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Stochastic Processes and Random Matrices

Lecture Notes of the Les Houches Summer School: Volume 104, 6th – 31st July 2015

Stochastic Processes and Random Matrices

Edited by Grégory Schehr, Alexander Altland, Yan V. Fyodorov, Neil O’Connell, Leticia F. Cugliandolo

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Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford.

It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries c Oxford University Press 2017

The moral rights of the authors have been asserted First Edition published in 2017 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2017943724 ISBN 978–0–19–879731–9 Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.

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Quantum mechanics. Quantum field theory Quantum mechanics. Statistical mechanics. Nuclear physics Quantum mechanics. Solid state physics. Statistical mechanics. Elementary particle physics Quantum mechanics. Collision theory. Nucleon-nucleon interaction. Quantum electrodynamics Quantum mechanics. Non equilibrium phenomena. Nuclear reactions. Interaction of a nucleus with atomic and molecular fields Quantum perturbation theory. Low temperature physics.

Quantum theory of solids. Ferromagnetism Scattering theory. Recent developments in field theory. Nuclear and strong interactions. Experiments in high energy physics The many body problem The theory of neutral and ionized gases Elementary particles and dispersion relations Low temperature physics Geophysics; the earths environment Relativity groups and topology Quantum optics and electronics High energy physics High energy astrophysics Many body physics Nuclear physics Physical problems in biological systems Statistical mechanics and quantum field theory Particle physics Plasma physics Black holes Fluids dynamics Molecular fluids Atomic and molecular physics and the interstellar matter Frontiers in laser spectroscopy Methods in field theory Weak and electromagnetic interactions at high energy Nuclear physics with heavy ions and mesons Ill condensed matter Membranes and intercellular communication Physical cosmology Laser plasma interaction

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Physics of defects Chaotic behavior of deterministic systems Gauge theories in high energy physics New trends in atomic physics Recent advances in field theory and statistical mechanics Relativity, groups and topology Birth and infancy of stars Cellular and molecular aspects of developmental biology Critical phenomena, random systems, gauge theories Architecture of fundamental interactions at short distances Signal processing Chance and matter Astrophysical fluid dynamics Liquids at interfaces Fields, strings and critical phenomena Oceanographic and geophysical tomography Liquids, freezing and glass transition Chaos and quantum physics Fundamental systems in quantum optics Supernovae Particles in the nineties Strongly interacting fermions and high T c superconductivity Gravitation and quantizations Progress in picture processing Computational fluid dynamics Cosmology and large scale structure Mesoscopic quantum physics Fluctuating geometries in statistical mechanics and quantum field theory Quantum fluctuations Quantum symmetries From cell to brain Trends in nuclear physics, 100 years later Modeling the earths climate and its variability Probing the Standard Model of particle interactions Topological aspects of low dimensional systems Infrared space astronomy, today and tomorrow The primordial universe Coherent atomic matter waves Atomic clusters and nanoparticles New trends in turbulence Physics of bio-molecules and cells Unity from duality: Gravity, gauge theory and strings Slow relaxations and nonequilibrium dynamics in condensed matter

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Accretion discs, jets and high energy phenomena in astrophysics Quantum entanglement and information processing Methods and models in neurophysics Nanophysics: Coherence and transport Multiple aspects of DNA and RNA Mathematical statistical physics Particle physics beyond the Standard Model Complex systems Particle physics and cosmology: the fabric of spacetime String theory and the real world: From particle physics to astrophysics Dynamos Exact methods in low-dimensional statistical physics and quantum computing Long-range interacting systems Ultracold gases and quantum information New trends in the physics and mechanics of biological systems Modern perspectives in lattice QCD: quantum field theory and high performance computing Many-body physics with ultra-cold gases Quantum theory from small to large scales Quantum machines: Measurement control of engineered quantum systems Theoretical physics to face the challenge of LHC Advanced data assimilation for geosciences Soft interfaces Strongly interacting quantum systems out of equilibrium Post-Planck cosmology Quantum Optics and Nanophotonics Statistical Physics, Optimization, Inference and Message-Passing Algorithms From Molecules to Living Organisms: An Interplay Between Biology and Physics Topological Aspects of Condensed Matter Physics Stochastic Processes and Random Matrices

