Beta ensembles, stochastic Airy spectrum, and a diffusion
TLDR
In this paper, it was shown that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the low-lying eigen values of the random Schrodinger operator − d2 dx2 + x+ 2 √ β bx restricted to the positive half-line.Abstract:
We prove that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the low-lying eigenvalues of the random Schrodinger operator − d2 dx2 + x+ 2 √ β bx restricted to the positive half-line, where b ′ x is white noise. In doing so we extend the definition of the Tracy-Widom(β) distributions to all β > 0, and also analyze their tails. Last, in a parallel development, we provide a second characterization of these laws in terms of a one-dimensional diffusion. The proofs rely on the associated tridiagonal matrix models and a universality result showing that the spectrum of such models converge to that of their continuum operator limit. In particular, we show how Tracy-Widom laws arise from a functional central limit theorem.read more
Citations
More filters
Journal ArticleDOI
Shape Fluctuations and Random Matrices
TL;DR: In this article, the authors studied a random Groeth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process and showed that shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble.
Journal ArticleDOI
Continuum limits of random matrices and the Brownian carousel
Benedek Valkó,Bálint Virág +1 more
TL;DR: The authors showed that at any location away from the spectral edge, the eigenvalues of the Gaussian unitary ensemble and its general β siblings converge to Sine β, a translation invariant point process.
GUEs and queues
TL;DR: In this article, it was shown that the process D has the law of the process of the largest eigenvalues of the main minors of an infinite random matrix drawn from Gaussian Unitary Ensemble.
Journal ArticleDOI
Random matrix theory in statistics: A review
Debashis Paul,Alexander Aue +1 more
TL;DR: An overview of random matrix theory is given with the objective of highlighting the results and concepts that have a growing impact in the formulation and inference of statistical models and methodologies.
Journal ArticleDOI
Universality of Wigner random matrices: a survey of recent results
TL;DR: In this article, the authors studied the universality of spectral statistics for Wigner matrices with independent identically distributed entries, where the probability distribution of each matrix element is given by a measure with zero expectation and with subexponential decay.
References
More filters
Book
Markov Processes: Characterization and Convergence
TL;DR: In this paper, the authors present a flowchart of generator and Markov Processes, and show that the flowchart can be viewed as a branching process of a generator.
Journal ArticleDOI
On the distribution of the largest eigenvalue in principal components analysis
TL;DR: In this article, the authors derived the Tracey-Widom law of order 1 for large p and n matrices, where p is the largest eigenvalue of a p-variate Wishart distribution on n degrees of freedom with identity covariance.
Journal ArticleDOI
Level spacing distributions and the Airy kernel
Craig A. Tracy,Harold Widom +1 more
TL;DR: In this paper, the authors derived analogues for the Airy kernel of the following properties of the sine kernel: the completely integrable system of P.D.E., the expression of the Fredholm determinant in terms of a Painleve transcendent, the existence of a commuting differential operator, and the fact that this operator can be used in the derivation of asymptotics, for generaln, of the probability that an interval contains preciselyn eigenvalues.
Book
Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach
TL;DR: In this paper, the authors present an asymptotics for orthogonal polynomials in Riemann-Hilbert problems and Jacobi operators for continued fractions.
Book
Log-Gases and Random Matrices
TL;DR: Forrester as discussed by the authors presents an encyclopedic development of log-gases and random matrices viewed as examples of integrable or exactly solvable systems, and provides hundreds of guided exercises and linked topics.