Results 1  10
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16
Orthogonal polynomial ensembles in probability theory
 Prob. Surv
, 2005
"... Abstract: We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an orthogonal polynomial ensemble. The most prominent example is apparently the Hermite ensemble, the eigenvalue distribution of the Gaussian Unitary ..."
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Cited by 62 (1 self)
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Abstract: We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an orthogonal polynomial ensemble. The most prominent example is apparently the Hermite ensemble, the eigenvalue distribution of the Gaussian Unitary Ensemble (GUE), and other wellknown ensembles known in random matrix theory like the Laguerre ensemble for the spectrum of Wishart matrices. In recent years, a number of further interesting models were found to lead to orthogonal polynomial ensembles, among which the corner growth model, directed last passage percolation, the PNG droplet, noncolliding random processes, the length of the longest increasing subsequence of a random permutation, and others. Much attention has been paid to universal classes of asymptotic behaviors of these models in the limit of large particle numbers, in particular the spacings between the particles and the fluctuation behavior of the largest particle. Computer simulations suggest that the connections go even farther
Multicritical unitary random matrix ensembles and the general Painlevé II equation
, 2008
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Multivariate analysis and Jacobi ensembles: Largest eigenvalue, Tracy–Widom limits and rates of convergence
 ANN. STATIST
, 2008
"... Let A and B be independent, central Wishart matrices in p variables with common covariance and having m and n degrees of freedom, respectively. The distribution of the largest eigenvalue of (A+B) −1 B has numerous applications in multivariate statistics, but is difficult to calculate exactly. Suppos ..."
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Cited by 30 (2 self)
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Let A and B be independent, central Wishart matrices in p variables with common covariance and having m and n degrees of freedom, respectively. The distribution of the largest eigenvalue of (A+B) −1 B has numerous applications in multivariate statistics, but is difficult to calculate exactly. Suppose that m and n grow in proportion to p. We show that after centering and scaling, the distribution is approximated to secondorder, O(p −2/3), by the Tracy–Widom law. The results are obtained for both complex and then realvalued data by using methods of random matrix theory to study the largest eigenvalue of the Jacobi unitary and orthogonal ensembles. Asymptotic approximations of Jacobi polynomials near the largest zero play a central role.
Diffusion at the random matrix hard edge
 Communications in Mathematical Physics 288 (2009
"... We show that the limiting minimal eigenvalue distributions for a natural generalization of Gaussian samplecovariance structures (the “beta ensembles”) are described in by the spectrum of a random diffusion generator. By a Riccati transformation, we obtain a second diffusion description of the limit ..."
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Cited by 19 (6 self)
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We show that the limiting minimal eigenvalue distributions for a natural generalization of Gaussian samplecovariance structures (the “beta ensembles”) are described in by the spectrum of a random diffusion generator. By a Riccati transformation, we obtain a second diffusion description of the limiting eigenvalues in terms of hitting laws. This picture pertains to the socalled hard edge of random matrix theory and sits in complement to the recent work [15] of the authors and B. Virág on the general beta random matrix soft edge. In fact, the diffusion descriptions found on both sides are used here to prove there exists a transition between the soft and hard edge laws at all values of beta. 1
Orthogonal and symplectic matrix models: universality and other properties
 Comm. Math. Phys
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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 13 (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.
Edge Universality for Orthogonal Ensembles of Random Matrices
, 2008
"... We prove edge universality of local eigenvalue statistics for orthogonal invariant matrix models with real analytic potentials and one interval limiting spectrum. Our starting point is the result of [21] on the representation of the reproducing matrix kernels of orthogonal ensembles in terms of scal ..."
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Cited by 11 (3 self)
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We prove edge universality of local eigenvalue statistics for orthogonal invariant matrix models with real analytic potentials and one interval limiting spectrum. Our starting point is the result of [21] on the representation of the reproducing matrix kernels of orthogonal ensembles in terms of scalar reproducing kernel of corresponding unitary ensemble.
Edge Universality of Beta Ensembles
, 2013
"... We prove the edge universality of the beta ensembles for any β � 1, provided that the limiting spectrum is supported on a single interval, and the external potential is C 4. We also prove that the edge universality holds for generalized Wigner matrices for all symmetry classes. Moreover, our results ..."
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Cited by 10 (0 self)
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We prove the edge universality of the beta ensembles for any β � 1, provided that the limiting spectrum is supported on a single interval, and the external potential is C 4. We also prove that the edge universality holds for generalized Wigner matrices for all symmetry classes. Moreover, our results allow us to extend bulk universality for beta ensembles from analytic potentials to potentials in class C 4.