Stochastic Flips on Dimer Tilings

Details Stochastic Flips on Dimer Tilings Stochastic Flips on Dimer Tilings

2011-11-30

arxiv.org/abs/1111.7297v1

Mathematics

Probability

13 pages, 10 figures; DMTCS Proceedings, 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in

Scientific paper

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 {\em 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 bound quadratic in the number n 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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