Results 1  10
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67
High dimensional statistical inference and random matrices
 IN: PROCEEDINGS OF INTERNATIONAL CONGRESS OF MATHEMATICIANS
, 2006
"... Multivariate statistical analysis is concerned with observations on several variables which are thought to possess some degree of interdependence. Driven by problems in genetics and the social sciences, it first flowered in the earlier half of the last century. Subsequently, random matrix theory ..."
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Cited by 49 (1 self)
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Multivariate statistical analysis is concerned with observations on several variables which are thought to possess some degree of interdependence. Driven by problems in genetics and the social sciences, it first flowered in the earlier half of the last century. Subsequently, random matrix theory (RMT) developed, initially within physics, and more recently widely in mathematics. While some of the central objects of study in RMT are identical to those of multivariate statistics, statistical theory was slow to exploit the connection. However, with vast data collection ever more common, data sets now often have as many or more variables than the number of individuals observed. In such contexts, the techniques and results of RMT have much to offer multivariate statistics. The paper reviews some of the progress to date.
Continuum limits of random matrices and the Brownian carousel
, 2008
"... We show that at any location away from the spectral edge, the eigenvalues of the Gaussian unitary ensemble and its general β siblings converge to Sineβ, a translation invariant point process. This process has a geometric description in term of the Brownian carousel, a deterministic function of Brown ..."
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Cited by 40 (2 self)
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We show that at any location away from the spectral edge, the eigenvalues of the Gaussian unitary ensemble and its general β siblings converge to Sineβ, a translation invariant point process. This process has a geometric description in term of the Brownian carousel, a deterministic function of Brownian motion in the hyperbolic plane. The Brownian carousel, a description of the a continuum limit of random matrices, provides a convenient way to analyze the limiting point processes. We show that the gap probability of Sineβ is continuous in the gap size and β, and compute its asymptotics for large gaps. Moreover, the stochastic differential equation version of
Eigenvalue statistics for CMV matrices: From Poisson to clock via CβE
"... Abstract. We study CMV matrices (a discrete onedimensional Diractype operator) with random decaying coefficients. Under mild assumptions we identify the local eigenvalue statistics in the natural scaling limit. For rapidly decreasing coefficients, the eigenvalues have rigid spacing (like the numer ..."
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Cited by 26 (2 self)
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Abstract. We study CMV matrices (a discrete onedimensional Diractype operator) with random decaying coefficients. Under mild assumptions we identify the local eigenvalue statistics in the natural scaling limit. For rapidly decreasing coefficients, the eigenvalues have rigid spacing (like the numerals on a clock); in the case of slow decrease, the eigenvalues are distributed according to a Poisson process. For a certain critical rate of decay we obtain the circular beta ensembles of random matrix theory. The temperature β −1 appears as the square of the coupling constant. 1.
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
Nonintersecting Brownian walkers and YangMills theory on the sphere
 Nucl. Phys. B
"... We study a system of N nonintersecting Brownian motions on a line segment [0, L] with periodic, absorbing and reflecting boundary conditions. We show that the normalized reunion probabilities of these Brownian motions in the three models can be mapped to the partition function of twodimensional c ..."
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Cited by 19 (5 self)
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We study a system of N nonintersecting Brownian motions on a line segment [0, L] with periodic, absorbing and reflecting boundary conditions. We show that the normalized reunion probabilities of these Brownian motions in the three models can be mapped to the partition function of twodimensional continuum YangMills theory on a sphere respectively with gauge groups U(N), Sp(2N) and SO(2N). Consequently, we show that in each of these Brownian motion models, as one varies the system size L, a third order phase transition occurs at a critical value L = Lc(N) ∼ N in the large N limit. Close to the critical point, the reunion probability, properly centered and scaled, is identical to the TracyWidom distribution describing the probability distribution of the largest eigenvalue of a random matrix. For the periodic case we obtain the TracyWidom distribution corresponding to the GUE random matrices, while for the absorbing and reflecting cases we get the TracyWidom distribution corresponding to GOE random matrices. In the absorbing case, the reunion probability is also identified as the maximal height of N nonintersecting Brownian excursions (“watermelons ” with a wall) whose distribution in the asymptotic scaling limit is then described by GOE TracyWidom law. In addition, large deviation formulas for the maximum height are also computed. 1 ar
Limits of spiked random matrices
, 2013
"... Given a large, highdimensional sample from a spiked population, the top sample covariance eigenvalue is known to exhibit a phase transition. We show that the largest eigenvalues have asymptotic distributions near the phase transition in the rank one spiked real Wishart setting and its general β ana ..."
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Cited by 17 (2 self)
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Given a large, highdimensional sample from a spiked population, the top sample covariance eigenvalue is known to exhibit a phase transition. We show that the largest eigenvalues have asymptotic distributions near the phase transition in the rank one spiked real Wishart setting and its general β analogue, proving a conjecture of Baik, Ben Arous and Péche ́ (2005). We also treat shifted mean Gaussian orthogonal and β ensembles. Such results are entirely new in the real case; in the complex case we strengthen existing results by providing optimal scaling assumptions. One obtains the known limiting random Schrödinger operator on the halfline, but the boundary condition now depends on the perturbation. We derive several characterizations of the limit laws in which β appears as a parameter, including a simple linear boundary value problem. This PDE description recovers known explicit formulas at β = 2, 4, yielding in particular a new and simple proof of the Painleve ́ representations for these
GAUSSIAN FLUCTUATIONS FOR β ENSEMBLES.
, 2007
"... Abstract. We study the Circular and Jacobi βEnsembles and prove Gaussian fluctuations for the number of points in one or more intervals in the macroscopic scaling limit. 1. ..."
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Cited by 12 (0 self)
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Abstract. We study the Circular and Jacobi βEnsembles and prove Gaussian fluctuations for the number of points in one or more intervals in the macroscopic scaling limit. 1.
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.
Asymptotics of the spectral gap for the interchange process on large hypercubes
 arxiv:0802.1368v2 (2008). the electronic journal of combinatorics 16 (2009), #N29 7
"... sstarr at math dot rochester dot edu ..."
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