Uh... Something's fishy with the first graph of the 'drunken man's walk'. It's well known that the random walk veers away from the zero-line at a rate sqrt(N), where N is number of flips. Normalized by the number of flips, the random walk converges to the zero-line at the rate 1/sqrt(N). The graph does neither, so I'm not quite sure how it was generated...
Oops, I misinterpreted how the graph was generated... but the complaint still holds. The averaging of a million trials will cut the amplitude of the fluctuations by a factor of 1000, but the sqrt(N) effect should still be visible. (Note that if you multiply the Y-axis by sqrt(million) = 1000, then you get a scale of about 20, which is approx sqrt(500))
I'm not very good at this stuff, but I don't see anything wrong with the graph. It hovers around 0. A Brownian motion starting which starts at 0, will, at time T be normally distributed with mean 0 and variance T.
