a phase transition for the eigenvalues of $M$ in the large $n$ limit. Indeed, by assuming the matrix $A$ is diagonal with two eigenvalues $a$ and $-a$ of equal multiplicity, it follows from the work of Pastur [@Pastur] that if $a>1$, the eigenvalues are distributed on two disjoint intervals, while for $0<a<1$, the eigenvalues are distributed on a single interval. In the critical case when $a\to 1$ as $n\to \infty$, the gap closes at the origin and the limiting mean eigenvalue density exhibits a cusp-like singularity, i.e., the density vanishes like $|x|^{1/3}$ as $x\to 0$. Upon letting $n\to\infty$ and after proper scaling, a new local eigenvalue process characterized by the so-called Pearcey kernel emerges near the origin.

The Pearcey kernel $K^{\mathrm{Pe}}$ is defined as (see [@BH; @BH1]) $$\begin{aligned}
\label{eq: pearcey kernel}
K^{\mathrm{Pe}}(x,y;\rho)&=\int_0^{\infty}p(x+z)q(y+z){\,\mathrm{d}}z
\nonumber
\\
&=\frac{p(x)q''(y)-p'(x)q'(y)+p''(x)q(y)-\rho
p(x)q(y)}{x-y},\end{aligned}$$ where $\rho\in\mathbb{R}$, $$\label{eq:pearcey integral}
p(x)=\frac{1}{2\pi}\int_{-\infty}^\infty
e^{-\frac14s^4-\frac{\rho}{2}s^2+isx} {\,\mathrm{d}}s \qquad \text{and} \qquad
q(y)=\frac{1}{2\pi} \int_\Sigma e^{\frac14
t^4+\frac{\rho}{2}t^2+ity} {\,\mathrm{d}}t.$$ The contour $\Sigma$ in the definition of $q$ consists of the four rays $\arg t=\pi/4,3\pi/4,5\pi/4,7\pi/4$, where the first and the third rays are oriented from infinity to zero while the second and the last rays are oriented outwards; see Figure \[fig: sigma\] for an illustration. The functions $p$ and 