Explanation: Random walk in 2D has a unity probability of making it back to the starting point as the number of steps approach infinity but random walk in 3D only has ~0.34.
Explanation: Random walk in 2D has a unity probability of making it back to the starting point as the number of steps approach infinity but random walk in 3D only has ~0.34.
I don’t think that changes it actually. Even if the drunk guy lives on a floor that’s high up, he didn’t fly to get there, so there is some combo of random walking and moving that gets him back there. Like walking into an elevator that happens to be on the ground floor, and also happens to be going up to the right floor at that time. Or walking to the base of a staircase, and (50/50) randomly going up it. It definitely isn’t technically a 2d plane anymore, but there are still a finite number of 2d planes that he could be trying to get to ( the highest floor on the building with the most floors).