Airy limit for ${\hat{I}}^2$-additions through Dunkl operators

Abstract

It is well known that the edge limit of Gaussian/Laguerre Beta-ensembles, as well as a large class of ${\hat{I}}^2$-ensembles is given by the $\mathrm{Airy}({\hat{I}}^2)$ 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 ${\hat{I}}^2$-corners process. In order to make sense of the ${\hat{I}}^2$-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 $\mathrm{Airy}({\hat{I}}^2)$. 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}.

56 pages, v3: significantly revised based on the referee's comments