# Workshop Programme

## for period 17 - 21 September 2012

### Mathematics and Physics of Disordered Systems

17 - 21 September 2012

Timetable

 Monday 17 September 08:30-09:25 Registration 09:25-09:30 Welcome from John Toland (INI Director) Chair: T Spencer 09:30-10:10 Pastur, L (National Academy of Sciences of Ukraine) On Random Matrices Related to Quantum Informatics Sem 1 We consider two ensembles of random matrices related to certain problems of quantum informatics. The rst ensemble is a generalization of the Wishart Ensemble viewed as the sum of independent rank-one operators. However, in contrast to the Wishart Ensemble the corresponding independent random vectors are the tensor products of a xed number p  2 of other independent vectors. The second ensemble is also similar to the Wishart Ensemble viewed as the product of matrix with independent entries and the transposed (hermitian conjugate) matrix. However, in contrast to the Wishart Ensemble the corresponding matrix is triangular. We show that the limiting Normalized Counting Measures can be found from certain functional equations and discuss their new properties. 10:10-10:50 Finkelstein, A (Weizmann Institute and Texas A&M University) Effects of Interactions and Non-linearity for a Pulse Propagation in a Disordered Medium Sem 1 10:50-11:10 Morning Coffee 11:10-11:50 Imbrie, J (University of Virginia) Newton's Method and Localization Sem 1 We show how a scheme based on Newton's method can be used to diagonalize a random Hamiltonian. This leads to results on localization at strong disorder for Anderson tight-binding models and for many-body Hamiltonians. 11:50-12:30 Germinet, F (Université de Cergy-Pontoise) Localization for Schrödinger operators with Delone potentials Sem 1 We prove that a large family of Delone Schrödinger operators exhibit Anderson localization. The proof relies on the analysis of random Schrödinger operators with a Delone Background potential and a Delone-Bernoulli random potential. Joint work with P. Mueller et C. Rojas-Molina. 12:30-13:30 Lunch at Wolfson Court Chair: M Zirnbauer 14:20-15:00 Gruzberg, I (University of Chicago) Quantum Hall transitions and conformal restriction Sem 1 A spectacular success in the study of random fractal clusters and their boundaries in critical statistical mechanics systems using Schramm-Loewner Evolutions (SLE) naturally calls for extensions in various directions. Can this success be repeated for disordered and/or non-equilibrium systems? Naively, when one thinks about disordered systems and their average correlation functions one of the very basic assumptions of SLE, the so called domain Markov property, is lost. Also, in some lattice models of Anderson transitions (the network models) there are no natural clusters to consider. Nevertheless, in this talk I will argue that one can apply the so called conformal restriction, a notion of stochastic conformal geometry closely related to SLE, to study the integer quantum Hall transition and its variants. I will focus on the Chalker-Coddington network model and will demonstrate that its average transport properties can be mapped to a classical problem where the basic objects ar e geometric shapes (loosely speaking, the current paths) that obey an important restriction property. At the transition point this allows to use the theory of conformal restriction to derive exact expressions for mean point contact conductances in the presence of various non-trivial boundary conditions. 15:00-15:40 Bolthausen, E (Universität Zürich) On the localization-delocalization critical line for the random copolymer Sem 1 The random copolymer is a model for a long polymer chain at an interface of two liquids, say oil and water. The nodes of the polymer chain are either water repellent or oil repellent, placed in a random way or with random strengths. The model exhibits a phase transition between a localized phase and a delocalized phase. We present some new results on the phase separation line. 15:40-16:10 Afternoon Tea 16:10-16:50 Kupiainen, A (University of Helsinki) Critical Mandelbrot Cascades Sem 1 Mandelbrot's multiplicative cascade measures are random energy models with logarithmically correlated energies. They exhibit a freezing transition from high temperature phase with (singular) continuous Gibbs measure to an atomic one in low temperatures. I will discuss recent work proving that in the critical case these measures have no atoms. I also discuss the multi fractal spectrum and the KPZ formula of Liouville gravity at the critical point and low temperature phase. 16:50-17:30 Müller, P (Ludwig-Maximilians-Universität München) Localisation for random block operators Sem 1 We study spectral properties of random block operators that are common in the physical modelling of mesoscopic disordered systems such as dirty superconductors. Our results include ergodic properties, the location of the spectrum, existence and regularity of the integrated density of states (Wegner estimate), Lifshits tails and dynamical localisation in a neighbourhood of the internal band edges. Special attention is paid to the peculiarities arising from the block structure. Technically, one has to cope with a non-monotone dependence on the random couplings. 17:30-18:30 Welcome Drinks Reception 18:45-19:30 Dinner at Wolfson Court
