This volume celebrates the tenth edition of the Brazilian School of Probability (EBP), held at IMPA, Rio de Janeiro, from July 30 to August 4, 2006, jointly with the 69th Annual Meeting of the Institute of Mathematical Statistics. It was indeed an exceptional occasion for the local community working in this ?eld. The EBP, ?rst envisioned and organized in 1997, has since developed into an annual meeting with two or three advanced mini-courses and a high level conference. This volume grew up from invited or contributed articles by researchers that during the last ten yearshave been participating in the BrazilianSchool of Pro- bility. As a consequence, its content partially re?ects the topics that have pred- inated in the activities during the various editions of the School, with a strong - peal that comes from statistical mechanics and areasof concentrationthat include interacting particlesystems, percolation, random media anddisordered systems. All articles of this volume were peer-refereed.

The intersection of probability and physics has been a rich and explosive area of growth in the past two decades, specifically covering such subjects as percolation theory, random walks, interacting particle systems, and various topics related to statistical mechanics. In the last several years, substantial progress has been made in a number of directions: fluctuations of 2-dimensional growth processes, Wulf constructions in higher dimensions for percolation, Potts and Ising models, classification of random walks in random environments, the introduction of the stochastic Loewner equation, the rigorous proof of intersection exponents for planar Brownian motion, and finally the proof of conformal invariance for critical percolation on the triangular lattice. This volume consists of a collection of invited articles, written by some of the most distinguished probabilists in the above-mentioned areas, most of whom were personally responsible for advances in the various subfields of probability. All of the articles are an outgrowth of the Fourth Brazilian School of Probability, held in Mambucaba, Brazil, August 2000. Contributors: K. Alexander * J.M. AzaAs * J. van den Berg * T. Bodineau * F. Camia * N. Cancrini * G. Grimmett * P. Hiemer * A.E. Holroyd * H. Kesten * G.F. Lawler * T.M. Liggett * J. Lorinczi * F. Martinelli * C. M. Newman * J. Quastel * C.-E. Pfister * M. PrAhofer * C. Roberto * O. Schramm * V. Sidoravicius * H. Spohn * A. Toom * B. TA3th * D. Ueltschi * W. Werner * M. Wschebor * M. WA1/4thrich Graduate students and researchers in probability theory and math physics will find this book a useful reference.

This volume features selected and peer-reviewed articles from the Pan-American Advanced Studies Institute (PASI). The chapters are written by international specialists who participated in the conference. Topics include developments based on breakthroughs in the mathematical understanding of phenomena describing systems in highly inhomogeneous and disordered media, including the KPZ universality class (describing the evolution of interfaces in two dimensions), random walks in random environment and percolative systems. PASI fosters a collaboration between North American and Latin American researchers and students. The conference that inspired this volume took place in January 2012 in both Santiago de Chile and Buenos Aires. Researchers and graduate students will find timely research in probability theory, statistical physics and related disciplines.

The XV-th International Congress on Mathematical Physics took place in Rio de Janeiro, on August 5-11, 2006. I believe it was a very successful and enjoyable meeting. It is very fortunate for our community that Latin America, and especially Brazil, in decades has become the place where all ?elds of intellectual activity which are traditionally regarded as part of Mathematical Physics are developing with steady pace and remarkable quality. Another important aspect of this devel- mentis thatbesidesareaswhichalreadyhaveworldwiderecognitionandlongsta- ing tradition in Brazil such as Dynamical Systems, Statistical Mechanics, Probab- ity Theory, etc. , there is intensive growth in other directions such as String Theory and Algebraic Geometry. Brazil's major universities and research institutes such as IMPA and CBPF are successfully bringing up a new generation of researchers in our ?eld. Thus it was especially pleasant to see that among more than 500 participants of the Congress, the majority was constituted by young researchers and graduate students. Given the enormous range of subjects covered during the Congress, and the - versity of scienti?c contributions, it would be pointless to summarize the contents here-the quality is re? ected in these pages. Traditionally, besides its very intense scienti?c program, the Congress was the occasion for the award of the Henri Poincare Prizes of the IAMP, sponsored by the Daniel Iagolnitzer Foundation. The Laureates for the year 2006 were Ludwig Faddeev, David Ruelle and Edward Witten.

