Pradeep Mutalik has an interesting column in Quanta:
One of the most cherished mathematical learning moments of my youth came from an old and very funny math book, whose name I have sadly forgotten. It was through a cartoon in that book that I first learned how to figure out the distance covered by a completely random form of movement — the drunkard’s walk. The first panel of the cartoon showed a disheveled man near a lamppost. His future path was represented by a series of wild zigs and zags on the path in front of him, shown as a dotted line. “I know how to figure out how far I’ll be from this point on average,” he says. “All I have to do is to measure the average length of my zigs and zags, and multiply by the square root of their number.” Then, in the second panel, he pulls out a bottle from his coat pocket, lifts it toward his mouth, and says, “But first, I’ll have a little drink!” Now, where are the funny math books today?
Randomness is an inextricable and essential aspect of our world. Combined with selection, randomness can do incredible things: It has powered evolution and created the entire biological world. Yet randomness is commonly underestimated and misunderstood. Certain observed phenomena prompt many people to attribute magical causes to events and imbue people with supernatural abilities, when the workings of randomness are all we need to explain their observations. Of course, probability theorists have always known that randomness, to a large extent, rules our lives, as the author Leonard Mlodinow explains in his delightful book The Drunkard’s Walk. Recently, researchers have penetrated deeper into the intricacies of randomness, as Kevin Hartnett reports in the Quanta article “A Unified Theory of Randomness.” Hartnett’s article explains how this unified theory of randomness is informed by variants of the same random phenomenon we alluded to earlier: the random walk, or, as it is more colorfully named, the drunkard’s walk. This phenomenon explains the diffusion of fluids and also describes Brownian motion, which Einstein famously analyzed to determine the existence and size of atoms.
But back to our drunkard. When an object or a person moves randomly, the average distance it will be from its starting point can be predicted to be x times the square root of n, where x is the average length of each step and n is the number of steps. The more the drunkard walks, the farther he gets from his starting point. Why should this be when his steps are random? Here’s a very nice informal argument presented by Marty Green, that helps to understand this result for equal-size steps. In the figure above, which has been reproduced from Green’s blog, let us suppose that the drunkard started from the lamppost (center of the circle), took several one-meter steps, and by chance found himself on the circumference of the circle, say, five meters away. After the next step, he will be on the circumference of the smaller circle of radius one meter centered on the point where he had been. More than half of this circle (the green part) is outside the five-meter circle. So the next step will be more likely to take him farther away from the lamppost rather than closer, and this is true no matter where he is at a given time. How much farther will he go? In order to determine the new average distance we will have to integrate all his possible distances around the circumference of the circle. Some points on it are closer to the origin than before, and a larger number are farther. The most “neutral” step he could take is perpendicular to the big circle’s radius: one meter along the tangent. Coincidentally, considering just this one neutral direction gives the right answer. One meter along the tangent would place him, by Pythagoras’ theorem, . . .