A non-Hermitian generalisation of the Marchenko-Pastur distribution: from the circular law to multi-criticality. (English) Zbl 1460.60004
Summary: 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.
MSC:
| 60B20 | Random matrices (probabilistic aspects) |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 76D27 | Other free boundary flows; Hele-Shaw flows |
| 81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
| 20G30 | Linear algebraic groups over global fields and their integers |
Software:
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