Pfaffian structure of the eigenvector overlap for the symplectic Ginibre ensemble

Abstract

We study the integrable structure and scaling limits of the conditioned eigenvector overlap of the symplectic Ginibre ensemble of Gaussian non-Hermitian random matrices with independent quaternion elements. The average of the overlap matrix elements constructed from left and right eigenvectors, conditioned to $x$, are derived in terms of a Pfaffian determinant. Regarded as a two-dimensional Coulomb gas with the Neumann boundary condition along the real axis, it contains a kernel of skew-orthogonal polynomials with respect to the weight function ${\omega }^{(\mathrm{over})}(z)=|z-\overline{x}{|}^2(1+|z-x{|}^2)e^{-2|z{|}^2}$, including a non-trivial insertion of a point charge. The mean off-diagonal overlap is related to the diagonal (self-)overlap by a transposition, in analogy to the complex Ginibre ensemble. For $x$ conditioned to the real line, extending previous results at $x=0$, we determine the skew-orthogonal polynomials and their skew-kernel with respect to ${\omega }^{(\mathrm{over})}(z)$. This is done in two steps and involves a Christoffel perturbation of the weight ${\omega }^{(\mathrm{over})}(z)=|z-\overline{x}{|}^2{\omega }^{(\mathrm{pre})}(z)$, by computing first the corresponding quantities for the unperturbed weight ${\omega }^{(\mathrm{pre})}(z)$. Its kernel is shown to satisfy a differential equation at finite matrix size $N$. This allows us to take different large-$N$ limits, where we distinguish bulk and edge regime along the real axis. The limiting mean diagonal overlaps and corresponding eigenvalue correlation functions of the point processes with respect to ${\omega }^{(\mathrm{over})}(z)$ are determined. We also examine the effect on the planar orthogonal polynomials when changing the variance in ${\omega }^{(\mathrm{pre})}(z)$, as this appears in the eigenvector statistics of the complex Ginibre ensemble.

36 pages, 2 figures, final version published in Annales Henri Poincaré

References in corpus (34)

• Eigenvector statistics in non-Hermitian random matrix ensembles
• Random Matrices close to Hermitian or unitary: overview of methods and results
• Fluctuations of eigenvalues of random normal matrices
• Characteristic Polynomials of Complex Random Matrix Models
• Statistics of resonance width shifts as a signature of eigenfunction non-orthogonality
• Noninteracting fermions in a trap and random matrix theory
• Statistical properties of eigenvectors in non-Hermitian Gaussian random matrix ensembles
• Eigenvalue correlations in non-Hermitean symplectic random matrices
• The Complex Laguerre Symplectic Ensemble of Non-Hermitian Matrices
• Correlations of Eigenvectors for Non-Hermitian Random-Matrix Models
• On the statistics of resonances and non-orthogonal eigenfunctions in a model for single-channel chaotic scattering
• 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
• Experimental Width Shift Distribution: A Test of Nonorthogonality for Local and Global Perturbations
• Massive partition functions and complex eigenvalue correlations in Matrix Models with symplectic symmetry
• The random normal matrix model: insertion of a point charge
• Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem
• Analogies between random matrix ensembles and the one-component plasma in two-dimensions
• Probing non-orthogonality of eigenvectors in non-Hermitian matrix models: diagrammatic approach
• Condition numbers for real eigenvalues in the real Elliptic Gaussian ensemble
• A Note on Mixed Matrix Moments for the Complex Ginibre Ensemble
• Distribution of complex eigenvalues for symplectic ensembles of non-Hermitian matrices
• Full Dysonian dynamics of the complex Ginibre ensemble
• Universal eigenvector statistics in a quantum scattering ensemble
• Universal scaling limits of the symplectic elliptic Ginibre ensemble
• Skew-orthogonal polynomials in the complex plane and their Bergman-like kernels
• Symmetries of the Quaternionic Ginibre Ensemble
• Eigenvector statistics of the product of Ginibre matrices
• Strong Asymptotics of Planar Orthogonal Polynomials: Gaussian Weight Perturbed by Finite Number of Point Charges
• Universal eigenvector correlations in quaternionic Ginibre ensembles
• Wronskian structures of planar symplectic ensembles
• Fermionic Mapping For Eigenvalue Correlation Functions Of (Weakly) Non-Hermitian Symplectic Ensemble
• Explicit formulas concerning eigenvectors of weakly non-unitary matrices
• Condition numbers for real eigenvalues of real elliptic ensemble: weak non-normality at the edge