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

David J. Grabiner

To appear at Formal Power Series and Algebraic Combinatorics (FPSAC01), Tempe, Arizona (USA), May 20-26, 2001


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 the chambers 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).


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