Microscopic Parametric Correlations and Spectral Rigidity in the Non-Hermitian Threefold Way
Abstract
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→∞ with n fixed. 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=N|z1−z2|2+N(1−|z0|2)|t1−t2|. 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 Var𝒩X(Dy)=κX√y+ βX/√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.
Related
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BibTeX
@misc{fyodorov2026microscopic,
title = {{Microscopic Parametric Correlations and Spectral Rigidity in the Non-Hermitian Threefold Way}},
author = {Yan V Fyodorov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13434},
eprint = {2609.13434},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13434},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Microscopic Parametric Correlations and Spectral Rigidity in the Non-Hermitian Threefold Way
AU - Yan V Fyodorov
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.13434
UR - https://arxiv.org/abs/2609.13434
AB - 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→∞ 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=N|z_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 Var𝒩_X(D_y)=κ_X√y+ β_X/√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.
ER -