Diffusive propagation and disorder-scaling conjecture for ergodic Schrödinger potentials

Let $u$ be a bounded potential on $\mathbb{Z}^d$, let $\lambda>0$, and let $\psi_t$ solve the corresponding randomly perturbed Schrödinger equation. Define the lower and upper diffusion constants by

$$\underline{D}(\lambda):=\liminf_{t\rightarrow\infty}\frac{1}{t}\sum_{x\in\mathbb{Z}^d}|x|^2\mathbb{E}(|\psi_t(x)|^2),\qquad \overline{D}(\lambda):=\limsup_{t\rightarrow\infty}\frac{1}{t}\sum_{x\in\mathbb{Z}^d}|x|^2\mathbb{E}(|\psi_t(x)|^2).$$

Diffusive propagation and disorder-scaling conjecture. For every bounded potential $u$ and every $\lambda>0$, these constants are positive and finite, with $\underline{D}(\lambda)\leq\overline{D}(\lambda)$. If $\Delta+U$ exhibits ballistic motion, then $\underline{D}(\lambda),\overline{D}(\lambda)\sim O(\lambda^{-2})$ as $\lambda\sim0$; if $\Delta+U$ exhibits dynamical localization, then $\underline{D}(\lambda),\overline{D}(\lambda)\sim O(\lambda^2)$ as $\lambda\sim0$.

The conjecture asks for diffusive propagation for general ergodic or deterministic potentials and relates the small-coupling diffusion scale to the transport of the unperturbed operator. The source also raises the broader transport-exponent formulation, but that expectation is not included as a separate row because it is presented as a generalization of this conjecture.