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ProbabilityTheoryandRelatedFieldsmanuscriptNo. (willbeinsertedbytheeditor) Wetting of gradient fields: pathwise estimates YvanVelenik 8 0 0 2 n a Received:date/Accepted:date J 5 1 Abstract We consider the wetting transition in the framework of an effective interface model of gradient type, in dimension 2 and higher. We prove pathwise estimates show- ] R ing that the interface is localized in the whole thermodynamically-defined partial wetting regimeconsideredinearlierworks.Moreover,westudyhowtheinterfacedelocalizesasthe P wetting transition is approached. Our main tool is reflection positivity in the form of the . h chessboardestimate. t a m Keywords Interface Wetting Prewetting Reflectionpositivity · · · [ MathematicsSubjectClassification(2000) 60K35 82B41 · 4 v 2 1 Introductionandresults 3 6 1 1.1 Themodel . 6 Effectiveinterfacemodelsofgradienttypehavebeenaveryactivefieldofresearchinrecent 0 7 years.Inparticular,theunderstandingoftheinteractionofaninterfacewithvarioustypesof 0 external potentials (wall, pinning potential, etc.)has motivated numerous works, resulting : insubstantial progress onsuchissues.Wereferto[12,13,18]forreviewsoftheproblems v investigatedandreferences. i X Among such questions, theanalysis of theeffect of anattractive wall on thebehavior r ofaninterfaceisofparticularrelevance.Suchasituationiscommonlymodeledasfollows. a LetL = L/2 +1,..., L/2 d,L = L/2 ,..., L/2 +1 d and¶L =L L . L L L L L Icnotnevrfeaxcefuc{no−cnt⌈ifiognu,raw⌉tiiothnsVare⌊g0ivaennd⌋}bVy(j0)=={0j{.−i}Gi⌈∈ivLeLn∈⌉hR,lL L⌊.L0e,t⌋walesoi}nVtr:oRdu→ceRthebefoalnloe\wvienng, 6≡ ≥ SupportedinpartbyFondsNationalSuisse. Y.Velenik SectiondeMathe´matiques,Universite´deGene`ve 2-4,rueduLie`vre,Casepostale64 CH-1211Gene`ve4 E-mail:[email protected] 2 probabilitymeasureonRL L. m L0;l ,h (dj )= Z01 exp −81d (cid:229) (j i−j j)2−l (cid:229) V(j i) L;l ,h (cid:2) i,ij∈LjL i∈L L (cid:3) ∼ (cid:213) dj +hd (dj ) (cid:213) d (dj ), i 0 i 0 i × i∈L L(cid:0) (cid:1)i∈¶L L wheredj andd (dj )denoterespectivelyLebesguemeasureandtheDiracmassat0,and i 0 i i j means that i j =1. Writing W = j 0, i L , we then introduce the ∼ k − k1 + { i ≥ ∀ ∈ L} probabilitymeasure m L+;l,0,h (·)=m L0;l ,h (·|W +). Thisisthemeasureweshallbemostlyinterestedinthiswork.Weshalldenoteby Z+,0 =m 0 (W ) (1) L;l ,h L;l ,h + theassociatedpartitionfunction.Beforegoingon,letusbrieflydescribethephysicalmean- ingofallthepiecesenteringthedefinitionofm +,0 .Interpretingasusualj astheheightof L;l ,h i theinterfaceabovesitei,thepositivityconstraintW correspondstothepresenceofahard + wallatheight0,whichtheinterfacecannotcross.Theterm 1 (cid:229) (j j )2 i j 8d − i,ij∈LjL ∼ representstheinternalenergyassociatedtodeformationoftheinterfacefromthehorizontal plane.Theterm l (cid:229) V(j ) i i∈L L representsthecontributiontotheenergycomingfromthepresenceofanexternalpotential. A common choice ifV(x)=x2 (usually termed a mass term), but given the situation we want to model here a more natural choice isV(x)= x. The latter choice allows for the | | interpretationoftheinterfaceasseparatingathermodynamicallystablephase(above)from