Random Walks on Graphs Induced by Aperiodic Tilings

B. de Loynes

2017, v.23, №1, 103-124

ABSTRACT

In this paper, simple random walks on a class of graphs induced by quasi-periodic tilings of the Euclidean space $\mathbb{R}^d$ are investigated. Roughly speaking, these graphs are obtained by considering a $d$-dimensional slice of the Cayley graph of $\mathbb{Z}^N$. 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 asymptotic 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 a varying local curvature does not modify the global behavior of the simple random walk as long as the graph remains roughly globally flat.

Keywords: random walks, aperiodic tilings, aperiodic graphs, non homogeneous spaces, asymptotic entropy, Markov additive processes, random walk with random transition probabilities, recurrence, transience, isoperimetric inequalities

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