A Note on Optimal Soft Edge Expansions for the Gaussian $β$ Ensembles

TL;DR

This note surveys optimal soft edge expansions for Gaussian $\beta$ ensembles, detailing the GUE case and drawing parallels to GOE/GSE, while proposing a differential-equation framework to derive higher-order corrections for general $\beta$. For GUE, the soft edge limit is $\rho_{(1),N}^{\rm GUE,s}(y) \to - y Ai(y)^2 + (Ai'(y))^2$, with initial $N^{-2/3}$ and $N^{-4/3}$ corrections expressed as Airy-function combinations, and a similar Airy-structure governs the kernel. In GOE/GSE, a shifted scale $N' = N + (\beta-2)/(2\beta)$ yields an analogous $N'^{-2/3}$ expansion with a five-function basis, linked by a duality in moments, and extendable to Laguerre/Jacobi ensembles. The authors propose to extend the differential-equation approach to general even $\beta$ (and beyond) by exploiting third-order global equations and their soft-edge recasts, using hierarchical recurrences for correction terms and explicit Laplace-transform relations, with the aim of obtaining systematic higher-order soft-edge corrections for all Gaussian $\beta$ ensembles.

Abstract

We present some review material relating to the topic of optimal asymptotic expansions of correlation functions and associated observables for $β$ ensembles in random matrix theory. We also give an introduction to a related line of study that we are presently undertaking.