Publishers – Session VIII: Dunod, Wiley, Methuen – Sessions IX and X: Herman, Wiley – Session XI: Gordon and Breach, Presses Universitaires – Sessions XII–XXV: Gordon and Breach – Sessions XXVI–LXVIII: North Holland – Session LXIX–LXXVIII: EDP Sciences, Springer – Session LXXIX–LXXXVIII: Elsevier – Session LXXXIX– : Oxford University Press

Preface The field of stochastic processes and RMT has been a rapidly evolving subject during the last fifteen years where the continuous development and discovery of new tools, connections and ideas have led to an avalanche of new results. These breakthroughs have been made possible thanks, to a large extent, to the recent development of various new techniques in Random Matrix Theory (RMT). Matrix models have been playing an important role in theoretical physics for a long time and they are currently also a very active domain of research in mathematics. An emblematic example of these recent advances concerns the theory of growth phenomena in the Kardar-Parisi-Zhang (KPZ) universality class where the joint efforts of physicists and mathematicians during the last twenty years have unveiled the beautiful connections between this fundamental problem of statistical mechanics and the theory of random matrices, namely the fluctuations of the largest eigenvalue of certain ensemble of random matrices. These lecture notes not only cover this topic in detail but also present more recent developments that have emerged from these discoveries, for instance in the context of low dimensional heat transport (on the physics side) or in context of integrable probability (on the mathematical side). More widely, our goal, in organizing this school in Les Houches, was to present the latest developments on these topics at the interface between theoretical physics and mathematics, with a special emphasis on the large spectrum of techniques and applications of RMT. By following the lectures in this volume the reader will surely be able to appreciate the breadth and beauty of the subject.

The school was held in July 2015. It started with a historical introductory lectures on the applications of RMT given by H. Weidenmüller and consisted of more than fifty 90-minutes lectures, covering a wide range of topics. The school comprised, on the one hand, five long courses (five lectures each) on rather general subjects, and, on the other hand, ten shorter courses (two to three lectures each) on more specialized topics. This volume presents the lecture notes prepared by the speakers.

This volume starts with the lecture notes of three of the five long lectures: A. Borodin on Integrable Probability, A. Guionnet on Free Probability and H. Spohn on the Kardar-Parisi-Zhang equation (we regret that we were not able to include the lectures notes by P. Le Doussal and B. Virag in this volume). They are followed up by the lecture notes on more specialized topics: G. Akemann on RMT and quantum chromodynamics, J.-Ph. Bouchaud on RMT and (Big) data analysis, B. Eynard on Random matrices and loop equations, J. P. Keating on Random matrices and number theory, A. L. Moustakas on the applications of RMT to modern telecommunications, H. Schomerus on random matrix approaches to open quantum systems, Y. Tourigny (and A. Comtet) on impurity models and products of random matrices, V. Vargas (and R. Rhodes) Gaussian multiplicative chaos and Liouville Quantum Gravity and A. Zabrodin on Quantum spin chains and classical integrable systems. We also had

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the pleasure to listen to a seminar by J.-P. Eckmann on M. Hairer’s work and the Kardar-Parisi-Zhang equation.

The students, half of them physicists and the other half mathematicians, were full of enthusiasm both inside and outside the classroom. Many of them had the opportunity to present their work during two poster sessions. This school in Les Houches has certainly been a very good opportunity for them to interact scientifically with other students as well as with the lecturers, who were all of them extremely open to discussions with the students during their stay in Les Houches.

We are deeply grateful to the lecturers work in preparing the lecture notes, which will be useful in the future to the whole community (both physicists and mathematicians) working on stochastic processes and RMT. We also wish to thank Piotr Warchol for his pictures of the lecturers and Sunčana Dulić for her beautiful drawings of the mountains around Les Houches. Finally, we want to warmly thank the staff of the School of Physics in Les Houches, who made a fantastic work during this school, which was really a great moment for all of us.

