Self-consistent equations and quantum diffusion for the Anderson model

We consider the Anderson tight-binding model on ${\mathbb{Z}}^d$, $d\ge 2$, 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 $\mathrm{Im}z\sim {\lambda }^{2+{\kappa }_d}$, ${\kappa }_d>0$, with high probability. As consequences, we establish quantum diffusion (in a time-averaged sense) for the Schrödinger 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 $\mathrm{\parallel }R(z){\mathrm{\parallel }}_{{\mathrm{\ell }}^p\to {\mathrm{\ell }}^q}$ that control the recollisions.