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 th

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.