Drawings by Sunčana Dulić

Contents List of participants 1 History—an overview Oriol BOHIGAS and Hans A. WEIDENMÜLLER 1.1 Bohr’s concept of the compound nucleus 1.2 Spectral properties 1.3 Data 1.4 Many-body theory 1.5 Chaos 1.6 Number theory 1.7 Scattering theory 1.8 Replica trick and supersymmetry 1.9 Disordered solids 1.10 Interacting fermions and field theory Acknowledgements References 2 Integrable probability: stochastic vertex models and symmetric functions Alexei BORODIN and Leonid PETROV 2.1 Introduction 2.1.1 Preface 2.1.2 Our model in a quadrant 2.1.3 The main result 2.1.4 Symmetric rational functions 2.1.5 Cauchy identities 2.1.6 Organization of the chapter 2.2 Vertex weights 2.2.1 Higher spin six-vertex model 2.2.2 Vertex weights 2.2.3 Motivation 2.2.4 Conjugated weights and stochastic weights 2.3 The Yang–Baxter equation 2.3.1 The Yang–Baxter equation in coordinate language 2.3.2 The Yang–Baxter equation in operator language 2.3.3 Attaching vertical columns 2.4 Symmetric rational functions 2.4.1 Signatures 2.4.2 Semi-infinite operators A and B and definition of symmetric rational functions

xxiii 1 3 4 8 9 10 11 12 15 18 20 20 20

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2.4.3 2.4.4

Semi-infinite operator D Cauchy-type identities from the Yang–Baxter commutation relations 2.4.5 Symmetrization formulas 2.5 Stochastic weights and fusion 2.5.1 Stochastic weights Lu 2.5.2 Fusion of stochastic weights 2.5.3 Principal specializations of skew functions 2.6 Markov kernels and stochastic dynamics 2.6.1 Probability measures associated with the Cauchy identity 2.6.2 Four Markov kernels 2.6.3 Specializations 2.6.4 Interacting particle systems 2.6.5 Degeneration to the six-vertex model and the ASEP 2.6.6 Degeneration to q-Hahn and q-Boson systems 2.7 Orthogonality relations 2.7.1 Spatial biorthogonality 2.7.2 Plancherel isomorphisms and completeness 2.7.3 An integral representation for Gμ 2.8 q-correlation functions 2.8.1 Computing observables via the Cauchy identity 2.8.2 Computation of Gν ( , w1 , . . . , wk ) 2.8.3 Extracting terms by integrating over wi 2.8.4 q-correlation functions 2.8.5 Remark. From observables to duality, and back 2.9 q-moments of the height function 2.9.1 Height function and its q-moments 2.10 Degenerations of moment formulas 2.10.1 Moment formulas for the stochastic six-vertex model and the ASEP 2.10.2 Moment formulas for q-Hahn and q-Boson systems Acknowledgements References 3 Free probability Alice GUIONNET 3.1 Introduction 3.2 Free probability setting 3.2.1 Noncommutative probability space 3.2.2 Weak *-topology 3.2.3 Freeness 3.3 Asymptotic freeness of random matrices 3.3.1 Independent GUE matrices are asymptotically free 3.3.2 Matrices conjugated by Haar unitary matrices are asymptotically free

46 48 51 59 59 60 63 66 66 69 71 73 76 80 85 85 91 95 96 97 98 101 104 109 111 111 123 124 126 128 128 132 134 135 136 140 140 141 141 146

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3.4

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Free convolution 3.4.1 The R-transform 3.4.2 The free central limit theorem 3.5 Nonnormal operators 3.5.1 The circular law 3.5.2 The single ring theorem 3.6 Loop equations and topological expansions 3.6.1 The free difference quotient and the conjugate variable 3.6.2 Matrix model and integration by parts 3.6.3 Topological expansions 3.6.4 Formal topological expansions 3.6.5 Loop equations and asymptotic expansions 3.6.6 Extension to Haar distributed matrices 3.7 Applications to loop models 3.7.1 Application to planar algebras and loop models 3.7.2 Loop models and subfactors References

