Jamming and Tiling in Aggregation of Rectangles

D. S. Ben-Naim; E. Ben-Naim; P. L. Krapivsky

doi:10.1088/1751-8121/aae4c0

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arXiv · 1808.03714

Jamming and Tiling in Aggregation of Rectangles

D. S. Ben-Naim·E. Ben-Naim·P. L. Krapivsky

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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.

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D. S. Ben-Naim, E. Ben-Naim, P. L. Krapivsky. 2018-08-10. Jamming and Tiling in Aggregation of Rectangles. https://doi.org/10.1088/1751-8121%2Faae4c0

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