论文标题
几何和拓扑功能以及相关点过程的大偏差原理
Large deviation principle for geometric and topological functionals and associated point processes
论文作者
论文摘要
对于$ \ Mathbb r^d $，相对于连接性radii $ r_n \ to \ infty $，我们证明了与$ k $ - 元素连接的组件相关的点过程的大偏差原理。随机点是从均匀的泊松点过程中生成的，因此$（r_n）_ {n \ ge1} $满足$ n^kr_n^{d（k-1）} \ to \ infty $和$ nr_n^d \ to0 $ as as $ n \ to to \ nr_n^d \ to as $ n \ to \ n \ to \ infty $ infty $（i.e.e.e。获得的大偏差原理的速率函数可以表示为相对熵。作为一种应用，我们推断出出现在随机几何和拓扑中的各种功能和点过程的大偏差原理。作为拓扑不变的具体示例，我们考虑了持续的betti数量络合物数量以及最小距离函数的Morse临界点的数量。
We prove a large deviation principle for the point process associated to $k$-element connected components in $\mathbb R^d$ with respect to the connectivity radii $r_n\to\infty$. The random points are generated from a homogeneous Poisson point process, so that $(r_n)_{n\ge1}$ satisfies $n^kr_n^{d(k-1)}\to\infty$ and $nr_n^d\to0$ as $n\to\infty$ (i.e., sparse regime). The rate function for the obtained large deviation principle can be represented as relative entropy. As an application, we deduce large deviation principles for various functionals and point processes appearing in stochastic geometry and topology. As concrete examples of topological invariants, we consider persistent Betti numbers of geometric complexes and the number of Morse critical points of the min-type distance function.
