Random Walks and Plane Arrangements in Three Dimensions
Louis J. Billera, Kenneth S. Brown, Persi W. Diaconis · American Mathematical Monthly · 1999
This paper explains some modern geometry and probability in the course of solving a random walk problem. Consider n planes through the origin in three dimensional Euclidean space. Assume, for simplicity, that they are in "general position". They then divide space into n(n \\Gamma 1)+2 regions. We study a random walk on these regions. Suppose the walk is in region C. Pick a pair of the planes at random. These determine a line through the origin. Pick one of the two halves of the line with equal probability. The walk now moves to the region adjacent to the chosen half-line which is closest to C. We determine the long-term stationary distribution: All regions of i sides have stationary probability proportional to i \\Gamma 2. We further show that the walk is close to its stationary distribution after two steps if n is large.