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40
Smooth Analysis of the Condition Number and the Least Singular Value
 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation
, 2009
"... Abstract. Let x be a complex random variable with mean zero and bounded variance. Let Nn be the random matrix of size n whose entries are iid copies of x and M be a fixed matrix of the same size. The goal of this paper is to give a general estimate for the condition number and least singular value o ..."
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Abstract. Let x be a complex random variable with mean zero and bounded variance. Let Nn be the random matrix of size n whose entries are iid copies of x and M be a fixed matrix of the same size. The goal of this paper is to give a general estimate for the condition number and least singular value of the matrix M + Nn, generalizing an earlier result of Spielman and Teng for the case when x is gaussian. Our investigation reveals an interesting fact that the “core ” matrix M does play a role on tail bounds for the least singular value of M + Nn. This does not occur in SpielmanTeng studies when x is gaussian. Consequently, our general estimate involves the norm ‖M‖. In the special case when ‖M ‖ is relatively small, this estimate is nearly optimal and extends or refines existing results. 1.
On the Law of Addition of Random Matrices
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2000
"... Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U ∗ n BnUn) is studied in the limit of large matrix order n. Converg ..."
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Cited by 6 (1 self)
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Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U ∗ n BnUn) is studied in the limit of large matrix order n. Convergence in probability to a limiting nonrandom measure is established. A functional equation for the Stieltjes transform of the limiting measure in terms of limiting eigenvalue measures of An and Bn is obtained and studied.
Random matrices: A general approach for the least singular value problem, preprint
"... Abstract. Let x be a complex random variable with mean zero and bounded variance. Let Nn be the random matrix of size n whose entries are iid copies of x and M be an arbitrary matrix. We give a general estimate for the least singular value of the matrix Mn: = M + Nn. In various special cases, our es ..."
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Cited by 5 (4 self)
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Abstract. Let x be a complex random variable with mean zero and bounded variance. Let Nn be the random matrix of size n whose entries are iid copies of x and M be an arbitrary matrix. We give a general estimate for the least singular value of the matrix Mn: = M + Nn. In various special cases, our estimate extends or refines previous known results. 1.
Kuijlaars: A phase transition for nonintersecting Brownian motions, and the Painlevé II equation
, 809
"... We consider n nonintersecting Brownian motions with two fixed starting positions and two fixed ending positions in the large n limit. We show that in case of ‘large separation ’ between the endpoints, the particles are asymptotically distributed in two separate groups, with no interaction between t ..."
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Cited by 5 (2 self)
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We consider n nonintersecting Brownian motions with two fixed starting positions and two fixed ending positions in the large n limit. We show that in case of ‘large separation ’ between the endpoints, the particles are asymptotically distributed in two separate groups, with no interaction between them, as one would intuitively expect. We give a rigorous proof using the RiemannHilbert formalism. In the case of ‘critical separation’ between the endpoints we are led to a model RiemannHilbert problem associated to the HastingsMcLeod solution of the Painlevé II equation. We show that the Painlevé II equation also appears in the large n asymptotics of the recurrence coefficients of the multiple Hermite polynomials that are associated with the RiemannHilbert problem.
A simple approach to global regime of the random matrix theory
 Mathematical Results in Statistical Mechanics. Singapore: World Scientific
, 1999
"... Abstract. We discuss a method of the asymptotic computation of moments of the normalized eigenvalue counting measure of random matrices of large order. The method is based on the resolvent identity and on some formulas relating expectations of certain matrix functions and the expectations including ..."
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Abstract. We discuss a method of the asymptotic computation of moments of the normalized eigenvalue counting measure of random matrices of large order. The method is based on the resolvent identity and on some formulas relating expectations of certain matrix functions and the expectations including their derivatives or, equivalently, on some simple formulas of the perturbation theory. In the framework of this unique approach we obtain functional equations for the Stieltjes transforms of the limiting normalized eigenvalue counting measure and the bounds for the rate of convergence for the majority known random matrix ensembles. 1.
Operatorvalued semicircular elements: solving a quadratic matrix equation with positivity constraints
 ID rnm086
"... Abstract. We show that the quadratic matrix equation V W + η(W)W = I, for given V with positive real part and given analytic mapping η with some positivity preserving properties, has exactly one solution W with positive real part. Also we provide and compare numerical algorithms based on the iterati ..."
