Return to Revisions

This is not an answer, just an observation. The probability of returning to the origin eventually is 1 (approaches 1 as the number of steps approaches infinity). This is Pólya's famous 1921 result. The same is true for reaching any fixed point $(x,y)$: the probability of reaching it is 1. My guess is that the probability of reaching $(x,y)$ before hitting the origin (or any other fixed point) is likely still 1. But this is only a guess. As pointed out in the comments, this was a terrible guess!