Jamming and Tiling in Aggregation 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.

arXiv comment9 pages, 11 figures

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Cited by in corpus (3)

• Non-extensive super-cluster states in aggregation with fragmentation
• Jamming and Tiling in Fragmentation of Rectangles
• Charged Aggregation