  Asymptotic results are derived for the number of random walks in alcoves of
affine Weyl groups (which are certain regions in $n$-dimensional Euclidean
space bounded by hyperplanes), thus solving problems posed by Grabiner [J.
Combin. Theory Ser. A 97 (2002), 285-306]. These results include asymptotic
expressions for the number of vicious walkers on a circle, and as well for the
number of vicious walkers in an interval. The proofs depart from the exact
results of Grabiner [loc. cit.], and require as diverse means as results from
symmetric function theory and the saddle point method, among others.
