论文标题
多参数上层模型中随机简单复合物的同源性
The homology of random simplicial complexes in the multi-parameter upper model
论文作者
论文摘要
我们研究多参数上层模型中的随机简单复合物。在此模型中，各个维度的简单是随机和独立进行的，然后将我们的随机简单复杂$ y $ $ $ $ $ y $ y $ y $，然后是包含此简单集合的最小简单综合体。 我们研究了$ y $的同源性的渐近行为，因为顶点数量为$ \ infty $。我们几乎可以肯定地渐近地观察到以下现象。采用简单的给定概率确定一系列尺寸$ \ ell \ ell \ leq k \ leq \ ell'$带有$ \ ell'\ leq 2 \ ell +1 $，在其同源$ y $ nasishes之外。在此范围内，同源物从维度到维度大大减少。特别是，关键维度$ \ ell $中的同源性是最大的。
We study random simplicial complexes in the multi-parameter upper model. In this model simplices of various dimensions are taken randomly and independently, and our random simplicial complex $Y$ is then taken to be the minimal simplicial complex containing this collection of simplices. We study the asymptotic behavior of the homology of $Y$ as the number of vertices goes to $\infty$. We observe the following phenomenon asymptotically almost surely. The given probabilities with which the simplices are taken determine a range of dimensions $\ell \leq k \leq \ell'$ with $\ell' \leq 2\ell +1$, outside of which the homology of $Y$ vanishes. Within this range, the homologies diminish drastically from dimension to dimension. In particular, the homology in the critical dimension $\ell$ is significantly the largest.
