Ivars Peterson's MathTrek
One consequence of this type of erratic movement back and forth is that a random walker is certain to return to its starting pointeventually.
In two dimensions, a walker can take steps east, west, north, or south, randomly choosing a direction with equal probability. It's like going from vertex to vertex on an infinite checkerboard grid. If such a walk continues for an arbitrarily long time, the walker is certain to touch every vertex. This includes the case of returning to the starting point.
For his Intel Science Talent Search math project, Yi Sun of the The Harker School in San Jose, Calif., worked out the expected number of steps it takes a walker on a two-dimensional grid to encircle a given point (origin). One such circuit corresponds to a winding number of 1.
Yi discovered that the expected number of steps to complete such a circuit in two dimensions is infinite. He also derived an explicit (very complicated) formula for the expected value of the winding number after n steps.
Yi says that his result has applications to modeling polymer behavior around a rod and may lead to a deeper understanding of the structure and properties of random walks.
For his work, Sun won second place in the 2006 STS competition for high school seniors.
Copyright © 2006 by Ivars Peterson
For a profile of Yi Sun, go to http://www.sciserv.org/sts/65sts/Sun.asp.
Information about the Intel Science Talent Search can be found at http://www.sciserv.org/sts/.