Random Sequential Covering

TL;DR

This paper studies the statistical properties of random sequential covering processes, analyzing congestion, coverage, and dynamics in finite and infinite systems, including intervals, segments, sticks, and continuous spaces.

Contribution

It provides exact calculations of average coverage, cumulants, and congestion probabilities for various covering models, advancing understanding of random sequential deposition.

Findings

Average number of dimers deposited is determined.

All higher cumulants of the distribution are computed.

Probabilities of minimal and maximal congestion are established.

Abstract

In random sequential covering, identical objects are deposited randomly, irreversibly, and sequentially; only attempts increasing the coverage are accepted. A finite system eventually gets congested, and we study the statistics of congested configurations. For the covering of an interval by dimers, we determine the average number of deposited dimers, compute all higher cumulants, and establish the probabilities of reaching minimally and maximally congested configurations. We also investigate random covering by segments with $\ell$ sites and sticks. Covering an infinite substrate continues indefinitely, and we analyze the dynamics of random sequential covering of $\mathbb{Z}$ and $\mathbb{R}^d$.