# Random walk on a Penrose tiling

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.

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