Random sequential covering of a one-dimensional lattice by $k$-mers

TL;DR

This paper analyzes the process of randomly and irreversibly covering an infinite one-dimensional lattice with k-mers, providing explicit solutions for small k and numerical insights for larger k.

Contribution

The authors introduce a comprehensive method to solve the dynamics of random sequential covering of a 1D lattice by k-mers, including explicit solutions for small k.

Findings

Explicit solutions for trimers, tetramers, and pentamers.

Numerical analysis for k > 5.

Insights into the coverage dynamics over time.

Abstract

In random sequential covering, identical objects are deposited randomly, irreversibly, and sequentially; only attempts that increase coverage are accepted. The process continues indefinitely on an infinite substrate, and we analyze the dynamics of random sequential covering of $\mathbb{Z}$ using $k$-mers. We introduce a method that provides a comprehensive solution to the dynamics of this process. We derive explicit solutions for trimers, tetramers, and pentamers; we study numerically random sequential covering by longer polymers ($k>5$).