Facilitated spin models recent and new results

ThematerialpresentedhereisanexpandedversionofaseriesoflecturesdeliveredbyF.MartinelliatthePraguesummerschool2006onMathematicalStatisticalMechanics.2

†

ThisworkwaspartiallysupportedbygdreCNRS-INdAM224,GREFIMEFI.

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N.Cancrini,F.Martinelli,C.Roberto†,,andC.Toninelli†,

1Introductionandmotivations

Considerthefollowingsimpleinteractingparticlesystem.AteachsiteofthelatticeZthereisadynamicalvariableσx,calledinthesequel“spin”,takingvaluesin{0,1}.Withrateoneeachspinattemptstochangeitscurrentvaluebytossingacoinwhichlandsheadwithprobabilityp∈(0,1)andsettingthenewvalueto1ifheadand0iftail.Howeverthewholeoperationisperformedonlyifthecurrentvalueonitsrightneighboris0.SuchamodelisknownunderthenameoftheEastmodel[18]anditiseasilycheckedtobereversiblew.r.t.theproductBernoulli(p)measure.AcharacteristicfeatureoftheEastmodelisthat,whenq:=1−p≈0,therelaxationtothereversiblemeasureisextremelyslow[12]:

1

Trelax≈(1/q)

1+eβ

ecβasβ→∞,abehaviorthatisreferedtoasasuper-Arrheniuslawinthephysicsliterature.

TheEastmodelisoneofthesimplestexamplesofageneralclassofinter-actingparticlesmodelswhichareknowninphysicalliteratureasfacilitatedorkineticallyconstrainedspinmodels(KCSM).

ThecommonfeaturetoallKCSMisthateachdynamicalvariable,oneforeachvertexofaconnectedgraphGandwithvaluesinafinitesetS,waitsanexponentialtimeofmeanoneandthen,ifthesurroundingcurrentconfigurationsatisfiesasimplelocalconstraint,isrefreshedbysamplinganewvaluefromSaccordingtosomeapriorispecifiedmeasureν.Thesemodelshavebeenintroducedinthephysicalliterature[19,20]tomodeltheliquid/glasstransitionandmoregenerallytheslow“glassy”dynamicswhichoccursindifferentsystems(see[32,10]forrecentreview).Inparticular,theyweredevisedtomimicthefactthatthemotionofamoleculeinadenseliquidcanbeinhibitedbythepresenceoftoomanysurroundingmolecules.Thatexplainswhy,inallphysicalmodels,S={0,1}(emptyoroccupiedsite)andtheconstraintsspecifythemaximalnumberofparticles(occupiedsites)oncertainsitesaroundagivenoneinordertoallowcreation/destructiononthelatter.Asaconsequence,thedynamicsbecomesincreasinglyslowasthedensityofparticles,p,isincreased.Moreoverthereusuallyexistblockedconfigurations,namelyconfigurationswithallcreation/destructionratesidenticallyequaltozero.Thisimpliestheexistenceofseveralinvariantmeasures(see[26]forasomewhatdetaileddiscussionofthisissueinthecontextoftheNorth-Eastmodel),theoccurrenceofunusuallylongmixingtimescomparedtostandardhigh-temperaturestochasticIsingmodelsandmayinducethepresenceofergodicitybreakingtransitionswithoutanycounterpartatthelevelofthereversiblemeasure[17].

2

thenTrelax≈

Facilitatedspinmodels:recentandnewresults3

Becauseofthepresenceoftheconstraintsamathematicalanalysisofthesemodelshavebeenmissingforalongtime,withthenotableexceptionoftheEastmodel[3],untilafirstrecentbreakthrough[12,13].

Inthisworkwepartlyreviewtheresultsandthetechniquesof[12]butwealsoextendthemintwodirections.Firstlyweshowthatthemaintechniquecanbeadaptedtodealwithaweakinteractionamongthevariablesobtainedbyreplacingthereversibleproductmeasurewithageneralhigh-temperatureGibbsmeasure.Secondly,motivatedbysomeunpublishedconsiderationsofD.Aldous[2],weanalyzeaspecialmodel,thesocalledFA-1fmodel,onageneralconnectedgraphandrelateitsrelaxationtimetothatoftheEastmodel.

