Self-consistent equations and quantum diffusion for the Anderson model

Abstract

We consider the Anderson tight-binding model on ${\mathbb{Z}}^d$, $d\ge 2$, with Gaussian noise and at low disorder $\hat{I}»>0$. We derive a diffusive scaling limit for the entries of the resolvent $R(z)$ at imaginary part $\mathrm{Im}z\sim \hat{I}{»}^{2+\hat{I}{º}_d}$, $\hat{I}{º}_d>0$, 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 $\Vert R(z){\Vert }_{{\ell }^p\to {\ell }^q}$ that control the recollisions.

Corrected the exponent $\hat{I}{º}_d$ in high dimensions