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

TL;DR

The paper develops a probabilistic Feynman-Kac framework for random vector-valued Schrödinger operators $\hat{H} = -\tfrac{1}{2}\partial_x^2 + (V+\xi)$ acting on $\mathbb{F}^r$-valued functions with matrix white noise perturbations. By combining a matrix-ordered-exponential representation with a coupled Brownian motion and a random walk on $\{1,\dots,r\}$, it derives detailed trace-moment formulas for $\mathbb{E}[\prod_{k=1}^n \mathrm{Tr}[e^{-t_k \hat{H}}]]$, including a rich combinatorial structure arising from self-intersections. A key corollary shows that, under linear growth of the deterministic potential, the spectrum is number rigid, thereby completing the rigidity picture for soft-edge limits of Gaussian $\beta$-ensembles and their rank-$r$ spikes. The results are specialized to the multivariate stochastic Airy operator, providing new tools for studying critical spike phenomena and potential SPDE connections, with a roadmap for future asymptotic and intermittency analyses. Overall, the work advances both the spectral theory of vector-valued random Schrödinger operators and the probabilistic understanding of rigidity and edge fluctuations in random matrix theory.

Abstract

We study the semigroups of random Schrödinger operators of the form $\widehat{H}f=-\frac12f''+(V+ξ)f$, 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 $ξ$ 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 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 $\widehat{H}$'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 $β$-ensembles and their finite-rank spiked versions.

Figures (8)

• Figure 1: This figure shows a realization $A=(U,Z)$. The $y$-axis represents the output of $Z$, and the colors represent the value of the process $U$ (the jump times $\tau_k$ therein are represented by the symbol $\otimes$).
• Figure 2: This figure shows a realization of the càdlàg concatenated path $A^{\boldsymbol a,\boldsymbol b}_{\boldsymbol t}$ with $\boldsymbol a=((1,1),(3,0))$, $\boldsymbol b=((1,1),(3,0))$, and $\boldsymbol t=(1,1)$ (i.e., the first path goes from $(i,x)=(1,1)$ to itself in time $1$, and the second path goes from $(i,x)=(3,0)$ to itself in time $1$).
• Figure 3: In this figure, the arrows represent the first six jumps of $U$ by time $t$, namely, $(1,2),(2,3),(3,1),(1,2),(2,3),(3,1)$. The partition above the arrows serve to highlight which jumps were paired by the perfect matching $p=\{\{1,4\},\{2,5\},\{3,6\}\}$. With this in hand, we conclude that this particular combination of $(p,U)$ is such that $\mathfrak C_t(p,U)\neq0$ when $\mathbb F=\mathbb R,\mathbb H$, since $J_1=J_4$, $J_2=J_5$, and $J_3=J_6$. However, $\mathfrak C_t(p,U)=0$ when $\mathbb F=\mathbb C$ because paired jumps are not reversed.
• Figure 4: In this figure, the six jumps by time $t$ are $(1,2),(2,1),(1,3),(3,1),(1,2),(2,1)$, and $p=\{\{1,6\},\{2,5\},\{3,4\}\}$. In this case, $\mathfrak C_t(p,U)\neq0$ for all $\mathbb F=\mathbb R,\mathbb C,\mathbb H$, since every paired jumps are reversed (i.e., $J_1=J_6^*$, $J_2=J_5^*$, and $J_3=J_4^*$).
• Figure 5: In this figure, the black arrows are the steps $J_k$ until $N(t)$, the blue arrows are the steps of a binary sequence $m\in\mathcal{B}^{0,0}_{N(t)}$, and the partition above all that represents the pairs given by $p$. In order to help decipher whether this choice of $m$ contributes to $\mathfrak D_t(p,U)$, the matched pairs of steps such that $J_{\ell_1}=J_{\ell_2}$ are colored in green, and the matched pairs such that $J_{\ell_1}=J_{\ell_2}^*$ are colored in red. In this case, $m$ does not respect $(p,J)$, and thus it does not contribute to $\mathfrak D_t(p,U)$. (E.g., $J_4$ and $J_5$ are matched and in opposite directions, but the corresponding steps in the binary sequence, $((0,0),(0,1))$, are not among those listed in Definition \ref{['Definition: Combinatorial Constant']}-(3.2))

Theorems & Definitions (80)

• Remark 1.1
• Proposition 1.2
• Remark 1.3
• Theorem 1.4
• Definition 1.5
• Theorem 1.6: Guneysu
• Definition 1.8
• Theorem 1.9: Informal
• Definition 1.10
• Proposition 1.11: Informal