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73
Poisson statistics for the largest eigenvalues of Wigner random matrices with heavy tails
 Elec. Commun. Probab
, 2004
"... We study large Wigner random matrices in the case when the marginal distributions of matrix entries have heavy tails. We prove that the largest eigenvalues of such matrices have Poisson statistics. 1 Introduction and Formulation of Results. The main goal of this paper is to study the largest eigenva ..."
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We study large Wigner random matrices in the case when the marginal distributions of matrix entries have heavy tails. We prove that the largest eigenvalues of such matrices have Poisson statistics. 1 Introduction and Formulation of Results. The main goal of this paper is to study the largest eigenvalues of Wigner real symmetric and Hermitian random matrices in the case when the matrix entries have heavy tails of distribution. We remind that a real symmetric Wigner random matrix is defined as a square symmetric n × n matrix with i.i.d. entries up from the diagonal
Dyson's Nonintersecting Brownian motions with a few outliers
, 2008
"... Consider n nonintersecting Brownian particles on R (Dyson Brownian motions), all starting from the origin at time t = 0, and forced to return to x = 0 at time t = 1. For large n, the average mean density of particles has its support, for each 0 < t < 1, on the interval ± √ 2nt(1 −t). The Air ..."
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Cited by 28 (5 self)
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Consider n nonintersecting Brownian particles on R (Dyson Brownian motions), all starting from the origin at time t = 0, and forced to return to x = 0 at time t = 1. For large n, the average mean density of particles has its support, for each 0 < t < 1, on the interval ± √ 2nt(1 −t). The Airy process A (τ) is defined as the motion of these nonintersecting Brownian motions for large n, but viewed from the curve C: y = √ 2nt(1 −t) with an appropriate spacetime rescaling. Assume now a finite number r of these particles are forced to a different target point, say a = ρ0 n/2> 0. Does it affect the Brownian fluctuations along the curve C for large n? In this paper, we show that no new process appears as long as one considers points (y,t) ∈ C, such that 0 < t < (1 + ρ2 0)−1, which is the tcoordinate of the point of tangency of the tangent to the curve passing through (ρ0 n/2,1). At this point of tangency the fluctuations obey a
PDEs for the Gaussian ensemble with external source and the Pearcey distribution
 MOTIONS, INTEGRABLE SYSTEMS AND ORTHOGONAL POLYNOMIALS 395
, 2007
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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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Cited by 23 (3 self)
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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.
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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Cited by 23 (5 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 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.
Central limit theorem for linear eigenvalue statistics of orthogonally invariant matrix models
, 2007
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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.
A phase transition for nonintersecting Brownian motions, and the Painlevé II equation
, 2008
"... 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 th ..."
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Cited by 16 (5 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.
Random matrices: The universality phenomenon for Wigner ensembles
, 2012
"... In this paper, we survey some recent progress on rigorously etablishing the universality of various spectral statistics of Wigner Hermitian random matrix ensembles, focusing on the Four Moment Theorem and its refinements and applications, including the universality of the sine kernel and the Centr ..."
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In this paper, we survey some recent progress on rigorously etablishing the universality of various spectral statistics of Wigner Hermitian random matrix ensembles, focusing on the Four Moment Theorem and its refinements and applications, including the universality of the sine kernel and the Central limit theorem of several spectral parameters. We also take the opportunity here to issue some errata for some of our previous papers in this area.
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 14 (4 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.