2024/01/04 by Brennecken, Dominik, Rösler, Margit
#22E66 (Secondary) #33C67 (Primary) 33C52 #43A90 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2401.02515
We study the asymptotic behaviour of Bessel functions associated of root systems of type An-1 and type Bn with positive multiplicities as the rank n 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 A case, this gives a new and very natural approach to recent results by Assiotis and Najnudel in the context of β-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 \mathbb F = \mathbb R, \mathbb C, \mathbb H (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.