Configuration space partitioning in tilings of a bounded region of the plane

Eduardo J. Aguilar, Valmir C. Barbosa, Raul Donangelo and Sergio R. Souza
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Eduardo J. Aguilar: Instituto de Ciência e Tecnologia, Universidade Federal de Alfenas, Poços de Caldas, MG 37715-400, Brazil
Valmir C. Barbosa: Programa de Engenharia de Sistemas e Computação, COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ 21941-914, Brazil3Programa de Pós-Graduação em Ciências Computacionais, IME, Universidade do Estado do Rio de Janeiro, Rio de Janeiro, RJ 20550-900, Brazil
Raul Donangelo: Instituto de Física, Facultad de Ingeniería, Universidad de la República, Montevideo 11.300, Uruguay5Instituto de Física, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ 21941-909, Brazil
Sergio R. Souza: Instituto de Física, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ 21941-909, Brazil6Departamento de Engenharia Nuclear, Universidade, Federal de Minas Gerais, Belo Horizonte, MG 31270-901, Brazil7Departamento de Física, ICEx, Universidade Federal Fluminense, Volta Redonda, RJ 27213-145, Brazil

International Journal of Modern Physics C (IJMPC), 2025, vol. 36, issue 02, 1-26

Abstract: Given a finite collection of two-dimensional tile types, we study the tiling of a rectangular region of the plane when the available tile types are all rectangular. Unlike the case of tiling the whole, unbounded plane, the additional complications imposed by the boundary conditions tend to constrain progress to mostly indirect results, such as recurrence relations. The tile types we use are squares, dominoes and straight tetraminoes, in which case not even recurrence relations are available. We seek to characterize this complex system through some fundamental physical quantities and follow two parallel tracks: One fully analytical for what seems to be the most complex special case still amenable to such approach, the other based on the Wang–Landau method for state-density estimation. Given a simple energy function based solely on tile contacts, we have found either approach to lead to illuminating depictions of entropy, temperature, and above all partitions of the configuration space. A configuration, in this context, refers to how many tiles of each type are used. We have found that certain partitions help bind together different aspects of the system in question and conjecture that future applications will benefit from the possibilities they afford.

Keywords: Tilings of bounded plane regions; combinatorics of tilings; scaling laws of tilings; entropic sampling of tiling spaces; Wang–Landau method (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1142/S0129183124501882

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