Airy limit for $\beta$-additions through Dunkl operators

It is well known that the edge limit of Gaussian/Laguerre Beta-ensembles, as well as a large class of $\beta$-ensembles is given by the $\mathrm{Airy}(\beta)$ point process. We extend this universality result to a general class of additions of Gaussian and Laguerre ensembles, which were identified in \cite{AN} as projection of the ergodic measures of the $\beta$-corners process. In order to make sense of the $\beta$-addition, we introduce the Type-A Bessel function as the characteristic function of our matrix ensemble, following the approach of \cite{GM}, \cite{BCG}. Then we extract its moment information through the action of Dunkl operators, and obtain certain limiting functional expressed via conditional Brownian bridges for the Laplace transform of $\mathrm{Airy}(\beta)$. Our limit expression is universal up to proper rescaling among all of our additions, and agrees with the single-time Laplace transform expression from the concurrent work \cite{GXZ}.