Self-Intersections of Two-Dimensional Equilateral Random Walks

Start: 02/08/2012 - 1:15pm
End  : 02/08/2012 - 2:15pm

Applied Math Seminar

Nicholas Pippenger, HMC


We consider the classical "drunkard's walk" as a polygonal path of n steps in the plane, in which each side has unit length and the angles between successive sides are independent random variables uniformly distributed on the circle. How may self-intersections does such a walk have? We show that the mean number of self-intersections is $ (2/\pi^2) n log n+O(n) $, and that the variance is $ O(n^2) $ (which implies that the distribution is concentrated around its mean). We also obtain analogous results for closed polygons (walks conditioned to return to their origin after n steps), and discuss its possible extensions to other models.

This work was done jointly with Max Kutler and Maggie Rogers (both HMC Math '11).

RN 103