2Themodels

2.1Settingandnotation

Themodelsconsideredherearedefinedonalocallyfinite,boundeddegree,connectedgraphG=(V,E)withvertexsetVandedgesetE.Theassociatedgraphdistancewillbedenotedbyd(·,·)andthedegreeofavertexxbydx.Thesetofneighborsofx,i.e.y∈Vsuchthatd(y,x)=1,willbedenotedbyNx.ForeverysubsetV′⊂Vwedenoteby∂V′thesetofverticesinV\\V′withoneneighborinV′.InmostcasesthegraphGwilleitherbethed-dimensionallatticeZdorafiniteportionofitandinbothcasesweneedsomeadditionalnotationthatwefixnow.Foranyvertexx∈Zdwedefinethe∗,thworientedandthe∗-orientedneighborhoodofxas

󰀐󰀐d

∗Nx={y∈Zd:y=x+i=1αiei,αi=±1,0andiα2i=0}󰀐d

Kx={y∈Nx:y=x+i=1αiei,αi≥0}

󰀐d

∗∗Kx={y∈Nx:y=x+i=1αiei,αi=1,0}

whereeiarethebasisvactorsofZd.Accordingly,theorientedand*-oriented

∗

ΛofafinitesubsetΛ⊂Zdaredefinedas∂+Λ:=neighborhoods∂+Λ,∂+

∗∗

{∪x∈ΛKx}\\Λ,∂+Λ:={∪x∈ΛKx}\\Λ.ArectangleRwillbeasetofsitesoftheform

R:=[a1,b1]×···×[ad,bd]whilethecollectionoffinitesubsetsofZdwillbedenotedbyF.2.2Theprobabilityspace

Let(S,ν)beafiniteprobabilityspacewithν(s)>0foranys∈S.G⊂SwilldenoteadistinguishedeventinS,oftenreferredtoasthesetof“goodstates”,andq≡ν(G)itsprobability.

Given(S,ν)wewillconsidertheconfigurationspaceΩ≡ΩV=SVwhoseelementswillbedenotedbyGreekletters(ω,η...).IfG′=(V′,E′)is

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N.Cancrini,F.Martinelli,C.Roberto†,,andC.Toninelli†,

asubgraphofGandω∈ΩVwewillwriteωV′foritsrestrictiontoV′.Wewillalsosaythatavertexxisgoodfortheconfigurationωifωx∈G.

OnΩequippedwewillconsidertheproduct󰀑withthenatural′σ-algebra′′

measureµ:=x∈Vνx,νx≡ν.IfG=(V,E)isasubgraphofGwewillwriteµV′orµG′fortherestrictionofµtoΩV′.Finally,foranyf∈L1(µ),wewillusetheshorthandnotationµ(f)todenoteitsexpectedvalueandVar(f)foritsvariance(whenitexists).2.3TheMarkovprocess

ThegeneralinteractingparticlemodelsthatwillbestudiedhereareGlaubertypeMarkovprocessesinΩ,reversiblew.r.t.themeasureµandcharacterizedbyafinitecollectionofinfluenceclasses{Cx}x∈V,whereCxisjustacollectionofsubsetsofV(oftenoftheneighborsofthevertexx)satisfyingthefollowinggeneralhypothesis:

Hp1Forallx∈VandallA∈CxthevertexxdoesnotbelongtoA.Hp2r:=supxsupA∈Cxd(x,A)<+∞.

InturntheinfluenceclassestogetherwiththegoodeventGarethekeyingredientstodefinetheconstraintsofeachmodel.

Definition1.Givenavertexx∈Vandaconfigurationω,wewillsaythattheconstraintatxissatisfiedbyωiftheindicator

󰀉

1ifthereexistsasetA∈Cxsuchthatωy∈Gforally∈A

cx(ω)=

0otherwise

(1)

isequaltoone.

Remark1.Thetwogeneralhypothesesabovetellusthatinordertocheckwhethertheconstraintissatisfiedatagivenvertexwedonotneedtocheckthecurrentstateofthevertexitselfandweonlyneedtochecklocallyaroundthevertex.Thislastrequirementcanactuallybeweakenedandindeed,inordertoanalyzecertainspinexchangekineticallyconstrainedmodels[11],averyefficienttoolistoconsiderlongrangeconstraints!

Theprocessthatwillbestudiedinthesequelcanthenbeinformallydescribedasfollows.Eachvertexxwaitsanindependentmeanoneexponentialtimeandthen,providedthatthecurrentconfigurationωsatisfiestheconstraintatx,thevalueωxisrefreshedwithanewvalueinSsampledfromνandthewholeprocedurestartsagain.

ThegeneratorLoftheprocesscanbeconstructedinastandardway(seee.g.[27,26])anditisanon-positiveself-adjointoperatoronL2(Ω,µ)withdomainDom(L)andDirichletformgivenby

Facilitatedspinmodels:recentandnewresults5

D(f)=

x∈V

󰀒󰀂󰀒󰀁2

HereVarx(f)≡dν(ωx)f2(ω)−dν(ωx)f(ω)denotesthelocalvariancewithrespecttothevariableωxcomputedwhiletheothervariablesareheldfixed.TothegeneratorLwecanassociatetheMarkovsemigroupPt:=etLwithreversibleinvariantmeasureµ.

