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


1 Answer 1


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! $\endgroup$ Commented Apr 12, 2014 at 0:06

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.