• Corpus ID: 245335046

Rigidity of the Stochastic Airy Operator

  title={Rigidity of the Stochastic Airy Operator},
  author={Pierre Yves Gaudreau Lamarre and Promit Ghosal and Wenxuan Li and Yuchen Liao},
We prove that the spectrum of the stochastic Airy operator is rigid in the sense of Ghosh and Peres [19] for Dirichlet and Robin boundary conditions. This proves the rigidity of the Airy-β point process and the softedge limit of rank-1 perturbations of Gaussian β-Ensembles for any β > 0, and solves an open problem mentioned in [8]. Our proof uses a combination of the semigroup theory of the stochastic Airy operator and the techniques for studying insertion and deletion tolerance of point… 


Definition and Self-Adjointness of the Stochastic Airy Operator
In this note, it is shown that the stochastic Airy operator, which is the "Schr\"odinger operator" on the half-line whose potential term consists of Gaussian white noise plus a linear term tending to
Universality of the Stochastic Airy Operator
We introduce a new method for studying universality of random matrices. Let Tn be the Jacobi matrix associated to the Dyson beta ensemble with uniformly convex polynomial potential. We show that
Stochastic Airy semigroup through tridiagonal matrices
We determine the operator limit for large powers of random tridiagonal matrices as the size of the matrix grows. The result provides a novel expression in terms of functionals of Brownian motions for
On Number Rigidity for Pfaffian Point Processes
Our first result states that the orthogonal and symplectic Bessel processes are rigid in the sense of Ghosh and Peres. Our argument in the Bessel case proceeds by an estimate of the variance of
Beta ensembles, stochastic Airy spectrum, and a diffusion
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 +
Rigidity of determinantal point processes with the Airy, the Bessel and the Gamma kernel
A point process is said to be rigid if for any bounded domain in the phase space, the number of particles in the domain is almost surely determined by the restriction of the configuration to the
Edge of spiked beta ensembles, stochastic Airy semigroups and reflected Brownian motions
We access the edge of Gaussian beta ensembles with one spike by analyzing high powers of the associated tridiagonal matrix models. In the classical cases beta=1, 2, 4, this corresponds to studying
Schrödinger Semigroups
Let H = \L + V be a general Schrödinger operator on R" (v~> 1), where A is the Laplace differential operator and V is a potential function on which we assume minimal hypotheses of growth and
Universality of the Stochastic Bessel Operator
We establish universality at the hard edge for general beta ensembles assuming that: the background potential V is a polynomial such that $$x \mapsto V(x^2)$$x↦V(x2) is strongly convex, $$\beta \ge
Spectral Rigidity of Random Schrödinger Operators via Feynman–Kac Formulas
We develop a technique for proving number rigidity (in the sense of Ghosh and Peres in Duke Math J 166(10):1789–1858, 2017) of the spectrum of general random Schrödinger operators (RSOs). Our method