TITLE: Anderson localization for fractional Laplacians
QUESTION [5 upvotes]: There is a vast literature on Anderson localization, namely, the study of decay of eigenfunctions of operators on $l^2(\mathbb{Z}^d)$ such as
$$
-\Delta+\lambda V
$$
where $\Delta$ is the discrete lattice Laplacian and the potential $V$ is random say given by a vector $(V_{\mathbf{x}})_{\mathbf{x}\in\mathbb{Z}^d}$ of iid standard normal variables. The constant $\lambda$ is the strength of disorder.
Did anyone study similar random operators
$$
(-\Delta)^{\alpha}+\lambda V
$$
with a fractional Laplacian? 
I am in particular interested in references from the physics literature which provide some heuristics, e.g., a scaling theory à la Abrahams et al.

REPLY [2 votes]: Since fractional Laplacians describe diffusion on a fractal, for a physics application I would focus on Anderson localization on a fractal. The scaling theory mentioned in the OP has been applied to a fractal in An attractive critical point from weak antilocalization on fractals.