Front cover -- Lecturers who contributed to this volume -- Title page -- Copyright page -- Previous sessions -- Organizers -- Lecturers -- Participants -- Preface -- Informal seminars -- Table of contents -- Course 1 Random matrices and determinantal processes -- Introduction -- Point processes -- General theory -- Determinantal processes -- Measures defined by products of several determinants -- Non-intersecting paths and the Aztec diamond -- Non-intersecting paths and the LGV theorem -- The Aztec diamond -- Relations to other models -- Asymptotics -- Double contour integral formula for the correlation kernel -- Asymptotics for the Aztec diamond -- Asymptotics for random permutations -- The corner growth model -- Mapping to non-intersecting paths -- The Schur and Plancherel measures -- A discrete polynuclear growth model -- Proof of theorem 5.1 -- References -- Course 2 Some recent aspects of random conformally invariant systems -- Some discrete models -- Self-avoiding walks and polygons -- Random walk loops -- Site-percolation -- The Ising model -- The Potts models -- FK representations of Potts models -- The O(N) models -- Conformal invariance -- A "conformal Haar measure" on self-avoiding loops -- Preliminaries -- A conformal invariance property -- Uniqueness -- Existence -- Schramm-Loewner Evolutions -- Definition -- Computing with SLE -- Conformal loop-ensembles -- Definition -- First properties -- The loop-soup construction -- The Gaussian free field -- Definition -- "Cliffs" as level lines -- References -- Course 3 Conformal random geometry -- Preamble -- Introduction -- A brief conformal history -- Conformal geometrical structures -- Quantum gravity -- Stochastic Löwner evolution -- Recent developments -- Synopsis -- Intersections of random walks -- Non-intersection probabilities -- Quantum gravity -- Random walks on a random lattice.

Non-intersections of packets of walks -- Mixing random & self-avoiding walks -- General star configurations -- Quantum gravity for SAW's & RW's -- RW-SAW exponents -- Brownian hiding exponents -- Percolation clusters -- Cluster hull and external perimeter -- Harmonic measure of percolation frontiers -- Harmonic and path crossing exponents -- Quantum gravity for percolation -- Multifractality of percolation clusters -- Conformally invariant frontiers and quantum gravity -- Harmonic measure and potential near a fractal frontier -- Calculation of multifractal exponents from quantum gravity -- Geometrical analysis of multifractal spectra -- Higher multifractal spectra -- Double-sided spectra -- Higher multifractality of multiple path vertices -- Winding of conformally invariant curves -- Harmonic measure and rotations -- Exact mixed multifractal spectra -- Conformal invariance and quantum gravity -- Rotation scaling exponents -- Legendre transform -- O(N) & Potts models and the Stochastic Löwner Evolution -- Geometric duality in O(N) and Potts cluster frontiers -- Geometric duality of SLEkappa -- Quantum gravity duality and SLE -- Dual dimensions -- KPZ for SLE -- Short distance expansion -- Multiple paths in O(N), Potts models and SLE -- SLE(kappa, rho) and quantum gravity -- Multifractal exponents for multiple SLE's -- References -- Course 4 Random motions in random media -- Introduction -- RWRE -- RCM -- Back to RWRE -- Diffusions in random environment -- References -- Course 5 An introduction to mean field spin glas theory: methods and results -- Introduction -- The mean field ferromagnetic model. Convexity and cavity methods -- The mean field spin glass model. Basic definitions -- The interpolation method and its generalizations -- The thermodynamic limit and the variational bounds -- The Parisi representation for the free energy.

