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 {\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.
Keywords
Cite
@article{arxiv.1111.7297,
title = {Stochastic Flips on Dimer Tilings},
author = {Thomas Fernique and Damien Regnault},
journal= {arXiv preprint arXiv:1111.7297},
year = {2011}
}
Comments
13 pages, 10 figures; DMTCS Proceedings, 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10), 2010