athermodynamicallyunstablephase(below),thelatterbeingstabilizedlocallybecauseitis favoredbythewall;l thenmeasuresthedifferenceoffreeenergiesbetweenthestableand unstablephases(bothbeingstablewhenl =0);see[17,18]foramoredetailedexplanation. Finally,forh >0,themeasure (cid:213) dj +hd (dj ) i 0 i i∈L L(cid:0) (cid:1) models the local attractivity of the interface/wall interaction, by rewarding each contact between theinterface and thewall. Oneway toseethis better (which alsoturns out tobe technicallyusefullater)istorealizethatm 0 canbeseenastheweaklimitofthemeasures L;l ,h m L0;,l(e,h)(dj )= ZL01,;(le,)h exp(cid:2)−81di,ij(cid:229)∈LjL(j i−j j)2−l i∈(cid:229) L LV(j i)−i∈(cid:229) L LUh(e)(j i)(cid:3) ∼ (cid:213) dj (cid:213) d (dj ), i 0 i × i∈L L i∈¶L L wthheewreeaek−Ulih(me)i(tj,i)as=e1+0,2hoef1{|j i|≤e}, ase ↓0. Similarly, m L+;l,0,h iseasilyseentobegivenby ↓ m +,0,(e)=m 0,(e) ( W ). (2) L;l ,h L;l ,2h ·| + 3 1.2 Earlierresults Variousaspectsofthismodelhavebeenstudiedinseveralpapers.Letusbrieflyreviewear- lierworks relevant tothepresent contribution. Manyoftheresultsquoted below arevalid inthemoregeneralcontextofgradientfieldwithuniformlystrictlyconvexinteractions,i.e. thoseforwhichtheterm(j j )2inthedefinitionofthemeasureisreplacedbyU(j j ) i j i j withU :R Ranevenfun−ction withsecondderivative uniformly bounded awayfr−om0 and¥ .Tok→eepthediscussionshort,weshallnotdiscussthishere(norshallwediscussthe caseofnon-nearest-neighbor interactions),andrefertothecitedpapers,andtothereviews mentionedatthebeginning,formoreinformation.Letusjustremarkthatmostofouranal- ysisactuallyextendstothiscaseaswell,theGaussiancharacterofthemeasurebeingused in an essential way only in very few places. However, most earlier results about the free energyinthewettingproblem, uponwhichourwholeapproach rests,concernexclusively theGaussiansetting(orLipschitzinteractions). 1.2.1 Freeinterface Weverybrieflyrecallwhatisknownwhenl =h =0,forthemeasurewithoutthepositivity constraint,i.e.,forthemeasure m 0 .Inthatcase,themeasureisGaussian,andtherefore L;0,0 amenabletoexplicitcomputations.Manythingsareknown,butforourpurposeshere,itis enoughtosaythatthevarianceofthefieldsatisfies1 (g(1)+o (1))L (d=1), L j 2 0 = (g(2)+o (1))logL (d=2), h 0iL;0,0  L g(d)+oL(1) (d 3), ≥ for explicit constants g(d)>0, whichshows that this measuredescribes adelocalized interface, with unbounded fluctuations, in dimensions 1 and 2, and a localized interface in dimension3andhigher.Inthelattercase,althoughlocalized,theinterfaceisstronglycor- related, Llim¥ hj ij ji0L;0,0=(a(d)+oki−jk2(1))ki−jk22−d, → witha(d)>0,d 3. ≥ 1.2.2 Interfaceandpinningpotential Settingh >0,keepingeverythingasbefore,changesdramaticallythebehaviorofthefield however small h is chosen. More precisely, it is known that the interface is localized in any dimension [11,2,10], and has exponentially decaying covariances [2,15]. Moreover, detailedinformationonthecriticalbehaviorash 0isavailable[6],showingforexample ↓ that 1h 2+o(h 2) (d=1), Ll→im¥ hj 02i0L;0,h =(2p1|l−ogh |+O−(log|logh |) (d=2), andthattheratem(h )ofexponentialdecayoflimL→¥ hj ij ji0L;0,0satisfies 1h 2+o(h 2) (d=1), 2 m(h )=O(h 1/2/ logh 3/4) (d=2), | | O(h 1/2) (d 3). ≥ 1 Wewriteoℓ(1)todenoteafunctionsuchthatlimℓ ¥ oℓ(1)=0. → 4 1.2.3 Interfaceandhard-wall The measure with hard-wall constraint, but no external potentials, i.e. m +,0 has been the L;0,0 subject of numerous works, focusing on the associated entropic repulsion phenomenon. Amongtheresultsthathavebeenobtained,wehighlightthetwomostrelevantinthepresent context.Letd 3;then[4,8] ≥ j +,0 lim h 0iL;0,0 2 g(d) =0. L ¥ √logL − → (cid:12) p (cid:12) The corresponding result in dime(cid:12)nsion 2, whose proo(cid:12)f is substantially more intricate, is provedin[3]andtakestheform j +,0 lim h 0iL;0,0 2 g(2) =0. L ¥ logL − → (cid:12) p (cid:12) (Actually,thestatementin[3]has(cid:12)onlybeenprovedw(cid:12)henthepositivityconstraint actson thesub-boxL d L,0<d <1,butitisclearthatthepreviousresultistrue,andthatitshould beprovableinthesameway,withsomeadditional,butminor,complications.) Themain thing toobserve hereis thefact that theinterface isrepelled by thewall, at adistancethatismuchlargerthanitstypicalfluctuations(whichareoforder√logLwhen d=2,andoforder1whend 3).Thisisthephenomenonofentropicrepulsion.Ofcourse, ≥ thisdoesnothappenwhend=1,sincethepinnedrandomwalkconditionedtobepositive convergesunderdiffusivescalingtotheBrownianexcursion. 1.2.4 Interfaceandattractivehard-wall:wettingtransition Wewanttodescribethebehaviorofthefieldwhenbothahard-wallandapinningpotential arepresent, m +,0 . In this situation, there is acompetition between the entropic repulsion L;0,h duetothehard-wallconstraintandthelocalizingeffectofthepinningpotential. Letusintroducethefinite-volumeaveragedensityofpinnedsites r L(h )=h|L L|−1i∈(cid:229) L L1{j i=0}iL+;,00,h , anditslimitr (h )=limL ¥ r L(h ).Itiseasytoshowthatr isnon-decreasinginh ,sothe followingcriticalvalueis→well-defined, h =inf h : r (h )>0 . c { } This critical point can be given an equivalent definition (the equivalence is proved, e.g., in[7]).Letusintroducethefreeenergy(orsurfacetension,orwallfreeenergy) Z+,0 fL(l ,h )=|L L|−1logZL+;l,0,h , L;l ,0 and f(l ,h )=limL ¥ fL(l ,h ).Then → h =inf h : f(0,h )>0 . c { } The sets h h , resp. h >h , are called regimes of complete wetting, resp. partial c c { ≤ } { } wetting.Theyaresupposedtocorrespond toregimesinwhichtheinterfaceisdelocalized, 5 resp.localized.Thephasetransitiontakingplaceath isknownasthewettingtransition.It c isknownthath =0whend 3[5],whileh >0whend=2[7]2.Thefactthath >0in c c c ≥ dimension1iseasilychecked,andhasbeenprovedlongagobyphysicists. Contrarily to the results described above, there are only very few pathwise results in thissetting,exceptindimension1,wherespecificfeatures(inparticular,anaturalrenewal structure)makesitpossibletofullydescribetheprocess[9].Beforethepresentpaper,the onlypathwiseresultsavailablearethoseof[17],andstatethat,indimension2, – Forallh sufficientlylarge, theinterface is localized, and covariances decay exponentially. – Forallh <h c,theinterfacedelocalizes,inthesensethatlimL→¥ hj 0iL+;,00,h =¥ . – Forh sufficientlysmall3, j +,0 logL. h 0iL;0,h ≍ Noticethath =0indimensions3andhigher,andthustheanalogueofthelaststatement c reducestotheentropicrepulsionestimategivenabove. Themaingoal of thepresent paper istoprovide detailedpathwiseinformation onthe localizedregimeinthewholepartialwettingregimeandinanydimensions.Moreover,we shallgivesomeinformationontherateofdivergenceoftheheightasthewettingtransition isapproached (implying among otherthings thatdivergencedoes occuralsoindimension 2). 