Resonant delocalization for random Schrödinger operators on tree graphs

Michael Aizenman, Simone Warzel

Research output: Contribution to journalArticlepeer-review

54 Scopus citations


We analyse the spectral phase diagram of Schrödinger operators T + λV on regular tree graphs, with T the graph adjacency operator and 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 (λ * 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.

Original languageEnglish (US)
Pages (from-to)1167-1222
Number of pages56
JournalJournal of the European Mathematical Society
Issue number4
StatePublished - 2013

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • Applied Mathematics


  • Absolutely continuous spectrum
  • Anderson localization
  • Cayley tree
  • Mobility edge


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