Pfaffian structure of the eigenvector overlap for the symplectic Ginibre ensemble

TL;DR

This work advances the understanding of non-Hermitian eigenvector statistics in the symplectic Ginibre ensemble by establishing Pfaffian formulas for conditioned mean overlaps, expressed through planar skew-orthogonal polynomials with a specially perturbed weight $\omega^{(\mathrm{over})}$. The analysis proceeds via a two-step construction from a pre-weight $\omega^{(\mathrm{pre})}$ using Christoffel perturbation, yielding a finite-$N$ differential equation for the pre-kernel and a transposition-based link between diagonal and off-diagonal overlaps. The authors derive comprehensive large-$N$ asymptotics, including explicit bulk and edge limits for the conditional diagonal overlap and the associated eigenvalue correlation functions, culminating in universal limiting skew-kernels and densities along the real axis. Collectively, the results provide an integrable framework for planar skew-orthogonal polynomials in the GinSE and deliver concrete, scalable descriptions of eigenvector statistics with potential applications to quantum chaotic scattering and related Coulomb gas representations of non-Hermitian matrices.

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 $ω^{(\mathrm{over})}(z)=|z-\overline{x}|^2(1+|z-x|^2)e^{-2|z|^2}$, 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 $ω^{(\mathrm{over})}(z)$. This is done in two steps and involves a Christoffel perturbation of the weight $ω^{(\mathrm{over})}(z)=|z-\overline{x}|^2ω^{(\mathrm{pre})}(z)$, by computing first the corresponding quantities for the unperturbed weight $ω^{(\mathrm{pre})}(z)$. 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 $ω^{(\mathrm{over})}(z)$ are determined. We also examine the effect on the planar orthogonal polynomials when changing the variance in $ω^{(\mathrm{pre})}(z)$, as this appears in the eigenvector statistics of the complex Ginibre ensemble.

Figures (8)

• Figure 1: Bulk
• Figure 2: Edge
• Figure 4: bulk $R_1^{(\mathrm{over})}(x+iy)$ at $\chi=0.5$
• Figure 5: bulk $R_{N,1}^{(\mathrm{over})}(0.5+iy)$ at $\chi=0.5$
• Figure 6: bulk $R_{N,1}^{(\mathrm{over})}(-2+iy)$ at $\chi=0.5$

Theorems & Definitions (26)

• Lemma 2.1: Pfaffian structure of the mean diagonal overlap
• Lemma 2.2: Relation between mean off-diagonal overlap and mean diagonal overlap
• Theorem 2.3: Construction of SOPs associated with pre-overlap weight \ref{['PreOverlapWeight']}
• Proposition 2.4: SOPs and skew-kernel associated with overlap weight \ref{['OverlapWeight']} via Christoffel perturbation AEP22
• Proposition 2.5: Conditional mean diagonal overlap at finite-$N$
• Theorem 2.6: Differential equation for the skew-kernel with pre-overlap weight \ref{['PreOverlapWeight']}
• Theorem 2.7: Scaling limits of the conditional expectation of the diagonal overlap
• Theorem 2.8: Scaling limits of eigenvalue correlation functions
• Proposition 3.1
• proof : Proof of Proposition \ref{['prop_pre_originkernel_Limit']}