This volume features selected and peer-reviewed articles from the Pan-American Advanced Studies Institute (PASI). The chapters are written by international specialists who participated in the conference. Topics include developments based on breakthroughs in the mathematical understanding of phenomena describing systems in highly inhomogeneous and disordered media, including the KPZ universality class (describing the evolution of interfaces in two dimensions), random walks in random environment and percolative systems. PASI fosters a collaboration between North American and Latin American researchers and students. The conference that inspired this volume took place in January 2012 in both Santiago de Chile and Buenos Aires. Researchers and graduate students will find timely research in probability theory, statistical physics and related disciplines.

1. N. Anantharaman, H. Koch, S. Nonnenmacher, Entropy of Eigenfunctions 2. G. Benfatto, Rigorous construction of Luttinger liquids through Ward identities 3. C. Bergbauer, D. Kreimer, New algebraic aspects of perturbative and non-perturbative Quantum Field Theory 4. P. Bleher, Exact solution of the Six-vertex model with domain wall boundary conditions 5. M. Bojowald, mathematical issues in loop quantum cosmology 6. L. Bertini, S. Brassesco, P. Butta, Boundary effects on the interface dynamics for the stochastic Allen-Cahn equation. 7. J. Buzzi, Dimensional entropies and semi-uniform hyperbolicity 8. F. Camia, The scaling limit of (Near-) critical 2D percolation 9. G.L. Cardoso, Black hole entropy function and duality 10. J. Colliander, Weak turbulence for periodic NLS 11. S. Dain, Angular momentum-mass inequality for axisymmetric black holes 12. D. Damanik, Almost everything about the Fibonacci operator 13. I. Davetak, T. Brun, M.-H. Hsieh, Entanglement-assisted quantum error-correcting codes 14. G. Delfino, Particle decay in Ising field theory with magnetic field 15. B. Derrida, Fluctuations and large deviations in non-equilibrium systems 16. L. Diaz, Robust heterodimensional cycles and tame dynamics 17. B. Dubrovin, Hamiltonian perturbations of hyperbolic PDEs: from classification results to the properties of solutions 18. P. Fendley, K. Schoutens, Lattice supersymmetry from the ground up 19. L.R.G. Fontes, P.H.S. Lima, Convergence of symmetric trap models in the hypercube 20. F. Guerra, Spontaneous replica symmetry breaking in the mean field spin glass model 21. S. Gukov, Surface Operators and Knot Homologies 22. Y. Kawahigashi, Conformal Field Theory and Operator Algebras. 23. A. Kiselev, Diffusion and mixing in fluid flow: a review 24. F. Germinet, A. Klein, Random Schrodingeroperators: Localization and delocalization, and all that 25. A. Kleinschmidt, Unifying R-symmetry in M-theory 26. O. Kozlovski, S. van Strien, Stable maps are dense in dimension one 27. I. Krasovsky, large gap asymptotics for random matrices 28. G. Landi, Noncommutatiove manifolds and quantum groups 29. M. Marinho, Topological strings on local curves 30. M. Merkli, Repeated interaction quantum systems 31. J. Mund, String-localized quantum fields, modular localization, and gauge theories 32. G. Mussardo, Kinks and particles in non-integrable Quantum Field Theory 33. A. Jensen, G. Nenciu, Exponential decay laws in perturbation theory of threshold an embedded eigenvalues 34. S. Olla, Energy diffusion and superdiffusion in oscillators lattice networks 35. E.R. Pujals, Trying to characterize robust and generic dynamics 36. B. Schlein, Dynamics of Bose-Einstein condensates 37. B. Nachtergaele, R. Sims, Locality estimates for quantum spin systems 38. A. Soshnikov, On reseolvent identities in Gaussian ensembles at the edge of the spectrum 39. H. Spohn, Energy current correlations for weakly anharmonic lattices 40. C.M. Newman, D.L. Stein, Metastates, translation ergodicity, and simplicity of thermodynamic states in disordered systems: an illustration 41. M.Sh. Birman, T.A. Suslina, Homogenization of periodic differential operators as a spectral threshold effect 42. P. Dorey, C. Dunning, D. Masoero, J. Suzuki, R. Tateo, ABCD and ODEs 43. J. Teschner, Nonrational conformal field theory 44. C. A. Tracy, H. Widom, The distributions of random matrix theory and their applications 45. J. Kappeli, S. Theisen, P. Vanhove, Hybrid formalism and topological amplitudes 46. M. Aizenman, E.H. Lieb, R. Seiringer, J.P. Solovej, J. Yngvason, Quantum phases of cold bosons in an optical lattice 47. O. Zeitouni, R

This volume consists of a collection of invited articles, written by some of the most distinguished probabilists, most of whom were personally responsible for advances in the various subfields of probability. Graduate students and researchers in probability theory and math physics will find this book a useful reference.