1.2.5 Interface,attractivehard-wall,awayfromcoexistence:prewetting Finally,lettingl >0introduces anothersourceoflocalizationoftheinterface.Ofcourse, underourassumptionsonthispotential,itisnotsurprisingthatitalwaysyieldslocalization oftheinterface,whichcorresponds totheimpossibilityofgrowingalargefilmofthermo- dynamically unstablephase.Themainquestion hereistounderstand whathappens asthe systemisbroughtclosetophasecoexistence,i.e.whenl 0. Thesituationstudiedin[14,17]isthefollowing:Fix0↓ h <h ,indimension1or2,or c takeh =0indimension3andlarger.Setalsol >0.Then≤,asl 0,thesystemgetscloser ↓ andclosertotheregimeofphasecoexistence,andinthatregime,becauseofthechoicefor h , the interface is delocalized. The problem was then to determine the rate at which this delocalization takes place.Themain resultof[17]canbestatedasfollows: Foralll >0 sufficientlysmall, logl (d=2), Llim¥ hj 0i+L;,l0,h ≍(|logl |1/2 (d 3). → | | ≥ Thisresultisvalidforanyeven,convex,notidenticallyzero,externalpotentialV satisfying some mild growth condition (e.g.,any polynomial growth is fine). In dimension 1, on the otherhand,thecriticalbehaviordoesdependonthechoiceofV,see[14];inthiscase,ithas alsobeenpossibletoproveexponentialdecayofcovariances. 1.3 Newresults We consider the measure m +,0 . Let h >h and l >0. When l =0 (i.e., at phase co- L;l ,h c existence), the system is in the partial wetting regime, and the interface is expected to be 2 Actually, itisinteresting toobservethatitisalsoprovedin[7]thath c>0inanydimensions ifthe interactionterm(j i j j)2isreplacedby,say, j i j j. − | − | 3 a bmeaninghereandintherestofthispaper,thatthereexistsaconstantc>0,dependingonnothing ≍ exceptpossiblythedimension,suchthatac b a/c. ≤ ≤ 6 localized.Thenexttheoremshowsthatthisisindeedthecase.Moreover,itrelatestherate ofvanishingofthefreeenergytothedivergencerateoftheinterfaceheight,ash issentto h . c Theorem1 There exist T >0, h¯ >h , l >0, a >0 and C <¥ such that, for any 0 c 0 d d T >T ,h (h ,h¯),l (0,l )andL 1, 0 c 0 ∈ ∈ ≥ m +,0 j T logf(l ,h ) C exp a T2 logf(l ,h )2/(logT+ logf(l ,h )) , L;l ,h i≥ | | ≤ 2 − 2 | | | | ford=2(cid:0),and (cid:1) (cid:0) (cid:1) m +,0 j T logf(l ,h ) C exp a T2 logf(l ,h ) , L;l ,h i≥ | | ≤ d − d | | ford 3.Inparticu(cid:0)lar,therpeexistc <¥ s(cid:1)uchthat,fo(cid:0)rallh (h ,h¯)and(cid:1)allL 1, ≥ ′d ∈ c ≥ hj 0i+L;,00,h ≤c′2|logf(0,h )|, whend=2,while hj 0i+L;,00,h ≤c′d |logf(0,h )|, whend 3. p ≥ Remark1 Since j +,0 isnon-increasinginh andinl (byFKGinequality),itfollows, h 0iL;l ,h forexample,that sup j +,0 <¥ , l 0h 0iL;l ,h L≥1 ≥ foranyh >h andinanydimensiond 2. c ≥ Weexpect thatthe logf(0,h ) and logf(0,h ) upperbounds areofthecorrect order. | | | | Wenowstatelowerboundsofthistype. p Theorem2 Leta >1.Thereexistsh¯ >h andc =c (a )>0suchthat,forallh (h ,h¯) andallL L (h ,a ), c ′d′ ′d′ ∈ c 0 ≥ hj 0i+L;,00,h ≥c′2′|logf(0,ah )|, whend=2,while hj 0i+L;,00,h ≥c′d′ |logf(0,ah )|, whend 3. p ≥ Remark2 