Jamming and Tiling in Aggregation of Rectangles

TL;DR

This paper investigates a random rectangle aggregation process leading to jammed states and tilings, revealing a power-law scaling for the number of frozen rectangles and characterizing the statistical properties of these states.

Contribution

It introduces a novel model of rectangle aggregation with a detailed analysis of jammed states and tilings, including a new growth exponent and scaling laws.

Findings

Average frozen rectangles scale as N^0.229

Jammed states exhibit a specific growth exponent

Tiling of the domain with linear scaling of rectangles

Abstract

We study a random aggregation process involving rectangular clusters. In each aggregation event, two rectangles are chosen at random and if they have a compatible side, either vertical or horizontal, they merge along that side to form a larger rectangle. Starting with $N$ identical squares, this elementary event is repeated until the system reaches a jammed state where each rectangle has two unique sides. The average number of frozen rectangles scales as $N^\alpha$ in the large-$N$ limit. The growth exponent $\alpha=0.229\pm 0.002$ characterizes statistical properties of the jammed state and the time-dependent evolution. We also study an aggregation process where rectangles are embedded in a plane and interact only with nearest neighbors. In the jammed state, neighboring rectangles are incompatible, and these frozen rectangles form a tiling of the two-dimensional domain. In this case, the final number of rectangles scales linearly with system size.