The two-dimensional Anderson model is the model $$ H = T + \lambda V_\omega $$ where $T$ is nearest-neighbor hopping on $\mathbb{Z}^2$ and $V_\omega$ is a random potential. $\lambda > 0$ is the disorder strength. Taking the normalization of $T$ so that the spectrum of it is $[0,8]$, it is conjectured (yet not proven if I'm not mistaken) that for arbitrarily small $\lambda>0$ the system exhibits localization at all energies. For concreteness we pick $$T_{x,y} = 4\delta_{x,y}-\delta_{1,\|x-y\|} $$ and $$ V_{\omega xy} = \delta_{x,y}\omega(x) $$ where $\omega:\mathbb{Z}^2\to[0,1]$ is some sequence of independent and uniformly distributed random variables.
Furthermore it is conjectured that this complete localization is different in nature than that of one dimension, in the sense that it takes a particle in an infinite box "an infinite amount of time" to feel localized (I suppose beforehand it undergoes stages of ballistic and then diffusive motion).
My question is what kind of signature this "different" kind of localization leaves on the Greens function when compared with the one-dimensional complete localization or the any-dimensional localization of strong disorder?
For the "usual" type of localization we know that the localization length is finite, meaning that in a certain sense, $$ G(x,y;E) \sim \exp(-\mu \|x-y\|)\qquad (x,y\in\mathbb{Z}^d)\tag{1} $$ where $G$ is the Greens function from $x$ to $y$ at energy $E$, and $1/\mu$ is the localization length.
What kind of behavior should one expect for the Greens function of $2D$ systems for arbitrarily small $\lambda$ and not necessarily at the edges of the spectrum? Supposedly there has to be a difference in this kind of special localization in $2D$.
- The sense in which (1) is meant is really $$ \sup_{\eta>0}\eta\langle |G(x,y;E+\mathrm{i}\eta)|^2\rangle \leq C \exp(-\mu\|x-y\|)\qquad(E\in\mathbb{R};\,x,y\in\mathbb{Z}^d) $$ where $\langle\cdot\rangle$ is the disorder average.
