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

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.

20 pages + 4 pages of references, 12 figs; v2: typos corrected, refs added; v3: more explanatory

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Cited by in corpus (15)

• The distribution of overlaps between eigenvectors of Ginibre matrices
• On statistics of bi-orthogonal eigenvectors in real and complex Ginibre ensembles: combining partial Schur decomposition with supersymmetry
• Condition numbers for real eigenvalues in the real Elliptic Gaussian ensemble
• From synaptic interactions to collective dynamics in random neuronal networks models: critical role of eigenvectors and transient behavior
• Full Dysonian dynamics of the complex Ginibre ensemble
• Top Eigenpair Statistics for Weighted Sparse Graphs
• Eigenfunction non-orthogonality factors and the shape of CPA-like dips in a single-channel reflection from lossy chaotic cavities
• Universal eigenvector correlations in quaternionic Ginibre ensembles
• Explicit formulas concerning eigenvectors of weakly non-unitary matrices
• Products of random matrices from fixed trace and induced Ginibre ensembles
• Computing the spectral action for fuzzy geometries: from random noncommutative geometry to bi-tracial multimatrix models
• Determinantal structure and bulk universality of conditional overlaps in the complex Ginibre ensemble
• Pfaffian structure of the eigenvector overlap for the symplectic Ginibre ensemble
• Condition numbers for real eigenvalues of real elliptic ensemble: weak non-normality at the edge
• A path integral approach to sparse non-Hermitian random matrices