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36
Universality Of The Local Eigenvalue Statistics For A Class Of Unitary Invariant Random Matrix Ensembles
, 1997
"... The paper is devoted to the rigorous proof of the universality conjecture of the random matrix theory, according to which the limiting eigenvalue statistics of n \Theta n random matrices within spectral intervals of the order O(n \Gamma1 ) is determined by the type of matrices (real symmetric, Her ..."
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Cited by 85 (4 self)
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The paper is devoted to the rigorous proof of the universality conjecture of the random matrix theory, according to which the limiting eigenvalue statistics of n \Theta n random matrices within spectral intervals of the order O(n \Gamma1 ) is determined by the type of matrices (real symmetric, Hermitian or quaternion real) and by the density of states. We prove this conjecture for a certain class of the Hermitian matrix ensembles that arose in the quantum field theory and have the unitary invariant distribution defined by a certain function (the potential in the quantum field theory) satisfying some regularity conditions. Key words: random matrices, local asymptotic regime, universality conjecture, orthogonal polynomial technique. 1 Introduction. Problem and results. The random matrix theory (RMT) has been extensively developed and used in a number of areas of theoretical and mathematical physics. In particular the theory provides quite satisfactory description of fluctuations in s...
Generic Behavior of the Density of States in Random Matrix Theory and Equilibrium Problems in the Presence of Real Analytic External Fields
, 2000
"... 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 pos ..."
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Cited by 55 (15 self)
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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 EulerLagrange 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 oneparameter 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.
Hilbert analysis for orthogonal polynomials
 Orthogonal Polynomials and Special Functions (E. Koelink and W. Van Assche eds.) Lecture Notes in Mathematics 1817 (2003
"... Summary. This is an introduction to the asymptotic analysis of orthogonal polynomials based on the steepest descent method for RiemannHilbert problems of Deift and Zhou. We consider in detail the polynomials that are orthogonal with respect to the modified Jacobi weight (1 − x) α (1 + x) β h(x) on ..."
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Cited by 24 (9 self)
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Summary. This is an introduction to the asymptotic analysis of orthogonal polynomials based on the steepest descent method for RiemannHilbert problems of Deift and Zhou. We consider in detail the polynomials that are orthogonal with respect to the modified Jacobi weight (1 − x) α (1 + x) β h(x) on [−1, 1] where α, β> −1 and h is real analytic and positive on [−1, 1]. These notes are based on joint work with
Convergence analysis of Krylov subspace iterations with methods from potential theory
 SIAM Review
"... Abstract. Krylov subspace iterations are among the bestknown and most widely used numerical methods for solving linear systems of equations and for computing eigenvalues of large matrices. These methods are polynomial methods whose convergence behavior is related to the behavior of polynomials on t ..."
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Cited by 16 (2 self)
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Abstract. Krylov subspace iterations are among the bestknown and most widely used numerical methods for solving linear systems of equations and for computing eigenvalues of large matrices. These methods are polynomial methods whose convergence behavior is related to the behavior of polynomials on the spectrum of the matrix. This leads to an extremal problem in polynomial approximation theory: how small can a monic polynomial of a given degree be on the spectrum? This survey gives an introduction to a recently developed technique to analyze this extremal problem in the case of symmetric matrices. It is based on global information on the spectrum in the sense that the eigenvalues are assumed to be distributed according to a certain measure. Then depending on the number of iterations, the Lanczos method for the calculation of eigenvalues finds those eigenvalues that lie in a certain region, which is characterized by means of a constrained equilibrium problem from potential theory. The same constrained equilibrium problem also describes the superlinear convergence of conjugate gradients and other iterative methods for solving linear systems. Key words. Krylov subspace iterations, Ritz values, eigenvalue distribution, equilibrium measure, contrained equilibrium, potential theory AMS subject classifications. 15A18, 31A05, 31A15, 65F15 1. Introduction. Krylov
A weighted energy problem for a class of admissible weights
 Houston J. of Math
"... Abstract. We study the minimization problem for weighted logarithmic energy integrals over the set of probability Borel measures supported on a closed subset of the extended complex plane. The weight is a nonnegative upper semicontinuous function that behaves like 1/z  at infinity. We show that t ..."
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Cited by 15 (0 self)
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Abstract. We study the minimization problem for weighted logarithmic energy integrals over the set of probability Borel measures supported on a closed subset of the extended complex plane. The weight is a nonnegative upper semicontinuous function that behaves like 1/z  at infinity. We show that there exists a unique measure that minimizes the energy integral and we give a characterization of this measure in terms of a weighted logarithmic potential. 1. Statement of the problem and main results Definition 1. Let Σ be a closed subset of C. Let w: Σ → [0, ∞) be a function that is (i) upper semicontinuous, (ii) the set Σ0: = {z ∈ Σ: w(z)> 0} has positive logarithmic capacity, (iii) zw(z) → γ> 0 as z  → ∞ if Σ is unbounded set. Then, w is called weakly admissible weight. A function w: Σ → [0, ∞) is called admissible weight if it satisfies (i) and (ii), and if Σ is unbounded, zw(z) → 0 as z  → ∞. Definition 2. The external field Qw of a weight w is defined by (1.1) Qw(z): = log(1/w(z)), z ∈ Σ.
The Szegő curve, zero distribution and weighted approximation
 TRANS. AMER. MATH. SOC
, 1997
"... In 1924, Szegő showed that the zeros of the normalized partial sums, sn(nz), of ez tended to what is now called the SzegőcurveS,where S: = � z ∈ C: ze 1−z  =1andz  ≤1 �. Using modern methods of weighted potential theory, these zero distribution results of Szegő can be essentially recovered, al ..."
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Cited by 14 (7 self)
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In 1924, Szegő showed that the zeros of the normalized partial sums, sn(nz), of ez tended to what is now called the SzegőcurveS,where S: = � z ∈ C: ze 1−z  =1andz  ≤1 �. Using modern methods of weighted potential theory, these zero distribution results of Szegő can be essentially recovered, along with an asymptotic formula for the weighted partial sums {e−nzsn(nz)} ∞ n=0. We show that G: = Int S is the largest universal domain such that the weighted polynomials e−nzPn(z) are dense in the set of functions analytic in G. As an example of such results, it is shown that if f(z) isanalyticinGand continuous on G with f(1) = 0, then there is a sequence of polynomials {Pn(z)} ∞ n=0,withdegPn≤n, such that lim n→ ∞ �e−nzPn(z) − f(z)�G =0, where �·�G denotes the supremum norm on G. Similar results are also derived for disks. Let sn(z):= n� k=0 zk � ∞ k! n=0 1.
A graphbased equilibrium problem for the limiting distribution of nonintersecting Brownian . . .
, 2009
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Weighted polynomial approximation in the complex plane
 CONSTR. APPROX
, 1996
"... Given a pair (G, W) of an open bounded set G in the complex plane and a weight function W (z) which is analytic and different from zero in G, we consider the problem of the locally uniform approximation of any function f(z), which is analytic in G, by weighted polynomials of the form {W n (z)Pn(z) ..."
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Cited by 7 (6 self)
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Given a pair (G, W) of an open bounded set G in the complex plane and a weight function W (z) which is analytic and different from zero in G, we consider the problem of the locally uniform approximation of any function f(z), which is analytic in G, by weighted polynomials of the form {W n (z)Pn(z)} ∞ n=0, where deg Pn ≤ n. The main result of this paper is a necessary and sufficient condition for such an approximation to be valid. We also consider a number of applications of this result to various classical weights, which give explicit criteria for these weighted approximations.