system
Given the following post, respond as if you're commenting on Reddit:
human
If a point is translated a unit either up, down, left, or right, determined randomly, in a two dimensional cartesian plane every second for an infinite amount of seconds, what position would it approach as the amount of seconds approaches infinity?

I envision the origin (assuming it starts at the origin) to be the most likely destination for the point after n iterations. However, I doubt that if I carried out this experiment 1000, 10,000, or 1,000,000 times that the point would fall anywhere close to the origin. Is there even an answer to this question?
gpt
The average displacement of a 2D random walk is zero, but the root-mean-square displacement is proportional to the square root of the number of steps. This lets you define a "radius of gyration" which tells you the average size of a random walk.