# Modern random matrix theory and connections to many – body – localization

Prof Vladimir Kravtsov (ICTP, Trieste)

Course description

Program

1. Application of random matrices in physics

2. Invariant and non-invariant random matrix ensembles

2.1. Probability distribution of invariant and Gaussian-non-invariant ensemble

2.2. Strong and weak confinement

2.3. Special cases of non-invariant ensembles: Rosenzweig-Porter and Power-law banded

random matrices

3. Joint probability distribution of eigenvalues and eigenvectors of Wigner-Dyson ensemble and

level repulsion

3.1. The Jacobian and Vandermond determinant

3.2. Level repulsion (poor men derivation)

4. Dyson symmetry classes and their extension

4.1 Time-reversal symmetry and the Dyson symmetry classes

4.2. Particle-hole symmetry and the 10-fold way Cartan symmetry classes

5. Level statistics of Wigner-Dyson RMT as 1D gas of quantum and classical particles

5.1. Classical plasma of logarithmically-interacting particles

5.2. Calogero-Sutherland models of interacting fermions

6. Wigner semi-circle and the probability of hole creation in a 1D log-interacting plasma

6.1. Wigner semi-circle from the solution of integral equation for equilibrium profile of log-

interacting 1D plasma

6.2. Probability of a hole in log-interacting plasma and the tail of the level spacing

distribution

6.3. Hole production at weak confinement

7. Level compressibility, normalization sum rule and normalization anomaly

7.1. Dos correlation function

7.2. The level number variance

7.3. The sum rule and the normalization anomaly

8. Solution of invariant ensembles by orthogonal polynomials

8.1. Orthogonal polynomials and there-term recursive relation

8.2. Expression for DoS correlation function in terms of orthogonal polynomials

8.3. The Wigner-Dyson correlation kernel and higher-order correlation functions

9. WKB quasi-classical approximation for orthogonal polynomials and the one-dimensional

Wigner crystal of energy levels

10. Logarithmic confinement and DoS correlation function

10.1. Dos at logarithmic confinement

10.2. Unfolding and the DoS correlation function: emergence of a “ghost correlation hole”

10.3. Analogy with a black hole: singular metric and Hawking temperature

11. Luttinger liquid of energy levels and DoS correlation at \beta=2,4

11.1. Luttinger liquid from the Calogero-Sutherland model

11.2. DoS correlation function for \beta=2,4

11.3. Power-law banded random matrices as deformation of the Wigner-Dyson theory

11.4. Levels statistics for the Power-Law banded random matrices as a Luttinger liquid at a

finite temperature

12. Localization and multifractality

12.1. What is multifractality of wave functions?

12.2. Multifractal measures: moments of IPR and f(alpha)

13. Multifractal wave functions in Power-law banded random matrices

14. Multifractal wave functions in Rosenzweig-Porter random matrix theory

14.1. Anderson localization and Mott’s delocalization criteria for full random matrices

14.2. Phase diagram for the Gaussian and log-normal Rosenzweig-Porter modelsc.

14.3. Multifractal dimensions

15. Localization, diffusion and sub-diffusion in log-normal Rosenzweig-Porter model

15.1. Return/survival probability

15.2. Mini-bands

15.3. Wigner-Weisskopf approximation and average return probability

15.4. Stretch-exponential dynamics of return probability and sub-diffusion

15.5. Dynamical phase diagram

16. Anderson model on random regular graph and the log-normal Rosenzweig-model

16.1. The Cayley tree and RRG

16.2. Statistics of Green’s functions on the Cayley tree

16.3. Abou-Chakra-Thouless-Anderson (A-CTA) duality

16.4. Symmetry of the moments of Green’s functions

16.5. Construction of the Rosenzweig-Porter model associated withRRG

16.6. The log-normal Rosenzweig-Porter model with A-CTA symmetry