随机递归单纯复形
Random Recursive Simplicial Complexes
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中文总结 AI 辅助
本文研究随机递归单纯复形的生长规律，证明特定条件下单纯形数量为渐近自平均随机变量，探究度分布、极端结果概率及首顶点特征，确定单纯形数量平均值的渐近生长律。
中文摘要 AI 辅助
我们研究随机递归单纯复形，其生长方式为每一步添加一个顶点，同时添加一个由新顶点与随机选取的现有单纯形连接而成的单纯形；我们还添加新单纯形的所有面，以确保所得对象仍为单纯复形。若现有单纯形的选择在维度小于 m 的单纯形中是均匀的，则任意容许维度 d≤m 的单纯形数量 S_d 是渐近自平均的随机变量。这一特性使我们能够确定顶点数发散时 S_d 平均值的渐近生长律。我们还探究了度分布、考察了各种极端结果的概率，并分析了第一个顶点的特征。
英文摘要
We investigate random recursive simplicial complexes growing by adding, at each step, a vertex together with a simplex formed by joining the new vertex with a randomly chosen existing simplex. We also add all faces of the new simplex to ensure that the resulting object remains a simplicial complex. If the choice of an existing simplex is uniform among simplices of dimension $<m$, the number $S_d$ of simplices of any admissible dimension $d\leq m$ is an asymptotically self-averaging random variable. This feature allows us to determine the asymptotic growth law of the average of $S_d$ when the number of vertices diverges. We also probe the degree distribution, examine the probabilities of various extreme outcomes, and analyze the characteristics of the first vertex.
