Standard deviation
How fast does the typical deviation of a random walk grow with time?
Goal: estimate the spread from more and more paths and watch the empirical standard deviation settle onto the ±√n envelope.
5 paths
The green ±√n curve is the theoretical standard deviation; the dashed curve is the std measured from the cloud at every step. With few paths the estimate is noisy; add paths and it settles onto √n.
Linear
Empirical std√n law
On log–log axes every power law is a straight line whose slope is the exponent. The √n law has slope 1/2.