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.