Resonant delocalization for random Schr\"odinger operators on tree graphs

We analyse the spectral phase diagram of Schr"odinger operators $ T +\lambda V$onregulartreegraphs,with$T$thegraphadjacencyoperatorand$V$ a random potential given by iid random variables. The main result is a criterion for the emergence of absolutely continuous (ac) spectrum due to fluctuation-enabled resonances between distant sites. Using it we prove that for unbounded random potentials ac spectrum appears at arbitrarily weak disorder $(\lambda \ll 1)$ in an energy regime which extends beyond the spectrum of $T$. Incorporating considerations of the Green function’s large deviations we obtain an extension of the criterion which indicates that, under a yet unproven regularity condition of the large deviations’ ‘free energy function’, the regime of pure ac spectrum is complementary to that of previously proven localization. For bounded potentials we disprove the existence at weak disorder of a mobility edge beyond which the spectrum is localized.