\end{split}$$ the real roots of (\[eq:quartic1\]) are important in the determination of $\mathrm{Supp}(F)$. In particular, we have the following result. (See Theorem \[thm:cuts\])

\[thm:main1\] Let $\Delta$ be the discriminant of the quartic polynomial (\[eq:quartic1\]). If $\Delta<0$, then the support of $F$ consists of a single intervals.

In our previous paper [@Mo], we have shown that $\mathrm{Supp}(F)$ consists of 2 disjoint intervals if and only if $\Delta>0$ and have obtained the asymptotic eigenvalue statistics in that case. In this paper, we consider the case when $\Delta<0$ and together with [@Mo], we have proven the universality in this class of complex Wishart ensemble for all $\Delta\neq 0$. When $\Delta=0$, a phase transition takes place and the support of the eigenvalues splits into 2 disjoint intervals.

When $\Delta<0$, we also have the following expression for the density function of $F$ (See Theorem \[thm:density\]).

\[thm:side1\] Let $\Delta$ be the discriminant of the quartic polynomial (\[eq:quartic1\]). Suppose $\Delta<0$ and let $\gamma_1<\gamma_2$ be the 2 real roots to (\[eq:quartic1\]) and $\gamma_3$, $\gamma_4$ be the 2 complex roots. Let $\lambda_k$, $k=1,\ldots,4$ be the following $$\lambda_k=-\frac{1}{\gamma_k}+c\frac{1-\beta}{1+\gamma_k}+c\frac{a\beta}{1+a\gamma_k},\quad
k=1,\ldots,4.$$ Then all the $\lambda_k$ are distinct and the p.d.f $F$ is supported on $[\lambda_1,\lambda_2]$ with the following density $d
F(z)=\rho(z)dz$ $$\label{eq:rho}
