Robust extended states in Anderson model on partially disordered random regular graphs

In this work we analytically explain the origin of the mobility edge in the ensemble of random regular graphs (RRG), with the connectivity d and the fraction β of disordered nodes, the location of which is under control. It is shown that the mobility edge in the spectrum survives in a certain range of parameters (d,β) at infinitely large uniformly distributed disorder. The critical curve separating extended and localized states is derived analytically and confirmed numerically. The duality in the localization properties between the sparse and extremely dense RRG has been found and understood. The mobility edge physics has been analyzed numerically for the above partially disordered RRG, perturbed by the non-reciprocity parameter of node as well as by the enhanced number of short cycles, usually almost absent on RRG.