Renormalization Group approach to the Anderson model in finite and infinite dimensions

In this talk, I will present a renormalization group analysis of the problem of Anderson localization in finite spacial dimensions d and on Regular Random Graphs (RRGs). I will first review and extend the finite-dimensional analysis of Abrahams, Anderson, Licciardello, and Ramakrishnan in terms of spectral observables, and discuss how to take the large-d limit. I will then motivate that the infinite-dimensional case, relevant also in the context of Many-Body Localization, recovers the Anderson model on RRGs. In this case, the renormalization group β-function necessarily involves two parameters, but the one-parameter scaling hypothesis is recovered for sufficiently large system sizes. I will also discuss how to understand this change in behavior in terms of the geometrical properties of the graphs. The talk will be based on arXiv:2306.14965 and ongoing work.