Directed Polymers in Random Environments: École d'Été de Probabilités de Saint-Flour XLVI – 2016 PDF
Preview Directed Polymers in Random Environments: École d'Été de Probabilités de Saint-Flour XLVI – 2016
Lecture 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 ©SpringerInternationalPublishingAG2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland 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
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