On Spatial Conditioning of the Spectrum of Discrete Random Schrödinger Operators

Abstract

Consider a random Schrödinger-type operator of the form $H:=-H_X+V+\xi$ acting on a general graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$, where $H_X$ is the generator of a Markov process $X$ on $\mathcal{G}$, $V$ is a deterministic potential with sufficient growth (so that $H$ has a purely discrete spectrum), and $\xi$ is a random noise with at-most-exponential tails. We prove that $H$'s eigenvalue point process is number rigid in the sense of Ghosh and Peres (Duke Math. J. 166 (2017), no. 10, 1789--1858); that is, the number of eigenvalues in any bounded domain $B\subset \mathbb{C}$ is determined by the configuration of eigenvalues outside of $B$. Our general setting allows to treat cases where $X$ could be non-symmetric (hence $H$ is non-self-adjoint) and $\xi$ has long-range dependence. Our strategy of proof consists of controlling the variance of the trace of the semigroup ${\mathrm{e}}^{-tH}$ using the Feynman-Kac formula.

32 pages; revised version incorporating referee comments and fixing a few misprints and errors. Accepted in the Journal of Spectral Theory

References in corpus (3)

• From Random Matrices to Stochastic Operators
• Rigidity of Determinantal Point Processes with the Airy, the Bessel and the Gamma Kernel
• Rigidity and tolerance for perturbed lattices