Beta ensembles, stochastic Airy spectrum, and a diffusion

By J. Ramírez, Brian Rider, B'alint Vir'ag. 2006

291 citations recorded by the source corpus.

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 Schroedinger operator -d^2/dx^2 + x + (2/beta^{1/2}) b_x' restricted to the positive half-line, where b_x' is white noise. In doing so we extend the definition of the Tracy-Widom(beta) distributions to all beta>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.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/84f8421bac29d8df66d01a15aa4e773e8f886047

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Eigenvalue statistics for CMV matrices: From Poisson to clock via random matrix ensembles (2009, unknown): “The Prüfer phase ψ(x;E) and amplitude R(x;E) are defined by u(x;E) = R(x;E) sin(ψ(x;E)), u′(x;E) = R(x;E) cos(ψ(x;E)) and ψ(0;E) = 0.”
Spectral Radii of Large Non-Hermitian Random Matrices (2014, background): “Ramı́rez et al. (2011) obtain the asymptotic distribution of the largest eigenvalues of beta-Hermite ensemble.”
Approximation of Rectangular Beta-Laguerre Ensembles and Large Deviations (2013, unknown): “[52].”
Small deviations for beta ensembles (2009, background): “The proof of the above hinges on the following version of integration by parts (as in fact does the basic convergence result in [19]).”
GAUSSIAN FLUCTUATIONS OF YOUNG DIAGRAMS AND STRUCTURE CONSTANTS OF JACK CHARACTERS (2014, background): “n converges in probability towards 2/ √ α and the joint vector n1/3 (λ (n))j √ n − 2 √ α 1≤j≤k converges in law towards the β-Tracy-Widom distribution, which has been introduced and studied in [RRV11] to study edge ﬂuctuations of β-ensemble. Naturally, a similar conjecture can be formulated for t”
Universality of the Stochastic Airy Operator (2013, methodology): “As for convergence of Hn to H , the needed result from [20], as extended in [3], reads:”
Random matrix theory in statistics: A review (2014, background): “Ramírez et al. (2011) described such limit laws in terms of the eigenstates of a stochastic Airy operator introduced by Edelman and Sutton (2007) and Sutton (2005).”
Continuum limits of random matrices and the Brownian carousel (2007, background): “ss given by (1), and let µn be a sequence so that n 1/6(2 √ n− µn) → a ∈ R. Then n1/6(Λ n − µn) ⇒ Airyβ +a HereAiryβ canbedeﬁnedas−1timesthepointprocessofeigenvaluesofthestochastic Airy operator, see Ramirez et al. (2007) for more details. A straightforward diagonalization argument gives the followi”
Limits of spiked random matrices I (2010, background): “[47], who in particular established joint convergence of the largest eigenvalues.”