Beta ensembles, stochastic Airy spectrum, and a diffusion

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 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.Comment: Revised content, new results. In particular, Theorems 1.3 and 5.1 are ne

The Tracy–Widom distribution is not infinitely divisible

“…The β-Tracy-Widom distribution is defined as the limiting distribution of the largest eigenvalue of a β-Hermite ensemble. In the case β = 1, 2, 4 the β-Tracy-Widom distribution coincides with the classical Tracy-Widom, [21].…”

“…This stochastic operator approach to random matrix theory was conjectured by Edelman and Sutton [12], and was proved by Ramírez, Rider and Virág [21], who in particular established convergence of the largest eigenvalue of a β-Hermite ensemble for any β > 0. Let λ max = λ N H β N , with H β N defined as in (2), in [21] it is shown the existence of a β-Tracy-Widom random variable T W β such that…”

Spiking the random matrix hard edge

“…For r = 1 these are the same random tridiagonal matrices introduced by Dumitriu-Edelman [10], which subsequently formed the basis for the conjectured Stochastic Airy Operator formulation of the Tracy-Widom(β) laws in [11,26] (and again proved in [19]). In particular consider the XΣ 1/2 for X as above (comprised of iid real/complex/quaternion Gaussians) but specify Σ = σ ⊕ I m−1 for a scalar σ.…”

“…Again recall that −λ 0 (β, +∞) has the Tracy-Widom(β) distribution [19]. Likewise, for r > 1, the process (1.16) is replaced by the joint diffusion p = (p 1 , .…”

“…At r = 1 the result holds for all β > 0. In that case the noise term reduces to 2 √ β b ′ x for x → b x a standard Brownian motion, and H is recognized as the Stochastic Airy Operator from [19], but with a simple Robin f ′ (0) = wf (0) rather than Dirichlet boundary condition at the origin. Hence, for r = 1 the single parameter c interpolates between the general beta Tracy-Widom law at w = +∞ (which forces f (0) = 0) and the Gaussian law in the supercritical regime, w = −∞ (which requires a little perturbation argument to make precise, see [3]).…”