# Workshop Programme

## for period 15 - 19 December 2008

### Classical and Quantum Dynamics in the Presence of Disorder

15 - 19 December 2008

Timetable

Monday 15 December | ||||

09:00-09:55 | Registration | |||

09:55-10:00 | Welcome from David Wallace (INI Director) | |||

10:00-11:00 | Spencer, T (Princeton) |
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Quasi-Diffusion in a 3D SUSY Hyperbolic Sigma Model | Sem 1 | |||

We analyze a SUSY lattice field model which qualitatively reflects Anderson localization and delocalization. Correlations in this model may be described in terms of a random walk in a highly correlated random environment. We prove this model exhibits a "diffusive" phase in 3 or more dimensions. This is joint work with M. Disertori and M. Zirnbauer. |
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11:00-11:30 | Coffee and posters | |||

11:30-12:30 | Bolthausen, E (Zürich) |
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On a perceptron version of the Generalized Random Energy Model | Sem 1 | |||

We consider a macroscopically large family of cavity variables whose covariances are given in a hierarchical way, similar to Derrida's generalized random energy model. The model is similar to the perceptron where the cavity variables have a covariance structure given by the overlaps of two spin configurations. We therefore call the model a perceptron version of the GREM. In contrast to the GREM itself, the model has a rich asymptotic behavior. The free energy is given by a formula which closely resembles the Parisi formula for the SK model. The limiting ultrametric structure depends in a complicated way on the the temperature parameter. Also, in contrast to the GREM, the model has the chaos property (in temperature). |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Hislop, P (Kentucky) |
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Correlation functions for random Schrodinger operators | Sem 1 | |||

Correlation functions are the expectations of moments of the spectral measures for random Schrodinger operators. They play a role in describing the transport properties of the system. This talk will review progress in understanding these moments with an emphasis on the first and second moments. These include the density of states, the current-current correlation functions, and second-order moments involved in the Minami estimate. The talk will present results that are joint work with J. Bellissard, J. M. Combes, F. Germinet, F. Klopp, O. Lenoble, P. Muller, and G. Stolz. |
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15:00-15:30 | Tea and posters | |||

15:30-16:30 | Wang, WM (Université Paris-Sud 11) |
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Long time Anderson localization for the nonlinear random Schroedinger equation | Sem 1 | |||

We prove long time Anderson localization for the nonlinear random Schroedinger equation, giving an answer to a widely debated question in the physics community. I will conjecture Arnold type diffusion in the infinite time limit, which is also supported by my work on the time dependent linear Schroedinger equation on the circle. |
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16:30-17:30 | Erdos, L (Ludwig-Maximilians-Universität München) |
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Wegner estimate and level repulsion for Wigner random matrices | Sem 1 | |||

We consider N x N Hermitian random matrices with independent identically distributed entries (Wigner matrices). The matrices are normalized so that the average spacing between consecutive eigenvalues is of order 1/N. Under suitable assumptions on the distribution of the single matrix element, we first prove that, away from the spectral edges, the empirical density of eigenvalues concentrates around the Wigner semicircle law on energy scales of order 1/N. This result establishes the semicircle law on the optimal scale and it removes a logarithmic factor from our previous result. We then show a Wegner estimate, i.e. that the averaged density of states is bounded. Finally, we prove that the eigenvalues of a Wigner matrix repel each other, in agreement with the universality conjecture. |
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17:30-18:30 | Welcome wine reception | |||

18:30-19:00 | Dinner at Wolfson Court (residents only) |

Tuesday 16 December | ||||

09:00-10:00 | Simon, B (CALTECH) |
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Lubinsky universality and clock spacing for ergodic orthogonal polynomials | Sem 1 | |||

We present background and the proof of universality for ergodic Jacobi matrices with a.c. spectrum. This includes cases where the support of the measure is a positive measure Cantor set. This describes joint work with Artur Avila and Yoram Last. |
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10:00-11:00 | Merkl, F (Ludwig-Maximilians-Universität München) |
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The 'magic formula' for linearly edge-reinforced random walks | Sem 1 | |||

Linearly edge-reinforced random walk on a finite graph is a mixture of reversible Markov chains with an explicitly known mixing measure. We give a new proof of this fact. (Joint work with Silke Rolles). |
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11:00-11:30 | Coffee and posters | |||

11:30-12:30 | Warzel, S (Technischen Universität München) |
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Localization bounds for multiparticle systems | Sem 1 | |||

We consider the spectral and dynamical properties of quantum systems of N particles on the lattice $Z^d$, of arbitrary dimension, with a Hamiltonian which in addition to the kinetic term includes a random potential with iid values at the lattice sites and a finite-range interaction. Two basic parameters of the model are the strength of the disorder and the strength of the interparticle interaction. It is established here that for all N there are regimes of high disorder, and/or weak enough interactions, for which the system exhibits spectral and dynamical localization. The localization bounds are expressed in terms of exponential decay in the Hausdorff distance in the configuration space. The results are derived through the analysis of fractional moments of the N-particle Green function, and related bounds on the eigenfunction correlators. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Kirsch, W (FernUniversität Hagen) |
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Spectral properties of some Hamiltonians modeling superconductors | Sem 1 | |||

