I had forgotten that one. You inspired me to visit Wikipedia where I learned that the probability of a 3D random walk returning to the origin (given an unbound

I had forgotten that one. You inspired me to visit Wikipedia where I learned that the probability of a 3D random walk returning to the origin (given an unbounded number of steps) is about .34 (in 2D, as you note, this probability is 1). The probability of eventual return goes down as dimensionality increases. As you say, get lost forever.