I have troubles with the following problem:
Consider the grid of equilateral triangles (triangular lattice). Is random walk recurrent or transient?
I was thinking of computing $\mathbf{E}(\text{the particle returns to the origin})$ and then depending whether the value if infinity or not to decide wether the walk is recurrent or transient. The only problem is to calculate $\mathbf{P}(S_k = o)$, where $S_k$ the position of the particle at moment $k$ and $o$ denotes the origin. So I splitted the cases when $k$ is even and odd. When $k$ is even, the the number of horizontal and non-horizontal steps should be even, but this doesn't work as we do not know the direction of such steps. Personally, I think the walk is recurrent, but I can be mistaken. Maybe we can relate this problem to Pólya problem, but I don't know how. Any suggestions?
You can try to show that the random walk has mean zero, and that it is irreducible and aperiodic, i.e. for every $x,y$ there exists $N$ such that $\mathbf{P}_x(S_n = y) >0$ for $n\geq N$. Since the dimension is $2$, the random walk will be recurrent. See e.g. Theorem 4.1.1 in Random Walk: A Modern Introduction by Greg Lawler and Vlada Limic here.