1. V.E.Kravtsov ICTP, Trieste and Landau Institute Collaboration: Ivan Khaymovich, Aaalto Emilio Cuevas, Murcia

3. rr site disorder: branching number K Energy E = 0

5. Giulio Biroli, 2011: ET - YES De Luca et al. 2014: ET – No, only non- ergodic phase Mirlin & Fyodorov: 1991-1997: ET: No, only ergodic phase Extrapolation from N=2K- 32K

6. Structural disorder: WD (Uzy Smiliansky) On-site energy disorder: P  

7. GOE: Special diagonal: Rosenzweig- Porter ensemble Adding diagonal disorder

8. Two-level correlation function (a generalization of Kunz and Shapiro) R(    R(    Level compressibility 

9. GUE Diagonal RM Izikson-Zuber formula of integrating over unitary matrices diagonalizing H

10. Rescaling and N-> infinity Depends on the distribution function of the diagonal matrix A=diag{a}

11. 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  

12.  1 2 localized Extended, non-ergodic Extended ergodic

15. 1/lnN

18. There is some curvature!

19. ideal insulator GOE

20. Rosenzweig-Porter two transition points

21. only one transition pointy K=2

22.  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.