The drunken man is moving away from where he started (any point in time can be labelled as "start") at a speed of about the square root of his linear speed (speed from his point of view). And the direction is random. This can also be 1D motion. Actually in 1 and 2D it is likely that the drunken man hits his starting point again at some point, it goes to 0 fast in 3D.

For 1D It's trivial to see that a side scroller with side-strafing controls (Space Invaders not Asteroids) will always return to the center if L and R appear with equal probability, in euclidean geometry

As the parent poster quotes, drunk looks like:

> if the sequence was of n steps, almost surely its distance from the starting point would be close to √n.

Of course that raises the question "What is close?"