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Stochastic Flips on Dimer Tilings

Abstract : This paper introduces a Markov process inspired by the problem of quasicrystal growth. It acts over dimer tilings of the triangular grid by randomly performing local transformations, called $\textit{flips}$, which do not increase the number of identical adjacent tiles (this number can be thought as the tiling energy). Fixed-points of such a process play the role of quasicrystals. We are here interested in the worst-case expected number of flips to converge towards a fixed-point. Numerical experiments suggest a $\Theta (n^2)$ bound, where $n$ is the number of tiles of the tiling. We prove a $O(n^{2.5})$ upper bound and discuss the gap between this bound and the previous one. We also briefly discuss the average-case.
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Submitted on : Thursday, August 20, 2015 - 4:34:06 PM
Last modification on : Friday, October 22, 2021 - 3:31:23 AM
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Thomas Fernique, Damien Regnault. Stochastic Flips on Dimer Tilings. pp.205-218, ⟨10.46298/dmtcs.2803⟩. ⟨hal-01185602⟩



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