Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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GENERALIZED DOMINOES TILING'S MARKOV CHAIN MIXES FAST

  • KAYIBI, K.K. (School of Mathematics, University of Bristol) ;
  • SAMEE, U. (Department of Mathematics, Islamia College of Sciences and Commerce) ;
  • MERAJUDDIN, MERAJUDDIN (Department of Applied Mathematics, AMU) ;
  • PIRZADA, S. (Department of Mathematics, University of Kashmir)
  • Received : 2019.01.11
  • Accepted : 2019.08.05
  • Published : 2019.09.30
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Abstract

A generalized tiling is defined as a generalization of the properties of tiling a region of ${\mathbb{Z}}^2$ with dominoes, and comprises tiling with rhombus and any other tilings that admits height functions which can be ordered into a distributive lattice. By using properties of the distributive lattice, we prove that the Markov chain consisting of moving from one height function to the next by a flip is fast mixing and the mixing time ${\tau}({\epsilon})$ is given by ${\tau}({\epsilon}){\leq}(kmn)^3(mn\;{\ln}\;k+{\ln}\;{\epsilon}^{-1})$, where mn is the area of the grid ${\Gamma}$ that is a k-regular polycell. This result generalizes the result of the authors (T-tetromino tiling Markov chain is fast mixing, Theor. Comp. Sci. (2018)) and improves on the mixing time obtained by using coupling arguments by N. Destainville and by M. Luby, D. Randall, A. Sinclair.

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References

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