Abstract : We study Bessel and Dunkl processes $\left(X_{t, k}\right)_{t>0}$ on $\mathbb{R}^{N}$ with possibly multivariate coupling constants $k \geq 0$. These processes describe interacting particle systems of Calogero-Moser-Sutherland type with $N$ particles. For the root systems $A_{N-1}$ and $B_{N}$ these Bessel processes are related with $\beta$-Hermite and $\beta$-Laguerre ensembles. Moreover, for the frozen case $k=\infty$, these processes degenerate to deterministic or pure jump processes. We use the generators for Bessel and Dunkl processes of types $\mathrm{A}$ and $\mathrm{B}$ and derive analogues of Wigner's semicircle and Marchenko-Pastur limit laws for $N \rightarrow \infty$ for the empirical distributions of the particles with arbitrary initial empirical distributions by using free convolutions. In particular, for Dunkl processes of type $\mathrm{B}$ new non-symmetric semicircle-type limit distributions on $\mathbb{R}$ appear. Our results imply that the form of the limiting measures is already completely determined by the frozen processes. Moreover, in the frozen cases, our approach leads to a new simple proof of the semicircle and Marchenko-Pastur limit laws for the empirical measures of the zeroes of Hermite and Laguerre polynomials respectively. (based on joint work with Michael Voit)

Keywords : Bessel and Dunkl processes; interacting particle systems; Hermite ensembles; Laguerre ensembles; free convolution