## Discrete Models: Combinatorics, Computation, and Geometry, DM-CCG 2001

### Robert Cori and Jacques Mazoyer and Michel Morvan and Rémy Mosseri (eds.)

### DMTCS Conference Volume AA (2001), pp. 1-22

author: | Nicolas Destainville |
---|---|

title: | Mixing Times of Plane Random Rhombus Tilings |

keywords: | Random tilings, Discrete dynamical systems, Markovian processes, Quasicrystals |

abstract: | We address the question of single flip discrete dynamics in sets of two-dimensional random rhombus tilings with fixed polygonal boundaries. Single flips are local rearrangements of tiles which enable to sample the configuration sets of tilings via Markov chains. We determine the convergence rates of these dynamical processes towards the statistical equilibrium distributions and we demonstrate that the dynamics are rapidly mixing: the ergodic times are polynomial in the number of tiles up to logarithmic corrections. We use an inherent symmetry of tiling sets which enables to decompose them into smaller subsets where a technique from probability theory, the so-called coupling technique, can be applied. We also point out an interesting occurrence in this work of extreme-value statistics, namely Gumbel distributions. |

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reference: | Nicolas Destainville (2001),
Mixing Times of Plane Random Rhombus Tilings, in Discrete
Models: Combinatorics, Computation, and Geometry, DM-CCG
2001, Robert Cori and Jacques Mazoyer and Michel Morvan
and Rémy Mosseri (eds.), Discrete Mathematics and
Theoretical Computer Science Proceedings AA, pp.
1-22 |

bibtex: | For a corresponding BibTeX entry, please consider our BibTeX-file. |

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