Althoughtheseresultsareinteresting,itwouldbemoreinformativetohaveesti- matesoftheheightthatareexpresseddirectlyintermsofthemicroscopicparameterh ,and not intermsofthefreeenergy. Todothis,oneneedstounderstand thedependence ofthe latteronh closetothewettingtransition,ataskthat seemstoohardforthemoment. Itis howeverpossibletoextractalowerbound ofthistypefromtheproofin[5],whichshows that,ford 3, ≥ f(0,h ) c1(d)e−c2/h , (3) ≥ forsomeconstants0<c ,c <¥ .This,combinedwiththeaboveestimates,impliesthat 1 2 hj 0i+L;,00,h ≤c3h −1/2, forsomeconstant c <¥ .Observethatif,aswebelieve,theestimate(3)isofthecorrect 3 order,thentherateofdivergenceoftheinterfaceheightismuchfasterthanthelogarithmic divergences seenintheresultsdescribedintheprevious subsection. This wouldofcourse beduetotheverylowdensityofpinnedsitesash getsclosetoh . c 7 Remark3 Observealsothattheintroductionoftheparametera inthelowerboundshould beirrelevant.Indeed,ifthelogarithmofthefreeenergybehaves(asindicatedbythelower boundof[5]ford 3)likeapolynomial functionof1/h ,ash h ,ourupperandlower c ≥ ↓ boundswouldactuallydifferonlybyamultiplicativeconstant. 1.4 Openproblems Eventhoughtheresultspresentedheresubstantiallyimprovethedescriptionofthewetting transitionintheseeffectivemodels,anumberofopenproblemsremain. – Itwouldalsobedesirabletoremovethefactora inTheorem2.Asremarkedabove,we expectthat thelatterplays norole,but theverificationofthishinges onthenext open problem. – Obtain information on the behavior of the free energy as a function of h close to the wettingtransition.Thisseemstoohardatthepresenttimewhend=2,buttheremight besomewaytoproveupperbounds whend 3.Atafuturestage,itwouldofcourse ≥ beextremelyinterestingtodeterminethecriticalexponentdescribingthedivergenceof theheight. – Provethatthecovariances areexponentially decaying withthedistance.Itisnot clear how this should be tackled. The only nonperturbative methods to prove this type of resultweareawareofapplyonlywhenasuitablegraphicalrepresentationisavailable (arandomwalkrepresentation,forexample).However,alltherepresentationsavailable for this model only apply to 2-point functions, not covariances, and therefore do not seemveryhelpful. – Thepresentwork dealsonlywiththepartialwettingregime. Thesituationconcerning thecompletewettingregimeisstillnotassatisfactoryaswewouldlike.Inparticular,it wouldbequitedesirabletoprovethat,inthewholecompletewettingregime(oratleast intheinteriorofthisdomain),theinterfaceheightdivergeslikelogLunderthemeasure m +,0 indimension2.Thisisonlyknowntoholdfarfromthecriticalpoint.Ofcourse, L;0,h one expects even more, namely that the fields under m +,0 and m +,0 should be very L;0,h L;0,0 close. Acknowledgements Theauthorwouldliketoexpresshiswarmestthankstoananonymousrefereeforpoint- ingoutnumerousmistakesandimprecisionsinthesubmittedversionofthiswork,andforsuggestingmany improvementstotheexposition.DiscussionswithErwinBolthausen,Jean-DominiqueDeuschelandDima Ioffearealsogratefullyacknowledged. 