  Multivariate Bessel processes, otherwise known as radial Dunkl processes, are
stochastic processes defined in a Weyl chamber that are repelled from the
latter's boundary by a singular drift with a strength given by the multiplicity
function $k$. It is a well-known fact that when $k$ is sufficiently small,
these processes hit the Weyl chamber's boundary almost surely, and it was
recently shown by the authors that the collision times for the process of type
$A$, also known as the Dyson model (a one-dimensional multiple-particle
stochastic system), have a Hausdorff dimension that depends on $k$. In this
paper, we use the square of the alternating polynomial, which corresponds to
the reflection group of the process, to extend this result to all multivariate
Bessel processes of rational type, and we show that the Hausdorff dimension of
collision times is a piecewise-linear function of the minimum of $k$, but is
independent of the dimension of the space where the process lives. This implies
that the Hausdorff dimension is independent of the particle number for
processes with a particle system representation.
