Results 1  10
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19
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 54 (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 36 (14 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 16 (8 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
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 10 (6 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.
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 9 (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
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 6 (5 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.
A problem of Totik on fast decreasing polynomials
, 1998
"... this paper is to solve the following problem posed by V. Totik in [4, Section 13.2, p.202]. ..."
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Cited by 3 (3 self)
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this paper is to solve the following problem posed by V. Totik in [4, Section 13.2, p.202].
Polynomial approximation with varying weights on compactsetsofthecomplex plane
 Proc. Amer. Math. Soc
"... Abstract. For a compact set E with connected complement, let A(E) bethe uniform algebra of functions continuous on E and analytic interior to E. We describe A(E,W), the set of uniform limits on E of sequences of the weighted polynomials {W n (z)Pn(z)} ∞ n=0, as n →∞, where W ∈ A(E) is a nonvanishin ..."
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Cited by 2 (1 self)
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Abstract. For a compact set E with connected complement, let A(E) bethe uniform algebra of functions continuous on E and analytic interior to E. We describe A(E,W), the set of uniform limits on E of sequences of the weighted polynomials {W n (z)Pn(z)} ∞ n=0, as n →∞, where W ∈ A(E) is a nonvanishing weight on E. If E has empty interior, then A(E,W) is completely characterized byazerosetZW ⊂ E. However, if E is a closure of Jordan domain, the description of A(E,W) also involves an inner function. In both cases, we exhibit the role of the support of a certain extremal measure, which is the solution of a weighted logarithmic energy problem, played in the descriptions of A(E,W). 1.