Learn More
We 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 of each sample become large. When all but finitely many, say r, eigenvalues of the covariance matrix are equal to 1, the dependence of the limiting distribution of the largest eigenvalue(More)
The purpose of this paper is to investigate the limiting distribution functions for a polynuclear growth model with two external sources, which was considered by Prähofer and Spohn in [13]. Depending on the strength of the sources, the limiting distribution functions are either the Tracy-Widom functions of random matrix theory, or a new explicit function(More)
We consider a spiked population model, proposed by Johnstone, whose population eigenvalues are all unit except for a few fixed eigenvalues. The question is to determine how the sample eigenvalues depend on the non-unit population ones when both sample size and population size become large. This paper completely determines the almost sure limits for a(More)
In this paper, we obtain optimal uniform lower tail estimates for the probability distribution of the properly scaled length of the longest up/right path of the last passage site percolation model considered by Johansson in [12]. The estimates are used to prove a lower tail moderate deviation result for the model. The estimates also imply the convergence of(More)
We compute the pointwise asymptotics of orthogonal polynomials with respect to a general class of pure point measures supported on finite sets as both the number of nodes of the measure and also the degree of the orthogonal polynomials become large. The class of orthogonal polynomials we consider includes as special cases the Krawtchouk and Hahn classical(More)
We obtain an identity between Fredholm determinants of two kinds of operators, one acting on functions on the unit circle and the other acting on functions on a subset of the integers. This identity is a generalization of an identity between a Toeplitz determinant and a Fredholm determinant that has appeared in the random permutation context. Using this(More)