Trace Moments for Schr\"odinger Operators with Matrix White Noise and the Rigidity of the Multivariate Stochastic Airy Operator

TL;DR

This paper develops Feynman-Kac formulas for trace moments of matrix-valued Schrödinger operators with white noise, demonstrating spectrum rigidity for multivariate stochastic Airy operators and advancing understanding of eigenvalue fluctuations in random matrix models.

Contribution

It introduces trace moment formulas for matrix Schrödinger operators with white noise, including the multivariate stochastic Airy operator, and proves spectrum rigidity under linear growth conditions.

Findings

Established Feynman-Kac formulas for trace moments.

Proved spectrum number rigidity for multivariate stochastic Airy operators.

Connected rigidity results to eigenvalue fluctuations in random matrix ensembles.

Abstract

We study the semigroups of random Schr\"odinger operators of the form $\widehat{H}f=-\frac12f''+(V+\xi)f$, 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 $\mathbf E[\prod_{k=1}^n\mathrm{Tr}[\mathrm e^{-t_k\widehat{H}}]]$ ($n\in\mathbb N$, $t_k>0$). One notable example covered by our main result consists of the multivariate stochastic Airy operator (SAO) of Bloemendal and Vir\'ag (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 $\widehat{H}$'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.