Contents Entropy of Eigenfunctions ........................................ 1 Nalini Anantharaman, Herbert Koch and Stéphane Nonnenmacher 1 Motivations.............................................. 1 2 MainResult.............................................. 4 3 Outline of the Proof . 7 3.1 Definition of the Metric Entropy . 7 3.2 From Classical to Quantum Dynamical Entropy . 9 3.3 Entropic Uncertainty Principle. 13 3.4 Applying the Entropic Uncertainty Principle to the Laplacian Eigenstates . 14 References . 21 Stability of Doubly Warped Product Spacetimes ..................... 23 Lars Andersson 1 Introduction . 23 2 Warped Product Spacetimes . 24 2.1 AsymptoticBehavior .............................. 26 3 Fuchsian Method . 27 3.1 VelocityDominatedEquations....................... 28 3.2 VelocityDominatedSolution........................ 29 4 Stability . 30 References . 31 Rigorous Construction of Luttinger Liquids Through Ward Identities ... 33 Giuseppe Benfatto 1 Introduction . 33 2 The Tomonaga Model with Infrared Cutoff . 34 3 TheRGAnalysis.......................................... 35 4 TheDysonEquation....................................... 37 5 The First Ward Identity . 39 6 The Second Ward Identity . 40 xxvii xxviii Contents 7 The Euclidean Thirring Model . 41 References . 43 New Algebraic Aspects of Perturbative and Non-perturbative Quantum Field Theory ................................................... 45 Christoph Bergbauer and Dirk Kreimer 1 Introduction . 45 2 Lie and Hopf Algebras of Feynman Graphs . 46 3 From Hochschild Cohomology to Physics . 50 4 Dyson-Schwinger Equations . 51 5 Feynman Integrals and Periods of Mixed (Tate) Hodge Structures................................................ 55 References . 57 Exact Solution of the Six-Vertex Model with Domain Wall Boundary Conditions ..................................................... 59 Pavel M. Bleher 1 Six-Vertex Model . 59 2 Phase Diagram of the Six-Vertex Model . 62 3 Izergin-Korepin Determinantal Formula . 63 4 The Six-Vertex Model with DWBC and a Random Matrix Model ................................................... 63 5 Asymptotic Formula for the Recurrence Coefficients . 65 6 Previous Exact Results . 67 7 Zinn-Justin’sConjecture................................... 70 8 Large N Asymptotics of ZN in the Ferroelectric Phase . 71 References . 71 Mathematical Issues in Loop Quantum Cosmology ................... 73 Martin Bojowald 1 Introduction . 73 2 Quantum Representation and Dynamical Equations . 75 2.1 Quantum Reduction . 75 2.2 Dynamics . 76 3 Quantum Singularity Problem . 78 4 Examples for Properties of Solutions . 79 5 Effective Theory . 81 6 Summary................................................ 84 References . 84 Boundary Effects on the Interface Dynamics for the Stochastic Allen–Cahn Equation ............................................ 87 Lorenzo Bertini, Stella Brassesco and Paolo Buttà 1 Introduction . 87 2 ResultsandStrategyofProofs .............................. 89 References . 92 Contents xxix Dimensional Entropies and Semi-Uniform Hyperbolicity .............. 95 Jérôme Buzzi 1 Introduction . 95 2 LowDimension........................................... 97 2.1 IntervalMaps..................................... 97 2.2 SurfaceTransformations............................ 98 3 Dimensional Entropies . 99 3.1 Singular Disks . 99 3.2 Entropy of Collections of Subsets . 100 3.3 Definitions of the Dimensional Entropies . 101 4 Other Growth Rates of Submanifolds . 102 4.1 VolumeGrowth...................................102 4.2 ResolutionEntropies...............................106 5 Properties of Dimensional Entropies . 107 5.1 Link between Topological and Resolution Entropies . 107 5.2 Gap Between Uniform and Ordinary Dimensional Entropies ........................................108 5.3 Continuity Properties . 109 6 Hyperbolicity from Entropies . 110 6.1 A Ruelle-Newhouse Type Inequality . 110 6.2 Entropy-Expanding Maps . 110 6.3 Entropy-Hyperbolicity . 112 6.4 Examples of Entropy-Hyperbolic Diffeomorphisms . 113 7 FurtherDirectionsandQuestions............................113 7.1 Variational Principles . 113 7.2 Dimensional Entropies of Examples . 113 7.3 Other Types of Dimensional Complexity . 114 7.4 Necessity of Topological Assumptions . 114 7.5 Entropy-Hyperbolicity . 114 7.6 Generalized Entropy-Hyperbolicity . 115 8 Cr Sizes.................................................115 References . 116 The Scaling Limit of (Near-)Critical 2D Percolation .................. 117 Federico Camia 1 Introduction . 117 1.1 CriticalScalingLimitsandSLE.....................117 1.2 Percolation.......................................120 2 The Critical Loop Process . 121 2.1 General Features . 121 2.2 ConstructionofaSingleLoop.......................122 3 TheNear-CriticalScalingLimit.............................124 References . 125 xxx Contents Black Hole Entropy Function and Duality ........................... 127 Gabriel Lopes Cardoso 1 Introduction . 127 2 Entropy Function and Electric/Magnetic Duality Covariance . 128 3 Application to N = 2 Supergravity . 130 4 Duality Invariant OSV Integral . 133 References . 133 Weak Turbulence for Periodic NLS ................................ 135 James Colliander 1 Introduction . 135 2 NLSasanInfiniteSystemofODEs..........................137 3 Conditions on a Finite Set Λ ⊂ Z2 ..........................138 4 Arnold Diffusion for the Toy Model ODE . 139 5 Construction of the Resonant Set Λ ..........................140 References . 142 Angular Momentum-Mass Inequality for Axisymmetric Black Holes .... 143 Sergio Dain 1 Introduction . 143 2 Variational Principle for the Mass . 144 References . 147 Almost Everything About the Fibonacci Operator .................... 149 David Damanik 1 Introduction . 149 2 TheTraceMap...........................................150 3 The Cantor Structure and the Dimension of the Spectrum . 152 4 The Spectral Type . 154 5 Bounds on Wavepacket Spreading . 156 References . 158 Entanglement-Assisted Quantum Error-Correcting Codes ............. 161 Igor Devetak, Todd A. Brun and Min-Hsiu Hsieh 1 Introduction . 161 2 Notations................................................162 3 Entanglement-Assisted Quantum Error-Correcting Codes . 163 3.1 The Channel Model: Discretization of Errors . 164 3.2 The Entanglement-Assisted Canonical Code . 165 3.3 The General Case . 167 3.4 Distance . 169 3.5 Generalized F4 Construction........................169 3.6 Bounds on Performance . 170 4 Conclusions ..............................................171 References . 171 Contents xxxi Particle Decay in Ising Field Theory with Magnetic Field .............. 173 Gesualdo Delfino 1 Ising Field Theory . 173 2 Evolution of the Mass Spectrum . ..