2 Proofs Themaintools usedintheproofs below areFKGinequality andthechessboardestimate. Bothholdforthemeasuresconsideredhere,becausetheydofortheGaussianmeasure,and areinsensitivetoperturbationoftheform(cid:213) eU(j i)(aftersuitablysmoothingourpotential– i inparticularthepositivityconstraintandthepinningpotential–andtakingweaklimits).We refer,e.g.,toAppendixBof[13]foradditional information andreferencesonthevalidity anduseofFKGinequalityinthecontextofgradientfields,andto[1]foranicereviewon reflectionpositivity(and,inparticular,thechessboardestimate).Letusalsoemphasize,to makethefollowingargumentsclearer,thatinalltheapplicationsofthechessboardestimates inthepresentwork,weareusingreflectionthroughplanesbetweenlatticesites. 8 In order to use reflection positivity, we shall need to work with periodic boundary condition. Let us quickly recall the corresponding definitions. We denote by Td the torus L Zd/(LZd).Configurations arethengivenbyj ∈RTdL.Themeasures m Lp;elr,h and m L+;l,p,ehr are definedpreciselyasbefore, m per (dj )= 1 exp 1 (cid:229) (j j )2 l (cid:229) V(j ) L;l ,h Zper −8d i− j − i L;l ,h i,j Td i Td (cid:2) i∈jL ∈ L (cid:3) ∼ (cid:213) dj +hd (dj ) , i 0 i × i Td ∈ L(cid:0) (cid:1) m +,per( )=m per ( W ), L;l ,h · L;l ,h ·| + reinterpretingi jtomeanthatiand jareneighboringverticesonTd.Noticethatforthese measurestobe∼well-defined,itisnecessarythatl >0. L WedenotebyZper andZ+,per thecorrespondingpartitionfunctions,andby L;l ,h L;l ,h fLper(l ,h )=L−dlog(ZL+;,lp,ehr/ZL+;,lp,e0r) the corresponding free energy. In the thermodynamic limit, this free energy and the one definedwith0-boundaryconditionagree. Laexneidmstismnacanred1acsFoinoingrciaindllehsl aw≥nidth0l,fh.(Ml≥,ohr0e,)o.tvheer,lifmoritalfl(ll ,>h )0,=h l≥im0L→,t¥hefLl0i(mli,thli)meLx→is¥tsfLapenrd(lis,hc)onavlseox Proof TheexistenceoflimL ¥ fL0(l ,h )follows,forexample,byFKGinequalityandcom- pletelystandardsuperadditiv→ityarguments.Itsmonotonicityinl isalsoimmediate ¶ ¶l fL0(l ,h )=L−d (cid:229) hV(j i)iL+;,l0,0−hV(j i)iL+;,l0,h ≥0, i∈L L(cid:8) (cid:9) byFKGinequality,sinceV isincreasingonR+.Tocheckthemonotonicityinh ,itsuffices to observe that h¶ /¶h f0(l ,h ) is simply the density of pinned sites, and thus positive. L Convexityfollowssimilarly,sincesecondderivativesyieldvariances. LetusdenotebyZ[A]therestrictionofthepartitionfunctionZtoconfigurationssatis- fyingtheconditionA.Withthisnotation,wehave, 12Z+L;,lp,ehr ≤Z+L;,lp,ehr j i≤V−1((2d/l )logL),∀i∈TdL ≤ZL+;,lp,ehr, (4) (cid:2) (cid:3) forallLlargeenough.Indeed, Z+L;,lp,ehr(cid:2)j i≤V−Z1+L(;,l2lpd,ehrlogL),∀i∈TdL(cid:3) =m L+;l,p,ehr(j i≤V−1(2ldlogL),∀i∈TdL), whichprovesthesecondinequality,and,byFKGinequality, m L+;l,p,ehr(j i≤V−1(2ldlogL),∀i∈TdL))≥ (cid:213) m L+;l,p,ehr(j i≤V−1(2ldlogL)). i Td ∈ L 9 ButanotherapplicationofFKG,andthechessboardestimateyield 2d 2d m L+;l,p,ehr(j i>V−1( l logL))≤m L+;l,p,e0r(j i>V−1(l logL)) ≤m L+;l,p,e0r(j j>V−1(2ldlogL),∀j∈TdL)1/|TdL| 2d ≤exp −l V(V−1(l logL)) =L−2(cid:0)d. (cid:1) Therefore, m L+;l,p,ehr(j i≤V−1((2d/l )logL),∀i∈TdL))≥e−2L−d ≥ 12, forallLlargeenough.Thisproves(4).Noticenowthatthesamealsoholdsfor0-boundary condition.Indeed,theupperboundisagaintrivial,andforthelowerbound,wecanuseFKG inequalitytoget 2d 2d m L+;l,0,h (j i≤V−1( l logL),∀i∈TdL))≥m L++,p1e;lr,h (j i≤V−1(l logL),∀i∈TdL)). It isthus sufficient tocompare thepartition function withperiodic and0-boundary conditions,undertheconstraintthatallspinssatisfyj <V 