Trace Moments for Schrödinger Operators with Matrix White Noise and the Rigidity of the Multivariate Stochastic Airy Operator

Abstract

We study the semigroups of random Schrödinger operators of the form $\hat{H}f=-\frac{1}{2}{f}^{\prime \prime }+(V+\xi )f$, where $f:I\to {\mathbb{F}}^r$ ($\mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H}$) are vector-valued functions on a possibly infinite interval $I\subset \mathbb{R}$ that satisfy a mix of Robin and Dirichlet boundary conditions, $V$ is a deterministic diagonal potential with power-law growth at infinity, and $\xi$ is a matrix white noise. Our main result consists of Feynman-Kac formulas for trace moments of the form $\mathbf{E}[{\prod }_{k=1}^n\mathrm{Tr}[{\mathrm{e}}^{-t_k\hat{H}}]]$ ($n\in \mathbb{N}$, $t_k>0$). One notable example covered by our main result consists of the multivariate stochastic Airy operator (SAO) of Bloemendal and Virág (Ann. Probab., 44(4):2726-2769, 2016), which characterizes the soft-edge eigenvalue fluctuations of critical rank-$r$ spiked Wishart and GO/U/SE random matrices. As a corollary of our main result, we prove that if $V$'s growth is at least linear (this includes the multivariate SAO), then $\hat{H}$'s spectrum is number rigid in the sense of Ghosh and Peres (Duke Math. J., 166(10):1789-1858, 2017). Together with the rigidity of the scalar SAO, this completes the characterization of number rigidity in the soft-edge limits of Gaussian $\beta$-ensembles and their finite-rank spiked versions.

93 pages, 8 figures. This preprint supersedes arXiv:2311.08564, along with some important bibliographical updates. This second version corrects typos and improved the presentation (incorporating referee comments)