23 research outputs found
Random Walk in an Alcove of an Affine Weyl Group, and Non-Colliding Random Walks on an Interval
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 the circle.Comment: v.2, 22 pages; correction in a definition led to changes in many
formulas, also added more background, references, and example
Walks confined in a quadrant are not always D-finite
We consider planar lattice walks that start from a prescribed position, take
their steps in a given finite subset of Z^2, and always stay in the quadrant x
>= 0, y >= 0. We first give a criterion which guarantees that the length
generating function of these walks is D-finite, that is, satisfies a linear
differential equation with polynomial coefficients. This criterion applies,
among others, to the ordinary square lattice walks. Then, we prove that walks
that start from (1,1), take their steps in {(2,-1), (-1,2)} and stay in the
first quadrant have a non-D-finite generating function. Our proof relies on a
functional equation satisfied by this generating function, and on elementary
complex analysis.Comment: To appear in Theoret. Comput. Sci. (special issue devoted to random
generation of combinatorial objects and bijective combinatorics
Brownian motion in a truncated Weyl chamber
We examine the non-exit probability of a multidimensional Brownian motion
from a growing truncated Weyl chamber. Different regimes are identified
according to the growth speed, ranging from polynomial decay over
stretched-exponential to exponential decay. Furthermore we derive associated
large deviation principles for the empirical measure of the properly rescaled
and transformed Brownian motion as the dimension grows to infinity. Our main
tool is an explicit eigenvalue expansion for the transition probabilities
before exiting the truncated Weyl chamber