The authors consider the length, l N , of the length of the longest increasing subsequence of a random permutation of N numbers. The main result in this paper is a proof that the distribution function for l N , suitably centered and scaled, converges to the Tracy-Widom distribution [TW1] of the largest eigenvalue of a random GUE matrix. The authors also prove convergence of moments. The proof is based on the steepest decent method for Riemann-Hilbert problems, introduced by Deift and Zhou in 1993 [DZ1] in the context of integrable systems. The applicability of the Riemann-Hilbert technique depends, in turn, on the determinantal formula of Gessel [Ge] for the Poissonization of the distribution function of l N .

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 in each sample become large. When all but finitely many, say r, eigenvalues of the covariance matrix are the same, the dependence of the limiting distribution of the largest eigenvalue of the sample covariance matrix on those distinguished r eigenvalues of the covariance matrix is completely characterized in terms of an infinite sequence of new distribution functions that generalize the Tracy-Widom distributions of the random matrix theory. Especially a phase transition phenomena is observed. Our results also apply to a last passage percolation model and a queuing model. Y := 1 M ( y 1 + · · · + y M ) and we set X = [ y 1 − Y , · · · , y M − Y ] to be the (centered) N × M sample matrix. Let S = 1 M XX ′ be the sample covariance matrix. The eigenvalues of S, called the 'sample eigenvalues', are denoted by λ 1 > λ 2 > · · · > λ N > 0. (The eigenvalues are simple with probability 1.) The probability space of λ j 's is sometimes called the Wishart ensemble (see e.g. [29]).Contrary to the traditional assumptions, it is of current interest to study the case when N is of same order as M . Indeed when Σ = I (null-case), several results are known. As N, M → ∞ such that M/N → γ 2 ≥ 1, the following holds.• Density of eigenvalues [27]: For any real x,

We consider a spiked population model, proposed by Johnstone, in which all the population eigenvalues are one 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 of the sample eigenvalues in a spiked model for a general class of samples.

We present a number of results relating partial Cauchy-Littlewood sums, integrals over the compact classical groups, and increasing subsequences of permutations. These include: integral formulae for the distribution of the longest increasing subsequence of a random involution with constrained number of fixed points; new formulae for partial Cauchy-Littlewood sums, as well as new proofs of old formulae; relations of these expressions to orthogonal polynomials on the unit circle; and explicit bases for invariant spaces of the classical groups, together with appropriate generalizations of the straightening algorithm.