Nicolas Destainville - Mixing Times of Plane Random Rhombus Tilings

dmtcs:2300 - Discrete Mathematics & Theoretical Computer Science, January 1, 2001, DMTCS Proceedings vol. AA, Discrete Models: Combinatorics, Computation, and Geometry (DM-CCG 2001) - https://doi.org/10.46298/dmtcs.2300
Mixing Times of Plane Random Rhombus Tilings

Authors: Nicolas Destainville

    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.


    Volume: DMTCS Proceedings vol. AA, Discrete Models: Combinatorics, Computation, and Geometry (DM-CCG 2001)
    Section: Proceedings
    Published on: January 1, 2001
    Imported on: November 21, 2016
    Keywords: Random tilings,Discrete dynamical systems,Markovian processes,Quasicrystals,[INFO] Computer Science [cs],[INFO.INFO-CG] Computer Science [cs]/Computational Geometry [cs.CG],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM],[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]

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