Wetting of gradient fields: pathwise estimates

0.15 MB·English

by  Yvan Velenik

ProbabilityTheoryandRelatedFieldsmanuscriptNo.
(willbeinsertedbytheeditor)
Wetting of gradient fields: pathwise estimates
YvanVelenik
8
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0
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n
a Received:date/Accepted:date
J

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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 in-
terface, 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 exponen-
tially.
– 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 condi-
tions,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).