150 150 153 154 157 157 157 158 158 159 160 165 171 171 171 174 175

4 The Kardar–Parisi–Zhang equation: a statistical physics perspective Herbert SPOHN 4.1 Stable-metastable interface dynamics 4.2 Scaling properties, KPZ equation as weak drive limit 4.3 Eden-type growth models 4.4 The KPZ universality class 4.4.1 One-point distributions 4.4.2 Multipoint distributions 4.4.3 Stationary covariance 4.5 Directed polymers in a random medium 4.6 Replica solutions 4.7 Statistical mechanics of line ensembles 4.8 Noisy local conservation laws 4.9 One-dimensional fluids and anharmonic chains 4.10 Discrete nonlinear Schrödinger equation 4.11 Linearized Euler equations 4.12 Linear fluctuating hydrodynamics 4.13 Second-order expansion, nonlinear fluctuating hydrodynamics 4.14 Coupling coefficients, dynamical phase diagram 4.15 Low-temperature DNLS 4.16 Mode-coupling theory 4.16.1 Decoupling hypothesis 4.16.2 One-loop, diagonal, and small overlap approximations 4.17 Molecular dynamics and other missed topics Acknowledgements References

177 179 182 184 185 187 188 189 190 192 194 195 196 202 204 207 211 212 214 216 216 217 222 222 223

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5 Random matrix theory and quantum chromodynamics Gernot AKEMANN 5.1 Introduction and motivation 5.2 Orthogonal polynomial approach to the Dirac operator spectrum 5.2.1 The eigenvalue model and definition of its correlation functions 5.2.2 Properties of orthogonal polynomials with general weights 5.2.3 All correlation functions with masses in terms of Laguerre polynomials 5.2.4 The large-N limit at the hard edge 5.3 Symmetries of QCD and its relation to RMT 5.3.1 The Dirac operator and global symmetries of QCD 5.3.2 Chiral symmetry breaking and chiral perturbation theory 5.3.3 The limit to RMT: the epsilon regime 5.4 Recent developments 5.4.1 RMT and QCD at finite lattice spacing—the Wilson Dirac operator 5.4.2 RMT and QCD with chemical potential 5.5 Summary Acknowledgements References

228 230 233 234 237 242 247 251 252 254 257 262 262 267 276 277 277

6 Random matrix theory and (big) data analysis Jean-Philippe BOUCHAUD 6.1 Introduction 6.2 Eigenvalue spectrum and eigenvector overlaps 6.2.1 Relation among resolvent, spectrum, and overlaps 6.2.2 Free additive noise 6.2.3 Free multiplicative noise and empirical covariance matrices 6.3 Rotationally invariant estimators 6.3.1 The oracle estimator and the overlaps 6.3.2 Optimal rotational invariant estimator 6.4 Rectangular correlation matrices 6.5 Conclusion and (some) open problems 6.6 Appendix A: Reminder on transforms in RMT 6.7 Appendix B: Resolvents and replicas 6.7.1 The replica method 6.7.2 Free additive noise References

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7 Random matrices and loop equations Bertrand EYNARD 7.1 Introduction 7.1.1 Random matrices 7.1.2 A detailed example: random partitions as random matrices 7.2 Coulomb gas and saddle-point approximation

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7.2.1 Coulomb gas of eigenvalues 7.2.2 Saddle-point approximation 7.2.3 Functional saddle-point 7.2.4 1-cut case, Joukovski transformation 7.2.5 Examples 7.3 Loop equations 7.3.1 Basis of moments and observables 7.3.2 Integrating by parts 7.3.3 Perturbative topological expansion 7.3.4 Higher orders References 8 Random matrices and number theory: some recent themes Jon P. KEATING 8.1 The Riemann zeta function 8.1.1 Definition and basic properties 8.1.2 Explicit formulae 8.1.3 L-functions 8.1.4 Counting zeros 8.1.5 General explicit formula 8.1.6 Connection with random matrices: Montgomery’s pair-correlation conjecture 8.2 Value distribution of the zeta function on its critical line 8.2.1 Selberg’s theorem

8.2.2 Moments of ζ 12 + it 8.2.3 Characteristic polynomials of random matrices 8.2.4 Extreme values 8.3 Function Fields 8.3.1 Background 8.3.2 Zeta functions of curves 8.3.3 Spectral interpretation 8.3.4 Spectral statistics 8.3.5 Arithmetic statistics in function fields Acknowledgements References 9 Modern telecommunications: a playground for physicists?

