Airy limit for $β$-additions through Dunkl operators

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

💡 Research Summary

The paper investigates the edge behavior of a broad class of “β‑additions”, which are generalized matrix sums defined for any positive β by means of Type‑A Bessel generating functions rather than concrete random matrices. Starting from the observation that classical Gaussian (GβE) and Laguerre (LβE) β‑ensembles admit a multivariate Bessel function as a characteristic function, the authors define the β‑addition of several such ensembles as the product of their Bessel generating functions. This definition coincides with the projection of the ergodic measures of the β‑corners process, as identified in earlier work.

To study the fluctuations of the largest eigenvalues, the authors employ a moment method based on Dunkl operators of type A. These differential‑difference operators act naturally on the Bessel functions and allow one to express moments of power sums (\sum_{i=1}^N \lambda_i^k) as repeated applications of Dunkl operators to the generating function. By translating the action of the operators into a combinatorial picture of “walks and blocks”, they connect the algebraic expressions to conditional Brownian bridges that stay non‑negative. The walks are classified according to local configurations; delicate cancellations among high‑order terms are proved, and precise tail estimates are obtained.

The central result (Theorem 1.2) states that, under mild regularity assumptions on the parameters (including positivity of certain linear combinations (\kappa_\ell)), the rescaled eigenvalues \