Random Walks

A random walk is often described as thinking of the path a drunkard takes going along a sidewalk from one lamp post to another. Each step that he takes could either be forwards or backwards with some probability. In fact you can generalize this to say that at each step , the drunk moves by a distance , from where he was before. is drawn from some arbitrary distribution, not necessarily a Gaussian.

So how do you describe this random walk? Using probability of course. Suppose for the moment that $\langle x_i \rangle = 0$ , that is he's equally likely to go forwards or backwards. How far on average will he have gone after steps? That is what's $\langle X\rangle$ where is the sum of all steps, that is the total displacement of the drunk as we define before in eqn. 1.14. Since the average of the sum is the sum of the average, $\langle X\rangle = 0$ in this case. So as expected, on average, the drunk doesn't move anywhere.

This is a rather incomplete description of the behavior of the drunk. How often does the drunk wake up in his vomit, just at the point where became completely wasted? He's still likely to have wandered quite far away from this point of complete inebriation, but whether it's up the sidewalk or down it, is impossible to tell.

To describe this, we also give the variance . As we just saw, section 1.6.1, variance is $N\sigma^2$ , where $\sigma^2$ is the variance of a single step.

This tells you that after steps, the drunk is likely to be found somewhere in a window of width roughly $\sqrt{N}\sigma$ . Note as noted above, the is proportional to $\sqrt{N}$ . Why?

If he were sober and walking forwards down the street, then after steps he would be a distance proportional to away from his starting point. For example if his step length was a yard, after 1024 steps, he'd be 1024 yards away. If he were drunk executing a random walk of the same number of steps, who knows where precisely he'd be. But he'd likely have wandered only $\sqrt{1024}= 32 yards$ away from his initial location. That's a lot less, as to be expected. Unlike steroids, drinking alcohol doesn't normally help you win any races, but despite that it's still more popular at parties.

This is a plot of how a typical random walk looks

$\begin{figure}\begin{center} \epsfig{width=.4\textwidth,file=1drw.eps} \end{center}\end{figure}$

which shows the distance gone versus

, the number of steps taken.

Here's a link showing a nice applet of a one dimensional random walk.

Keywords:

random walk theory

josh 2010-10-20