\begin{split}
\rho(z)=\frac{3}{2\pi}\left|\left(\frac{r(z)+\sqrt{-\frac{1}{27a^4z^4}D_3(z)}}{2}\right)^{\frac{1}{3}}-\left(\frac{r(z)-\sqrt{-\frac{1}{27a^4z^4}D_3(z)}}{2}\right)^{\frac{1}{3}}\right|,
\end{split}$$ where $D_3(z)$ and $r(z)$ are given by $$\begin{split}
D_3(z)&=(1-a)^2\prod_{j=1}^4(z-\lambda_j),\\
r(z)&=\frac{1}{27}\Bigg(-\frac{2B_2^3}{a^3}z^{-3}+\left(\frac{9B_1B_2}{a^2}-\frac{6A_2B_2^2}{a^3}\right)z^{-2}+\left(\frac{9B_2}{a^2}+\frac{9B_1A_2}{a^2}-\frac{27}{a}-\frac{6A_2^2B_2}{a^3}\right)z^{-1}\\
&+\left(\frac{9A_2}{a^2}-\frac{2A_2^3}{a^3}\right)\Bigg).
\end{split}$$ The constants $A_1$, $B_1$ and $B_2$ in the above equation are defined by $$A_2=(1+a),\quad B_2=a(1-c),\quad B_1=1-c(1-\beta)+a(1-c\beta).$$ The cube root in (\[eq:density\]) is chosen such that $\sqrt[3]{A}\in\mathbb{R}$ for $A\in\mathbb{R}$ and the square root is chosen such that $\sqrt{A}>0$ for $A>0$.

\[re:sqrt\] Since $D_3(z)$ can be written as $$\frac{D_3(z)}{a^4z^4}=-27\left(r(z)\right)^2-4\left(p(z)\right)^3,$$ for some polynomial $p(z)$ in $z^{-1}$. We see that if $r(z)$ vanishes at any of the $\lambda_k$, then $D_3(z)$ will have at least a double root at these points, hence $r(\lambda_k)\neq0$. From this and (\[eq:rho\]), we see that the density $\rho(z)$ vanishes like a square root at the points $\lambda_k$, $k=1,2$. $$\label{eq:rhok}
\rho(z)=\frac{\rho_k}{\pi}|z-\lambda_k|^{\frac{1}{2}}+O\left((z-\lambda_k)\right),\quad
z\rightarrow\lambda_k.$$

An open problem in the studies of Wishart ensembles is the universality and the distribution of the largest eigenvalue. Although the Wishart distribution is known for a long time, results in the universality and the largest eigenvalue distribution were only obtained recently and only for spiked models [@J] whose covariance matrices are finite perturbations of the identity matrix [@BaikD], [@Baik95], [@Baikspike], [@DF], [@El03], [@Fo93], [@J], [@Jo], [@W1], [@W2]. The result in this paper is one of the few results obtained for models with covariance 