Found this interesting random walk simulation. I thought random walks tended to stay close to the initial state, regardless of the number of steps/step size, but this simulation ends up making some pretty cool random walk art.
An interesting fact: a 1D random walk will almost always return to the origin (as in, given an infinite number of runs, only finitely many will not return to the origin), wheras a 2D random walk will almost never return to the origin (of an infinite number of runs, only finitely many will return to the origin). Of course, this is directly saying much about the average distance from the origin, which slowly increases with steps as mentioned by others.
It'll gradually creep away*, proportionally to sqrt(t).
Got a timestamp in that video that says differently?
* in the absence of (restoring) forces, it's unwise to rely on things staying where you left them, as that'd be betting against entropy. Lagniappe: https://www.youtube.com/watch?v=i6rVHr6OwjI
Think of it this way: if randomness is evenly distributed, the steps to the left and right will balance out, just as the steps up and down will. This balance tends to keep the random walk close to its starting point.
You're right. (I'd count enforcement of "without replacement" as a force; at least for classical walks it'd require some kind of Maxwell daemon?)
I guess what I want to get at: a Galton board is easily understood by elementary school students (and probably constructible by those in the upper grades), and it doesn't take much time playing with one to notice that all the balls don't wind up with their paths balanced out, in the middle column.
(even the normal distribution is misleading, because centrality is a 1D phenomenon; given enough dimensions observing the exact average becomes exceedingly unlikely)
If you take the same Galton board and focud on the paths taken by all balls that ended up in a particular bin you get the hypergeometric distribution for the steps they have taken. So the draws without replacement lurk in the Galton board too.
It does, in other words, not require a demon, only a restricted viewing angle.
Another way to think about it: the more steps, the more likely it is you get a run which is imbalanced for long enough to make it some given distance from the origin.
