Airy edge universality for general invariant ensemble additions

Fix $\beta>0$ and deterministic sequences of $N$-tuples $Vec{a}_i(N)$, for $i=1,2,\ldots,k$, and let

$$Vec{\lambda}(N)=Vec{a}_1(N)+\cdots+Vec{a}_k(N).$$

Assume that, for each $i$, the empirical measure of $Vec{a}_i(N)$ converges as $N\to\infty$ to a deterministic probability measure $mu_i$ on $\mathbb{R}$. Let $boxplus$ denote free convolution and set

$$mu=mu_1\boxplusmu_2\boxplus\cdots\boxplusmu_k.$$

Assume without loss of generality that $|\mu_+|\geq|\mu_-|$.

Airy edge universality conjecture. Under some mild conditions on the summands, the upper edge limit of $Vec{\lambda}(N)$ near $mu_+$ is still given by $\operatorname{Airy}(\beta)$ under proper rescaling.

This conjecture proposes universality of the upper-edge limit beyond the Gaussian and Laguerre $\beta$-ensemble additions treated in the paper, suggesting that the limiting edge behavior does not depend on the individual eigenvalue distributions of the summands. The precise mild conditions remain unspecified in the source.