Aris L. MOUSTAKAS 9.1 Introduction 9.1.1 Outline 9.2 Information theory basics 9.2.1 Information capacity 9.2.2 Linear precoders 9.3 Wireless communications: replicas and mobility 9.3.1 Capacity of correlated antennas 9.3.2 Effect of mobility on energy consumption

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317 318 322 325 326 330 330 332 336 340 347 348 350 350 351 353 354 355 356 360 360 361 362 365 369 370 371 372 372 373 377 377 382 384 385 385 385 386 387 388 390

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9.3.3 Discussion Optical communications: moments and tails 9.4.1 Character expansions in communications 9.4.2 Tails of the mutual information 9.4.3 Discussion 9.5 Conclusions and outlook Acknowledgments References 9.4

10 Random matrix approaches to open quantum systems Henning SCHOMERUS 10.1 Introduction 10.1.1 Welcome 10.1.2 Primer 10.1.3 Open systems 10.1.4 Preview 10.2 Foundations of random-matrix theory 10.2.1 Random Hamiltonians and Gaussian Hermitian ensembles 10.2.2 Time-reversal symmetry and the Wigner–Dyson ensembles 10.2.3 Chiral symmetry 10.2.4 Charge-conjugation symmetry 10.2.5 Random time-evolution operators and circular ensembles 10.2.6 Positive-definite matrices and Wishart–Laguerre ensembles 10.2.7 Jacobi ensembles 10.2.8 Non-Hermitian matrices 10.3 The scattering matrix 10.3.1 Points of interest 10.3.2 Definition of the scattering matrix 10.3.3 Preliminary answers 10.3.4 Effective scattering models 10.3.5 Stroboscopic scattering approach 10.3.6 Continuous-time scattering theory 10.3.7 Merits 10.4 Decay, dynamics, and transport 10.4.1 Scattering poles 10.4.2 Mode nonorthogonality 10.4.3 Delay times 10.4.4 Transport 10.5 Localization, thermalization, and entanglement 10.5.1 Anderson localization 10.5.2 Thermalization and localization in many-body systems 10.6 Conclusions 10.7 Appendix A: Eigenvalue densities of matrices with large dimensions 10.7.1 Gaussian Hermitian ensembles 10.7.2 Appendix A.1: Wishart–Laguerre ensembles

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Contents

10.7.3 Appendix A.2: Jacobi ensembles 10.7.4 Appendix A.3: Ginibre ensembles Acknowledgements References 11 Impurity models and products of random matrices Alain COMTET and Yves TOURIGNY 11.1 Introduction 11.1.1 Product of matrices 11.1.2 Disordered systems 11.1.3 Outline 11.1.4 Recommended reading 11.2 Some impurity models 11.2.1 The vibrating string and Dyson’s random chain model 11.2.2 The Frisch–Lloyd model 11.2.3 The Anderson model 11.2.4 Further motivations 11.3 The spectral problem 11.3.1 The spectral measure 11.3.2 The integrated density of states per unit length 11.3.3 The Weyl coefficient 11.3.4 The Riccati equation 11.3.5 Classification in terms of limit-circle and limit-point types 11.3.6 Dyson’s disordered strings of Type I 11.4 The complex Lyapunov exponent 11.4.1 The imaginary part 11.4.2 Some deterministic examples 11.4.3 A disordered example: Kotani’s Type II string 11.4.4 Calculation of the Lyapunov exponent 11.4.5 The relationship between Ω(λ) and w(λ) 11.4.6 Application to the Frisch–Lloyd model 11.4.7 Notation 11.5 Further remarks on the Frisch–Lloyd model 11.5.1 Prüfer variables: the phase formalism 11.5.2 Riccati analysis: a qualitative picture 11.5.3 The Rice formula 11.6 The white noise limit 11.6.1 Lévy processes 11.6.2 The Lyapunov exponent 11.6.3 Riccati analysis 11.7 Lifshitz tails and Lifshitz singularities 11.7.1 Short-range, repulsive potentials 11.7.2 Strings of Type I 11.7.3 Supersymmetric potentials 11.7.4 Further reading

