We investigate some topological properties of random geometric complexes and random geometric graphs on Riemannian manifolds in the thermodynamic limit. In particular, for random geometric complexes we prove that the normalized counting measure of connected components, counted according to isotopy type, converges in probability to a deterministic measure. More generally, we also prove similar convergence results for the counting measure of types of components of each $k$-skeleton of a random geometric complex. As a consequence, in the case of the $1$-skeleton (i.e. for random geometric graphs) we show that the empirical spectral measure associated to the normalized Laplace operator converges to a deterministic measure.

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热力学极限下黎曼流形的随机几何复形和图

我们研究了热力学极限下黎曼流形上随机几何复形和随机几何图的一些拓扑性质。特别是，对于随机几何复合体，我们证明了连接分量的归一化计数度量，根据同位素类型计数，在概率上收敛到确定性度量。更一般地说，我们还证明了对随机几何复合体的每个 $k$-骨架的组件类型的计数度量的类似收敛结果。因此，在$1$-骨架的情况下（即对于随机几何图），我们表明与归一化拉普拉斯算子相关联的经验谱度量收敛到确定性度量。