We discuss the spectral theory of some Hamiltonians which arise in the theory of superconductors. These operators are given by operator valued two-by-two matrices which describe the evolution of both electrons and holes as well as the interaction between them. We investigate the behavior of the density of states of such operators as well as localization properties. This is joint work with Bernd Metzger and Peter Muller. |
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15:00-15:30 | Tea and posters | |||

15:30-16:30 | Klopp, F (Université Paris 13 Nord) |
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Resonances for large ergodic systems | Sem 1 | |||

In this talk, we consider Schrödinger operators where the potential is the restriction of an ergodic potential to a large cube. We study the resonances i.e. the poles of the scattering matrix of this operator in the limit when the size of the cube goes to infinity. Depending on the characteristics of the limit ergodic Schrödinger operator, the resonances, in particular, the resonances widths, exhibit very different behaviors. We will concentrate on the case of the dimension one and on two types of ergodic potentials, a periodic one and an homogeneous random one. The work presented is still in progress. |
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16:30-17:30 | Keating, J (Bristol) |
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Quantum chaotic resonance eigenfunctions | Sem 1 | |||

I will review some recent results concerning the semiclassical structure of resonance eigenfunctions in chaotic scattering system. |
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18:30-19:00 | Dinner at Wolfson Court (residents only) |

Wednesday 17 December | ||||

09:00-10:00 | Tchulaevsky, V (Université de Reims Champagne-Ardenne) |
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Simple proofs of simple facts in localization theory | Sem 1 | |||

We give an introduction to the Multi-Particle Multi-Scale Analysis, based on simple geometrical ideas and simple proofs. |
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10:00-11:00 | Mueller, M (Université de Genève) |
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Collective electronic transport close to the metal-insulator or superconductor insulator transitions. | Sem 1 | |||

Insulators close to the transition to a metal or a superconductor exhibit interesting collective electronic phenomena which are prominently reflected in transport properties. An important feature in systems close to the metal-insulator transition is the apparently purely electronic nature of activated transport seen in experiments. This does not fit into the standard theory of phonon-assisted hopping conduction and has remained an unexplained puzzle for decades. It also seems to contradict recent theories of “many body localization” (localization in the Fock space of interacting systems), which have predicted a finite temperature metal insulator transition for interacting, Anderson localized electrons. I will address this problem for Anderson insulators with a single-particle localization length much larger than the mean distance between electrons. I will argue that under these circumstances Coulomb interactions drive the electrons into a strongly correlated quantum glass phase with a gapless spectrum of delocalized collective excitations which act as a bath with which individual electrons can exchange energy. However, the same reasoning does not necessarily apply close to a insulator-to-superconductor transition, where electrons are bound into preformed Cooper pairs and Coulomb interactions are weak. I will argue that these systems are promising candidates to exhibit strong remnants of “many body localization”, which may be the key to an explanation for their unusual transport properties. |
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11:00-11:30 | Coffee and posters | |||

11:30-12:30 | Zeitouni, O (Weizmann Institute) |
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Exit measures for random walks in random environments | Sem 1 | |||

I will describe recent work with Erwin Bolthausen, concerning exit measures from large balls for isotropic (in law) random walks in random environments. ``Almost local'' limit theorems are derived. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Smilansky, U (Weizmann Institute) |
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Quantum chaos on discrete graphs | Sem 1 | |||

The spectral statistics of the discrete Laplacian on random d-regular graphs (in the limit of large graphs), will be discussed. It will be shown that in this limit some spectral statistics follow the predictions of Random Matrix Theory. Counting statistics of cycles on the graphs play an important role in the analysis. The level sets of eigenvectors will be shown to display a percollation transition which can be proved by assuming that eigenvectors distribute normally, with a covariance which can be computed using the special properties of the random ensemble of large d-regular graphs. |
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15:00-15:30 | Tea and posters | |||

15:30-16:30 | Kamenev, A (Minnesota) |
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Rare events and phase transitions in reaction-diffusion systems | Sem 1 | |||

I shall discuss a way to evaluate tails of the probability distribution functions in stochastic reaction-diffusion systems. The method is based on the semi-classical treatment of a proper ``quantum'' field theory, which may be associated with the reaction-diffusion models. The same set of ideas may be applied to a classification of non-equilibrium phase transitions, taking place in these models. |
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18:30-19:00 | Dinner at Wolfson Court (residents only) |

Thursday 18 December | ||||

09:00-10:00 | Sznitman, AS (ETH Zürich) |
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Disconnection of discrete cylinders and random interlacements | Sem 1 | |||

