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Contribution à l'étude des lacets markoviens

Abstract : We are interested in Markov laces defined in the framework of the theory of Markov chains in continuous time on a discrete state space. This particular subject has been studied by Le Jan [LJ11] and Sznitman [Szn12]. In contrast to these references, we do not assume the reversibility of the chain and we are mostly interested in the case of countable state space. All the results are presented in terms of the generator of semigroup. In comparison with [LJ11], some demonstration has been detailed or improved.We also provide some results on the loop clusters (see [LJL12] in the reversible case). In particular, we study the example of discrete circle. We also study the spanning tree algorithm defined by Wilson in the non-symmetric case.In the last part, we consider the proportion of loops covering the whole space. Using the limit of the spectrums, we give a general expression for the limit of this ratio for a sequence of graphs. As an application, we give two examples in which a phase transition occurs.
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Yinshan Chang. Contribution à l'étude des lacets markoviens. Mathématiques générales [math.GM]. Université Paris Sud - Paris XI, 2013. Français. ⟨NNT : 2013PA112069⟩. ⟨tel-00846462⟩

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