Self-consistent equations and quantum diffusion for the Anderson model

We consider the Anderson tight-binding model on $\mathbb{Z}^d$, $d\geq 2$, with Gaussian noise and at low disorder $\lambda>0$. We derive a diffusive scaling limit for the entries of the resolvent $R(z)$ at imaginary part $\operatorname*{Im} z\sim\lambda^{2+\kappa_d}$, $\kappa_d>0$, with high probability. As consequences, we establish quantum diffusion (in a time-averaged sense) for the Schr\"{o}dinger propagator at the longest timescale known to date and improve the best available lower bounds on the localization length of eigenfunctions. Our results for $d=2$ are the first quantum diffusion results for the Anderson model on $\mathbb{Z}^2$. The proof avoids the use of diagrammatic expansions and instead proceeds by analyzing certain self-consistent equations for $R(z)$. This is facilitated by new estimates for $\|R(z)\|_{\ell^p\rightarrow \ell^q}$ that control the recollisions.