Drunken birds vs drunken ants: random walks between two and three dimensions It's well known that a random walk in the two dimensional grid will return to the origin with probability 1. It's also known that the same random walk in THREE dimensions has a probability strictly less than 1 of returning to the origin. 

My question is:


  Is there something in between ? For example, suppose my space was
  actually a bounded region of the plane extruded out to infinity in the
  z-direction. (what's often called 2.5 dimensional). Do the
  two-dimensional results apply, or the three dimensional ?


This came up in discussions, and one heuristic argument saying that it behaves two dimensionally is that since the finite region of the plane will be covered eventually, the only nontrivial part of the walk is the 1 dimensional ray along the z direction, and so return to the origin will happen. 

Are there other shapes that interpolate between the two-D and the three-D case ? 

Update (pulled in from comments): a related question was asked on MO - a short summary is that if the walk is even (2+ϵ) dimensional, then uncertain return follows loosely from a diverging series. However, the above question is slightly different IMO since I'm asking about other kinds of shapes that might admit certain return.
Probability on Trees and Networks by Peres and Lyons mentions this in Chapter 2 (page 50):


  One way to make sense of this is to ask about the type of spaces intermediate between $\mathbb{Z}^2$ and $\mathbb{Z}^3$. For example, consider the wedge
  
  $$ W_f := \{ (x, y, z) : |z| \leq f(|x|) \} $$
  
  where $f : \mathbb{N} \rightarrow \mathbb{N}$ is an increasing function. The number of edges that leave $W_f \cap \{(x, y, z) : |x| \text{ or } |y| \leq n\}$ is of the order $n(f(n)+1)$, so that according to the Nash-Williams Criterion, 
  
  $$\sum_{n \geq 1}\frac{1}{n(f(n)+1)} = \infty$$
  
  is sufficient for recurrence.