 Tuesday 18 September Chair: D Khmelnitski 09:30-10:10 Ryu, S (University of Illinois at Urbana-Champaign) Topological phases of matter, Anderson localization/delocalization and anomalous transport law Sem 1 Topological phases of matter are a very quantum, highly entangled state of matter. While topological phases and Anderson localization are a quite big research field by themselves and can be discussed independently, they have a very close connection. I will discuss a cross fertilization of both fields by focusing on, in particular, physics at a boundary of topological phases. Due to the "bulk-boundary correspondence", well known in the quantum Hall effect, quantum transport phenomena mediated by modes appearing at the boundary have a strong stability against impurities. I will extend this idea to a wider class of topological phases in higher dimensions and with the Altland-Zirnbauer discrete symmetries. In topological superconductors in two and three spatial dimensions, I will derive a new thermodynamic relation ("Streda-like" formula) due to the non-zero thermal Hall conductance, and a thernal analogue of the axion electrodynamics. 10:10-10:50 Vassiliev, D (University College London) Recent progress in the spectral theory of first order elliptic systems Sem 1 The talk deals with the distribution of eigenvalues of a linear self-adjoint elliptic operator. The eigenvalue problem is considered in the deterministic setting, i.e. the coefficients of the operator are prescribed smooth functions. The objective is to derive a two-term asymptotic formula for the counting function (number of eigenvalues between zero and a positive lambda) as lambda tends to plus infinity. There is an extensive literature on the subject (see, for example, [1]), mostly dealing with scalar operators. It has always been taken for granted that all results extend in a straightforward manner to systems. However, the author has recently discovered [2,3] that all previous publications on first order systems give formulae for the second asymptotic coefficient that are either incorrect or incomplete (i.e. an algorithm for the calculation of the second asymptotic coefficient rather than an explicit formula). The aim of the talk is to explain the spectral theoretic difference between scalar operators and systems and to present the correct formula for the second asymptotic coefficient. [1] Yu.Safarov and D.Vassiliev, The asymptotic distribution of eigenvalues of partial differential operators, American Mathematical Society, 1997 (hardcover), 1998 (softcover). [2] Preprint arXiv:1204.6567. [3] Preprint arXiv:1208.6015. 10:50-11:10 Morning Coffee 11:10-11:50 Mirlin, A (Universität Karlsruhe (TH)) Classification and symmetry properties of scaling dimensions at Anderson transitions Sem 1 Multifractality of wave functions is a remarkable property of Anderson transition critical points in disordered systems. We develop a classification of gradientless composite operators (that includes the leading multifractal operators but is much broader) representing correlation functions of local densities of states (or wave function amplitudes) at Anderson transitions. Our classification is based on the Iwasawa decomposition for the underlying supersymmetric sigma-model field: the operators are represented by "plane waves" in terms of the corresponding "radial" coordinates. We present also an alternative (but equivalent) construction of scaling operators that uses the notion of highest-weight vectors. We further show that the invariance of the sigma-model manifold with respect to a Weyl group leads to numerous exact symmetry relations between the scaling dimensions of the composite operators. 11:50-12:30 Keating, J (University of Bristol) Freezing Transition, Characteristic Polynomials of Random Matrices, and the Riemann Zeta-Function Sem 1 We argue that the freezing transition scenario, previously explored in the statistical mechanics of 1/f-noise random energy models, also determines the value distribution of the maximum of the modulus of the characteristic polynomials of large N x N random unitary (CUE) matrices. We postulate that our results extend to the extreme values taken by the Riemann zeta-function zeta(s) over sections of the critical line s=1/2+it of constant length and present the results of numerical computations in support. Our main purpose is to draw attention to possible connections between the statistical mechanics of random energy landscapes, random matrix theory, and the theory of the Riemann zeta function. 