Random walks on quasi-periodic tilings

Basile de Loynes (1)

1 IRMA - Institut de Recherche Mathématique Avancée

In this paper, we study random walks on quasi-periodic graphs induced by tiling the standard real vector space R d using the cut-and-project method. We first show a dichotomy of Pólya type, namely we prove that the simple random walk on a cut-and-project graph is recurrent if d ≤ 2 and transient otherwise. Nonetheless the aperiodic graphs we consider here are no longer the Cayley graph of a group but of a groupoid. Secondly, we prove the asymptotic entropy of such random walks is zero.

Mots clés: markov additive processes; entropy; semigroupoids; random walks

Domaines: Probabilités [math.PR]

Basile de Loynes. Random walks on quasi-periodic tilings. 2015. ⟨hal-01134965v2⟩