Pólya proved that a random walk on $\mathbb{Z}^2$ almost surely returns to the origin, or, equivalently, returns to the origin infinitely often. It was subsequently established that in $\mathbb{Z}^3$, a random walk returns to the origin with probability about 34%.

My question is:

Q. What is the probability that a random walk on the edges of a Penrose tiling returns to its starting point?

Each step from a vertex of the tiling follows one of the incident edges chosen randomly.

          Penrose Tiling
          (Image from this link)
I am wondering to what extent this aperiodic but "quasiregular" tiling mimics the $\mathbb{Z}^d$ lattice from the viewpoint of random walks.

One could ask the same question of a notion of 3D Penrose tiles, e.g.,

Dietl, Ruth Maria Katharina, and Jost-Hinrich Eschenburg. "Three-Dimensional Penrose Tilings and Self-Similarity." (PDF download link)
          3D Penrose Tiling


Presumably (though I don't have references handy) the behaviour of a quasiregular tiling in ${\mathbb Z}^d$ is essentially the same as that of ${\mathbb Z}^d$ itself. Essentially this should depend on the fact that the number of sites within $r$ steps of your starting point grows as $r^d$.

  • $\begingroup$ Thanks, Robert, especially for the high-level viewpoint, which makes my potentially complex question actually quite simple. So you cut through the complexity to the essence! $\endgroup$ – Joseph O'Rourke Apr 12 '14 at 0:06

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