Dispersion models on a circle: universal properties and asymptotic results

TL;DR

This paper studies dispersion models on a circle, revealing universal properties, asymptotic behaviors, and connections to coalescent processes, with implications for physical phenomena and combinatorial models.

Contribution

It introduces a general framework for dispersion on a circle, proving universal spacing and distribution properties, and links these models to additive coalescent processes.

Findings

Number of free connected components follows a binomial distribution.

Sizes of free connected components follow a Dirichlet distribution.

Asymptotic behavior of the covered space is characterized as the number of masses grows.

Abstract

Consider a sequence of masses $m_0,m_1,...$ arriving uniformly at random at some points $u_0,u_1,...$ on the unit circle $\mathbb{R}/\mathbb{Z}$ (or on $\mathbb{Z}/n\mathbb{Z}$, in the discrete version). Upon arrival, each mass undergoes a relaxation phase during which it is dispersed, possibly also at random. This process can model many physical phenomena, such as the diffusion of liquid in a porous medium. In the discrete case, it can model parking (related to additive coalescence and hashing with linear probing) in which the cars are permitted to follow random displacement policies. The dispersion policies considered in the paper ensure that at time $k$, after the successive dispersions of $m_0,\cdots,m_{k-1}$, the total covered region has Lebesgue measure $m_0+\cdots+m_{k-1}$. Furthermore, during the dispersion of a given mass, the covered domain increases continuously, except when it merges with another covered connected component (CC). We show a very general exchangeability property for the sequence of covered CC. Additionally, we demonstrate a universal spacing property between these CC, and a notable general result: if the $(u_i)$ are independent and rotationally invariant, then the number of free (not covered) CC follows a binomial distribution whose parameters depend solely on the number and total mass of arrived particles. Furthermore, conditional on the number of CC, the sizes of the free CC follow a simple Dirichlet distribution in the continuous case, regardless of the dispersion policy considered and the values of the masses. We also characterize the distribution of the occupied space. In the second part of the paper, we study the total cost associated with these models for various cost models, and establish connections with the additive coalescent. We also provide an asymptotic representation of the limiting covered space as the number of masses goes to infinity.