# Workshop Programme

## for period 14 - 25 July 2008

### Anderson Localization Transition: Introductory Training Course

14 - 25 July 2008

Timetable

 Monday 14 July 08:30-09:25 Registration 09:25-09:30 David Wallace - welcome 09:30-10:30 Pastur, L (Kharkov) Introduction to spectral theory (1) Sem 1 Basic notions of Hilbert spaces, linear operators (bounded and not), selfadjoint and essentially selfadjoint operators, resolvent, criteria of selfadjointess, spectral theorem, spectral types(absolutely continuous, singular continuous, pure point). 10:30-11:00 Coffee 11:00-12:30 Fyodorov, Y (Nottingham) Introduction to supermatrix techniques (1) Sem 1 Background and motivation. Inverse of the spectral determinant as a Gaussian integral. Reducing negative moments of the GUE spectral determinant to matrix integrals by exploiting the singular value ("polar") parametrization 12:30-13:30 Lunch at Churchill College 14:00-15:30 Pastur, L (Kharkov) Introduction to spectral theory (2) Sem 1 Finite-difference and differential operators, the Schrödinger operator (discrete and continuous), one-dimensional operators of second order, examples (operators with constant and periodic coefficients), Weyl's theory. Generalized eigenfunctions and their polynomial boundedness. 15:30-16:00 Tea 16:00-16:45 Benjamin, C (Leeds) Effect of disorder on positive shot noise cross-correlations in superconducting hybrids Sem 1 Disorder can suppress as well as enhance quantum interference effects. In this work we discuss the effects of disorder on positive shot noise cross-correlations seen in normal metal-superconductor-normal metal structures. There is an enhancement in the cross-correlation signal due to disorder and this is related with the enhanced crossed conductance. 17:30-18:10 Welcome Wine Reception 18:15-19:00 Dinner at Churchill College
 Tuesday 15 July 09:00-10:30 Pastur, L (Kharkov) Introduction to spectral theory (3) Sem 1 General frameworks of ergodic theory, ergodic theorems (additive and subadditive). Selfadjoint ergodic operators and their basic spectral properties (selfadjointess, non-randomness of the spectrum and its components). Examples and heuristics. 10:30-11:00 Coffee 11:00-12:30 Brydges, D (Vancouver) Statistical mechanics of phase transitions (1) Sem 1 Gaussian measures on a lattice. Main examples: massive and massless Gaussian, Ising model equals perturbation of massless Gaussian by Stratonovich transformation. Wick theorem and Feynman diagrams. Normal ordering = orthogonal functions = hermite polynomials. 12:30-13:30 Lunch at Churchill College 14:00-15:30 Pastur, L (Kharkov) Introduction to spectral theory (4) Sem 1 Integrated Density of States: existence, continuity, Wegner lemma, asymptotic behavior on spectrum edges, including the Lifshitz tails. Correlation functions (moments of spectral measure) and their spectral and physical meaning. 15:30-16:00 Tea 16:00-16:45 Winn, B (Loughborough) Quasimodes of Seba billiards Sem 1 We construct quasimodes for the Seba billiard and related systems. The Seba billiard is a rectangle billiard with a delta-function potential. By making further assumptions on the spectrum of the rectangle billiard, we are able to prove that the quasimodes, in fact, approximate a sequence of true eigenmodes. The additional hypothesis required is related to the Berry-Tabor conjecture for energy levels of integrable quantum systems 18:15-19:00 Dinner at Churchill College
 Wednesday 16 July 09:00-10:30 Pastur, L (Kharkov) Introduction to spectral theory (5) Sem 1 One dimensional ergodic operators of second order with ergodic coefficients, existence of the Lyapunov exponents, their positivity, typical asymptotic behavior, and the role in the spectral analysis. Pure point spectrum (complete localization) for one dimensional operators with random coefficients. 10:30-11:00 Coffee 11:00-12:30 Fyodorov, Y (Nottingham) Introduction to supermatrix techniques (2) Sem 1 Exploiting hyperbolic symmetry for asymptotic analysis of the negative moments. Example of a zero dimensional hyperbolic nonlinear sigma model (n.s.m). Failure of the naïve Hubbard-Stratonovich transformation (HST). HST on Schäfer-Wegner domain and reduction to the U(1,1) n.s.m. 12:30-13:30 Lunch at Churchill College 14:00-15:30 Goldsheid, I (Queen Mary, London) Rigorous techniques for 1D systems (1) Sem 1 Measure spaces, measure preserving transformations, ergodic transformations, Birkhoff's ergodic theorem, Kingman's sub-additive ergodic theorem. Examples. Products of matrices and Lyapunov exponents: elementary properties. Stationary sequences of random matrices. Products of stationary random matrices: examples. Lyapunov exponents of products of stationary sequences of matrices; the Oseledets's multiplicative ergodic theorem (MOT). The idea of the proof of the MOT; the MOT for products of 2x2 matrices. 15:30-16:00 Tea 16:00-17:30 Pastur, L (Kharkov) Introduction to spectral theory (6) Sem 1 18:15-19:00 Dinner at Churchill College
