1. Spectral and Wavefunction Statistics (II) V.E.Kravtsov, Abdus Salam ICTP

2. Wavefunction statistics  Porter-Thomas distribution  Wavefunction statistics in one-dimensional Anderson model  Weak multifractality in 2D disordered conductors

3. Wavefunction statistics and Coulomb peaks heights Contact area Bunching

4. Porter-Thomas distribution -follows from random matrix theory -describes local distribution of wavefunction intensity in chaotic systems -fails to describe the “web” pattern

5. The Anderson model 1D: all states are localized with localization length W 3D: Anderson localization transition at W=16.5 2D: all states are localized with exponentially large localization radius ~exp(-a/W ) 2 2

6. Distribution of eigenfunction amplitude for 1D Anderson model Porter-Thomas: Take =2then the distribution for 1D Anderson model can be considered as limit of the Porter-Thomas distribution.

7. Poisson distribution as a  of the Wigner surmise   No surprise that the limit of Porter-Thomas gives the distribution of localized functions in one dimension.

8. Distribution of wavefunction amplitude in 2D conductors (L x Porter- Thomas Porter-Thomas with corrections Log-normal

9. Sample dimension from local measurement 2D Quasi 1D Porter- Thomas Porter-Thomas with corrections Log-normal in 2D: Stretch- exponential in quasi-1D Zero- dimensional quantum dot For g>>1 RMT behavior for a typical wavefunction Dimension-specific behavior for large amplitudes

10. Where the RMT works Porter- Thomas Porter-Thomas with corrections g>>1 For dynamic phenomena g Diffusion displacement for time 1/

11. Weak multifractality in 2D conductors 2=dimensionality of space q-dependent multifractal dimensionality Magnetic field makes fractality weaker

12. Fractal dimension of this map decreases with increasing the level=“multi”-fractality Multifractality: qualitative picture Map of the regions where  exceeds the chosen level  Arbitrary chosen level Weight of the dark blue regions scales like