Cluster Statistics in Expansive Combinatorial Structures

TL;DR

This paper introduces a unified approach to analyze cluster size distributions in expansive combinatorial structures, providing limiting distributions, moments, and local limit theorems.

Contribution

It combines saddle-point methods and inclusion/exclusion principles to comprehensively study cluster statistics in expansive sets, a novel integration of techniques.

Findings

Limiting distribution of smallest and largest clusters determined

All moments of the cluster count distribution established

A local limit theorem for the cluster count distribution proved

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.