V.E.Kravtsov ICTP, Trieste and Landau Institute Collaboration: Ivan Khaymovich, Aaalto Emilio Cuevas, Murcia rr site disorder: branching number K Energy E = 0 Giulio Biroli, 2011: ET - YES De Luca et al. 2014: ET No, only non- ergodic phase Mirlin & Fyodorov: : ET: No, only ergodic phase Extrapolation from N=2K- 32K Structural disorder: WD (Uzy Smiliansky) On-site energy disorder: P GOE: Special diagonal: Rosenzweig- Porter ensemble Adding diagonal disorder Two-level correlation function (a generalization of Kunz and Shapiro) R( R( Level compressibility GUE Diagonal RM Izikson-Zuber formula of integrating over unitary matrices diagonalizing H Rescaling and N-> infinity Depends on the distribution function of the diagonal matrix A=diag{a} tend to a finite limit independent of cancels out Poisson due to infinitely fast oscillations regular oscillations 1< no oscillations jump to the GUE form 1 2 localized Extended, non-ergodic Extended ergodic 1/lnN There is some curvature! ideal insulator GOE Rosenzweig-Porter two transition points only one transition pointy K=2 Rosenzweig-Porter RM model contains both Anderson and Ergodic transitions They are seen in the rigorous theory of the two-level correlation function Perturbative treatment of the eigenfunction statistics gives the three phases: localized, non-ergodic extended, ergodic extended Numerics confirm existence of three phases moments (Shannon entropy) vs lnN has a curvature which changes sign at the transitions.