Cluster Statistics in Expansive Combinatorial Structures

2022/08/01 by Panagiotou, Konstantinos, Ramzews, Leon
#Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2208.00925

Abstract

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.

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