Random Walk in an Alcove of an Affine Weyl Group, and Non-colliding Random Walks on an Interval

✍ Scribed by David J. Grabiner

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 159 KB
• Volume: 97
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.2001.3216

✦ Synopsis

We use a reflection argument, introduced by Gessel and Zeilberger, to count the number of k-step walks between two points which stay within a chamber of a Weyl group. We apply this technique to walks in the alcoves of the classical affine Weyl groups. In all cases, we get determinant formulas for the number of k-step walks. One important example is the region m > x 1 > x 2 > • • • > x n > 0, which is a rescaled alcove of the affine Weyl group C ˜n. If each coordinate is considered to be an independent particle, this models n non-colliding random walks on the interval (0, m). Another case models n non-colliding random walks on a circle.