The Tracy-Widom β distribution is the large dimensional limit of the top eigenvalue of β random matrix ensembles. We use the stochastic Airy operator representation to show that as a → ∞ the tail of the Tracy Widom distribution satisfiesFlowlines for the ODE of X − 2 √ β B

We study the Sine β process introduced in [B. Valkó and B. Virág. Invent. math. 177 463-508 (2009)] when the inverse temperature β tends to 0. This point process has been shown to be the scaling limit of the eigenvalues point process in the bulk of β-ensembles and its law is characterized in terms of the winding numbers of the Brownian carrousel at different angular speeds. After a careful analysis of this family of coupled diffusion processes, we prove that the Sine β point process converges weakly to a Poisson point process on R. Thus, the Sine β point processes establish a smooth crossover between the rigid clock (or picket fence) process (corresponding to β = ∞) and the Poisson process. arXiv:1407.5402v2 [math.PR]

Abstract. We investigate the marginal distribution of the bottom eigenvalues of the stochastic Airy operator when the inverse temperature β tends to 0. We prove that the minimal eigenvalue, whose fluctuations are governed by the Tracy-Widom β law, converges weakly, when properly centered and scaled, to the Gumbel distribution. More generally we obtain the convergence in law of the marginal distribution of any eigenvalue with given index k. Those convergences are obtained after a careful analysis of the explosion times process of the Riccati diffusion associated to the stochastic Airy operator. We show that the empirical measure of the explosion times converges weakly to a Poisson point process using estimates proved in [L. Dumaz and B. Virág. Ann. Inst. H. Poincaré Probab. Statist. 49, 4, 915-933, (2013)]. We further compute the empirical eigenvalue density of the stochastic Airy ensemble on the macroscopic scale when β → 0. As an application, we investigate the maximal eigenvalues statistics of βN -ensembles when the repulsion parameter βN → 0 when N → +∞. We study the double scaling limit

We study the bottom of the spectrum of the Anderson Hamiltonian HL := −∂ 2 x + ξ on [0, L] driven by a white noise ξ and endowed with either Dirichlet or Neumann boundary conditions. We show that, as L → ∞, the point process of the (appropriately shifted and rescaled) eigenvalues converges to a Poisson point process on R with intensity e x dx, and that the (appropriately rescaled) eigenfunctions converge to Dirac masses located at independent and uniformly distributed points. Furthermore, we show that the shape of each eigenfunction, recentered around its maximum and properly rescaled, is given by the inverse of a hyperbolic cosine. We also show that the eigenfunctions decay exponentially from their localization centers at an explicit rate, and we obtain very precise information on the zeros and local maxima of these eigenfunctions. Finally, we show that the eigenvalues/eigenfunctions in the Dirichlet and Neumann cases are very close to each other and converge to the same limits.