Noticethattheconstraintscx(ω)areincreasingfunctionsw.r.tthepartial

′

orderinΩforwhichω≤ω′iffωx∈Gwheneverωx∈G.HoweverthatdoesnotimplyingeneralthattheprocessgeneratedbyLisattractiveinthesenseofLiggett[27].

Duetothefactthatingeneralthejumpratesarenotboundedawayfromzero,irreducibilityoftheprocessisnotguaranteedandthereversiblemeasureµisusuallynottheonlyinvariantmeasure(typicallythereexistinitialconfigurationsthatareblockedforever).AninterestingquestionwhenGisinfiniteisthereforewhetherµisergodic/mixingfortheMarkovprocessandwhetherthereexistotherergodicstationarymeasures.Tothispurposeitisusefultorecallthefollowingwellknownresult(seee.g.Theorem4.13in[27]).

Theorem2.1Thefollowingareequivalent,(a)limt→∞Ptf=µ(f)inL2(µ)forallf∈L2(µ).(b)0isasimpleeigenvalueforL.

Clearly(a)impliesthatlimt→∞µ(fPtg)=µ(f)µ(g)foranyf,g∈L2(µ),i.e.µismixing.

Remark2.Evenifµismixingtherewillexistingeneralinfinitelymanysta-tionarymeasures,i.e.probabilitymeasuresµ˜satisfyingµ˜Pt=µ˜forallt≥0.Asanexample,assumecxnotidenticallyequal󰀂󰀁tooneandtakeanarbitraryVprobabilitymeasureµ˜suchthatµ˜{S\\G}=1.Aninterestingproblemisthereforetoclassifyallthestationaryergodicmeasuresµ˜of{Pt}t≥0,whereergodicitymeansthatPtf=f(˜µa.e.)forallt≥0impliesthatfisconstant(˜µa.e.).Aswewillseelater,whenG=Z2andforaspecificchoiceoftheconstraintknownastheNorth-Eastmodel,aratherdetailedanswerisnowavailable[26].

WhenGisfiniteconnectedsubgraphofaninfinitegraphG∞=(V∞,E∞),theergodicityissueoftheresultingcontinuoustimeMarkovchaincanbeattackedintwoways.

Thefirstoneistoanalyzethechainrestrictedtoasuitablydefinedergodiccomponent.Althoughsuchanapproachisfeasibleandnaturalinsomecases(seesection6foranexample),thewholeanalysisbecomesquitecumbersome.

Anotherpossibility,whichhasseveraltechnicaladvantagesoverthefirstone,istounblockcertainspecialverticesofGbyrelaxingtheirconstraintsandrestoreirreducibilityofthechain.Anaturalwaytodothatistoimaginetoextendtheconfigurationω,aprioridefinedonlyinV,totheverticesin

󰀖

µ(cxVarx(f)),f∈Dom(L)

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N.Cancrini,F.Martinelli,C.Roberto†,,andC.Toninelli†,

V∞\\Vandtokeepittherefrozenandequaltosomereferenceconfigurationτthatwillbereferredtoastheboundarycondition.IfenoughverticesinV∞\\Varegoodforτ,thenenoughverticesofGwillbecomeunblockedandthewholechainergodic.

Morepreciselywecandefinethefinitevolumeconstraintswithboundaryconditionτas

cτ(2)x,V(ω):=cx(ω·τ)wherecxaretheconstraintsforG∞definedin(1)andω·τ∈Ωdenotes

theconfigurationequaltoωinsideVandequaltoτinV∞\\V.Noticethat,foranyx∈V,theratecτx,V(ω)(2)dependsonτonlythroughtheindicators{1Iτz∈G}z∈B,whereBistheboundarysetB:=(V∞\\V)∩(∪z∈VCz).Therefore,insteadoffixingτ,itisenoughtochooseasubsetM⊂B,calledthegoodboundaryset,anddefine

τ

cM(3)x,V(ω):=cx,V(ω)whereτisanyconfigurationsatisfyingτz∈Gforallz∈Mandτz∈/Gfor

z∈B\\M.WewillsaythatachoiceofMisminimalifthecorrespondingchaininGwiththerates(3)isirreducibleanditisnon-irreducibleforanyotherchoiceM′⊂M.ThechoiceM=Bwillbecalledmaximal.ForconveniencewewillwriteLmax(LminΛΛ)forthecorrespondinggenerators.Remark3.Withoutanyotherspecificationfortheinfluenceclassesofthemodelitmayverywellbethecasethatthereexistsnoboundaryconditionsforwhichthechainisirreducibleand/ortheirexistencemaydependonthechoiceofthefinitesubgraphG.However,aswewillseelater,foralltheinterestingmodelsdiscussedintheliteraturealltheseissueswillhavearathersimplesolution.

