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this post was submitted on 05 Aug 2023
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Showerthoughts
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Expanding on what OP is talking about:
In this context, a random walk happens on a 2D coordinate plane. Your drunk person starts at the origin, (0, 0), and for a "random walk" they move either left, right, up, or down by exactly 1 unit each step. It's a mathematical fact that this process, taken to its limit where infinitely many random steps are taken, will always have the drunk return to the origin - in fact, for any given integer coordinate on the plane there's a 100% chance the drunk will eventually visit that coordinate following a random walk.
This doesn't work in 3D though, where there's an x, y, and z axis. A random walk there won't always return to the origin - it only will about 34% of the time. If the drunk gets too far away the probability of ever finding their way back at random quickly drops to 0.
That doesn't make sense to me. Sure, the probability in 3D is gonna get really low. Never 0 though since there is a chance the previously taken steps will be done in reverse. And since we talk about infinity here .. the drunk bird should also find home.
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.
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