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), $AI^\dagger$ (complex symmetric), and $AII^\dagger$ (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 $N\to\infty$ 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 $AI^\dagger$ and $AII^\dagger$ 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 $\mbox{Var}\mathcal N_X(D_y)=\kappa_X\sqrt y+ \beta_X/\sqrt y+o(y^{-1/2})$, 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 $AII^\dagger$. Direct simulations agree quantitatively with its conjectured inverse-gamma law.