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- W3017000404 abstract "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 $Ntimes (N+nu)$ are correlated through a non-Hermiticity parameter $tauin[0,1]$. 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 $mathcal{D}=begin{pmatrix} 0 & X_1 X_2^* & 0 end{pmatrix},$ 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." @default.
- W3017000404 created "2020-04-24" @default.
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- W3017000404 date "2020-10-29" @default.
- W3017000404 modified "2023-10-17" @default.
- W3017000404 title "A Non-Hermitian Generalisation of the Marchenko–Pastur Distribution: From the Circular Law to Multi-criticality" @default.
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- W3017000404 doi "https://doi.org/10.1007/s00023-020-00973-7" @default.
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