To answer the question of the person ever getting back to the original starting point of the walk, this is the 2-dimensional equivalent of the level-crossing problem discussed above. In 1921 George Pólya proved that the person almost surely would in a 2-dimensional random walk, but for 3 dimensions or higher, the probability of returning to the origin decreases as the number of dimensions increases. In 3 dimensions, the probability decreases to roughly 34%. The mathematician Shizuo Kakutani was known to refer to this result with the following quote: "A drunk man will find his way home, but a drunk bird may get lost forever".
Another variation of this question which was also asked by Pólya is: "if two people leave the same starting point, then will they ever meet again?" It can be shown that the difference between their locations (two independent random walks) is also a simple random walk, so they almost surely meet again in a 2-dimensional walk, but for 3 dimensions and higher the probability decreases with the number of the dimensions. Paul Erdős and Samuel James Taylor also showed in 1960 that for dimensions less or equal than 4, two independent random walks starting from any two given points have infinitely many intersections almost surely, but for dimensions higher than 5, they almost surely intersect only finitely often.