article published yes Ji Lee author Kevin Schnelli author 434AD0AE-F248-11E8-B48F-1D18A9856A870000-0003-0954-3231 LaEr department Random matrices, universality and disordered quantum systems project We consider sample covariance matrices of the form Q = ( σ1/2X)(σ1/2X)∗, where the sample X is an M ×N random matrix whose entries are real independent random variables with variance 1/N and whereσ is an M × M positive-definite deterministic matrix. We analyze the asymptotic fluctuations of the largest rescaled eigenvalue of Q when both M and N tend to infinity with N/M →d ϵ (0,∞). For a large class of populations σ in the sub-critical regime, we show that the distribution of the largest rescaled eigenvalue of Q is given by the type-1 Tracy-Widom distribution under the additional assumptions that (1) either the entries of X are i.i.d. Gaussians or (2) that σ is diagonal and that the entries of X have a sub-exponential decay. Institute of Mathematical Statistics2016 eng 1409.4979 00039124010001610.1214/16-AAP1193 2663786 - 3839 Lee, Ji, and Kevin Schnelli. “Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population.” <i>Annals of Applied Probability</i>. Institute of Mathematical Statistics, 2016. <a href="https://doi.org/10.1214/16-AAP1193">https://doi.org/10.1214/16-AAP1193</a>. J. Lee, K. Schnelli, Annals of Applied Probability 26 (2016) 3786–3839. Lee J, Schnelli K. 2016. Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population. Annals of Applied Probability. 26(6), 3786–3839. Lee J, Schnelli K. Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population. <i>Annals of Applied Probability</i>. 2016;26(6):3786-3839. doi:<a href="https://doi.org/10.1214/16-AAP1193">10.1214/16-AAP1193</a> Lee, Ji, and Kevin Schnelli. “Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population.” <i>Annals of Applied Probability</i>, vol. 26, no. 6, Institute of Mathematical Statistics, 2016, pp. 3786–839, doi:<a href="https://doi.org/10.1214/16-AAP1193">10.1214/16-AAP1193</a>. J. Lee and K. Schnelli, “Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population,” <i>Annals of Applied Probability</i>, vol. 26, no. 6. Institute of Mathematical Statistics, pp. 3786–3839, 2016. Lee, J., &#38; Schnelli, K. (2016). Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population. <i>Annals of Applied Probability</i>. Institute of Mathematical Statistics. <a href="https://doi.org/10.1214/16-AAP1193">https://doi.org/10.1214/16-AAP1193</a> 11572018-12-11T11:50:27Z2025-09-22T09:55:43Z