The disconnection by random walk of a discrete cylinder with a large finite connected base has been a recent object of interest. It has to do with the way paths of random walks can create interfaces. In this talk we describe some current results and explain how this problem is related to questions of percolation and to the model of random interlacements. |
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10:00-11:00 | Altland, A (Köln) |
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Interpretation of the Altshuler Andreev saddle point | Sem 1 | |||

In recent years, progress has been made in understanding the status of random matrix theory in the physics of classically chaotic quantum systems by methods of semiclassical analysis. We owe much of this progress to the linkage between semiclassical expansions in terms of periodic orbits on the one hand, and field theoretical methods (integration over certain sigma model manifolds) on the other hand. I will review this connection, special emphasis put on the role of a stationary point in the domain of integration, the Altshuler Andreev saddle point. This point had long been reckognized as instrumetal in understanding the non-perturbative aspects of spectral statistics (the sine-kernel, for instance.) The recently constructed semiclassical analogies have offered the possibility to understand its role in more intuitive terms, to be discussed in the talk. |
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11:00-11:30 | Coffee and posters | |||

11:30-12:30 | Last, Y (Hebrew University of Jerusalem) |
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On level spacings for one-dimensional Schroedinger operators | Sem 1 | |||

The talk would discuss some results, conjectures, and open problems concerning the spacings of eigenvalues for restrictions of one-dimensional Schroedinger operators to large "boxes" (a.k.a. zeroes of orthogonal polynomials). These include results on "clock behavior" for decaying potentials, as well as for absolutely continuous spectrum of ergodic potentials. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Kupiainen, A (Helsinki) |
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Nonlinear diffusion in a random environment | Sem 1 | |||

We discuss dynamics of coupled map lattices where local dynamics consists of one conserved quantity ("energy") and other degrees of freedom are chaotic. Upon coupling only total energy is conserved and should diffuse. The model can be mapped to a nonlinear version of random walks in a random environment. We use renormalization group to establish the diffusive behavior. |
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15:00-15:30 | Tea and posters | |||

15:30-16:30 | Nazarenko, S (Warwick) |
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Wave Turbulence: solved and open problems | Sem 1 | |||

Wave Turbulence (WT) refers to a statistical state of many dispersive modes which are weakly nonlinear on average. I will present the fundamental building blocks of the WT theory, assumptions and their justifications, physical examples, solved problems and open questions. I will describe some recent work which goes beyond the traditional WT consideration of the wave spectra, and deals with wave PDFs, non-gaussianity, intermittency. I will outline picture of the WT cycle in which weak (on average) random waves can get transformed into strong coherent structures, which in turn partially dissipate their energy and partially return it to the incoherent waves. My main aim will be to reach out to the stat mech community, in hope to find common points and approaches which could be employed for better understanding of the WT systems. |
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16:30-17:30 | Goldstein, M (Toronto) |
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Fluctuations and growth of the magnitude of the Dirichlet determinants of Anderson Model at all disorders | Sem 1 | |||

We consider the Schrodinger operator of the Anderson model in a quasi-one-dimensional domain with Dirichlet boundary condition. We show the exponential growth of the characteristic determinant of the problem. In particular, we give an effective, finite number of factors lower bound for the upper Lyapunov exponent of the product of the corresponding symplectic matrices. We explain the mechanism responsible for exponentially large magnitude of the Dirichlet determinant. The central part of this mechanism consists of the fact that the logarithm of the determinant has large fluctuations. |
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19:30-23:00 | Conference dinner at Emmanuel College (The Gardner Room) |

Friday 19 December | ||||

09:00-10:00 | Rolles, SWW (Technische Universität München) |
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Edge-reinforced random walk on a two-dimensional graph | Sem 1 | |||

We consider linearly edge-reinforced random walk on a class of two-dimensional graphs with constant initial weights. The graphs are obtained from Z^2 by replacing every edge by a sufficiently large, but fixed number of edges in series. We prove that linearly edge-reinforced random walk on these graphs is recurrent. |
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10:00-11:00 | Combes, JM (Université du Sud Toulon-Var) |
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Eigenvalue statistics for the discrete and continuous Anderson model | Sem 1 | |||

We review some recent results obtained with F.Germinet and A.Klein on Minami estimate for the Anderson model and application to Poisson statistics of rescaled eigenvalues in the localisation regime. |
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11:00-11:30 | Coffee and posters | |||

11:30-12:30 | Germinet, F (Université de Cergy-Pontoise) |
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Localization and delocalization in quantum hall systems | Sem 1 | |||

We shall present recent results on localization and delocalization in quantum Hall systems. Anderson potentials are known to be relevant for a mathematical proof of the existence of plateaux in the integer quantum Hall effect. Moreover, for such models, a dynamical statement of delocalization can be proved near the Landau levels, implying a dynamical Anderson transition for such models. Last results include the case where the Landau gaps are closed. We may then mention recent results concerning quantum Hall currents. |
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12:30-13:30 | Lunch at Wolfson Court | |||

18:30-19:00 | Dinner at Wolfson Court (residents only) |