Abstract not found

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 define a class of “algebraic” random matrices. These are random matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic, i.e., it satisfies a (bivariate) polynomial equation. The Wigner and Wishart matrices whose limiting eigenvalue distributions are given by the semi-circle law and the Marčenko-Pastur law are special cases. Algebraicity of a random matrix sequence is shown to act as a certificate of the computability of the limiting eigenvalue density function. The limiting moments of algebraic random matrix sequences, when they exist, are shown to satisfy a finite depth linear recursion so that they may often be efficiently enumerated in closed form. In this article, we develop the mathematics of the polynomial method which allows us to describe the class of algebraic matrices by its generators and map the constructive approach we employ when proving algebraicity into a software implementation that is available for download in the form of the RMTool random matrix “calculator” package. Our characterization of the closure of algebraic probability distributions under free additive and multiplicative convolution operations allows us to simultaneously establish a framework for computational (non-commutative) “free probability ” theory. We hope that the tools developed allow researchers to finally harness the power of the infinite random matrix theory.

We investigate the support of the equilibrium measure associated with a class of nonconvex, nonsmooth external fields on a finite interval. Such equilibrium measures play an important role in various branches of analysis. In this paper we obtain a sufficient condition which ensures that the support consists of at most two intervals. This is applied to external fields of the form −c sign(x)|x | α with c>0, α ≥ 1 and x ∈ [−1, 1]. If α is an odd integer, these external fields are smooth, and for this case the support was studied before by Deift, Kriecherbauer and McLaughlin, and by Damelin and Kuijlaars. 1. Introduction. In recent years, equilibrium measures with external fields have found an increasing number of applications in a variety of areas. We refer to [2, 3, 4, 5, 8, 10, 14, 15] for these relations, ranging from classical topics as

LetΛR denote the linear space overRspanned by zk, k∈Z. Define the real inner product (with varying exponential weights)〈·,·〉L:ΛR×Λ R→R, ( f, g)↦ → ∫ f (s)g(s) exp(−N V(s)) ds, N∈N, where R the external field V satisfies: (i) V is real analytic onR\{0}; (ii) lim|x|→∞(V(x) / ln(x2 +1))=+∞; and (iii) lim|x|→0(V(x) / ln(x−2 +1))=+∞. Orthogonalisation of the (ordered) base{1, z−1, z, z−2, z2,...,z −k, zk,...} with respect to〈·,·〉L yields the even degree and odd degree orthonormal Laurent polynomials (OLPs){φm(z)} ∞ m=0:φ2n(z) = ∑n k=−nξ(2n) z k k,ξ (2n) n> 0, andφ2n+1(z) = ∑n k=−n−1ξ(2n+1) z k k,ξ (2n+1)

LetΛR denote the linear space overRspanned by zk, k∈Z. Define the real inner product (with varying exponential weights)〈·,·〉L:ΛR×Λ R→R, ( f, g)↦ → ∫ f (s)g(s) exp(−N V(s)) ds, N∈N, where R the external field V satisfies: (i) V is real analytic onR\{0}; (ii) lim|x|→∞(V(x) / ln(x2 +1))=+∞; and (iii) lim|x|→0(V(x) / ln(x−2 +1))=+∞. Orthogonalisation of the (ordered) base{1, z−1, z, z−2, z2,...,z −k, zk,...} with respect to〈·,·〉L yields the even degree and odd degree orthonormal Laurent polynomials {φm(z)} ∞ (2n):φ2n(z)=ξ m=0 −n z−n +···+ξ (2n) n zn,ξ (2n) n>0, andφ2n+1(z)=ξ (2n+1) −n−1 z−n−1 +···+ξ (2n+1) n zn,ξ (2n+1)>0. Define

We analyze a continuum limit of the finite non-periodic Toda lattice through an associated constrained maximization problem over spectral density functions. The maximization problem was derived by Deift and McLaughlin using the Lax-Levermore approach, initially developed for the zero dispersion limit of the KdV equation. It encodes the evolution of the continuum limit for all times, including evolution through shocks. The formulation of gaps in the support of the maximizer is indicative of oscillations in the Toda lattice and the lack of strong convergence of the continuum limit. For large times, the maximizer tends to have zero gaps, which is the continuum analogue of the sorting property of the finite lattice. Using methods from logarithmic potential theory, we show that this behavior depends crucially on the initial data. We exhibit initial data for which the zero gap ansatz 1 holds uniformly in the spatial parameter (at large times), and other initial data for which this uniformity fails at all times. We then construct an example of C 1 smooth initial data generating, at a later time, infinitely many gaps in the support of the maximizer, while for even larger times, all gaps have closed. 1

We prove the bulk universality of the β-ensembles with non-convex regular analytic potentials for any β> 0. This removes the convexity assumption appeared in the earlier work [6]. The convexity condition enabled us to use the logarithmic Sobolev inequality to estimate events with small probability. The new idea is to introduce a “convexified measure ” so that the local statistics are preserved under this convexification.