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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.

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  • $\begingroup$ I remember Kamil Kaleta and Katarzyna Pietruska-Pałuba did some research on non-local operators with random potentials. Not sure if this is relevant to your question (and certainly not on the physics side), but anyway you may like to have a look at their arXiv:1601.05597. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Dec 14, 2018 at 20:06
  • $\begingroup$ @MateuszKwaśnicki: thank you for this reference. $\endgroup$
    – Abdelmalek Abdesselam
    Commented Dec 18, 2018 at 11:56

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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.

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    $\begingroup$ Thanks for the reference. Although, I am not sure about the validity of the equivalence between fractional Laplacian on the square lattice versus ordinary Laplacian on fractal lattices. I would prefer references on the former. $\endgroup$
    – Abdelmalek Abdesselam
    Commented Nov 24, 2018 at 20:56

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