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Eigenvector correlations in the complex Ginibre ensemble. (English) Zbl 1498.60029

Summary: The complex Ginibre ensemble is the distribution of an \(N\times N\) non-Hermitian random matrix over \(\mathbb{C}\) with i.i.d. complex Gaussian entries normalized to have mean zero and variance \(1/N\). Unlike the Gaussian unitary ensemble, for which the eigenvectors are distributed according to Haar measure on the compact group \(U(N)\), independently of the eigenvalues, the geometry of the eigenbases of the Ginibre ensemble are not particularly well understood. In this paper we systematically study properties of eigenvector correlations in this matrix ensemble. In particular, we uncover an extended algebraic structure which describes their asymptotic behavior (as \(N\) goes to infinity). Our work extends previous results of J. T. Chalker and B. Mehlig [“Eigenvector statistics in non-Hermitian random matrix ensembles”, Phys. Rev. Lett. 81, No. 16 3367–3370 (1998; doi:10.1103/PhysRevLett.81.3367)], in which the correlation for pairs of eigenvectors was computed.

MSC:

60B20 Random matrices (probabilistic aspects)
60F99 Limit theorems in probability theory

Software:

Eigtool

References:

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