We consider the complex eigenvalues of a Wishart type random matrix model \(X=X_1 X_2^*\), where two rectangular complex Ginibre matrices \(X_{1,2}\) of size \(N\times (N+\nu )\) are correlated through a non-Hermiticity parameter \(\tau \in [0,1]\). For general \(\nu =O(N)\) and \(\tau \), we obtain the global limiting density and its support, given by a shifted ellipse. It provides a non-Hermitian generalisation of the Marchenko–Pastur distribution, which is recovered at maximal correlation \(X_1=X_2\) when \(\tau =1\). The square root of the complex Wishart eigenvalues, corresponding to the nonzero complex eigenvalues of the Dirac matrix \(\mathcal {D}=\begin{pmatrix} 0 &{} X_1 \\ X_2^* &{} 0 \end{pmatrix},\) are supported in a domain parametrised by a quartic equation. It displays a lemniscate type transition at a critical value \(\tau _c,\) where the interior of the spectrum splits into two connected components. At multi-criticality, we obtain the limiting local kernel given by the edge kernel of the Ginibre ensemble in squared variables. For the global statistics, we apply Frostman’s equilibrium problem to the 2D Coulomb gas, whereas the local statistics follows from a saddle point analysis of the kernel of orthogonal Laguerre polynomials in the complex plane.

我们考虑Wishart型随机矩阵模型\（X = X_1 X_2 ^ * \）的复特征值，其中两个矩形复数Ginibre矩阵\（X_ {1,2} \）的大小为\（N \ times（N + \ nu ）\）通过非赫米特参数\（\ tau \ in [0,1] \）进行关联。对于一般\（\ nu = O（N）\）和\（\ tau \），我们获得了整体极限密度及其支持，由移动的椭圆给出。它提供了Marchenko–Pastur分布的非Hermitian概括，当\（\ tau = 1 \）时，可以在最大相关\（X_1 = X_2 \）的情况下进行恢复。。复数Wishart特征值的平方根，对应于Dirac矩阵的非零复数特征值\（\ mathcal {D} = \ begin {pmatrix} 0＆{} X_1 \\ X_2 ^ *＆{} 0 \ end {pmatrix }，\）在由四次方程参数化的域参数中受支持。它在临界值\（\ tau _c，\）处显示出双体类型转换，其中频谱内部分为两个相连的分量。在多临界情况下，我们获得平方变量中Ginibre集合的边缘核所给定的有限局部核。对于全局统计，我们将Frostman平衡问题应用于二维库仑气体，而局部统计则来自对复杂平面中正交Laguerre多项式的核的鞍点分析。