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, Phys. Rev. Lett. 81 (1998) 3367] in the case of the complex Ginibre ensemble.
Highlights
Most of the studied properties of non-normal random operators dealt with the eigenvalues
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
In this work we focus on statistical ensembles of complex non-Hermitian matrix models, the probability density of which is invariant under the action of the unitary group P (X) = P (U XU †)
Summary
)−1 does not yield the correct result inside the spectrum, as one would naively expect. The reason for this failure is that differentiation and taking the ensemble average are not interchangeable. This phenomenon was termed the spontaneous breaking of holomorphic symmetry [42]. In. The expression on the right hand side provides a prescription for how the resolvent in the spectrum of X should be regularized. The expression on the right hand side provides a prescription for how the resolvent in the spectrum of X should be regularized Having this hint in mind, one defines g(z, z, w, w) =. Is the (regularized) electrostatic potential of charges interacting via repulsive central force
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
