非厄米三分类中的微观参数关联与谱刚性

中文摘要 AI 辅助

我们研究了三个非厄米高斯体类中两个邻近参数值之间的微观谱关联：$A$类（无约束）、$AI^\dagger$类（复对称）和$AII^\dagger$类（复自对偶，每个简并双态计为一次）。参数依赖性通过平稳矩阵Ornstein-Uhlenbeck演化建模，其微观时间尺度为$N^{-1}$。对于每个正整数副本数$n$，我们推导了联合特征多项式矩的精确有限$N$辅助场积分表示，并在$N\to\infty$且$n$固定时获得其体渐近行为。将谱归一化到单位圆盘后，在体点$z_0$附近，时间$t_1,t_2$处谱位置$z_1,z_2$之间的关联仅依赖于$s=N|z_1-z_2|^2+N(1-|z_0|^2)|t_1-t_2|$。采用静态厄米/非厄米副本延拓可得到显式的双时核；对于$AI^\dagger$和$AII^\dagger$类，这些是副本猜想，并且据我们所知，是这些类别中首次提出的微观双时核。它们决定了密度关联、跨时间特征值计数协方差和等时数方差。对于平均特征值计数为$y$的圆盘，后者满足$\mbox{Var}\mathcal N_X(D_y)=\kappa_X\sqrt y+ \beta_X/\sqrt y+o(y^{-1/2})$，其中每个类别的常数均明确给出。精确微扰理论进一步将特征向量非正交性识别为短时谱扩散的机制，并为$AII^\dagger$类中的二维Kramers特征空间导出了基无关的条件数。直接模拟与其猜想的逆伽马定律定量一致。

英文摘要

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 $\mbox{Var}\mathcal N_X(D_y)=κ_X\sqrt y+ β_X/\sqrt y+o(y^{-1/2})$, 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 $AII^\dagger$. Direct simulations agree quantitatively with its conjectured inverse-gamma law.

发表机构

• King’s College London（伦敦国王学院）

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