Thus, natural models of population structure predict that most of the eigenvalues of the theoretical covariance will be “small,” nearly equal, and arise from sampling noise, while just a few eigenvalues will be “large,” reflecting past demographic events. This is exactly the situation that Johnstone's application of Tracy–Widom theory addresses. We also note that on real data (as we show below), the TW theory works extremely well, which shows that the model will be a reasonable approximation to “truth” in many cases.