12:30-13:30 Lunch at Wolfson Court Chair: Y Fyodorov 14:20-15:00 Khanin, K (University of Toronto) Space-time stationary solutions for the random forced Burgers equation We construct stationary solutions for Burgers equation with random forcing in the absence of periodicity or any other compactness assumptions. In particular, for the forcing given by a homogeneous Poissonian point field in space-time we prove that there is a unique global solution with any prescribed average velocity. We also discuss connections with the theory of directed polymers in dimension 1+1 and the KPZ scalings. 15:00-15:40 O'Connell, N (University of Warwick) An interacting particle system related to the quantum Toda chain I will describe an interacting particle system which can be `solved' via a connection with the quantum Toda chain. The particle system can be regarded as a discretisation of the KPZ (or stochastic heat) equation and is closely related to a semi-discrete directed random polymer model in 1+1 dimensions. The connection to the quantum Toda chain yields an explicit integral formula for the Laplace transform of the distribution of the partition function associated with the polymer model. 15:40-16:20 Afternoon Tea 16:20-17:00 Sabot, C (Université Claude Bernard Lyon 1) Edge reinforced random walks, Vertex reinforced jump process, and the SuSy hyperbolic sigma model (I) Edge-reinforced random walk (ERRW), introduced by Coppersmith and Diaconis in 1986, is a random process which takes values in the vertex set of a graph G, and is more likely to cross edges it has visited before. We show that it can be represented in terms of a Vertex-reinforced jump process (VRJP) with independent gamma conductances: the VRJP was conceived by Werner and first studied by Davis and Volkov (2002,2004), and is a continuous-time process favouring sites with more local time. Then we prove that the VRJP is a mixture of time-changed Markov jump processes and calculate the mixing measure, which we interpret as a marginal of the supersymmetric hyperbolic sigma model introduced by Disertori, Spencer and Zirnbauer. This enables us to deduce that VRJP and ERRW are strongly recurrent in any dimension for large reinforcement (in fact, on graphs of bounded degree), using a localisation result of Disertori and Spencer (2010). 17:00-17:40 Ossipov, A (University of Nottingham) Critical properties of the long-range random models In this talk we consider random matrix ensembles and non-linear sigma models characterized by a long-range random hopping. At criticality, eigenstates of such systems are multifractal and the corresponding multifractal dimensions can be calculated analytically in the limit of strong or weak multifractality. Using explicit results for the multifractal dimensions and the level number variance, we discuss critical properties of various models. Among them are universal and non-universal features of the multifractality spectrum and unusual critical behavior of the two-dimensional power-law random matrix model. 19:30-22:00 Conference Dinner at Trinity College
 Wednesday 19 September Chair: B Khoruzhenko 09:30-10:10 Tarres, P (University of Oxford) Edge reinforced random walks, Vertex reinforced jump process and the SuSy hyperbolic sigma model (II) Sem 1 Edge-reinforced random walk (ERRW), introduced by Coppersmith and Diaconis in 1986, is a random process which takes values in the vertex set of a graph G, and is more likely to cross edges it has visited before. We show that it can be represented in terms of a Vertex-reinforced jump process (VRJP) with independent gamma conductances: the VRJP was conceived by Werner and first studied by Davis and Volkov (2002,2004), and is a continuous-time process favouring sites with more local time. Then we prove that the VRJP is a mixture of time-changed Markov jump processes and calculate the mixing measure, which we interpret as a marginal of the supersymmetric hyperbolic sigma model introduced by Disertori, Spencer and Zirnbauer. This enables us to deduce that VRJP and ERRW are strongly recurrent in any dimension for large reinforcement (in fact, on graphs of bounded degree), using a localisation result of Disertori and Spencer (2010). 