Microscopic Parametric Correlations and Spectral Rigidity in the Non-Hermitian Threefold Way

The gist: This study derives exact finite-N auxiliary-field integral representations for joint characteristic-polynomial moments in three non-Hermitian Gaussian bulk classes, yielding microscopic two-time kernels that govern spectral correlations and rigidity in these systems.

Microscopic Parametric Correlations

The research investigates microscopic spectral correlations between two nearby parameter values in the three non-Hermitian Gaussian bulk classes: A (unconstrained), AI† (complex symmetric), and AII† (complex self-dual). Parameter dependence is modeled by stationary matrix Ornstein–Uhlenbeck evolution, whose microscopic time scale is N −1. For every positive integer replica number n, exact finite-N auxiliary-field integral representations for joint characteristic-polynomial moments are derived and their bulk asymptotics as N → ∞ with n fixed are obtained. With the spectrum normalized to the unit disk, correlations near a bulk point z0, between spectral positions z1, z2 at times t1, t2, depend only on s = Nz1 − z22 + N(1 − z02) t1 − t2.

Key Results and Kernels

The study identifies explicit two-time kernels for AI† and AII†, which are replica conjectures, representing the first microscopic two-time kernels proposed for these classes. These kernels determine density correlations, cross-time eigenvalue-count covariances, and equal-time number variances. For a disk of mean eigenvalue count y, the latter obey VarNX(Dy) = κX√y + βX/√y + o(y −1/2), with specific relations like κAI† > κA > κAII†. Exact perturbation theory further identifies eigenvector nonorthogonality as the mechanism of short-time spectral diffusion and yields a basis-independent condition number for a two-dimensional Kramers eigenspace in class AII†.

Unified Counting Formalism

The central conjecture is the unified relation cXij(u) = d2du2/u2KX(u + a), where KX is the eigenvector-memory kernel obtained explicitly for all three symmetry classes in the generic spectral bulk. This relation describes how spectral separation and time enter universally through the single combination u + a. At equal times, it gives the static distinct-level correlation, while at distinct times it describes the progressive loss of spectral memory. The resulting formula provides an exact decomposition of the endpoint covariance into a singular mode and a regular remainder.

Rigidity Coefficients and Asymptotics

The leading long-range coefficient remembers the Hermitian Dyson index, as KX(s) ∼ 2/βXs2 for β = 1, 2, 4. The study establishes the rigidity coefficients κX = Z ∞0 u32KX(u) du and finds their values: κAI† ≈ 0.7142944933, κA ≈ 0.5641895835, and κAII† ≈ 0.4740979713. These coefficients are shown to be an integrated record of eigenvector memory.

Physical Interpretation and Checks

The study distinguishes between the endpoint quantity Ga, which depends only on the joint distribution of the two endpoint spectra, and the pathwise self intensity Sa, which requires matching eigenvalues along a chosen matrix evolution. The velocity calculation identifies the underlying physical mechanism of overlap-amplified matrix noise. Numerical checks confirm that for class AII†, the rescaled Kramers-plane overlap distribution converges to a specific inverse-gamma law, which is consistent with the conjectured limiting density.

Conclusion

The work establishes that the microscopic spectral diffusion in non-Hermitian systems is governed by eigenvector nonorthogonality. The derived kernels provide quantitative benchmarks for extensions of these theories to other ensembles and regimes, while the velocity calculation identifies the underlying physical mechanism of overlap-amplified matrix noise. This research provides a unified counting formalism with explicit subleading rigidity coefficients, and the intrinsic Kramers-plane diffusivity.

How it works

The analysis proceeds by modeling parameter dependence via stationary matrix Ornstein–Uhlenbeck evolution. The core of the derivation involves deriving exact finite-N auxiliary-field integral representations for joint characteristic-polynomial moments in the three non-Hermitian Gaussian bulk classes. These representations lead to microscopic two-time kernels, which are conjectured to be the first such kernels for AI† and AII†.

Key Components

1. The parameter separation enters through the additive shift u → u + a in the large-N saddle-point analysis.

2. The microscopic spectral separation is defined as s = u + a, where a = v2χ0.

3. The memory kernel KX(s) is determined by the second derivative of the replica generating function, KX(s) = d2ΦX(s)/ds2.

4. The three kernels are explicitly defined as KA(s), KAI† (s), and KAII† (s).

5. The kernel asymptotics at large parabolic separation show KX(s) ∼ 2/(βXs2) for the Hermitian class Dyson index βX.

Observables

The derived formulas allow for the calculation of density correlations, cross-time eigenvalue-count covariances, and equal-time number variances. The disk covariance is expressed as an Abel transform (9.5), which relates it to the kernel KX(u + a). The study also provides the intrinsic Kramers-plane diffusivity, which is related to the overlap distribution law.

