发表机构
Zewail City of Science and Technology（齐维尔科学与技术城）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出用柯西核作为谱探针，通过Pick矩阵正定性和dressed鞍点约束，解析推导四次和六次矩阵模型的平面结果，揭示矩空间中填充分数几何与法向结构。
AI 中文摘要
厄米单矩阵模型的平面圈方程留下有限多个低阶矩未确定。Hankel正定性约束了这些矩，但并未揭示固定的多割族如何嵌入矩空间。我们引入一种使用柯西核$f_n(x)=(z_n-x)^{-1}$的有限平面诊断方法，并将其用作谱探针。它们的Gram矩阵是半正定的Pick矩阵$P_{mn}$，其正定性定义了一个类似于Hankel自举的有限节点自举。对于固定的正则多割拓扑，谱判别式的非分支双零点（我们称之为“ dressed鞍点”）对相应的矩空间轨迹施加横向约束，而其剩余方向则是填充分数。记$z=E+i\eta$，$E$选择谱区域，$\eta>0$设定连续分辨率尺度。我们将$P(z,z)$用作有限平面谱响应，并将$\nabla_{\mathbf m}P(z,z)$解释为矩空间磁化率。在正则实dressed鞍点附近，该磁化率以$\eta^{-2}$的阶增强，并与填充分数流形的法向对齐，而沿切向形变的领先增强则被抵消。因此，多个独立鞍点附近的响应可以重构法向空间，并通过其公共核重构切向空间。将Pick正定性与预解式的实性和解析性相结合，我们在四次和六次模型中获得了已知平面结果的替代解析推导。非对称四次模型使一维填充分数几何变得显式，而六次模型则表现出独立的法向方向。这些结果阐明了在全局周期匹配和平衡变分不等式选择平衡测度之前，局部一致性条件如何约束低阶矩。
英文摘要
The planar loop equations of a Hermitian one-matrix model leave finitely many low moments undetermined. Hankel positivity constrains these moments but does not reveal how a fixed multicut family is embedded in moment space. We introduce a finite-plane diagnostic using the Cauchy kernels $f_n(x)=(z_n-x)^{-1}$ which we use as spectral probes. Their Gram matrix is the positive-semidefinite Pick matrix $P_{mn}$, whose positivity defines a finite-node bootstrap analogous to the Hankel bootstrap. For a fixed regular multicut topology, the nonbranching double zeros of the spectral discriminant, which we call \emph{dressed saddles}, impose transverse constraints on the corresponding moment-space locus, while its remaining directions are filling fractions. Writing $z=E+iη$, $E$ selects a spectral region and $η>0$ sets a continuous resolution scale. We use $P(z,z)$ as a finite-plane spectral response and interpret $\nabla_{\mathbf m}P(z,z)$ as a moment-space susceptibility. Near a regular real dressed saddle, this susceptibility is enhanced as $η^{-2}$ and aligns with a conormal to the filling-fraction manifold, whereas the leading enhancement cancels along tangent deformations. Responses near several independent saddles can therefore reconstruct the conormal space and, through their common kernel, its tangent space. Combining Pick positivity with reality and analyticity of the resolvent, we obtain alternative analytic derivations of known planar results in quartic and sextic models. The asymmetric quartic makes the one-dimensional filling-fraction geometry explicit, while sextic models exhibit independent conormal directions. These results clarify how local consistency conditions constrain the low moments before global period matching and the equilibrium variational inequality select the equilibrium measure.
Comments53 pages, 1 figure; accompanying code and numerical data available at https://doi.org/10.5281/zenodo.22959518
