Table Of ContentClay 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.
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2012017930
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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
