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Title: 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 Nα in the large-N limit. The growth exponent α = 0.229 ± 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. Here in this case, the final number of rectangles scales linearly with system size.

Authors:
 [1]; ORCiD logo [2];  [3]
+ Show Author Affiliations
  1. Univ. of California, Santa Barbara, CA (United States). Dept. of Computer Science
  2. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
  3. Boston Univ., MA (United States). Dept. of Physics
Publication Date:
Research Org.:
Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
Sponsoring Org.:
USDOE Laboratory Directed Research and Development (LDRD) Program
OSTI Identifier:
1492668
Report Number(s):
LA-UR-18-26798
Journal ID: ISSN 1751-8113
Grant/Contract Number:  
89233218CNA000001
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Physics. A, Mathematical and Theoretical
Additional Journal Information:
Journal Volume: 51; Journal Issue: 45; Journal ID: ISSN 1751-8113
Publisher:
IOP Publishing
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; aggregation; random processes; tiling

Citation Formats

Ben-Naim, Daniel S, Ben-Naim, Eli, and Krapivsky, Paul L. Jamming and Tiling in Aggregation of Rectangles. United States: N. p., 2018. Web. doi:10.1088/1751-8121/aae4c0.
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Ben-Naim, Daniel S, Ben-Naim, Eli, & Krapivsky, Paul L. Jamming and Tiling in Aggregation of Rectangles. United States. https://doi.org/10.1088/1751-8121/aae4c0
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Ben-Naim, Daniel S, Ben-Naim, Eli, and Krapivsky, Paul L. Tue . "Jamming and Tiling in Aggregation of Rectangles". United States. https://doi.org/10.1088/1751-8121/aae4c0. https://www.osti.gov/servlets/purl/1492668.
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@article{osti_1492668,
title = {Jamming and Tiling in Aggregation of Rectangles},
author = {Ben-Naim, Daniel S and Ben-Naim, Eli and Krapivsky, Paul L},
abstractNote = {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 Nα in the large-N limit. The growth exponent α = 0.229 ± 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. Here in this case, the final number of rectangles scales linearly with system size.},
doi = {10.1088/1751-8121/aae4c0},
journal = {Journal of Physics. A, Mathematical and Theoretical},
number = 45,
volume = 51,
place = {United States},
year = {Tue Oct 16 00:00:00 EDT 2018},
month = {Tue Oct 16 00:00:00 EDT 2018}
}
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Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
https://doi.org/10.1088/1751-8121/aae4c0
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Cited by: 2 works
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Figures / Tables:

FIG. 1 FIG. 1: Illustration of the aggregation process (2a).
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    Jamming and tiling in fragmentation of rectangles
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      Figures / Tables found in this record:

      FIG. 1 (p. 2)figure
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      TABLE I (p. 3)table
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      TABLE III (p. 9)table
      TABLE IV (p. 9)table
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