Results 1  10
of
50
Random matrices: Universality of local eigenvalue statistics up to the edge
, 2009
"... This is a continuation of our earlier paper [25] on the universality of the eigenvalues of Wigner random matrices. The main new results of this paper are an extension of the results in [25] from the bulk of the spectrum up to the edge. In particular, we prove a variant of the universality results ..."
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Cited by 156 (18 self)
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This is a continuation of our earlier paper [25] on the universality of the eigenvalues of Wigner random matrices. The main new results of this paper are an extension of the results in [25] from the bulk of the spectrum up to the edge. In particular, we prove a variant of the universality results of Soshnikov [23] for the largest eigenvalues, assuming moment conditions rather than symmetry conditions. The main new technical observation is that there is a significant bias in the Cauchy interlacing law near the edge of the spectrum which allows one to continue ensuring the delocalization of eigenvectors.
Nonintersecting paths, random tilings and random matrices
 Probab. Theory Related Fields
, 2002
"... Abstract. We investigate certain measures induced by families of nonintersecting paths in domino tilings of the Aztec diamond, rhombus tilings of an abchexagon, a dimer model on a cylindrical brick lattice and a growth model. The measures obtained, e.g. the Krawtchouk and Hahn ensembles, have the s ..."
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Cited by 125 (11 self)
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Abstract. We investigate certain measures induced by families of nonintersecting paths in domino tilings of the Aztec diamond, rhombus tilings of an abchexagon, a dimer model on a cylindrical brick lattice and a growth model. The measures obtained, e.g. the Krawtchouk and Hahn ensembles, have the same structure as the eigenvalue measures in random matrix theory like GUE, which can in fact can be obtained from nonintersecting Brownian motions. The derivations of the measures are based on the KarlinMcGregor or LindströmGesselViennot method. We use the measures to show some asymptotic results for the models. 1.
Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process
 Acta. Math
, 2005
"... Consider the zero set of a random power series ∑ anz n with i.i.d. complex Gaussian coefficients an. We show that these zeros form a determinantal process: more precisely, their joint intensity can be written as a minor of the Bergman kernel. We show that the number of zeros in a disk of radius r ab ..."
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Cited by 55 (5 self)
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Consider the zero set of a random power series ∑ anz n with i.i.d. complex Gaussian coefficients an. We show that these zeros form a determinantal process: more precisely, their joint intensity can be written as a minor of the Bergman kernel. We show that the number of zeros in a disk of radius r about the origin has the same distribution as the sum of independent indicators Xk where P(Xk = 1) = r −2k. The repulsion between zeros can be studied via a dynamic version where the coefficients perform Brownian motion; we show that this dynamics is conformally invariant. 1
Fluctuations of eigenvalues and second order Poincaré inequalities
, 2007
"... Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems. The proofs of such results are usually rather difficult, involving hard computations specific to the model in question. In this article we attempt to formulate a unified t ..."
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Cited by 49 (5 self)
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Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems. The proofs of such results are usually rather difficult, involving hard computations specific to the model in question. In this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments. In the process, we introduce a notion of ‘second order Poincaré inequalities’: just as ordinary Poincaré inequalities give variance bounds, second order Poincaré inequalities give central limit theorems. The proof of the main result employs Stein’s method of normal approximation. A number of examples are worked out; some of them are new. One of the new results is a CLT for the spectrum of gaussian Toeplitz matrices.
Gaussian limits for random measures in geometric probability
 Ann. Appl. Probab
, 2005
"... We establish Gaussian limits for measures induced by binomial and Poisson point processes in ddimensional space. The limiting Gaussian field has a covariance functional which depends on the density of the point process. The general central limit theorems are applied to measures induced by random gr ..."
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Cited by 27 (7 self)
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We establish Gaussian limits for measures induced by binomial and Poisson point processes in ddimensional space. The limiting Gaussian field has a covariance functional which depends on the density of the point process. The general central limit theorems are applied to measures induced by random graphs (nearest neighbor, Voronoi, and sphere of influence graph), random sequential packing models (ballistic deposition and spatial birth growth models), and statistics of germgrain models. 1
Determinantal processes and independence
, 2006
"... We give a probabilistic introduction to determinantal and permanental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees). They have the striking property that the number of points i ..."
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Cited by 22 (1 self)
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We give a probabilistic introduction to determinantal and permanental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees). They have the striking property that the number of points in a region D is a sum of independent Bernoulli random variables, with parameters which are eigenvalues of the relevant operator on L 2 (D). Moreover, any determinantal process can be represented as a mixture of determinantal projection processes. We give a simple explanation for these known facts, and establish analogous representations for permanental processes, with geometric variables replacing the Bernoulli variables. These representations lead to simple proofs of existence criteria and central limit theorems, and unify known results on the distribution of absolute values in certain processes with radially symmetric distributions.
Limiting laws of linear eigenvalue statistics for unitary invariant matrix models
 J. Math. Phys
, 2006
"... We study the variance and the Laplace transform of the probability law of linear eigenvalue statistics of unitary invariant Matrix Models of n ×n Hermitian matrices as n → ∞. Assuming that the test function of statistics is smooth enough and using the asymptotic formulas by Deift et al for orthogona ..."
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Cited by 17 (3 self)
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We study the variance and the Laplace transform of the probability law of linear eigenvalue statistics of unitary invariant Matrix Models of n ×n Hermitian matrices as n → ∞. Assuming that the test function of statistics is smooth enough and using the asymptotic formulas by Deift et al for orthogonal polynomials with varying weights, we show first that if the support of the Density of States of the model consists of q ≥ 2 intervals, then in the global regime the variance of statistics is a quasiperiodic function of n as n → ∞ generically in the potential, determining the model. We show next that the exponent of the Laplace transform of the probability law is not in general 1/2 × variance, as it should be if the Central Limit Theorem would be valid, and we find the asymptotic form of the Laplace transform of the probability law in certain cases.