 Thursday 17 July 09:00-10:30 Brydges, D (Vancouver) Statistical mechanics of phase transitions (2) Sem 1 Example: the generating function for an ensemble of self-avoiding loops equals a Gaussian integral. Grassmann integration versus differential forms. Examples (in order of whether I have time for them): 1. Self-avoiding walk as a Gaussian integral, 2. Matrix tree theorems. Supersymmetry in terms of forms (if I have time) 10:30-11:00 Coffee 11:00-12:30 Fyodorov, Y (Nottingham) Introduction to supermatrix techniques (3) Sem 1 Hubbard-Stratonovich transformation on the Pruisken-Schäfer domain and reduction to hyperbolic nonlinear sigma model. Grassmann numbers and their basic properties. Spectral determinant as a Gaussian integral over anticommuting variables. 12:30-13:30 Lunch at Churchill College 14:00-15:30 Disertori, M (Rouen) GUE density of states by SUSY Sem 1 15:30-16:00 Tea 16:00-18:00 Poster session 18:15-19:00 Dinner at Churchill College
 Friday 18 July 09:00-10:30 Spencer, T (IAS, Princeton) Anderson localisation: phenomenology and mathematics (1) Sem 1 Quantum mechanics, time evolution, Green's functions. Resolvent identity and finite rank perturbations. An intuitive semiclassical picture. Combes-Thomas estimates for Green's functions. 10:30-11:00 Coffee 11:00-12:30 Brydges, D (Vancouver) Statistical mechanics of phase transitions (3) Sem 1 Ideas of scaling.The Hierarchical Gaussian field. Renormalisation group and critical mass illustrated for the phi^4 lattice model within hierarchical approximation. Comments on RG for Euclidean models 12:30-13:30 Lunch at Churchill College 14:00-15:30 Goldsheid, I (Queen Mary, London) Rigorous techniques for 1D systems (2) Sem 1 Furstenberg's theorem about the positivity of the top Lyapunov exponent. The general theorem about the existence of distinct Lyapunov exponents. The Anderson model on a strip and products of symplectic matrices. Lyapunov exponents of products of symplectic matrices. One elementary application: randomness of the eigenvalues of a 1D and quasi-1D Anderson model. 15:30-16:00 Tea 16:00-16:45 Najar, H (Kairouan) Localization for random divergence operators Sem 1 In this work we study internal band edges localization of random operators in divergence form. Our results rely on the study of Lifshitz tails for the integrated density of states for random the operators of the form $H_{\omega}=-\sum_{i,j=1}^d\partial_{x_{i}}a_{i,j}(x,\omega)\partial_{x_j}$. Localization is then deduced by the standard multiscale argument. In dimension one and two we get results with weaker assumptions. 16:45-17:30 Rogers, T (King's College, London) Cavity approach to the spectral density of sparse symmetric random matrices Sem 1 In this talk the spectral density of the ensemble of sparse symmetric random matrices is analyzed using the cavity method. We consider two cases: matrices whose associated graphs are locally tree-like, and sparse covariance matrices. We derive a closed set of equations from which the density of eigenvalues can be efficiently calculated. Within this approach, the Wigner semicircle law for Gaussian matrices and the Marcenko-Pastur law for covariance matrices are recovered easily. Our results are compared with numerical diagonalization, finding excellent agreement. 18:15-19:00 Dinner at Churchill College