10:10-10:50 Kravtsov, V (Abdus Salam International Centre for Theoretical Physics) Anomalously localized states in the one-dimensional Anderson localization model at E=0 Sem 1 10:50-11:10 Morning Coffee 11:10-11:50 Knowles, A (New York University) Diffusion profile and delocalization for random band matrices Sem 1 I give a summary of recent progress in establishing the diffusion approximation for random band matrices. We obtain a rigorous derivation of the diffusion profile in the regime W > N^{4/5}, where W is the band width and N the dimension of the matrix. As a corollary, we prove complete delocalization of the eigenvectors. Our proof is based on a new self-consistent equation for the Green function. Joint work with L. Erdos, H.T. Yau, and J. Yin. 11:50-12:30 Shepelyansky, D (Université Paul Sabatier Toulouse III) Kolmogorov turbulence, Anderson localization and KAM integrability Sem 1 12:30-13:30 Lunch at Wolfson Court Chair: L Parnovski 14:20-15:00 Crawford, N (Technion - Israel Institute of Technology) Localization for Linearly Reinforced Random Walks and Vertex Reinforced Jump Processes (iii) Sem 1 15:00-15:40 Disertori, M (Université de Rouen) Supersymmetry in localization/delocalization problems Sem 1 Supersymmetric approach has proved to be a powerful tool in the analysis of models for quantum (but also classical) diffusion. I'll give an overview of the technique with some applications, known results and a few conjectures. 15:40-16:10 Afternoon Tea 16:10-16:50 De Roeck, W (Universität Heidelberg) Diffusion for a quantum particle coupled to a phonon gas Sem 1 16:50-17:30 Jitomirskaya, s (University of California, Irvine) Analytic quasiperiodic matrix cocycles: continuity and quantization of the Lyapunov exponents Sem 1 As, beginning with the famous Hofstadter's butterfly, all numerical studies of spectral and dynamical quantities related to quasiperiodic operators are actually performed for their rational frequency approximants, the questions of continuity upon such approximation are of fundamental importance. The fact that continuity issues may be delicate is illustrated by the recently discovered discontinuity of the Lyapunov exponent for non-analytic potentials. I will focus on work in progress, joint with Avila and Sadel, where we develop a new approach to continuity, powerful enough to handle matrices of any size and leading to a number of strong consequences. 18:45-19:30 Dinner at Wolfson Court
 Thursday 20 September Chair: V Tchoulaevski 09:30-10:10 Sodin, M (Tel Aviv University) New results on Rademacher Fourier and Taylor series Sem 1 This is a report on a joint work in progress with Fedor Nazarov and Alon Nishry. We prove that any power of the logarithm of Rademacher Fourier series (i.e. a square summable Fourier series with random independent signs) is integrable. This result has several applications to zeroes and value-distribution of random Talor series. One of this applications gives asymptotics for the counting function of zeroes of arbitrary Taylor series with random independent signs, and proves their angular equidistribution. Another application answers an old question by J.-P.Kahane. 10:10-10:50 Le Doussal, P (École Normale Supérieure) Universal statistics for directed polymers and the KPZ equation from the replica Bethe Ansatz Sem 1 10:50-11:10 Morning Coffee 11:10-11:50 Shcherbina, T (Institute for Advanced Study, Princeton) Universality of the second mixed moment of the characteristic polynomials of the 1D Gaussian band matrices Sem 1 We consider the asymptotic behavior of the second mixed moment of the characteristic polynomials of the 1D Gaussian band matrices, i.e. of the hermitian matrices $H_n$ with independent Gaussian entries such that $\langle H_{ij}H_{lk}\rangle=\delta_{ik}\delta_{jl}J_{ij}$, where $J=(-W^2\triangle+1)^{-1}$. Assuming that $W^2=n^{1+\theta}$, $0<\theta<1$, we show that this asymptotic behavior (as $n\to\infty$) in the bulk of the spectrum coincides with those for the Gaussian Unitary Ensemble. 11:50-12:30 Ostrovsky, P (Max Planck Institut für Festkörperforschung) Anderson localization, topology, and interaction Sem 1 Field-theoretical approach to Anderson localization in 2D disordered fermionic systems of chiral symmetry classes (BDI, AIII, CII) is developed. Important representatives of these symmetry classes are random hopping models on bipartite lattices at the band center. As was found by Gade and Wegner two decades ago within the sigma-model formalism, quantum interference effects in these classes are absent to all orders of perturbation theory. We demonstrate that the quantum localization effects emerge when the theory is treated nonperturbatively. Specifically, they are controlled by topological vortexlike excitations of the sigma models by a mechanism similar to the Berezinskii-Kosterlitz-Thouless transition. We derive renormalization-group equations including these nonperturbative contributions. Analyzing them, we find that the 2D disordered systems