This is a post about 1D random walks. For example, the first graphic is a plot of step number versus distance away from the origin for six independent chains. The code is in the first listing below. I realized when doing this plot that I need to look into generating "rainbow" colors in matplotlib, but I want to go ahead and put this up without that for now. I did a similar example in R in a previous post.

Now, the problem (as posed by Grinstead and Snell) is to investigate what happens at longer times. In the second code example, we run a random walk for a long time (1E7 steps) and record the step number each time the walk returns to 0. The plot shows the number of steps between returns to 0, as a histogram. Both axes are scaled as log

_{2}of the value.

The walk can wander quite far away from 0 (and for a long time). The list of the largest number of steps between successive visits to 0 is:

The log-log plot looks like some kind of exponential decay. And we always come back. The value for the last interval between visits to 0 is 4.

The code uses the Counter class from here.

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