Conclusion and outlook for future developments -- References -- Course 6 Short-range spin glasses: selected open problems -- Introduction -- The Fortuin-Kasteleyn random cluster representation and phase transitions -- Spin glass ground states and invasion percolation -- Ground state multiplicity in the 2D EA spin glass -- References -- Course 7 Computing the number of metastable states in infinite-range models -- Introduction -- The TAP equations -- A simple analysis of the solutions of the Bethe equations -- The direct approach: general considerations -- The supersymmetric formulation -- Spontaneous supersymmetry breaking -- An explicit computation: the complexity of the SK model -- A few words on quenched complexity -- Conclusions and open problems -- References -- Course 8 Dynamics of trap models -- Introduction -- Definition of the Bouchaud trap model -- Examples of trap models -- Natural questions on trap models -- References -- The one-dimensional trap model -- The Fontes-Isopi-Newman singular diffusion -- The scaling limit -- Time-scale change of Brownian motion -- Convergence of the fixed-time distributions -- A coupling for walks on different scales -- Scaling limit -- Aging results -- Subaging results -- Behaviour of the aging functions on different time scales -- References -- The trap model in dimension larger than one -- The fractional-kinetics process -- Scaling limit -- Aging results -- The coarse-graining procedure -- References -- The arcsine law as a universal aging scheme -- Aging on large complete graphs -- Deep traps -- Shallow traps -- Very deep traps -- Proof of Theorem 5.1 -- The alpha-stable subordinator as a universal clock -- Potential-theoretic characterisation -- Applications of the arcsine law -- Aging in the REM -- Short time scales -- Long time scales -- Open questions and conjectures -- Aging on large tori.

Appendix A. Subordinators -- References -- Course 9 Quantum entropy and quantum information -- Introduction -- Rudiments of Classical Information Theory -- Entropy in Classical Information Theory -- Entropy of a pair of random variables -- Shannon's Noiseless Channel Coding Theorem -- Asymptotic Equipartition Property (AEP) -- Consequences of the AEP -- Information transmission and Channel Capacity -- Introduction to Quantum Information Theory -- Open systems -- Properties of the density matrix -- Reduced density matrix and partial trace -- Time evolution of open systems -- Generalized measurements -- Implementation of a generalized measurement -- Quantum entropy -- Properties of the von Neumann entropy S(rho) -- Data compression in Quantum Information Theory -- Schumacher's Theorem for memoryless quantum sources -- Quantum channels and additivity -- The noise in the channel -- Capacities of a quantum channel -- Classical capacity of a quantum channel -- A sufficient condition for additivity -- Multiplicativity of the maximal p-norm -- Bibliography -- Course 10 Two lectures on iterative coding and statistical mechanics -- Introduction -- Codes on graphs -- A simple-minded bound and belief propagation -- Characterizing the code performances -- Bounding the conditional entropy -- A parenthesis -- Density evolution a.k.a. distributional recursive equations -- The area theorem and some general questions -- Historical and bibliographical note -- References -- Course 11 Evolution in fluctuating populations -- Introduction -- Some classical coalescent theory -- Kingman's coalescent -- Variable population size -- Introducing structure -- Fluctuations matter -- Balancing selection -- A second neutral locus -- The problem -- The ancestral recombination graph and local trees -- Back to the main plot -- The diffusion approximation -- Extensions.

Summary and notes of caution -- Spatial structure and the Malécot formula -- Branching process models -- The classical stepping stone model -- Duality -- Neutral evolution -- Random walks in random environments -- Models in continuous space -- Malécot's formula -- Spatial models -- Locally regulated populations -- Competing species -- Branching annihilating random walk -- A duality and a conjecture for Model II -- Conjectures for Model I -- Heteromyopia -- References -- Course 12 Multi-scale analysis of population models -- Spatial diffusion models of population genetics -- Duality and representation via coalescent processes -- The function-valued dual process -- The refined dual process -- Application: ergodic theorem -- Historical processes -- The neutral models: IMM and IFWD -- The concept of the historical process -- The concept of the look-down process -- Representation theorem via look-down -- Representation via coalescent -- Quasi-equilibria: elements of the multi-scale analysis -- Mean-field models and McKean-Vlasov limit -- The mean-field dual -- Longtime behavior of McKean-Vlasov process -- Large time-space scale behavior of mean-field model -- Punctuated equilibria and hierarchical mean-field limit -- Detailed specification of selection and mutation mechanisms -- Interpretation of the model from the point of view of evolutionary theory -- The basic scenario -- The hierarchical mean-field limit -- Two interaction chains -- Basic limit theorem -- Punctuated equilibria: emergence and take-over -- References -- Course 13 Elements of nonequilibrium statistical mechanics -- Elements of introduction -- The goal -- The plan -- Elements of an H-theorem -- Heuristics of an H-theorem -- H-theorem: the problem -- Macroscopic H-theorem -- Semigroup property -- Autonomy on a finite partition -- Reversibility -- Propagation of constrained equilibrium.

Pathwise H-theorem.