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11.8

Distribution of the ground state energy and statistics of energy levels 11.9 Scattering and hyperbolic geometry 11.9.1 Hyperbolic geometry 11.9.2 Decay of the transmission coefficient 11.9.3 Distribution of the reflexion phase 11.10 The Lyapunov exponent of a product of random matrices 11.10.1 The Cohen–Newman example 11.10.2 Application to the Frisch–Lloyd model 11.11 Furstenberg’s formula for the Lyapunov exponent 11.11.1 The projective space 11.11.2 A Markov chain and its stationary distribution 11.11.3 The case d = 2 11.12 Furstenberg’s Theorem 11.13 Concluding remarks Acknowledgements References

523 525 526 528 529 530 532 533 534 534 535 536 539 544 544 545

12 Gaussian multiplicative chaos and Liouville quantum gravity Rémi RHODES and Vincent VARGAS 12.1 Introduction 12.1.1 Notations 12.2 Gaussian multiplicative chaos 12.2.1 Reminder on Gaussian vectors and processes 12.2.2 Construction of the GMC measures 12.2.3 Main properties of the GMC measures 12.3 Liouville quantum gravity on the Riemann sphere 12.3.1 Elementary Riemannian geometry on the sphere 12.3.2 An introduction to CFT on the Riemann sphere 12.3.3 Introduction to LQFT on the Riemann sphere 12.3.4 Construction of LQFT 12.3.5 Properties of the theory 12.3.6 The Liouville measures 12.3.7 Conjectured relation with planar maps 12.3.8 On the Ising model at critical temperature 12.3.9 Final remarks and conclusion 12.4 Appendix 12.4.1 The conformal structure on planar maps Acknowledgments References

548 550 551 551 551 552 555 558 558 559 560 563 566 567 569 571 573 574 574 575 575

13 Quantum spin chains and classical integrable systems Anton ZABRODIN 13.1 Introduction 13.1.1 Organization of the paper 13.1.2 The notation 13.2 The master T-operator for spin chains 13.2.1 Quantum R-matrices

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13.2.2 Inhomogeneous XXX spin chains 13.2.3 The higher T-operators 13.2.4 The construction of the master T-operator 13.2.5 The master T-operator and the mKP hierarchy 13.3 From the master T-operator to the classical RS model and back 13.3.1 Eigenvalues of the spin chain Hamiltonians as velocities of the RS particles 13.3.2 Lax pair for the RS model from dynamics of poles 13.3.3 The BA function and the master T-operator 13.4 Spectrum of the spin chain Hamiltonians from the classical RS model 13.4.1 Twist parameters as eigenvalues of the Lax matrix 13.4.2 The QC correspondence 13.4.3 Algebraic equations for the spectrum 13.5 The QC correspondence via nested Bethe ansatz 13.5.1 The nested Bethe ansatz solution 13.5.2 The QC correspondence: a direct proof 13.6 Concluding remarks 13.7 Appendix A: The higher T-operators through characters 13.8 Appendix B: Hamiltonian formulation of the RS model Acknowledgements References

586 590 592 593 596 596 597 599 601 601 602 603 605 605 606 608 608 609 610 610

List of participants Organizers SCHEHR Grégory Université Paris-Sud, France ALTLAND Alexander University of Köln, Germany FYODOROV Yan V. King’s College London, Department of Mathematics, London WC2R 2LS, United Kingdom O’CONNELL University of Bristol, School of Mathematics, Howard House, Queen’s Ave, Bristol BS8 1SD, United Kingdom CUGLIANDOLO Leticia E. Sorbonnes Universités, France

