  This paper studies the asymptotic behavior of several central objects in
Dunkl theory as the dimension of the underlying space grows large. Our starting
point is the observation that a recent result from the random matrix theory
literature implies a large deviations principle for the hydrodynamic limit of
radial Dunkl processes. Using this fact, we prove a variational formula for the
large-$N$ asymptotics of generalized Bessel functions, as well as a large
deviations principle for the more general family of radial Heckman-Opdam
processes. As an application, we prove a theorem on the asymptotic behavior of
weight multiplicities of irreducible representations of compact or complex
simple Lie algebras in the limit of large rank. The theorems in this paper
generalize several known results describing analogous asymptotics for Dyson
Brownian motion, spherical matrix integrals, and Kostka numbers.
