This is an expository account of the edge eigenvalue distributions in random matrix theory and their application in multivariate statistics. The emphasis is on the Painlevé representations of these distribution functions.

We present a new and relatively elementary method for studying the solution of the initial-value problem for dispersive linear and integrable equations in the large-t limit, based on a generalization of steepest descent techniques for Riemann-Hilbert problems to the setting of ∂-problems. Expanding upon prior work [9] of the first two authors, we develop the method in detail for the linear and defocusing nonlinear Schrödinger equations, and show how in the case of the latter it gives sharper asymptotics than previously known under essentially minimal regularity assumptions on initial data. acting in L 2 (R; C 2 ) as described, for example, in [6].The modulus |α(z 0 )| of the complex amplitude α(z 0 ) as written in (4) was first obtained by Segur and Ablowitz [18] from trace formulae under the assumption that q(x, t) has the form (3) where E (x, t) is small for large t. Zakharov and Manakov [19] took the form (3) as an ansatz to motivate a kind of WKB analysis of the reflection coefficient r(z) and as a consequence were able to also calculate the phase of α(z 0 ), obtaining for the first time the phase as written in (5). Its [10] was the first to observe the key role played in the large-time behavior of q(x, t) by an "isomondromy" problem for parabolic cylinder functions; this problem has been an essential ingredient in all subsequent studies of the large-t limit and as we shall see Given the reflection coefficient r ∈ H 1,1 1 (R) associated with initial data q 0 ∈ H 1,1 (R) via the spectral transform R for the Zakharov-Shabat operator L, the solution of the Cauchy problem for the nonlinear Schrödinger equation (1) may be described as follows. Consider the following Riemann-Hilbert problem:Riemann-Hilbert Problem 1. Given parameters (x, t) ∈ R 2 , find M = M(z) = M(z; x, t), a 2 × 2 matrix, satisfying the following conditions:Analyticity: M is an analytic function of z in the domain C \ R. Moreover, M has a continuous extension to the real axis from the upper (lower) half-plane denoted M + (z) (M − (z)) for z ∈ R. Jump condition:The boundary values satisfy the jump condition

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