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Probing non-orthogonality of eigenvectors in non-Hermitian matrix models: diagrammatic approach. (English) Zbl 1395.81159

Summary: Using large \(N\) arguments, we propose a scheme for calculating the two-point eigenvector correlation function for non-normal random matrices in the large \(N\) limit. The setting generalizes the quaternionic extension of free probability to two-point functions. In the particular case of biunitarily invariant random matrices, we obtain a simple, general expression for the two-point eigenvector correlation function, which can be viewed as a further generalization of the single ring theorem. This construction has some striking similarities to the freeness of the second kind known for the Hermitian ensembles in large \(N\). On the basis of several solved examples, we conjecture two kinds of microscopic universality of the eigenvectors – one in the bulk, and one at the rim. The form of the conjectured bulk universality agrees with the scaling limit found by Chalker and Mehlig [J. T. Chalker and B. Mehlig, “Eigenvector statistics in non-Hermitian random matrix ensembles”, Phys. Rev. Lett. 81, 3367 (1998; doi:10.1103/PhysRevLett.81.3367; arXiv:cond-mat/9809090] in the case of the complex Ginibre ensemble.

MSC:

81T10 Model quantum field theories
62P35 Applications of statistics to physics

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References:

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