of chiral classes undergo a metal-insulator transition driven by topologically induced Anderson localization. We also show that the topological terms on surfaces of 3D topological insulators of chiral symmetry (in classes AIII and CII) overpower the vortex-induced localization. Similar vortex excitations also emerge in systems with strong spin-orbit interaction (symplectic symmetry class AII). Such systems may exhibit topological insulator state both in three and two dimensions. Interplay of nontrivial topology and Coulomb repulsion induces a novel critical state on the surface of a 3D topological insulator. Remarkably, this interaction-induced criticality, characterized by a universal value of conductivity, emerges without any adjustable parameters. Interaction also leads to a direct transition between trivial insulator and topological insulator in 2D (quantum-spin-Hall transition) via a similar critical state. The nature of this latter critical state is closely related to the effects of vortices within the Finkelstein sigma model. 12:30-13:30 Lunch at Wolfson Court 13:30-17:30 Discussion Time 18:45-19:30 Dinner at Wolfson Court
 Friday 21 September Chair: W Kirsch 09:30-10:10 Shcherbina, M (Institute for Low Temperatures, Kharkov) Statistical mechanics approach to the random matrix models Sem 1 We consider hermitian, real symmetric and symplectic matrix models with real analytic potentials and present some analogue of the mean-field approximation method to study their partition functions in the multi-cut regime. Then we discuss recent results on the asymptotic behavior of the characteristic functional of linear eigenvalue statistics, obtained by this method, in particular, non gaussian behavior of the characteristic functional in the multi-cut regime. The applications to the proof of the universality conjecture for real symmetric and symplectic matrix models will be also discussed. 10:10-10:50 Gornyi, I (Forschungszentrum Karlsruhe) Enhancement of Superconductivity by Anderson Localization Sem 1 We study the influence of disorder on the temperature of superconducting transition within the sigma-model renormalization-group framework. Electron-electron interaction in particle-hole and Cooper channels is taken into account and assumed to be short range. The approach takes into account mutual renormalization of disorder and all interaction constants (that, in particular, leads to mixing of different interaction channels). Two-dimensional systems in the weak localization and antilocalization regime, as well as systems near mobility edge are considered. We demonstrate that in all these regimes Anderson localization leads to strong enhancement of the critical temperature related to the multifractality of wave functions. Screening of the long-range Coulomb interaction thus opens a promising direction for searching novel materials for high-temperature superconductivity. 10:50-11:10 Morning Coffee 11:10-11:50 Bogomolny, E (Université Paris-Sud 11) Near integrable systems Sem 1 A two-dimensional circular quantum billiard with unusual boundary conditions introduced by Berry and Dennis (J Phys A 41 (2008) 135203) is considered in detail. It is demonstrated that most of its eigenfunctions are strongly localized and the corresponding eigenvalues are close to eigenvalues of the circular billiard with Neumann boundary conditions. Deviations from strong localization are also discussed. The results agree well with numerical calculations. 11:50-12:30 Klopp, F (Université Pierre & Marie Curie-Paris VI) The low lying spectrum of one-dimensional random models Sem 1 The talk will be devoted to the description of the low luing spectrum and the associated states of one-dimensional random models. The description will be applied to the study of one-dimensional fermions interacting in a random field. 12:30-13:30 Lunch at Wolfson Court Chair: I Goldsheid 14:20-15:00 Fishman, S (Technion - Israel Institute of Technology) Spreading in potentials random in space and time Sem 1 For time independent random potentials Anderson localization may take place. For time dependent potentials, Anderson localization is destroyed and hyper-transport, namely transport faster than ballistic, may take place. The spreading for time dependent random ppotentials is classified into universality classes characterized by the dependence of the diffusion coefficient on velocity, in the limit of high velocity. The most interesting behavior is found in one dimension. The work was motivated by experiments in optics and in atom optics. 15:00-15:40 Tchoulaevski, V (Université de Reims Champagne-Ardenne) Progress in rigorous multi-particle localization: Simpler proofs of stronger results Sem 1 15:40-16:20 Afternoon Tea 18:45-19:30 Dinner at Wolfson Court