Percy Deift

On the distribution of the length of the longest increasing subsequence of random permutations

Let SN be the group of permutations of 1,2,..., N. If 7r E SN, we say that 7(i1),... , 7F(ik) is an increasing subsequence in 7r if il < i2 < ... < ik and 7r(ii) < 7r(i2) < ...< 7r(ik). Let 1N(r) be the length of the longest increasing subsequence. For example, if N = 5 and 7r is the permutation 5 1 3 2 4 (in one-line notation: thus 7r(1) = 5, 7r(2) = 1, ... ), then the longest increasing subsequences are 1 2 4 and 1 3 4, and N() = 3. Equip SN with uniform distribution,

A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation

In this article we present a new and general approach to analyzing the asymptotics of oscillatory Riemann-Hilbert problems. Such problems arise, in particular, when evaluating the long-time behavior of nonlinear wave equations solvable by the inverse scattering method. We will restrict ourselves here exclusively to the modified Korteweg-de Vries (MKdV) equation

Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory

We consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn(x)dx = e−nV(x)dx on the line as n ∞. The potentials V are assumed to be real analytic, with sufficient growth at infinity. The principle results concern Plancherel-Rotach-type asymptotics for the orthogonal polynomials down to the axis. Using these asymptotics, we then prove universality for a variety of statistical quantities arising in the theory of random matrix models, some of which have been considered recently in [31] and also in [4]. An additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials (see also [4]). The orthogonal polynomial problem is formulated as a Riemann-Hilbert problem following [19, 20]. The Riemann-Hilbert problem is analyzed in turn using the steepest-descent method introduced in [12] and further developed in [11, 13]. A critical role in our method is played by the equilibrium measure dμV for V as analyzed in [8].

Pointwise bounds on eigenfunctions and wave packets in

AbstractWe describe several new techniques for obtaining detailed information on the exponential falloff of discrete eigenfunctions ofN-body Schrödinger operators. An example of a new result is the bound (conjectured by Morgan)

-body quantum systems. IV

\left| {\psi (x_1 \ldots x_N )} \right| \leqq C\exp ( - \sum\limits_1^N {\alpha _n r_n )}

Long-Time Asymptotics for Integrable Nonlinear Wave Equations

for an eigenfunction ω of

Universality at the edge of the spectrum for unitary, orthogonal, and symplectic ensembles of random matrices

H_N = - \sum\limits_{i = 1}^N {(\Delta _i - } \left. {\frac{Z}{{\left| {x_i } \right|}}} \right) + \sum\limits_{i< j} {\left| {x_i - x_j } \right|^{ - 1} }