$\beta$-ensembles

Let $B$ be standard Brownian motion on $[0, \infty)$ and interpret $B'$ as white noise. On $L^2((0, \infty))$, let the stochastic Airy operator with Dirichlet boundary condition at $0$ be

$$\mathcal A_\beta=-\frac{d^2}{dx^2}+x+\frac{2}{\sqrt{\beta}}B'(x).$$

This expression is understood through the corresponding random operator or quadratic form, not as multiplication by a pointwise-defined derivative of Brownian motion. Almost surely its spectrum is discrete; write its lowest eigenvalue as $\Lambda_0$. The operator in the question is $H_\beta=-\mathcal A_\beta$, so its largest eigenvalue is $\lambda_1=-\Lambda_0$. The question is whether, for $\beta=2$, one can derive directly from this operator the distribution function of $\lambda_1$, namely the Tracy–Widom $F_2$ formula. Ramírez–Rider–Virág, Beta ensembles, stochastic Airy spectrum, and a diffusion Deift, Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems (2007)

In the standard normalization, the formula at issue is

$$\mathbb P(\lambda_1\le s)=F_2(s)=\exp\!\left(-\int_s^\infty (t-s)u(t)^2\,dt\right),$$

where $u$ is the Hastings–McLeod solution of Painlevé II, characterized by $u''(t)=2u(t)^3+t u(t)$ and $u(t)\sim\operatorname{Ai}(t)$ as $t\to+\infty$. Bloemendal and Virág, Limits of spiked random matrices I (2013) Ramírez–Rider–Virág, Beta ensembles, stochastic Airy spectrum, and a diffusion

Background

The Gaussian Hermite $\beta$-ensemble has joint eigenvalue density proportional to $e^{-\beta\sum_j\lambda_j^2/4}\prod_{j<k}|\lambda_j-\lambda_k|^\beta$ for every $\beta>0$. At $\beta=1,2,4$ it agrees with the classical orthogonal, unitary, and symplectic ensembles; for general $\beta$ it is also the law of a one-dimensional logarithmic gas. Tridiagonal random-matrix models made these ensembles accessible beyond the classical symmetry classes. At the soft spectral edge, the tridiagonal matrices, viewed as discrete random Schrödinger operators, lead after rescaling to the stochastic Airy operator. Bloemendal and Virág, Limits of spiked random matrices I Ramírez–Rider–Virág, Beta ensembles, stochastic Airy spectrum, and a diffusion

Deift posed the operator-based question in 2007, against the background of new tridiagonal models and the emerging stochastic-operator approach. The central connection is not just a limiting eigenvalue law: the operator’s random potential can be transformed by a Riccati substitution into a one-dimensional diffusion. That provides a route from spectral events to distributional characterizations, including differential equations for the edge laws. Deift, Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems (2007) Ramírez–Rider–Virág, Beta ensembles, stochastic Airy spectrum, and a diffusion Bloemendal and Virág, Limits of spiked random matrices I (2013)

Known results

Ramírez, Rider, and Virág established the soft-edge convergence that underlies the operator formulation. For the Hermite $\beta$-ensemble, their theorem gives, for each fixed $k$, convergence of the top $k$ rescaled eigenvalues to the negatives of the first $k$ eigenvalues of the Dirichlet stochastic Airy operator. In particular, with the usual Hermite normalization, $n^{1/6}(2\sqrt n-\lambda_{\beta,j})$ converges jointly to $\Lambda_{j-1}$. They also derived a diffusion characterization of the limiting laws and, by combining it with the classical Tracy–Widom formulas, identified the $\beta=2$ law. Ramírez–Rider–Virág, Beta ensembles, stochastic Airy spectrum, and a diffusion

Bloemendal and Virág subsequently gave an operator-based characterization that answers the question in a concrete sense. For the Dirichlet case, their boundary-parameter distribution $F_{\beta,w}$ reduces at $w=+\infty$ to the stochastic Airy law. They characterize these laws through a linear boundary-value problem, obtained from the diffusion description; at $\beta=2$ they identify its solution with the Painlevé-II representation. In particular, their theorem gives $F_{2,\infty}(s)=F_2(s)$, recovering the formula above and providing a new, simple proof of its Painlevé representation. This is a derivation through the stochastic-operator characterization, rather than a formula read off from the operator without further analysis. Bloemendal and Virág, Limits of spiked random matrices I (2013)