Limit theorems for Bessel and Dunkl processes of large dimensions and free convolutions

Abstract We study Bessel and Dunkl processes ( X t , k ) t ≥ 0 on R N with possibly multivariate coupling constants k ≥ 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 β -Hermite and β -Laguerre ensembles. Moreover, for the frozen case k = ∞ , these processes degenerate to deterministic or pure jump processes. We use the generators for Bessel and Dunkl processes of types A and B and derive analogues of Wigner’s semicircle and Marchenko–Pastur limit laws for N → ∞ for the empirical distributions of the particles with arbitrary initial empirical distributions by using free convolutions. In particular, for Dunkl processes of type B new non-symmetric semicircle-type limit distributions on 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.