Table Of ContentLecture Notes in Mathematics 2175
École d'Été de Probabilités de Saint-Flour
Francis Comets
Directed
Polymers
in Random
Environments
École d'Été de Probabilités de Saint-
Flour XLVI – 2016
Lecture Notes in Mathematics 2175
Editors-in-Chief:
J.-M.Morel,Cachan
B.Teissier,Paris
AdvisoryBoard:
CamilloDeLellis,Zurich
MariodiBernardo,Bristol
MichelBrion,Grenoble
AlessioFigalli,Zurich
DavarKhoshnevisan,SaltLakeCity
IoannisKontoyiannis,Athens
GaborLugosi,Barcelona
MarkPodolskij,Aarhus
SylviaSerfaty,NewYork
AnnaWienhard,Heidelberg
Moreinformationaboutthisseriesathttp://www.springer.com/series/304
Francis Comets
Directed Polymers in
Random Environments
École d’Été de Probabilités de Saint-Flour
XLVI – 2016
123
FrancisComets
Mathematics,case7012
UniversitéParisDiderot-Paris7
Paris,France
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISBN978-3-319-50486-5 ISBN978-3-319-50487-2 (eBook)
DOI10.1007/978-3-319-50487-2
LibraryofCongressControlNumber:2017931071
MathematicsSubjectClassification(2010):Primary:60K37;secondary:82B44,60J10,60H05,60F10
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L’ordre est le plaisir de la raison,
mais le désordre
est le délice de l’imagination.
PaulClaudel,
Introductionof “Le soulierdesatin”(1924)
Preface
This monograph contains the notes of lectures I gave in Saint Flour Probability
SummerSchoolinJuly2016.ThetwoothercoursesweregivenbyPaulBourgade
andbyScottSheffieldandJasonMiller.Thematerialwasgrowninseveralcourses
overthelast15yearsintheUniversitiesofParisDiderotandCampinasandalsoat
CIRMinMarseille,MaxPlanckInstituteinLeipzig,andattheALEAConference.
Themodelofdirectedpolymersinrandomenvironmentisasimplifiedmodelfor
stretchedelasticchainswhicharepinnedbyrandomimpurities.Themainquestion
is:
Whatdoesarandomwalkpathlooklikeifrewardsandpenaltiesarerandomly
distributedinthespace?
In large generality, it experiences a phase transition from diffusive behavior to
localized behavior. In this monograph, we place a particular emphasis on the
localization phenomenon. The model can be mapped, or it relates, to many
other ones, including interacting particle systems, percolation, queueing systems,
randomly growing surfaces, and biological population dynamics. Even though it
attractsahugeresearchactivityinstochasticprocessesandinstatisticalphysics,it
still keepsmanysecrets, especiallyin spacedimensiontwo andlarger.Ithasnon-
Gaussianscalinglimits,anditbelongstotheso-calledKPZuniversalityclasswhen
thespacedimensionisone.Weadoptastatisticalmechanics(Gibbsian)approach,
usinggeneralandpowerfultoolsfromprobabilitytheory,tosuccessfullytacklethe
generalframework.Toobtainpreciseasymptotics,wealsostudyaparticularmodel
(theso-calledlog-gammapolymer)whichisexactlysolvable.
We consider the simplest discrete setup all through book. The only exception
is the KPZ equation—a stochastic partial differential equation—for the decisive
reason that it is the scaling limit of the discrete model in one dimension with
vanishing interaction, and therefore, it finds a natural place. I have chosen not to
includematerialcoveringothercontinuousorsemi-discreteversionsofthemodel;
this is the reason why I do not present all available techniques here. I hope that
my choice to stick to the simplest setup makes the purpose crystal clear and the
argumentstransparent.
vii
viii Preface
Givingthestateofartfromdifferentperspectives,themonographisdevisedfor
researchers interested in polymer models. Written in the format of a first course
onthesubject,itisalsoaccessibletomaster’sandPh.D.studentsinprobabilityor
statistical physics. In this perspective, I have deliberately preferred simple proofs
basedonfirst principlesandgeneralmethodsratherthanoptimalresultsrequiring
technical and specific proofs. The mathematician reader will understand how to
extend the proof to a wider framework and to get optimal constant. Researchers
beingnowadaysincentedforefficiencyreasonstobemoreandmorespecializedin
narrowsubjects,Iherebyadvocateforwideviewonthefield,fundamentalobjects,
androbusttechniques.Thewholetextcanbereadlinearly,andtechnicalitiesappear
inaprogressivemanner.
I was a juniormathematicianin the mid-1980swhen Erwin Bolthausenvisited
OrsayUniversity,justafterhebroughtpolymermodelsfromstatisticalphysicsinto
the realm of probability using martingale techniques in his paper [46]. He was
sharinghisenthusiasm,andindeedthechallengelookedgreat.Sincethebeginning
ofourcentury,themodelhasattractedmanydifferentmethodsfromvariousfieldsof
mathematicsincludingorthogonalpolynomials,randommatrices,semimartingales,
stochastic PDEs, integrable systems, tropical combinatorics, and representation
theory.Itisachallengingexperienceformetowritethesenotes,whichcoveronly
apartofthepicture.IfirstcametoSaintFlourProbabilitySummerSchoolin1981
as a student and, also many times since that time, found the format suitable for
learningnewtopicsinthemonasticatmosphereoftheGrandSéminaire.Iwarmly
thankChristopheBahadoran,ArnaudGuillin,andLaurentSerletfororganizingthe
school in a friendly and enjoyable ambiance, pursuing a long-standing, notorious
traditionoftraininginprobability,andgivingmeadefinitivereasontowritethese
notes.
