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arXiv · 2208.00925

Cluster Statistics in Expansive Combinatorial Structures

Konstantinos Panagiotou·Leon Ramzews
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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 Erd\H{o}s and Lehner that utilizes the elementary inclusion/exclusion principle.

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Konstantinos Panagiotou, Leon Ramzews. 2022-08-01. Cluster Statistics in Expansive Combinatorial Structures. https://arxiv.org/abs/2208.00925

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