黎曼流形上随机几何图的边界矩普适性与曲率修正
Boundary-Moment Universality and Curvature Corrections in Random Geometric Graphs on Riemannian Manifolds
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中文总结 AI 辅助
针对黎曼流形上随机几何图，推导对称三顶点边指示统计量的二阶展开，构造内在泛函的一致估计量并证明中心极限定理，还得到最大度阈值半径的对数生存律修正项。
中文摘要 AI 辅助
设$(M,g)$为光滑、闭、连通的$d$维黎曼流形，$X_1,\boldsymbol{\rmellipsis},X_n$为服从公共分布$f\boldsymbol{\rm dvol}_g$的独立同分布样本，其中$f \boldsymbol{\rmellipsis} C^4(M)$且严格为正。我们推导了支持连通构型的对称三顶点边指示统计量的一致内在二阶展开，包括诱导路径核与三角形核。二阶项分离了密度变化、法坐标雅可比行列式以及由曲率诱导的内弦边界运动。在该三顶点连通对称类中，普适的边界矩恒等式将依赖于核的贡献简化为包含$\boldsymbol{\rmint}_M f\boldsymbol{\rm}\boldsymbol{\rmgrad} f\boldsymbol{\rm}\boldsymbol{\rm}_g^2\boldsymbol{\rm dvol}_g$和$\boldsymbol{\rmint}_M f^3\boldsymbol{\rm Scal}_g\boldsymbol{\rm dvol}_g$的共同内在泛函。对归一化的路径-三角形对比，欧氏主导项相互抵消。我们构造了该内在泛函的一致估计量，在更密的带宽 regime 中，通过一阶 Hoeffding 投影证明了以精确期望为中心的根-$n$中心极限定理。对闭曲面上的均匀采样，该估计量可一致恢复欧拉示性数。我们还研究了二项随机几何图的最大度首次达到2时的阈值半径，利用活动三重强度展开和依赖图泊松近似，得到$d>6$时对数生存律的阶为$n^{-3/d}$的修正项。
英文摘要
Let $(M,g)$ be a smooth, closed, connected $d$-dimensional Riemannian manifold, and let $X_1,\ldots,X_n$ be i.i.d.\ with common law $f\,d\mathrm{vol}_g$, where $f\in C^4(M)$ is strictly positive. We derive a uniform intrinsic second-order expansion for symmetric three-vertex edge-indicator statistics supported on connected configurations, including the induced-path and triangle kernels. The second-order term separates density variation, normal-coordinate Jacobians, and the curvature-induced motion of the internal-chord boundary. Within this three-vertex connected symmetric class, a universal boundary--moment identity reduces the kernel-dependent contribution to a common intrinsic functional involving $\int_M f\|\grad f\|_g^2\,d\mathrm{vol}_g$ and $\int_M f^3\operatorname{Scal}_g\,d\mathrm{vol}_g$. For the normalized path--triangle contrast, the Euclidean leading term cancels. We construct a consistent estimator of this intrinsic functional and, in a denser bandwidth regime, we prove an exact-expectation-centered root-\(n\) central limit theorem via the first Hoeffding projection. For uniform sampling on a closed surface, the estimator consistently recovers the Euler characteristic. We also study the threshold radius at which the maximum degree of a binomial random geometric graph first reaches two. Using the active-triple intensity expansion and a dependency-graph Poisson approximation, we obtain the order-$n^{-3/d}$ correction to the log-survival law for $d>6$.