References

[1] G. Akemann, G. Angermann, N. Aygün, A. Mielke, P. Paßler, C. Raitzig, and T. Winkler

[2] G. Akemann, N. Aygün, M Kieburg, and P Paßler

[3] G Akemann, Y V Fyodorov, and D V Savin

[4] G Akemann, M Kieburg, A Mielke, and T Prosen

[5] C W J Beenakker, Brownian-motion model for parametric correlations in the spectra of disordered metals

[6] C W J Beenakker and B Rejaei, Random-matrix theory of parametric correlations in the spectra of disordered metals and chaotic billiards

[7] F A Berezin, Introduction to Superanalysis, D Reidel, Dordrecht

[8] M V Berry and J P Keating, Persistent current flux correlations calculated by quantum chaology

[9] P Bertelsen, C Ellegaard, T Guhr, M Oxborrow, and K Schaadt

[10] O Bohigas, M-J Giannoni, and C Schmit

[11] P Bourgade, G Cipolloni, and J Huang

[12] P Bourgade and G Dubach, The distribution of overlaps between eigenvectors of Ginibre matrices

[13] S-S Byun and P J Forrester, Progress on the Study of the Ginibre Ensembles, KIAS Springer Series in Mathematics, Vol. 3, Springer, Singapore

[14] J T Chalker and B Mehlig, Eigenvector statistics in non-Hermitian random matrix ensembles

[15] Z Chen, Z Xiao, Y Liu, and S Ryu, Duality between the level statistics of Hermitian and non-Hermitian random matrices

[16] N Cuppone and Y V Fyodorov, Right-eigenvector-resolved parametric correlations in the complex Ginibre ensemble

[17] G Dubach, Symmetries of the quaternionic Ginibre ensemble, Random Matrices Theory Appl.

1. Enhanced spectral statistics modeling for non-Hermitian systems: The AI system can now derive exact finite-N auxiliary-field integral representations for joint characteristic-polynomial moments across classes A, AI†, and AII†. This allows it to predict bulk asymptotics as N approaches infinity with fixed replica number n.

2. Microscopic spectral diffusion prediction: The system can calculate the two-time kernels for classes AI† and AII†, which are described as replica conjectures for the first time they were proposed in this work. These kernels determine density correlations, cross-time eigenvalue-count covariances, and equal-time number variances.

3. Eigenvector nonorthogonality identification: The AI system can identify eigenvector nonorthogonality as the mechanism which governs short-time spectral diffusion, a key physical insight for understanding how matrix noise amplifies in these systems.

4. Class-dependent rigidity coefficient calculation: The system can calculate the leading long-range coefficients of the spectral correlations, such as kappaA = 1/sqrt(pi) and kappaAII† = 3/sqrt(2) - log(1 + sqrt(2))/4, which are derived from eigenvector memory.

5. Disk-overlap covariance calculation: The system can compute the disk-count covariance using the disk Abel transform formula (Equation 9.5), relating it directly to the microscopic kernel KX(u + a). This provides a rigorous tool for analyzing eigenvalue counts within spectral windows of radius R.

6. Kramers-plane overlap law prediction: For classes AI† and AII†, the system can predict the limiting distribution of the rescaled Kramers-plane overlap mK using Conjecture 7.2, which states that Z infinity 0 m minus 1 e minus x over m mu dm equals 2 (1 + x) over m cubed for x greater than or equal to zero.

7. Finite-N consistency checking: The system can perform finite-N numerical sampling and compare the resulting disk-count covariances against the kernel predictions, providing a quantitative check for the proposed parameter normalization and class-dependent results.

We study microscopic spectral correlations between two nearby parameter values in the three non-Hermitian Gaussian bulk classes: A (unconstrained), AI (complex symmetric), and AII (complex self-dual, with each degenerate doublet counted once). Parameter dependence is modelled by stationary matrix Ornstein--Uhlenbeck evolution, whose microscopic time scale is N-1. For every positive integer replica number n, we derive exact finite- N auxiliary-field integral representations for joint characteristic-polynomial moments and obtain their bulk asymptotics as N to infinity with n fixed. With the spectrum normalized to the unit disk, correlations near a bulk point z 0, between spectral positions z 1,z 2 at times t 1,t 2, depend only on s=Nz 1-z 2 2+N(1-z 0 2)t 1-t 2. Adopting the static Hermitian/non-Hermitian replica continuation yields explicit two-time kernels; for AI and AII these are replica conjectures and, to our knowledge, the first microscopic two-time kernels proposed for these classes. They determine density correlations, cross-time eigenvalue-count covariances, and equal-time number variances. For a disk of mean eigenvalue count y, the latter obey Var N X(D y)=κ X sqrt y+ β X/sqrt y+o(y-1/2), with explicitly given constants for each class. Exact perturbation theory further identifies eigenvector nonorthogonality as the mechanism of short-time spectral diffusion and yields a basis-independent condition number for a two-dimensional Kramers eigenspace in class AII. Direct simulations agree quantitatively with its conjectured inverse-gamma law.