稀疏区域中几何泛函的适度偏差与迭代对数律
Moderate deviations and laws of the iterated logarithm for geometric functionals in the sparse regime
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- Joint Graduate School of Mathematics for Innovation, Kyushu University（九州大学数学创新联合研究生院）
- Faculty of Mathematics, Kyushu University（九州大学数学系）
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中文总结 AI 辅助
本文证明了稀疏区域中几何泛函的适度偏差原理和迭代对数律，并应用于随机几何图、Čech复形及Morse临界点的计数。
中文摘要 AI 辅助
我们证明了稀疏区域中与$k$点连通分量相关的几何和拓扑泛函的适度偏差原理和迭代对数律。这些极限定理在测度值和向量值两个层面上建立，适用于泊松和二项点过程。作为应用，我们推导了随机几何图和Čech复形中分量计数以及Morse临界点计数的相应结果。
英文摘要
We prove moderate deviation principles and laws of the iterated logarithm in the sparse regime for geometric and topological functionals associated with $k$-point connected components. These limit theorems are established at both the measure-valued and vector-valued levels, for Poisson and binomial point processes. As applications, we derive corresponding results for component counts in random geometric graphs and Čech complexes and for counts of Morse critical points.