Lecturers AKEMANN Gernot Bielefeld University, Germany BORODIN Alexei MIT, Cambridge, MA, USA BOUCHAUD Jean-Philippe CFM/Ecole polytechnique, Paris, France ECKMANN Jean-Pierre University of Geneva, Switzerland EYNARD Bertrand IPHT CEA Saclay, Gif sur Yvette, France GUIONNET Alice MIT, Cambridge, MA, USA Université de Lyon, Ecole Normale Supérieure, Lyon, France KEATING Jon University of Bristol, UK LE DOUSSAL Pierre ENS, Paris, France MOUSTAKAS Aris Universtiy of Athens, Greece

List of participants Organizers SCHEHR Grégory Université Paris-Sud, France ALTLAND Alexander University of Köln, Germany FYODOROV Yan V. King’s College London, Department of Mathematics, London WC2R 2LS, United Kingdom O’CONNELL University of Bristol, School of Mathematics, Howard House, Queen’s Ave, Bristol BS8 1SD, United Kingdom CUGLIANDOLO Leticia E. Sorbonnes Universités, France

Lecturers AKEMANN Gernot Bielefeld University, Germany BORODIN Alexei MIT, Cambridge, MA, USA BOUCHAUD Jean-Philippe CFM/Ecole polytechnique, Paris, France ECKMANN Jean-Pierre University of Geneva, Switzerland EYNARD Bertrand IPHT CEA Saclay, Gif sur Yvette, France GUIONNET Alice MIT, Cambridge, MA, USA Université de Lyon, Ecole Normale Supérieure, Lyon, France KEATING Jon University of Bristol, UK LE DOUSSAL Pierre ENS, Paris, France MOUSTAKAS Aris Universtiy of Athens, Greece

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SCHOMERUS Henning Lancaster University, UK SPOHN Herbert Technical University Munich, Germany TOURIGNY Yves University of Bristol, UK VARGAS Vincent ENS, Paris, France VIRAG Balint University of Toronto, Canada ¨ WEIDENMULLER Hans Max Planck Institut, Heidelberg, Germany ZABRODIN Anton ITEP, Moscow, Russia

Participants ALLEGRA Nicolas Université de Lorraine, Nancy, France ANDRAUS Sergio University of Tokyo, Japan ANOKHINA Alexandra ITEP, Moscow, Russia ASSIOTIS Theodoros University of Warwick, Coventry, UK AUGERI Fanny Université Paul Sabatier, Toulouse, France BALDWIN Christopher University of Washington, Seattle, WA, USA BENITO-MATIAS Enrique CFMAC, Madrid, Spain BENOIST Tristan McGill University, Montreal, Canada BUTEZ Raphaël Université Paris-Dauphine, Paris, France CHECINSKI Tomasz University of Bielefeld, Germany CUNDEN Fabio Deelan University of Bristol, UK De NARDIS Jacopo University of Amsterdam, NL

List of participants

DOERAENE Antoine Université Catholique de Louvain, Belgium DUCLUT Charlie LPTMC, Paris, France EMRAH Elnur University of Wisconsin, Madison, WI, USA FAHS Benjamin University of Louvain, Louvain-la-Neuve, Belgium FUKAI Yosuke Sano Laboratory, University of Tokyo, Japan GROUX Benjamin Laboratoire de Mathématiques de Versailles, France HOLCOMB Diane University of Arizona, Tucson, AZ, USA HUANG Jiaoyang Harvard University, Cambridge, MA, USA IPSEN Jesper Bielefeld University, Germany JANJIGIAN Christopher University of Wisconsin, Madison, WI, USA JOYNER Christopher Queen Mary University, London, UK JUNNILA Janne University of Helsinki, Finland KIEFF Max Columbia University, New York, USA LAMBERT Gaultier KTH Stockholm Sweden LI Yiting Brandeis University, Waltham, MA, USA LIU Dang-Zheng University of China, Hefei, China MARINO Ricardo LPTMS, Université Paris-Sud, Paris, France MUDUTE-NDUMBE Steve Imperial College, London, UK NEMISH Yuriy Université Paul Sabatier, Toulouse, France

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