论文标题
膨胀组合结构中的群集统计
Cluster Statistics in Expansive Combinatorial Structures
论文作者
论文摘要
我们开发了一种简单而统一的方法来研究随机膨胀（多）集群统计的几个方面。特别是，我们确定了最小和最大簇的尺寸的限制分布，我们确定了簇数的分布的所有矩，并且我们证明该分布的局部限制定理。我们的证明有效地结合了两种简单的成分：通过$ h $ admassibility的著名框架应用鞍点方法，以及Erdős和Lehner的巧妙想法，利用了基本包含/排除原则。
We develop a simple and unified approach to investigate several aspects of the cluster statistics of random expansive (multi-)sets. In particular, we determine the limiting distribution of the size of the smallest and largest clusters, we establish all moments of the distribution of the number of clusters, and we prove a local limit theorem for that distribution. Our proofs combine effectively two simple ingredients: an application of the saddle-point method through the well-known framework of $H$-admissibility, and an ingenious idea by Erdős and Lehner that utilizes the elementary inclusion/exclusion principle.
