Cluster Statistics in Expansive Combinatorial Structures
Probability
2022-08-02 v1 Combinatorics
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 -admissibility, and an ingenious idea by Erd\H{o}s and Lehner that utilizes the elementary inclusion/exclusion principle.
Cite
@article{arxiv.2208.00925,
title = {Cluster Statistics in Expansive Combinatorial Structures},
author = {Konstantinos Panagiotou and Leon Ramzews},
journal= {arXiv preprint arXiv:2208.00925},
year = {2022}
}