AI 中文总结
本文证明Bobrowski-Skraba强普适性猜想在随机Čech持续同调中失效：通过局部代理和Palm-Mecke计算，发现二维和三维泊松过程的归一化持续强度二阶量不同，中心化经验定律收敛到不同极限，且不属于左偏Gumbel族。
AI 中文摘要
Bobrowski和Skraba的强普适性猜想断言，在规定的依赖于数据的加性中心化下，随机几何复形的对数-对数变换后持续比率经验定律，在采样模型、维度、过滤和同调度上共享一个普适极限。他们猜想了一个左偏Gumbel极限，并将其用作检验拓扑显著性的零假设。我们证明，当未中心化的经验定律和中心化具有确定性极限时，该猜想迫使每个模型的极限定律的二阶量（由接近1的比率决定且不受加性平移影响）在模型和维度间一致。为计算一次Čech持续同调的这一量，我们绕过全局持续配对，采用基于等价alpha过滤中边及其首次关联三角形的局部代理。在二维和三维情形下，Palm-Mecke计算精确评估了平稳泊松过程的代理强度，且两侧拓扑界表明其与真实归一化持续强度在二阶上一致。转移到有限样本后，该量分别等于单位正方形和单位立方体中独立均匀样本极限定律的$\pi^2/32$和$4/\pi^2-1/4$。中心化经验定律依概率弱收敛到不同的确定性极限，其均值定律（这些随机定律的期望）也弱收敛到这些极限。因此，即使对均值定律，猜想也失效。两个极限定律都不属于左偏Gumbel位置-尺度族。
英文摘要
The strong universality conjecture of Bobrowski and Skraba asserts that, under a prescribed data-dependent additive centering, the empirical laws of log-log transformed persistence ratios of random geometric complexes share a universal limit across sampling models, dimensions, filtrations, and homological degrees. They conjectured a left-skewed Gumbel limit and used it as a null law for testing topological significance. We show that, when the uncentered empirical laws and the centerings have deterministic limits, the conjecture forces a second-order quantity of each model's limiting law, determined by ratios close to one and unchanged by additive shifts, to agree across models and dimensions. To compute this quantity for degree-one Čech persistence, we bypass the global persistence pairing with a local proxy based on edges and their first incident triangles in the equivalent alpha filtration. In dimensions two and three, a Palm-Mecke calculation evaluates the proxy intensity exactly for stationary Poisson processes, and two-sided topological bounds show that it agrees with the true normalized persistence intensity through second order. After transfer to finite samples, the quantity equals $π^2/32$ and $4/π^2-1/4$ for the limiting laws of independent uniform samples from the unit square and the unit cube, respectively. The centered empirical laws converge weakly in probability to distinct deterministic limits, and their mean laws, the expectations of these random laws, also converge weakly to these limits. Hence the conjecture fails even for mean laws. Neither limiting law belongs to the left-skewed Gumbel location-scale family.
Comments47 pages, 1 figure
