Do loop equations uniquely characterize the Airy1 point process?

Determine whether the hierarchy of microscopic loop equations at the spectral edge uniquely characterizes the Airy1 point process. Concretely, ascertain if any point process on the real line whose normalized Stieltjes transform S(w) satisfies the Airy1 loop equations (for example, the first-order equation E[(S(w) − √w)^2 + 2√w(S(w) − √w) + ∂_w S(w)] = 0 for w in the upper half-plane) must coincide in law with the Airy1 point process.

Background

In the paper, the authors derive and exploit microscopic loop equations at the spectral edge to prove edge universality and optimal eigenvalue rigidity for random d-regular graphs. They show that the Airy1 point process, which describes the edge scaling limit for GOE-type ensembles, satisfies a hierarchy of loop equations and that the same equations hold for the models they paper.

While these loop equations are necessary for Airy statistics, it remains an unresolved question whether they are also sufficient—namely, whether they uniquely pin down the Airy1 point process. Establishing such a uniqueness result would provide a characterization-by-equations principle for edge limits, potentially simplifying universality proofs and extending to other ensembles.