A non-Hermitian generalisation of the Marchenko-Pastur distribution: from the circular law to multi-criticality

Abstract

We consider the complex eigenvalues of a Wishart type random matrix model $X=X_1{X}_2^{\ast }$, 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 non-zero complex eigenvalues of the Dirac matrix $\mathcal{D}=\left(\begin{array}{cc}0& X_1\\ X_2^{\ast }& 0\end{array}\right),$ 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.

arXiv comment30 pages, 4 figures, v2: references added, typos corrected

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Cited by in corpus (8)

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• Universal scaling limits of the symplectic elliptic Ginibre ensemble
• Skew-orthogonal polynomials in the complex plane and their Bergman-like kernels
• Partition functions of determinantal and Pfaffian Coulomb gases with radially symmetric potentials
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