Correlation kernels for sums and products of random matrices. (English) Zbl 1330.15040
Summary: Let \(X\) be a random matrix whose squared singular value density is a polynomial ensemble. We derive double contour integral formulas for the correlation kernels of the squared singular values of \(GX\) and \(TX\), where \(G\) is a complex Ginibre matrix and \(T\) is a truncated unitary matrix. We also consider the product of \(X\) and several complex Ginibre/truncated unitary matrices. As an application, we derive the precise condition for the squared singular values of the product of several truncated unitary matrices to follow a polynomial ensemble. We also consider the sum \(H+M\) where \(H\) is a Gaussian unitary ensemble matrix and \(M\) is a random matrix whose eigenvalue density is a polynomial ensemble. We show that the eigenvalues of \(H+M\) follow a polynomial ensemble whose correlation kernel can be expressed as a double contour integral. As an application, we point out a connection to the two-matrix model.
MSC:
| 15B52 | Random matrices (algebraic aspects) |
| 60B20 | Random matrices (probabilistic aspects) |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 33C60 | Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions) |
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
Keywords:
sums of random matrices; products of random matrices; complex Ginibre matrix; truncated unitary matrix; determinantal point processes; polynomial ensembles; Meijer G-functions; squared singular value density; eigenvalue; Gaussian unitary ensembleSoftware:
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