Abstract not found

A lock step walk is a one-dimensional integer lattice walk in discrete time. Suppose that initially there are infinitely many walkers on the non-negative even integer sites. At each moment of time, every walker moves either to its left or to its right with equal probability. The only constraint is that no two walkers can occupy the same site at the same time. Hence we describe the walk as vicious. It is proved that as time tends to infinity, a certain limiting conditional distribution of the displacement of the leftmost walker is identical to the limiting distribution of the (scaled) largest eigenvalue of a random GOE matrix (GOE Tracy-Widom distribution). The proof is based on the bijection between path configurations and semistandard Young tableaux established recently by Guttmann, Owczarek and Viennot. The distribution of semistandard Young tableaux is analyzed using the Hankel determinant expression for the probability obtained from the work of Rains and the author. The asymptotics of the Hankel determinant are then obtained by applying the Deift-Zhou steepest-descent method to the Riemann-Hilbert problem for the related orthogonal polynomials. 1

Abstract: We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an orthogonal polynomial ensemble. The most prominent example is apparently the Hermite ensemble, the eigenvalue distribution of the Gaussian Unitary Ensemble (GUE), and other well-known ensembles known in random matrix theory like the Laguerre ensemble for the spectrum of Wishart matrices. In recent years, a number of further interesting models were found to lead to orthogonal polynomial ensembles, among which the corner growth model, directed last passage percolation, the PNG droplet, non-colliding random processes, the length of the longest increasing subsequence of a random permutation, and others. Much attention has been paid to universal classes of asymptotic behaviors of these models in the limit of large particle numbers, in particular the spacings between the particles and the fluctuation behavior of the largest particle. Computer simulations suggest that the connections go even farther

Abstract not found

The equilibrium measure in the presence of an external field plays a role in a number of areas in analysis, for example in random matrix theory: the limiting mean density of eigenvalues is precisely the density of the equilibrium measure. Typical behavior for the equilibrium measure is: 1. it is positive on the interior of a finite number of intervals, 2. it vanishes like a square root at endpoints, and 3. outside the support, there is strict inequality in the Euler-Lagrange variational conditions. If these conditions hold, then the limiting local eigenvalue statistics is loosely described by a "bulk" in which there is universal behavior involving the sine kernel, and "edge effects" in which there is a universal behavior involving the Airy kernel. Through techniques from potential theory and integrable systems, we show that this "regular" behavior is generic for equilibrium measures associated with real analytic external fields. In particular, we show that for any one-parameter family of external fields V=c the equilibrium measure exhibits this regular behavior, except for an at most countable number of values of c. We discuss applications of our results to random matrices, orthogonal polynomials and integrable systems.

We establish universality of local eigenvalue correlations in unitary random matrix ensembles Zn dM near the origin of the spectrum. If V is even, and if the recurrence coe#cients of the orthogonal polynomials associated with have a regular limiting behavior, then it is known from work of Akemann et al., and Kanzieper and Freilikher that the local eigenvalue correlations have universal behavior described in terms of Bessel functions. We extend this to a much wider class of confining potentials V . Our approach is based on the steepest descent method of Deift and Zhou for the asymptotic analysis of Riemann-Hilbert problems. This method was used by Deift et al. to establish universality in the bulk of the spectrum. A main part of the present work is devoted to the analysis of a local Riemann-Hilbert problem near the origin.

Abstract not found

The continuum limit of the Toda lattice was studied by Deift and McLaughlin in the spirit of the Lax-Levermore theory for the zero dispersion limit of the Kortewegde Vries equation. An important role is played by a quadratic minimization problem arising from an asymptotic analysis of the inverse spectral transform. The minimum is taken over density functions ψ in the spectral variable satisfying the constraints 0 ≤ ψ ≤ φ where φ is a function determined by the initial conditions. The finite gap ansatz is said to hold if the set where the two constraints are not effective consists of a finite union of intervals. If the finite gap ansatz hold, weak limits are described in terms of the endpoints of the intervals. Using techniques from logarithmic potential theory, we show that the finite gap ansatz holds for real analytic spectral data. This extends a previous result of Deift, Kriecherbauer and McLaughlin for the situation without upper constraint φ. 1

We give a new proof of universality properties in the bulk of spectrum of the hermitian matrix models, assuming that the potential that determines the model is globally C 2 and locally C 3 function (see Theorem 3.1). The proof as our previous proof in [21] is based on the orthogonal polynomial techniques but does not use asymptotics of orthogonal polynomials. Rather, we obtain the sin-kernel as a unique solution of a certain non-linear integro-differential equation that follows from the determinant formulas for the correlation functions of the model. We also give a simplified and strengthened version of paper [1] on the existence and properties of the limiting Normalized Counting Measure of eigenvalues. We use these results in the proof of universality and we believe that they are of independent interest.