Jamming and Tiling in Aggregation of Rectangles
D.S. Ben-Naim, E. Ben-Naim, and P.L. Krapivsky
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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