}]{} (40,40)[(1,-1)[20]{}]{}

    (30,50)[(1,-1)[1]{}]{} (30,40)[(1,0)[1]{}]{} (30,30)[(1,1)[1]{}]{} (50,50)[(1,1)[1]{}]{} (50,40)[(1,0)[1]{}]{} (50,30)[(1,-1)[1]{}]{}

    (39,36.3)[$0$]{}

    (20,15)[$\Sigma_4$]{} (20,63)[$\Sigma_2$]{} (3,40)[$\Sigma_3$]{} (60,15)[$\Sigma_5$]{} (60,63)[$\Sigma_1$]{} (75,40)[$\Sigma_0$]{}

    (22,44)[$\Theta_2$]{} (22,34)[$\Theta_3$]{} (55,44)[$\Theta_0$]{} (55,34)[$\Theta_5$]{} (38,53)[$\Theta_1$]{} (38,22)[$\Theta_4$]{} (40,40)

-   For $z\in \Sigma_k$, $k=0,1,\ldots,5$, the limiting values $$\Psi_+(z) = \lim_{\substack{\zeta \to z \\\zeta\textrm{ on $+$-side of }\Sigma_k}}\Psi(\zeta), \qquad
       \Psi_-(z) = \lim_{\substack{\zeta \to z \\\zeta\textrm{ on $-$-side of }\Sigma_k}}\Psi(\zeta),$$ exist, where the $+$-side and $-$-side of $\Sigma_k$ are the sides which lie on the left and right of $\Sigma_k$, respectively, when traversing $\Sigma_k$ 