we show that the Tracy-Widom limit law does not depend of how the sequence $(n,p)$ is embedded. As long as both dimensions go to infinity, the properly re-centered and re-scaled largest eigenvalue converges weakly to this law.\
We can compare this with the “classical" situation where $p$ is held fixed, in which case the limiting joint distribution is known, too (see e.g @anderson, corollary 13.3.2). In this case, the centering is done around $n$ and the scaling is $\sqrt{n}$; elementary computations show that $(l_1-\mu_{np})/\sigma_{np}$ also has a non-degenerate limiting distribution (possibly changing with each $p$). Nevertheless, even with the classical centering, it is hard to evaluate the marginals in this context and the results are therefore difficult to use in practice.

Our simulation results show that the Tracy-Widom approximation is reasonably good (for the upper quantiles) even when $p$ or $n$ are small. As remarked by @imj, Proposition 1.2, this implies that when doing PCA, one could develop (conservative) tests based on the Tracy-Widom distribution that could serve as alternatives to the scree plot or the Wachter plot.\
The paper is organized as follows: after presenting (Section 2) the main elements of the proof of Theorem \[ThJ\], we describe (Section 