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 was maybe a bit sloppy when I said it “quickly drops to 0” instead of it “quickly tends to 0”. It’ll of course always be positive - in fact if N is the sum of the absolute value of the three coordinates of its current position, the probability of returning to the origin is strictly greater than 1/6ᴺ.
But it does tend to 0 in such a way that the probability of its random walk ever returning to the starting position is not 100%. It has a 34% chance of ever getting back at the very start of its journey - but if it gets too far off track that probability is going to tend to 0 fast enough that it’s not likely to ever make it back, even with infinitely many steps. Here’s a youtube video (that I did not watch myself) that seems to go over the topic.
Here is an alternative Piped link(s): https://piped.video/watch?v=iH2kATv49rc
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