Limits of Bessel functions for root systems as the rank tends to infinity

We study the asymptotic behaviour of Bessel functions associated to root systems of type and type with positive multiplicities as the rank tends to infinity. In both cases, we characterize the possible limit functions and the Vershik–Kerov type sequences of spectral parameters for which such limits exist. In the type case, this gives a new and very natural approach to recent results by Assiotis and Najnudel in the context of -ensembles in random matrix theory. These results generalize known facts about the approximation of the positive-definite Olshanski spherical functions of the space of infinite-dimensional Hermitian matrices over (with the action of the associated infinite unitary group) by spherical functions of finite-dimensional spaces of Hermitian matrices. In the type B case, our results include asymptotic results for the spherical functions associated with the Cartan motion groups of non-compact Grassmannians as the rank goes to infinity, and a classification of the Olshanski spherical functions of the associated inductive limits.