Graphes et marches aléatoires

Abstract : The study of random walks demonstrates connections between their algebraic, combinatorial, geometric and stochastic properties. The first example of such a connection was given in a theorem of Pólya dealing with nearest neighbourhood random walks on the space of N-dimensional integers. Random walks on groups provide with many examples, however it should be interesting to have such rigid results in the case of weaker algebraic structures such that semigroupoids and groupoids. In this thesis, one example of semigroupoid and one example of groupoid are considered; they are both defined as constrained subgraphs of the Cayley graph of a group, the first one is genuinely directed contrary to the second one which is undirected. For this first example, it has been shown by Campanino and Petritis that the simple random walk is transient. Here, we refine this statement by determining the Martin boundary of this process and show its triviality. In the second example, we consider quasi-periodic tilings of the Euclidean spaces obtained with the help of the cut-and-project scheme. We have considered the simple random walk along the sides of the polytopes constituting the tiling and answered the question of its type, i.e. we determined whether the random walk is recurrent or transient. This result is a consequence of isoperimetric inequalities. This strategy allow us to obtain estimates of the rate of convergence of the heat kernel which could not have been done with the help of the Nash-Williams criterion.
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Submitted on : Thursday, August 30, 2012 - 12:14:03 PM
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Basile de Loynes. Graphes et marches aléatoires. Théorie des groupes [math.GR]. Université Rennes 1, 2012. Français. ⟨NNT : 2012REN1S041⟩. ⟨tel-00726483⟩



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