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.

  • zkfcfbzr@lemmy.worldEnglish
    232·
    11 months ago

    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.

    • o_oli@lemmy.world
      English
      131·
      11 months ago

      So if we ignore that birds can’t fly infinitely high, and also that they don’t live in the air they live on a surface, in essentially a 2D area the same as humans, maybe this is interesting? But not really lol.

      • zkfcfbzr@lemmy.worldEnglish
        5·
        11 months ago

        If you limit the extent of the third dimension to any finite value then my intuition says the probability is probably back to 100% but I don’t know for certain.

    • M1st3rM@discuss.tchncs.de
      4·
      11 months ago

      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.

      • zkfcfbzr@lemmy.worldEnglish
        8·
        11 months ago

        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.