$q$ in are solutions of the third order differential equations $$\begin{aligned}
p'''(x)&=xp(x)+\rho p'(x), \label{eq:Pearcey1}
\\
q'''(y)&=-yq(y)+\rho q'(y),\end{aligned}$$ respectively. Since $p$ and $q$ were first introduced by Pearcey in the context of electromagnetic fields [@Pear], the kernel $K^{\mathrm{Pe}}$ bears the name Pearcey kernel. To see how $K^{\mathrm{Pe}}$ describe the aforementioned phase transition, note that the eigenvalues of $M$ form a determinantal point process with a correlation kernel $K_n(x,y;a)$ depending on $a$ (see [@BH; @BH1; @Zinn]), it was established in [@BH; @BH1] (for $\rho=0$) and in [@BK3; @TW] (for general $\rho \in \mathbb{R}$) that $$\lim_{n\to\infty}\frac{1}{n^{3/4}}K_n\left(\frac{x}{n^{3/4}},\frac{y}{n^{3/4}}; 1+\frac{\rho}{2\sqrt{n}}\right)=K^{\mathrm{Pe}}(x,y;\rho),$$ i.e., the correlation kernel $K_n$ converges to the Pearcey kernel near the origin as $n\to \infty$ in a double scaling regime.

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Like the classical kernels (sine kernel and Airy kernel) arising from random matrix theory [@Forrester; @metha], the Pearcey kernel is a universal object as evidenced by its appearance in a variety of stochastic models. On one hand, the Pearcey statistics have been established in specific matrix models including large complex correlated Wishart matrices [@HHNa; @HHNb], a two-matrix model with special quartic potential [@GZ], and quite recently for general complex Hermitian Wigner-type 