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

2000/11/27 by David J. Grabiner, Grabiner, David J.
Mathematics · #05A15 (secondary) #60G50 (primary) #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR) #math.CO #math.PR #msc:05A15 #msc:60G50

paper · pdf · doi:10.48550/arxiv.math/0011218

v.2, 22 pages; correction in a definition led to changes in many formulas, also added more background, references, and examples

arxiv created 2001/06/04 · arxiv updated 2009/11/30

Abstract

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>x₁>x₂>...>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 the circle.

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