The paper by Jinho Baik, Gérard Ben Arous, and Sandrine Péché investigates the limiting distributions of the largest eigenvalue of a complex Gaussian sample covariance matrix as both the number of samples and the number of variables per sample become large. The authors focus on the case where all but finitely many eigenvalues of the covariance matrix are the same, and they characterize the dependence of the limiting distribution of the largest eigenvalue on these distinguished eigenvalues in terms of a sequence of new distribution functions that generalize the Tracy-Widom distributions from random matrix theory. A phase transition phenomenon is observed, where the fluctuation order changes from \(M^{2/3}\) to \(\sqrt{M}\) depending on the values of the distinguished eigenvalues. The results also apply to a last passage percolation model and a queuing model. The paper provides detailed asymptotic analysis and proofs for these findings, including the use of steepest-descent analysis and contour integration techniques.The paper by Jinho Baik, Gérard Ben Arous, and Sandrine Péché investigates the limiting distributions of the largest eigenvalue of a complex Gaussian sample covariance matrix as both the number of samples and the number of variables per sample become large. The authors focus on the case where all but finitely many eigenvalues of the covariance matrix are the same, and they characterize the dependence of the limiting distribution of the largest eigenvalue on these distinguished eigenvalues in terms of a sequence of new distribution functions that generalize the Tracy-Widom distributions from random matrix theory. A phase transition phenomenon is observed, where the fluctuation order changes from \(M^{2/3}\) to \(\sqrt{M}\) depending on the values of the distinguished eigenvalues. The results also apply to a last passage percolation model and a queuing model. The paper provides detailed asymptotic analysis and proofs for these findings, including the use of steepest-descent analysis and contour integration techniques.