论文标题
非均匀随机简单络合物的共同体学组的相变
Phase transition in cohomology groups of non-uniform random simplicial complexes
论文作者
论文摘要
我们考虑一个随机简单复合物的广义模型，该模型由随机超图引起。我们的模型是通过将非均匀二项式随机超图的向下闭合而生成的，其中每个$ k $，每组$ k+1 $ 1 $ vertices构成一个边缘，并独立使用一些概率$ p_k $。作为一种特殊情况，它包含了由Meshulam和Wallach引入的（均匀）随机简单复合物的广泛研究模型[随机结构和算法34（2009），否。 3，第408-417页]。 根据任意的Abelian集团$ r $的共同体学小组的消失，我们考虑了这种新模型上的连通性的更高维度概念。我们证明，这种连接性的概念显示了相变并确定阈值。我们还证明了自然过程解释的打击时间结果，其中简单及其向下闭合被一一添加。此外，我们确定了相变时临界窗口内的同胞组的渐近行为。
We consider a generalised model of a random simplicial complex, which arises from a random hypergraph. Our model is generated by taking the downward-closure of a non-uniform binomial random hypergraph, in which for each $k$, each set of $k+1$ vertices forms an edge with some probability $p_k$ independently. As a special case, this contains an extensively studied model of a (uniform) random simplicial complex, introduced by Meshulam and Wallach [Random Structures & Algorithms 34 (2009), no. 3, pp. 408-417]. We consider a higher-dimensional notion of connectedness on this new model according to the vanishing of cohomology groups over an arbitrary abelian group $R$. We prove that this notion of connectedness displays a phase transition and determine the threshold. We also prove a hitting time result for a natural process interpretation, in which simplices and their downward-closure are added one by one. In addition, we determine the asymptotic behaviour of cohomology groups inside the critical window around the time of the phase transition.
