TITLE: Random walk on a Penrose tiling
QUESTION [13 upvotes]: 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.

 
 
 
 
 


 
 
 
 
 
(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)

REPLY [15 votes]: 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 $r^d$.