Clay Mathematics Proceedings Volume 15 Probability and Statistical Physics in Two and More Dimensions Proceedings of the Clay Mathematics Institute Summer School and XIV Brazilian School of Probability Búzios, Brazil July 11–August 7, 2010 David Ellwood Charles Newman Vladas Sidoravicius American Mathematical Society Wendelin Werner Clay Mathematics Institute Editors Probability and Statistical Physics in Two and More Dimensions Clay Mathematics Proceedings Volume 15 Probability and Statistical Physics in Two and More Dimensions Proceedings of the Clay Mathematics Institute Summer School and XIV Brazilian School of Probability Búzios, Brazil July 11–August 7, 2010 David Ellwood Charles Newman Vladas Sidoravicius Wendelin Werner Editors American Mathematical Society Clay Mathematics Institute 2000 Mathematics Subject Classification. Primary 60–06, 60G60, 60K35, 60K37, 82B20, 82B27, 82B28, 82B41, 82B43, 82B44. Cover imagecourtesy of JasonMiller. Library of Congress Cataloging-in-Publication Data ClayMathematicsInstitute,SummerSchool(2010: Armac¸a˜odosBu´zios,Brazil) Probability and statisticalphysics in two and more dimensions : Clay MathematicsInstitute SummerSchoolandXIVBrazilianSchoolofProbability,Bu´zios,Brazil,July11–August7,2010/ DavidEllwood...[etal.],editors. p.cm. —(Claymathematicsproceedings;v.15) Includesbibliographicalreferences. ISBN978-0-8218-6863-8(alk.paper) 1. Probabilities—Congresses. I. Ellwood, D. (David), 1966– II. Brazilian School of Proba- bility(14th: 2010: Armac¸a˜odosBu´zios,Brazil) III.Title. QC174.85.P76C53 2010 519.2—dc23 2012017930 Copying and reprinting. Materialinthisbookmaybereproducedbyanymeansfor edu- cationaland scientific purposes without fee or permissionwith the exception ofreproduction by servicesthatcollectfeesfordeliveryofdocumentsandprovidedthatthecustomaryacknowledg- ment of the source is given. 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(cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ VisittheClayMathematicsInstitutehomepageathttp://www.claymath.org/ 10987654321 171615141312 Contents Preface vii Schramm-Loewner Evolution and other Conformally Invariant Objects 1 Vincent Beffara Noise Sensitivity and Percolation 49 Christophe Garban and Jeffrey E. Steif Scaling Limits of Random Trees and Planar Maps 155 Jean-Franc¸ois Le Gall and Gr´egory Miermont Conformal Invariance of Lattice Models 213 Hugo Duminil-Copin and Stanislav Smirnov Fractal and Multifractal Properties of Schramm-Loewner Evolution 277 Gregory F. Lawler Lectures on Random Polymers 319 Francesco Caravenna, Frank den Hollander and Nicolas P´etr´elis Lectures on Self-Avoiding Walks 395 Roland Bauerschmidt, Hugo Duminil-Copin, Jesse Goodman and Gordon Slade v Preface TheClayMathematicsInstitute2010SummerSchool,“ProbabilityandStatis- tical Physics in Two and More Dimensions” took place in Bu´zios, Brazil from July 11 to August 7. The final week was a joint event with the XIV Escola Brasileira da Probabilidade. In the past ten to fifteen years, various areas of probability theory related to statistical physics, disordered systems and combinatorics have undergone intensive development. A number of these developments deal with two-dimensional random structures at their critical points, and provide new tools and ways of coping with atleastsomeofthelimitationsofConformalFieldTheory(CFT)thathadbeenso successfully developed in the theoretical physics community to understand phase transitions of two-dimensional systems. One of the new ideas that emerged in the mathematics community just before thenewmilleniumistheStochasticLoewnerEvolution(SLE),introducedbyOded Schramm. This new approach is probabilistic in nature and focuses directly on non-local structuresthatcharacterizeagivensystem, such ascluster boundaries in Ising,Pottsandpercolationmodels,orloopsintheO(n)model. Atcriticality,these become,inthecontinuumlimit,randomcurveswhosedistributionscanbeuniquely identified thanks to their conformal invariance and a certain Markovian property. There is aone-parameter family of SLE’s indexed by apositive real number κ, and theyappear tobethe onlypossible candidatesforthe scalinglimitsof interfacesof two-dimensional critical systems that are conformally invariant. Acomplementaryapproachhasbeentounderstandandcontroldiscretemodels that exhibit discrete holomorphic features. These now include several important models,suchascriticalpercolationonthetriangularlattice,thecriticalIsingmodel and its