 Monday 21 July 09:00-10:30 Brydges, D (Vancouver) Statistical mechanics of phase transitions (4) Sem 1 The Ising Model. Curie-Weiss Mean field theory by steepest descent. Mean field theory as Kac limit. Mean field lower bound on pressure by Jensen 10:30-11:00 Coffee 11:00-12:30 Fyodorov, Y (Nottingham) Introduction to supermatrix techniques (4) Sem 1 Analysis of positive moments of spectral determinant for GUE by Hubbard-Stratonovich and by bosonisation identity. Compact version of the nonlinear sigma model. Resolvents as the main objects of interest. Basics of supercalculus. 12:30-13:30 Lunch at Churchill College 14:00-15:30 Goldsheid, I (Queen Mary, London) Rigorous techniques for 1D systems (3) Sem 1 The integrated density of states for the Anderson model on a strip and its regularity properties. Criteria for localization. Proof of localization for the Anderson model on a strip. 15:30-16:00 Tea 16:00-16:45 Krivolapov, Y (Technion) Anderson localisation and sub-diffusion for the nonlinear Schrodinger equtation: results and puzzles Sem 1 It is well known that transport is suppressed in disordered media in one dimension, which is known as Anderson localization. However, it is not known, even numerically, if adding nonlinearity destroys dynamical localization in the limit of long times. We have conducted an analytical and rigorous research that sheds some light on this subject. Using perturbation theory in the nonlinearity strength we have demonstrated that an initial wavepacket does not spread for short time scales and for long time scales it spreads at most logarithmically. These results provide better ground for understanding of the underlying processes of the competition between the randomness and the nonlinearity. 17:00-18:00 Anderson, PW (Princeton) The historical origins of localisation Sem 1 18:15-19:00 Dinner at Churchill College
 Tuesday 22 July 09:00-10:30 Brydges, D (Vancouver) Statistical mechanics of phase transitions (5) Sem 1 Breaking of continuous symmetries and the massless Gaussian Comment on Infra-red bounds and transfer matrix Example of breaking of symmetry in super-integrals Local time = square of Gaussian superfield. Luttinger's view of Donsker-Varadhan theory 10:30-11:00 Coffee 11:00-12:30 Spencer, T (IAS, Princeton) Anderson localisation: phenomenology and mathematics (2) Sem 1 Bound states, tunnelling. Wegner's estimate on the density of states, formal perturbation theory for the average Green's function and self energy. 12:30-13:30 Lunch at Churchill College 14:00-15:30 Ludwig, A (Santa Barbara) Critical phenomena in 2D disordered systems (1) Sem 1 The larger picture: Anderson localization and the "ten-fold way"; description of basic corresponding phenomena. Physics of the 2D Integer Quantum Hall plateau transition' (IQHT/class A): The Chalker-Coddington network model (CCNM) for the IQHT, supersymmetry formulation of the CCNM for the IQHT. Physics of the 2D Spin Quantum Hall Plateau transition' (SQHT/class C); a model for disordered superconductors 15:30-16:00 Tea 16:00-16:45 Brydges, D (Vancouver) Statistical mechanics of phase transitions (6) Sem 1 16:45-17:30 Halasan, F (British Columbia) Absolutely continuous spectrum for the Anderson model on more general trees Sem 1 We study the Anderson Model on trees that have a variation in their coordination number. Using geometric tools, we prove that the Anderson Hamiltonian has absolutely continuous spectrum for small disorder. 18:15-19:00 Dinner at Churchill College
 Wednesday 23 July 09:00-10:30 Ludwig, A (Santa Barbara) Critical phenomena in 2D disordered systems (2) Sem 1 Chalker-Coddington network model for the SQHT. Exact solution of the CCNM for the SQHT, and mapping to percolation. Discussion of some of the exact results: average density of states, average conductance in finite size, universal crossover function and critical conductivity. Physics of 2D `Thermal Quantum Hall transition' (TQHT/class D); topological (p_x+ip_y) superconductor; weak (topological) and strong pairing phases; connections with the 2D Ising model. The Chalker-Coddington network model for the TQHT: a richer variety of possible phenomena 10:30-11:00 Coffee 11:00-12:30 Fyodorov, Y (Nottingham) Introduction to supermatrix techniques (5) Sem 1 From the lattice of coupled GUEs to the lattice version of nonlinear sigma model. Brief overview of nonlinear sigma model beyond zero dimension. One-dimensional sigma-model: localisation. Delocalisation transition as a phenomenon of spontaneous symmetry breaking. 12:30-13:30 Lunch at Churchill College 14:00-15:30 Goldsheid, I (Queen Mary, London) Rigorous techniques for 1D systems (4) Sem 1 The Kotani theory. Stability and instability of localization. 