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arXiv · 1801.02526

Probing non-orthogonality of eigenvectors in non-Hermitian matrix models: diagrammatic approach

Maciej A. Nowak·Wojciech Tarnowski
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Abstract

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 [JT Chalker, B Mehlig, PRL, \textbf{81}, 3367 (1998)] in the case of the complex Ginibre ensemble.

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Maciej A. Nowak, Wojciech Tarnowski. 2018-01-08. Probing non-orthogonality of eigenvectors in non-Hermitian matrix models: diagrammatic approach. https://doi.org/10.1007/jhep06(2018)152

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