I am first of all indebted in exchanges on this particular subject with wonder-
ful collaborators over the years: Mike Cranston, Ryoki Fukushima, Quansheng
Liu, Shuta Nakajima, Vu-Lan Nguyen, Sergei Popov, Jeremy Quastel, Alejandro
Ramírez,TokuzoShiga,MarinaVachkovskaia,VincentVargas,andNobuoYoshida.
The works of many colleagues have had a profound impact on my research and
view of the subject, in particular Anton Bovier, Ivan Corwin, Bernard Derrida,
FrankdenHollander,PierreLeDoussal,KostyaKhanin,FirasRassoul-Agha,Timo
Seppäläinen,andHerbertSpohn.IacknowledgeKITPforitshospitalityduringthe
program on KPZ integrability and universality, which have been a great update
for me to write Chap.7. I owe special thanks to Quentin Berger, Chris Janjigian,
Oren Louidor, Gregorio Moreno, Makoto Nakashima, and Nikos Zygouras for
correcting parts of the manuscript and suggesting improvements. The notes have
benefitedfrom furtherexchangeswith RonfengSun and Attila Yilmaz. Questions
andcommentsbytheparticipants,includingRedaChhaibi,AserCortines,Clément
Cosco, and Giambattista Giacomin, helped me to improvethe matter. I apologize
forthe(numerous)remainingerrors;theyaremine!
Paris,France FrancisComets
October2016
Contents
1 Introduction .................................................................. 1
1.1 PolymerModels........................................................ 1
1.1.1 Random Walk in Random Environment:
AModelforDirectedPolymers .............................. 1
1.1.2 Modelization:Polymerin anEmulsionwith
RepulsiveImpurities........................................... 3
1.1.3 PolymersViewedasPercolationataPositive
Temperature.................................................... 4
1.1.4 ExponentsandLocalization................................... 5
1.2 ParticleinaRandomPotential......................................... 5
1.2.1 ParticleinDeadlyObstacles .................................. 6
1.2.2 BranchingRandomWalkPicture............................. 8
1.3 History,ExperimentsandRelatedModels............................ 9
1.4 Highlights............................................................... 11
2 ThermodynamicsandPhaseTransition................................... 13
2.1 Preliminaries............................................................ 13
2.1.1 MarkovPropertyandthePartitionFunction................. 14
2.1.2 ThePolymerMeasureasaMarkovChain ................... 14
2.2 FreeEnergy............................................................. 15
2.3 UpperBounds .......................................................... 18
2.3.1 TheAnnealedBound.......................................... 18
2.3.2 ImprovingtheAnnealedBound .............................. 20
2.4 Monotonicity ........................................................... 24
2.5 PhaseTransition........................................................ 27
3 TheMartingaleApproachandtheL2Region ............................ 31
3.1 AMartingaleAssociatedtothePartitionFunction................... 31
3.2 TheSecondMomentMethodandtheL2Region ..................... 35
3.3 DiffusiveBehaviorinL2 Region....................................... 38
3.4 LocalLimitTheoremintheL2Region................................ 44
ix
x Contents
3.5 Analytic Functions Method:Energy and Entropy
intheL2 Region ........................................................ 47
3.6 ConcentrationRevisited................................................ 51
3.6.1 InsidetheL2Region........................................... 51
3.6.2 GeneralCase:SublinearVarianceEstimate.................. 52
3.7 RateofMartingaleConvergence ...................................... 53
4 LatticeVersusTree .......................................................... 57
4.1 AMeanFieldApproximation.......................................... 57
4.2 MajorizingLatticePolymersbym-TreePolymers ................... 61
4.3 PhaseDiagramontheTree ............................................ 62
4.4 Free Energy is Strictly Convex for the Polymer
ontheLattice ........................................................... 67
4.5 ConclusionsandRelatedModels...................................... 72
5 SemimartingaleApproachandLocalizationTransition................. 75
5.1 SemimartingaleDecomposition ....................................... 75
5.2 WeakDisorderandDiffusiveRegime................................. 82
5.3 BoundsontheCriticalTemperaturebySize-Biasing................. 84
5.4 LocalizationVersusDelocalization.................................... 87
6 TheLocalizedPhase......................................................... 91
6.1 PathLocalization....................................................... 91
6.2 LowDimensions........................................................ 93
6.2.1 OverlapEstimates ............................................. 94
6.2.2 FractionalMomentsEstimates................................ 95
6.3 SimulationsoftheLocalizedPhase ................................... 96
6.4 LocalizationforHeavy-TailsEnvironment............................ 100
6.4.1 TheBigPicture................................................ 100
6.4.2 TheResults .................................................... 102
6.4.3 CharacteristicExponentsandFloryArgument............... 105
7 Log-GammaPolymerModel ............................................... 107
7.1 Log-GammaModelanditsStationaryVersion ....................... 107
7.2 FreeEnergyandFluctuationsfortheStationaryModel.............. 112
7.3 ModelWithoutBoundaries ............................................ 115
7.4 LocalizationintheLog-GammaPolymerwithBoundaries.......... 117
7.5 FinalRemarksandComplements...................................... 124
8 Kardar-Parisi-ZhangEquationandUniversality ........................ 127
8.1 Kardar-Parisi-ZhangEquation......................................... 128
8.2 Hopf-ColeSolution..................................................... 130
8.3 TheContinuumRandomPolymer(CRP)............................. 133
8.3.1 PropertiesofZˇ ............................................... 134
8.3.2 TheLawoftheContinuumRandomPolymer ............... 137
8.4 IntermediateDisorderRegimeforLatticePolymers ................. 139
8.5 FluctuationsandUniversality.......................................... 144
8.5.1 AdditionalRemarks........................................... 146
