Random walks on graphs induced by aperiodic tilings

Abstract : In this paper, simple random walks on a class of graphs induced by quasi-periodic tilings of the d-dimensional Euclidean space are investigated. In this sense, these graphs can be seen as perturbations of the Cayley graph of the N-dimensional integer lattice. The quasi-periodicity of the underlying tilings implies that these graphs are not space homogeneous (roughly speaking, there is no transitive group action). In this context, we prove that the symptotic entropy of the simple random walk is zero and characterize the type (recurrent or transient) of the simple random walk. These results are similar to the classical context of random walks on the integer lattice. In this sense, it suggests that the perturbation remains well controlled.
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Basile de Loynes. Random walks on graphs induced by aperiodic tilings. Markov Processes And Related Fields, Polymat Publishing Company, 2017, 23 (1), pp.103-124. ⟨hal-01134965v4⟩

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