RANDOM WALK in ONE-DIMENSION

The words, *Random Walk*, in their simplest incarnation, refer to this situation: The proverbial drunk is clinging to the lamppost. He decides to start walking. The road runs east and west. In his inebriated state he is as likely to take a step east (forward) as west (backward). (Probability 50% in either of the two directions.) From each new position he is again as likely to go forward as backward. Each of his steps are of the same length but of random direction - east or west. In the figure the drunk is a green rectangle which steps (bounces) either left or right, at random, starting at the lamppost in the center. Mouse-click on the start/stop button to begin a random walk. The same button resets the walker for another - different - random walk.

After having taken n steps (n can be one step, ten steps, fifty-three..) the walker is to be found standing at some position from which he makes a step to one of his neighboring positions. One can plot position (or displacement) against the number of steps taken for any particular random walk.

The next figure is a graph for one such random walk showing the position (the displacement) after having taken n steps. East (forward) is plotted here in the vertical direction (up) and west (backwards) is down on the graph. Clicking the *random walk generator* in the figure produces more random walk graphs. Superposing many such graphs brings out where a walker is likely to be statistically. He is most likely to be where the color is deepest because many random walks have covered that terrain. The green curve shows the expected root mean square displacement after n steps.

Mathematical Details:

Click here to procede to a fuller discussion of the random walk with the mathematical details included. It is given in motion graphic gallery format where the animated images stay put as you scroll the text until you call forth another image commensurate with the new text you are reading.

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November 2006

Marvin Chester

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© m chester 2006 Occidental CA