related random cluster model, loop-erased random walks and double-dimer models. Someoftheseresultsareveryrecent–orevenongoing–developments. They make it possible toprove thatindeed, these discrete models give rise toSchramm’s SLE curves in the large-scale limit, and toprovide a detailed description of various aspects of their large-scale behavior. Differentquestionscorrespondtothecasewhereoneconsidersthesesamemod- els from statistical physics on certain natural planar graphs that are themselves random (they are often called “planar maps”) - here conformality is not obvious toformulate, but the combinatorics of the problems turn out to be more tractable. This has led to spectacular recent progress, and the proof of several results in this discrete approach to what is often referred to as “quantum gravity”. It is inter- esting to note that another approach to quantum gravity builds on the Gaussian Free Field, which is another conformally invariant continuous model, that has also vii viii PREFACE been recently shown to be directly related to the SLE processes (and SLE(4) in particular). Wethusbelieveditwasagoodtimeforaschoolthatwouldprovideacomplete picture of the current state of the art in these topics and discuss the relations between them as well as other possible future directions. The School offered three long Foundational Courses: Beffara’s course provided an introduction to Schramm’s SLE processes and their properties. Garban and Steif’s course gave an account of recent results concerning the scaling limit of per- colation, and its relation to noise-sensitivity, and the course by Le Gall and Mier- mont focused on the description of planar maps via tree-like structures and their large-scale limits. Five advanced mini-courses covered further topics on this theme: Smirnov fo- cused on the conformal invariance of critical percolation and of the critical Ising model, whileKenyondescribedtheconformalinvarianceofanotherdiscretemodel, called the double-dimer model. Sheffield described aspects of the relation between theGaussianFreeFieldandSLEprocesses,whileLawlerfocusedonfinerstudiesof theSLE processes themselves. Di Francescoprovided an approach tothe combina- torial structures related to integrable systems. Courses by Slade (on self-avoiding walks) and by den Hollander (on polymers) were presented during the final week, jointly with the Brazilian School of Probability. In addition to all these courses, research seminars organized by young partici- pants and evening lectures by prominent senior researchers took place. Given the enormousrangeofsubjectscoveredduringtheSchoolandthediversityofscientific topics, it would be pointless to say more about the contents here, but we believe that the high quality of the lectures is reflected in these pages. Foundational Courses • SLE and other conformally invariant objects, Vincent Beffara. • Noise-sensitivity and percolation, Christophe Garban and Jeffrey Steif. • Largerandomplanarmapsandtheirscalinglimits, Jean-Fran¸coisLeGall and Gr´egory Miermont. Mini-Courses • Random geometry and Gaussian free field, Scott Sheffield. • Conformal invariance of lattice models, Stanislav Smirnov. • Integrable combinatorics, Philippe Di Francesco. • Fractal and multifractal properties of SLE, Gregory Lawler. • The double dimer model, Rick Kenyon. Courses joint with XIV Escola Brasileira da Probabilidade • Random polymers, Frank den Hollander. • Self-avoiding walks, Gordon Slade. A School of such scale could not have happened without the generous support of numerous sponsors and the efforts of many individuals. Besides CMI funding, IMPA,CNPq(Brazil)andNSF-PIRE,weremajorcontributorstothebudgetofthe school, we are grateful toall the other foundations who providedfinancial support. PREFACE ix All sponsors are listed at http://www.impa.br/opencms/pt/eventos/store_old/evento_1007 We would like to express our gratitude to Professors C. Camacho, C. Aragao and J. Palis for their constant support and also our many thanks to the whole administrative support teams of CMI (Amanda Battese and Katherine Brack) and DAC of IMPA, especially DACs coordinator Suely Lima and Pedro Faro for their personal efforts. Last but not least, the editors would like to give special recognition to CMI’s publications manager Vida Salahi for her work and dedication in managing the editorial process of this volume. David Alexandre Ellwood, Chuck Newman, Vladas Sidoravicius and Wendelin Werner April 2012