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.

  • M1st3rM@discuss.tchncs.de
    4·
    1 year 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
      7·
      1 year 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.