Wewillnowdescribesomeofthebasicmodelsandsolvetheproblemofboundaryconditionsforeachoneofthem.2.40-1Kineticallyconstrainedspinmodels

Inmostmodelsconsideredinthephysicalliteraturethefiniteprobabilityspace(S,ν)isasimple{0,1}BernoullispaceandthegoodsetGisconventionallychosenastheempty(vacant)state{0}.Anymodelwiththesefeatureswillbecalledinthesequela“0-1KCSM”(kineticallyconstrainedspinmodel).AlthoughinmostcasestheunderlyinggraphGisaregularlatticelikeZd,wheneverispossiblewewilltrytoworkinfullgenerality.

Givena0-1KCSM,theparameterq=ν(0)canbevariedin[0,1]whilekeepingfixedthebasicstructureofthemodel(i.e.thenotionofthegoodsetandtheconstraintscx)anditisnaturaltodefineacriticalvalueqcas

qc=inf{q∈[0,1]:0isasimpleeigenvalueofL}

Facilitatedspinmodels:recentandnewresults7

Aswewillprovebelowqccoincideswiththebootstrappercolationthresholdqbpofthemodeldefinedasfollows[34]5.Foranyη∈ΩdefinethebootstrapmapT:Ω→Ωas

(Tη)x=0

ifeither

ηx=0

orcx(η)=1.

(4)

Denotebyµ(n)theprobabilitymeasureonΩobtainedbyiteratingn-timestheabovemappingstartingfromµ.Asn→∞µ(n)convergestoalimitingmeasureµ(∞)[34]anditisnaturaltodefinethecriticalvalueqbpas

qbp=inf{q∈[0,1]:µ∞=δ0}

whereδ0istheprobabilitymeasureassigningunitmasstotheconstantcon-figurationidenticallyequaltozero.Inotherwordsqbpistheinfimumofthevaluesqsuchthat,withprobabilityone,thegraphGcanbeentirelyemptied.Usingthefactthatthecx’sareincreasingfunctionofηitiseasytocheckthatµ(∞)=δ0foranyq>qbp.

Proposition2.2([12])qc=qbpandforanyq>qc0isasimpleeigenvalueforL.

Remark4.In[12]thepropositionhasbeenprovedinthespecialcaseG=Zdbutactuallythesameargumentsapplytoanyboundeddegreeconnectedgraph.

Havingdefinedthebootstrappercolationitisnaturaltodividethe0-1KCSMintotwodistinctclasses.

Definition2.Wewillsaythata0-1KCSMisnoncooperativeifthereexistsafinitesetB⊂VsuchthatanyconfigurationηwhichisemptyinallthesitesofBreachestheemptyconfiguration(all0’s)underiterationofthebootstrapmapping.Otherwisethemodelwillbecalledcooperative.

Remark5.Noticethatforanon-cooperativemodelthecriticalvalueqcisobviouslyzerosincewithµ-probabilityoneaconfigurationwillcontaintherequiredfinitesetBofzeros.

Wewillnowillustratesomeofthemoststudiedmodels.

[1]Frederickson-Andersen(FA-jf)facilitatedmodels[19,20].Inthefacilitatedmodelstheconstraintatxrequiresthatatleastj≤dxneighborsarevacant.Moreformally

Cx={A⊂Nx:|A|≥j}

Whenj=1themodelisnon-cooperativeforanyconnectedgraphGandergodicityoftheMarkovchainisclearlyguaranteedbythepresenceofat

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N.Cancrini,F.Martinelli,C.Roberto†,,andC.Toninelli†,

leastoneunblockedvertex.Whenj>1ergodicityonageneralgraphismoredelicateandwerestrictourselvestofiniterectanglesRinZd.Inthatcaseandforthemostconstrainedcooperativecasej=damongtheirreducibleones,irreducibilityisguaranteedifweassumeaboundaryconfigurationidenticallyemptyon∂+R.Quiteremarkably,usingresultsfrombootstrappercolation[34]combinedwithproposition2.2,whenG=Zdand2≤j≤dtheergodicitythresholdqcalwaysvanishes.

