Some New Results on the Kinetic Ising Model in a P

arXiv:math-ph/0202013v18Feb2002SOMENEWRESULTSONTHEKINETICISINGMODELINAPUREPHASET.BODINEAUANDFABIOMARTINELLIAbstract.WeconsiderageneralclassofGlauberdynamicsreversiblewithrespecttothestandardIsingmodelinZdwithzeroexternalfieldandinversetemperatureβstrictlylargerthanthecriticalvalueβcindimension2orthesocalled“slabthreshold”ˆβcindimensiond3.WefirstprovethattheinversespectralgapinalargecubeofsideNwithplusboundaryconditionsis,apartfromlogarithmiccorrections,largerthanNind=2whilethelogarithmicSobolevconstantisinsteadlargerthanN2inanydimension.Sucharesultsubstantiallyimprovesoverallthepreviousexistingboundsandagreeswithasimilarcomputationsobtainedintheframeworkofaonedimensionaltoymodelbasedonmeancurvaturemotion.Theproof,basedonasuggestionmadebyH.T.Yausomeyearsago,explicitlyconstructsasubtletestfunctionwhichforcesalargedropletoftheminusphaseinsidetheplusphase.Therelevantboundsforgenerald≥2arethenobtainedviaacarefuluseoftherecentL1–approachtotheWulffconstruction.Finallyweprovethatind=2theprobabilitythattwoindependentinitialconfigurations,distributedaccordingtotheinfinitevolumeplusphaseandevolvingunderanycoupling,agreeattheoriginattimetisboundedfrombelowbyastretchedexponentialexp(−√t),againapartfromlogarithmiccorrections.Sucharesultshouldbeconsideredasafirststeptowardarigorousproofthat,asconjecturedbyFisherandHusesomeyearsago,theequilibriumtimeauto-correlationofthespinattheorigindecaysasastretchedexponentialind=2.2000MSC:82B10,82B20,60K35Keywordsandphrases:Isingmodel,Glauberdynamics,phaseseparation,spectralgap1.IntroductionInafinitedomain,thereversibleGlauberdynamicsassociatedtotheIsingmodelrelaxesexponentiallyfasttoitsequilibriummeasure.Nevertheless,thissimplestatementhidesawiderangeofbehaviorsdependingonthetemperature,thedomainandtheboundaryconditions.Intheuniquenessregime(whenthetemperatureislargeenough),thespeedofrelax-ationisuniformwithrespecttothedomainsandtheboundaryconditions.WerefertoMartinelli[Ma]foracompleteaccountofthistheory.Theoccurrenceofphasetransitiondrasticallymodifiesthebehaviorofthedynamicsandnewphysicalfeaturesslowdowntherelaxation;amongthose,thenucleationandtheinterfacemotions.Metastabilityis1991MathematicsSubjectClassification.Primary54C40,14E20;Secondary46E25,20C20.DuringthefirstpartofthisworkwehavebenefitfrominstructiveconversationswithD.Ioffewhomwewarmlythank.ThesecondpartofthisworkwasdonewhilebothauthorswerevisitingtheInstitutH.Poincar´eduringthespecialsemesterdevotedto“HydrodynamicLimits”.Wewouldliketowarmlythanktheorganizers,S.OllaandF.Golse,fortheirhospitalitythereandfortheverystimulatingscientificatmosphere.T.B.alsoacknowledgesUniversitiesofRomaIIandRomaIIIfortheirinvitationwhenthisworkstarted.Finally,wewouldliketothankY.VelenikandN.Yoshidaforusefulcomments.12T.BODINEAUANDF.MARTINELLIcharacteristicoftheseslowphenomenasincethesystemistrappedforaverylongperiodoftimeinalocalequilibrium.Inthiscase,therelaxationmechanismissoslowthatthetimeofnucleationcanbeexpressedintermsofequilibriumquantities.Inparticular,itwasprovenbyMartinelli(seeeg.[Ma]andreferencestherin)thatforfreeboundaryconditionstheasymptoticofthespectralgapwithrespecttothesizeofthedomainsisrelatedtothesurfacetensionandthemainmechanismdrivingthesystemtoequilibriumisnucleationofonephaseinsidetheother.AcompletepictureofthenucleationprocessinZ2intheframeworkofmetastabilitywasobtainedbySchonmannandShlosmanin[SS2].Inthispaper,weareinterestedinadifferentregimeinwhichtherelaxationtoequilib-riumisdrivenbytheslowmotionoftheinterfaces.ThisisthecaseoftheIsingmodelinalargeboxwithplusboundaryconditions.Whenadropletoftheminusphaseissurroundedbytheplusphase,ittendstoshrinkaccordingtoitscurvatureundertheactionofthenon-conservativedynamicsonthespinsclosetotheinterface.Thissubtlephenomenonhasbeenstudiedrigorouslyonlyinrareinstances:bySpohn[Sp]inthecaseofIsingmodelatzerotemperature(seealsoRezakhanlou,Spohn[RS]),byChayes,Schonmann,Swindle[CSS]foravariantofthismodelandbyDeMasi,Orlandi,Presutti,Triolo[DOPT1,DOPT2]fortheKac-Isingmodel.Noticealsothatthemotionbymeancurvatureplaysakeyroleinthecoarseningphenomenon,asithasbeenshownrecentlybyFontes,Schonmann,Sidoravicius[FSS].Forpositivetemperatures,amathematicalderivationofsimilarresultsseemstobemorechallenging.AwaytocapturesomeinsightsintotheslowrelaxationdrivenbyinterfacemotionistoestimatespectralquantitiesrelatedtothegeneratoroftheGlauberdynamics.Weprovethatforanydimensiond2,inthephasetransitionregimeandwithplusboundaryconditions,thelogarithmic-SobolevconstantforadomainoflinearsizeNdivergeatleastlikeN2(uptosomelogarithmiccorrections).Thiscanbeconsideredasafirstcharacteri-zationoftheslowdownofthedynamicsandisinagreementwiththeheuristicspredictedbythemotionbymeancurvature.Inthesamesettingbutd=2,weprovethattheinverseofthespectralgapgrowsatleastlikeN(uptologarithmiccorrections).Indimensiond3ourargumentfailstoproducearesultonthedivergenceofthespectralgap.Letusstressthatwehavenotbeenabletoderivematchingupperbounds;thebestexistingboundshavebeenprovedonlyind=2andareoftheformexp√N(logN)2
(see[YW]).However,anexactcomputationfo