This wouldn't be Hacker News if someone didn't comment that ACTUALLY a random walk of n steps will get you sqrt(n) in some direction, which is much better than log(n). I guess my special skill is being pedantic.
i realize this makes me picky, but it seems to be reasonable, as that incorrect usage often is a signal of sloppy thought.
Try it:
>>> n=1000; sorted(abs(sum(random.randrange(2) for i in range(n)) - n/2) for j in range(128))
As it happens, in two dimensions the distribution is radially symmetric, and its radius is still proportional to the square root of the number of steps. This accounts for the astounding isotropy of lattice-gas automata, whose particles are doing such random walks simultaneously, and for the depressing popularity of Gaussian filtering in image processing, because the Gaussian convolution kernel is the only anisotropic separable kernel, and can be closely approximated in each dimension very cheaply with a different application of the central limit theorem.