1(2dlogL).However,foraconfigu- i − l rationj onthetorussatisfyingthisconstraint,thechangeinenergyresultingfromsetting oneheightto0isboundedaboveby 1(V 1(2dlogL))2andboundedbelowby 2 − l 1 2d −2(V−1( l logL))2−2dlogL. Since ¶L =O(Ld 1),thisshowsthat,forallfixedl >0, L − | | Z+,0 j V 1(2dlogL), i Td) L−d(cid:12)(cid:12)logZ+LL+,;pl1e;,rlh ,(cid:2)h ij≤i≤V−−1l(2ldlogL)∀,∀∈i∈LTdL(cid:3) (cid:12)(cid:12)=oL(1), (cid:12) (cid:12) implying that thelim(cid:12)(cid:12)itingfreeene(cid:2)rgies coincide. Bytheabove(cid:3)(cid:12)(cid:12)considerations, this isalso true for the unrestricted partition functions and free energies. This concludes the proof of Lemma1. 2.1 Upperboundontheheight:proofofTheorem1 Inthissection,weprovetheuppertailestimatefortheheightattheorigin,andtheresulting upperboundontheheightoftheinterface. Intheproof,itwillbeconvenienttoassumefromthestartthatthefreeenergyissmall enough;moreprecisely,weshallalwaysassumethath¯ andl arechoseninsuchawaythat 0 f(l ,h¯) e 1,whichensuresthat logf(l ,h ) 1and f(l ,h ) 1/d 1,forallh (h ,h¯) 0 − − c andalll ≤ (0,l ). | |≥ ≥ ∈ 0 ∈ Let us first observe that FKG inequality implies that limL→¥ hj 0iL+;,l0,h exists in R∪ +¥ . Itisthereforesufficient forustorestrictourattentiontoboxes ofsizeL+1=2N, N{ 0},whend=2,andL=2N,N 0,whend 3.Thiswillbeusefultoensurethatthe ≥ ≥ ≥ sizesoftheblocksusedwhenapplyingthechessboardestimatedividethesizeofthetorus. 10 2.1.1 Thetwo-dimensionalcase Let us fix l >0 and h >h as above. Expanding over pinned sites (see [10,15,6], for c example),wehave m +,0 j T logf(l ,h ) L;l ,h 0≥ | | (cid:0) = (cid:229) (cid:229) (cid:1) z +,0 (A)m +,0 j T logf(l ,h ) j =0, i A , L;l ,h L;l ,0 0≥ | | i ∀ ∈ k≥1AA∩∩BBk−k6=1=0/0/ (cid:0) (cid:12)(cid:12) (cid:1) whereweusedthenotationBk= i∈TdL : kik¥ ≤k ,k≥0. ByFKGandLemma2below(providedthatT islargeenough),wecanfindc suchthat (cid:8) (cid:9) 1 m +,0 j T logf(l ,h ) j =0, i A L;l ,0 0≥ | | i ∀ ∈ (cid:0) sup(cid:12) m +,0 j T(cid:1)logf(l ,h ) j =0 ≤ (cid:12) L;0,0 0≥ | | i i ¥ =k kk (cid:0) (cid:12) (cid:1) exp(cid:12) c T2 logf(l ,h )2/logk , 1 ≤ − | | (cid:0) (cid:1) uniformlyinAsuchthatA B =0/ andA B =0/.Thisimpliesthat k 1 k ∩ − ∩ 6 m +,0 j T logf(l ,h ) (cid:229) exp c T2 logf(l ,h )2/logk (cid:229) z +,0 (A). L;l ,h 0≥ | | ≤ − 1 | | L;l ,h (cid:0) (cid:1) k≥1 (cid:0) (cid:1)A∩Bk−1=0/ But,forallk 1, ≥ Z+,0 (Bc ) (cid:229) z L+;l,0,h (A)= L;lZ,h+,0 k−1 , A∩Bk−1=0/ L;l ,h whereZ+,0 (Bc )isdefinedasin(1)butwiththepinningpotentialactingonlyonBc . L;l ,h k 1 k 1 To estimate−this last ratio, we would like to follow the idea in [2] and use reflex−ion positivityoftheGibbsmeasure(whichholds,sinceweareconsidering anearest-neighbor gradientfield,withon-sitepotentials).Ofcourse,thepinningpotentialisabitsingular,and makestheapplicationofthisinequalityawkward,sowefirstreplaceitbyitsmoreregular approximation(2).Theonlyremainingobstaclenowisthatwehave0-boundary condition insteadofperiodicboundary conditions. Toremovethisproblem, weuseoncemoreFKG inequalitytoobtain,foranyk 1, ≥ Z+L;,lZ0,,h(+Le;,l)0(,,h(Beck)−1) =hi∈(cid:213)Bk−1e−U2(he)(j i)iL+;,l0,,h(e)(Bck−1) ≥hi∈(cid:213)Bk−1e−U2(he)(j i)i+L+,p1e;rl,(,eh)(Bck−1)= ZL++,Zp1e;+Lrl,+(,,peh1e);rl(,(B,ehck)−1).