15:30-16:00 Tea 16:00-16:45 Sadel, C (Erlangen) Lyapunov exponents of randomly coupled wires: a perturbative calculation Sem 1 We consider L discrete wires with a random potential and random hopping terms across the wires with magnetic phases. The random terms are centered, independent, identically distributed along the wires and coupled with some small constant. Associated to such a Hamiltonian are 2L by 2L transfer matrices. To calculate the mean Lyapunov exponent, one has to consider the induced Markov process on a Lagrangian Grassmanian manifold which is diffeomorphic to the unitary group U(L). We obtain that the lowest order invariant measure for this Markov process on U(L) is given by the Haar measure for non-zero energies |E|<2 and some smooth density for E=0. Furthermore we get a perturbative expression for the mean Lyapunov exponent. In case L=1 the dynamics on U(1) is just the dynamics of the Pruefer phases and the model would be the one-dimensional Anderson model. The result away from E=0 is known as random phase approximation and the anomaly at the band-center, E=0, was first found by Kappus and Wegner. To get this result we first consider an abstract setting. Let be given a Markov process on some compact homogeneous space induced by the action of a random family of Lie group elements, where the random terms are coupled with some small constant. Under certain conditions we can prove, that the invariant measure for this process is unique to lowest order and given by a smooth density which is the ground state of a Fokker-Planck operator. Finally we will see, that this theorem can be used in the example described above. 19:30-23:00 Conference dinner - Emmanuel College (Old Library)
 Thursday 24 July 09:00-10:30 Ludwig, A (Santa Barbara) Critical phenomena in 2D disordered systems (3) Sem 1 (Thermal) Quantum Hall transition, and relationship to the 2D random bond Ising model / Nishimori fixed point; absence of metallic phase and relationship to Ising Kramers-Wannier duality. Metallic phase; metal-insulator transition. Scaling properties of the critical wavefunctions. Multifractality, in the bulk and at boundaries: the case of the 2D spin-orbit class metal insulator transition; direct numerical evidence of conformal symmetry. 2D SQHT: some exact results. 2D IQHT: multifractality. 10:30-11:00 Coffee 11:00-12:30 Fyodorov, Y (Nottingham) Introduction to supermatrix techniques (6) Sem 1 From the lattice of coupled GUEs to the lattice version of nonlinear sigma model. Brief overview of nonlinear sigma model beyond zero dimension. One-dimensional sigma-model: localisation. Delocalisation transition as a phenomenon of spontaneous symmetry breaking. 12:30-13:30 Lunch at Churchill College 14:00-15:30 Spencer, T (IAS, Princeton) Anderson localisation: phenomenology and mathematics (3) Sem 1 A statistical mechanics picture of localization. Finite volume criterion for localization. Simple multi-scale approach to localization. Fractional moment methods for proving localization. Relation of Greens functions and eigenfunctions. 15:30-16:00 Tea 16:00-16:45 Sidorova, N (UCL) A two cities theorem for the parabolic Anderson model Sem 1 The parabolic Anderson problem is the Cauchy problem for the heat equation u_t(t,z)=Delta u(t,z)+xi(z)u(t,z) on the d-dimensional integer lattice with random potential. We consider independent and identically distributed potentials, such that the corresponding distribution function converges polynomially at infinity. If the solution is initially localised in the origin we show that, as time goes to infinity, it will be completely localised in two points almost surely and in one point with high probability. We also identify the asymptotic behaviour of the concentration sites in terms of a weak limit theorem. 18:15-19:00 Dinner at Churchill College
 Friday 25 July 09:00-10:30 Spencer, T (IAS, Princeton) Anderson localisation: phenomenology and mathematics (4) Sem 1 Thouless scaling of conductance. A mathematical interpretation of scaling. Perturbation theory for conductivity. Ladder diagrams and crossed ladder graphs. A renormalization group approach. 10:30-11:00 Coffee 11:00-12:30 Goldsheid, I (Queen Mary, London) Rigorous techniques for 1D systems (5) Sem 1 Other 1D models. One-dimensional random Schroedinger operators with decaying potentials. Stationary non-hermitian Anderson model: properties of the eigenvalues and eigenfunctions. 12:30-13:30 Lunch at Churchill College 14:00-15:30 Ludwig, A (Santa Barbara) Critical phenomena in 2D disordered systems (4) Sem 1 15:30-16:00 Tea 18:15-19:00 Dinner at Churchill College