A Bose-Einstein Approach to the Random Partitioning of an Integer

TL;DR

This paper models the random partitioning of an integer using a Bose-Einstein framework, analyzing covering and parking probabilities, gap distributions, and asymptotic behaviors in different regimes, with applications to random graphs.

Contribution

It introduces a novel Bose-Einstein inspired approach to analyze discrete covering and partitioning problems, providing asymptotic results and insights into related random graph models.

Findings

Derived asymptotic probabilities for covering configurations.

Analyzed the distribution of gaps and connected components.

Linked the partitioning problem to a k-nearest neighbor random graph.

Abstract

Consider N equally-spaced points on a circle of circumference N. Choose at random n points out of $N$ on this circle and append clockwise an arc of integral length k to each such point. The resulting random set is made of a random number of connected components. Questions such as the evaluation of the probability of random covering and parking configurations, number and length of the gaps are addressed. They are the discrete versions of similar problems raised in the continuum. For each value of k, asymptotic results are presented when n,N both go to infinity according to two different regimes. This model may equivalently be viewed as a random partitioning problem of N items into n recipients. A grand-canonical balls in boxes approach is also supplied, giving some insight into the multiplicities of the box filling amounts or spacings. The latter model is a k-nearest neighbor random graph with N vertices and kn edges. We shall also briefly consider the covering problem in the context of a random graph model with N vertices and n (out-degree 1) edges whose endpoints are no more bound to be neighbors.