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Cited by 4 (2 self)
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Abstract. We show that the quadratic matrix equation V W + η(W)W = I, for given V with positive real part and given analytic mapping η with some positivity preserving properties, has exactly one solution W with positive real part. Also we provide and compare numerical algorithms based on the iteration underlying our proofs. This work bears on operatorvalued free probability theory, in particular on the determination of the asymptotic eigenvalue distribution of band or block random matrices. 1.
GAUSSIAN FLUCTUATIONS OF EIGENVALUES IN WIGNER RANDOM MATRICES
, 2009
"... We study the fluctuations of eigenvalues from a class of Wigner random matrices that generalize the Gaussian orthogonal ensemble. We begin by considering an n × n matrix from the Gaussian orthogonal ensemble (GOE) or Gaussian symplectic ensemble (GSE) and let xk denote eigenvalue number k. Under th ..."
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We study the fluctuations of eigenvalues from a class of Wigner random matrices that generalize the Gaussian orthogonal ensemble. We begin by considering an n × n matrix from the Gaussian orthogonal ensemble (GOE) or Gaussian symplectic ensemble (GSE) and let xk denote eigenvalue number k. Under the condition that both k and n − k tend to infinity as n → ∞, we show that xk is normally distributed in the limit. We also consider the joint limit distribution of eigenvalues (xk1,..., xkm from the GOE or GSE where k1, n − km and ki+1 − ki, 1 ≤ i ≤ m − 1, tend to infinity with n. The result in each case is an mdimensional normal distribution. Using a recent universality result by Tao and Vu, we extend our results to a class of Wigner real symmetric matrices with nonGaussian entries that have an exponentially decaying distribution and whose first four moments match the Gaussian moments.
Lectures on random matrix models. The RiemannHilbert approach
, 2008
"... This is a review of the RiemannHilbert approach to the large N asymptotics in random matrix models and its applications. We discuss the following topics: random matrix models and orthogonal polynomials, the RiemannHilbert approach to the large N asymptotics of orthogonal polynomials and its appli ..."
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Cited by 2 (0 self)
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This is a review of the RiemannHilbert approach to the large N asymptotics in random matrix models and its applications. We discuss the following topics: random matrix models and orthogonal polynomials, the RiemannHilbert approach to the large N asymptotics of orthogonal polynomials and its applications to the problem of universality in random matrix models, the double scaling limits, the large N asymptotics of the partition function, and random matrix models with external source.
Asymptotic Expansion of β Matrix Models in the Onecut Regime
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2012
"... We prove the existence of a 1/N expansion to all orders in β matrix models with a confining, offcritical potential corresponding to an equilibrium measure with a connected support. Thus, the coefficients of the expansion can be obtained recursively by the “topological recursion ” derived in Chekhov ..."
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We prove the existence of a 1/N expansion to all orders in β matrix models with a confining, offcritical potential corresponding to an equilibrium measure with a connected support. Thus, the coefficients of the expansion can be obtained recursively by the “topological recursion ” derived in Chekhov and Eynard (JHEP 0612:026, 2006). Our method relies on the combination of a priori bounds on the correlators and the study of SchwingerDyson equations, thanks to the uses of classical complex analysis techniques. These a priori bounds can be derived following (Boutet de Monvel et al. in J Stat Phys 79(3–4):585–611, 1995; Johansson in Duke Math J 91(1):151–204, 1998; Kriecherbauer and Shcherbina in Fluctuations of eigenvalues of matrix models and their applications, 2010) or for strictly convex potentials by using concentration of measure (Anderson et al. in An introduction to random matrices, Sect. 2.3, Cambridge University Press, Cambridge, 2010). Doing so, we extend the strategy of Guionnet and MaurelSegala (Ann Probab 35:2160–2212, 2007), from the hermitian models (β = 2) and perturbative potentials, to general β models. The existence of the first correction in 1/N was considered in Johansson (1998) and more recently in Kriecherbauer and Shcherbina (2010). Here, by taking similar hypotheses, we extend the result to all orders in 1/N.
Concentration of the Spectral Measure for Large Random Matrices with Stable Entries
, 2007
"... We derive concentration inequalities for functions of the empirical measure of large random matrices with infinitely divisible entries and, in particular, stable ones. We also give concentration results for some other functionals of these random matrices, such as the largest eigenvalue or the larges ..."
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We derive concentration inequalities for functions of the empirical measure of large random matrices with infinitely divisible entries and, in particular, stable ones. We also give concentration results for some other functionals of these random matrices, such as the largest eigenvalue or the largest singular value.