3 research outputs found

    Three lectures on random proper colorings of Zd\mathbb{Z}^d

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    A proper qq-coloring of a graph is an assignment of one of qq colors to each vertex of the graph so that adjacent vertices are colored differently. Sample uniformly among all proper qq-colorings of a large discrete cube in the integer lattice Zd\mathbb{Z}^d. Does the random coloring obtained exhibit any large-scale structure? Does it have fast decay of correlations? We discuss these questions and the way their answers depend on the dimension dd and the number of colors qq. The questions are motivated by statistical physics (anti-ferromagnetic materials, square ice), combinatorics (proper colorings, independent sets) and the study of random Lipschitz functions on a lattice. The discussion introduces a diverse set of tools, useful for this purpose and for other problems, including spatial mixing, entropy and coupling methods, Gibbs measures and their classification and refined contour analysis.Comment: 53 pages, 10 figures; Based on lectures given at the workshop on Random Walks, Random Graphs and Random Media, September 2019, Munich and at the school Lectures on Probability and Stochastic Processes XIV, December 2019, Delh

    Graph-indexed random walks on pseudotrees

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