This paper studies the phase transition of the largest eigenvalue for non-null complex sample covariance matrices. The authors compute the limiting distributions of the largest eigenvalue of a complex Gaussian sample covariance matrix when both the number of samples and the number of variables in each sample become large. They show that the dependence of the limiting distribution of the largest eigenvalue of the sample covariance matrix on the distinguished eigenvalues of the covariance matrix is characterized by an infinite sequence of new distribution functions that generalize the Tracy-Widom distributions of random matrix theory. A phase transition phenomenon is observed. The results also apply to a last passage percolation model and a queuing model. The paper considers M independent, identically distributed complex Gaussian samples, each of which is an N-dimensional vector. The sample covariance matrix is defined as the sample covariance of these vectors. The eigenvalues of the sample covariance matrix are denoted by λ₁ > λ₂ > ... > λ_N > 0. The authors show that when the number of samples and the number of variables in each sample become large, the limiting distribution of the largest eigenvalue of the sample covariance matrix depends on the distinguished eigenvalues of the covariance matrix. This dependence is characterized by an infinite sequence of new distribution functions that generalize the Tracy-Widom distributions of random matrix theory. The authors also show that when the covariance matrix has a non-null structure, the largest eigenvalue of the sample covariance matrix can be separated from the rest of the eigenvalues. This separation depends on the values of the distinguished eigenvalues of the covariance matrix. The authors find that the fluctuation order of the largest eigenvalue changes depending on the values of the distinguished eigenvalues. Specifically, when the distinguished eigenvalues are below a critical value, the fluctuation order is of order M^{2/3}, while when the distinguished eigenvalues are above the critical value, the fluctuation order is of order sqrt(M). The authors also show that the results apply to a last passage percolation model and a queuing model. The paper concludes with a conjecture that the results for real sample covariance matrices are similar to those for complex sample covariance matrices, with the same scaling.This paper studies the phase transition of the largest eigenvalue for non-null complex sample covariance matrices. The authors compute the limiting distributions of the largest eigenvalue of a complex Gaussian sample covariance matrix when both the number of samples and the number of variables in each sample become large. They show that the dependence of the limiting distribution of the largest eigenvalue of the sample covariance matrix on the distinguished eigenvalues of the covariance matrix is characterized by an infinite sequence of new distribution functions that generalize the Tracy-Widom distributions of random matrix theory. A phase transition phenomenon is observed. The results also apply to a last passage percolation model and a queuing model. The paper considers M independent, identically distributed complex Gaussian samples, each of which is an N-dimensional vector. The sample covariance matrix is defined as the sample covariance of these vectors. The eigenvalues of the sample covariance matrix are denoted by λ₁ > λ₂ > ... > λ_N > 0. The authors show that when the number of samples and the number of variables in each sample become large, the limiting distribution of the largest eigenvalue of the sample covariance matrix depends on the distinguished eigenvalues of the covariance matrix. This dependence is characterized by an infinite sequence of new distribution functions that generalize the Tracy-Widom distributions of random matrix theory. The authors also show that when the covariance matrix has a non-null structure, the largest eigenvalue of the sample covariance matrix can be separated from the rest of the eigenvalues. This separation depends on the values of the distinguished eigenvalues of the covariance matrix. The authors find that the fluctuation order of the largest eigenvalue changes depending on the values of the distinguished eigenvalues. Specifically, when the distinguished eigenvalues are below a critical value, the fluctuation order is of order M^{2/3}, while when the distinguished eigenvalues are above the critical value, the fluctuation order is of order sqrt(M). The authors also show that the results apply to a last passage percolation model and a queuing model. The paper concludes with a conjecture that the results for real sample covariance matrices are similar to those for complex sample covariance matrices, with the same scaling.