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Random Walks and Plane Arrangementsin Three
| Content Provider | Semantic Scholar |
|---|---|
| Author | Billera, D. J. Brown, Kenneth S. Abstract, Persi D. I. A. C. O. N. I. S. |
| Abstract | 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 ?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 ? 2. We further show that the walk is close to its stationary distribution after two steps if n is large. |
| File Format | PDF HTM / HTML |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
