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Theses

Pavages Aléatoires

Abstract : In this thesis we study two types of tilings : tilings by a pair of squares and tilings on the tri-hexagonal (Kagome) lattice. We consider different combinatorial and probabilistic problems. First, we study the case of 1x1 and 2x2 squares on infinite stripes of height k and get combinatorial results on proportions of 1x1 squares for k < 11 in plain and cylindrical cases. We generalize the problem for bigger squares. We consider questions about sampling and approximate counting. In order to get a random sample, we define Markov chains for square and Kagome tilings. We show ergodicity and find polynomial bounds on the mixing time for nxlog n regions in the case of tilings by 1x1 and sxs squares and for lozenge regions in the case of restrained Kagome tilings. We also consider weighted Markov chains where weights are put on the tiles. We show rapid mixing with conditions on for square tilings by 1x1 and sxs squares and for Kagome tilings. We provide simulations that suggest different conjectures, one of which existence of frozen regions in random tilings by squares and on the Kagome lattice of regions with non flat boundaries.
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Alexandra Ugolnikova. Pavages Aléatoires. Modélisation et simulation. Université Sorbonne Paris Cité, 2016. Français. ⟨NNT : 2016USPCD034⟩. ⟨tel-01889871⟩

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