[2]Spiralmodel[9,8]ThismodelisdefinedonZ2withthefollowingchoicefortheinfluenceclasses

Cx={NEx∪SEx;SEx∪SWx;SWx∪NWx;NWx∪NEx}

whereNEx=(x+e2,x+e1+e2),SEx=(x+e1,x+e1−e2),SWx=(x−e2,x−e2−e1)andNWx=(x−e1;x−e1+e2).InotherwordsthevertexxcanflipiffeitheritsNorth-East(NEx)oritsSouth-West(SWx)neighbours(orbothofthem)areemptyandeitheritsNorth-West(NWx)oritsSouth-East(SEx)neighbours(orbothofthem)areemptytoo.Themodelisclearlycooperativeandin[8]ithasbeenproventhatitscriticalpointqc

o

coincideswith1−poc,wherepcisthecriticaltresholdfororientedpercolation.Theinterestofthismodelliesonthefactthatitsbootstrappercolationisexpectedtodisplayapeculiarmixeddiscontinuous/criticalcharacterwhichmakesitrelevantasamodelfortheliquidglassandmoregeneraljammingtransitions[9,8].

[3]Orientedmodels.Orientedmodelsaresimilartothefacilitatedmodelsbuttheneighborsofagivenvertexxthatmustbevacantinorderforxtobecomefreetoflip,arechosenaccordingtosomeorientationofthegraph.Insteadoftryingtodescribeaverygeneralsettingwepresentthreeimportantexamples.

Example1.ThefirstandbestknownexampleisthesocalledEastmodel[18].HereG=Zandforeveryx∈ZtheinfluenceclassCxconsistsofthevertexx+1.Inotherwordsanyvertexcanflipiffitsrightneighborisempty.Theminimalboundaryconditionsinafiniteintervalwhichensureirreducibilityofthechainareofcourseemptyrightboundary,i.e.therightmostvertexisalwaysunconstrained.Themodelisclearlycooperativebutqc=0sinceinordertoemptyZitisenoughtostartfromaconfigurationforwhichanysitexhassomeemptyvertextoitsright.OnecouldeasilygeneralizethemodeltothecasewhenGisarootedtree(seesection6).Inthatcaseanyvertexdifferentfromtherootcanbeupdatediffitsancestorisempty.Therootitselfisunconstrained.

Example2.ThesecondexampleistheNorth-EastmodelinZ2[25].HereonechoosesCxastheNorthandEastneighborofx.Themodelisclearlycoop-o

erativeanditscriticalpointqccoincideswith1−poc,wherepcisthecritical

Facilitatedspinmodels:recentandnewresults9

thresholdfororientedpercolationinZ2[34].ForsuchamodelmuchmorecanbesaidaboutthestationaryergodicmeasuresoftheMarkovsemigroupPt.Theorem2.3([26])Ifq3Quantitiesofinterestandrelatedproblems

Backtothegeneralmodelwenowdefinetwomainquantitiesthatareofmathematicalandphysicalinterest.

ThefirstoneisthespectralgapofthegeneratorL,definedas

gap(L):=

f=const

inf

D(f)

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N.Cancrini,F.Martinelli,C.Roberto†,,andC.Toninelli†,

1.WhatisthebehaviorofF(t)forlargetimescales?

2.Fora0-1KCSMisitthecasethatF(t)decaysexponentiallyfastast→∞

′

foranyq>qc?

3.Iftheanswertothepreviousquestionispositive,isthedecayraterelatedtothespectralgapinasimplewayorthedecayrateofF(t)requiresadeeperknowledgeofthespectraldensityofL?

4.Isitpossibletoexhibitexamplesof0-1KCSMinwhichthepersistencefunctionshowsacrossoverbetweenastretchedandapureexponentialdecay?Unfortunatelytheabovequestionsarestillmostlyunansweredexceptforthefirsttwo.

3.1Someusefulobservationstoboundthespectralgap

Itisimportanttoobservethefollowingkindofmonotonicitythatcanbeexploitedinordertoboundthespectralgapofonemodelwiththespectralgapofanotherone.

′

Definition3.SupposethatwearegiventwoinfluenceclassesC0andC0,de-′notebycx(ω)andc′x(ω)thecorrespondingratesandbyLandLtheassociatedgeneratorsonL2(µ).If,forallω∈Ωandallx∈V,c′x(ω)≤cx(ω),wesaythatLisdominatedbyL′.

Remark7.Thetermdominationherehasthesamemeaningithasinthecontextofbootstrappercolation.ItmeansthattheKCSMassociatedtoL′ismoreconstrainedthantheoneassociatedtoL.

Clearly,ifLisdominatedbyL′,D′(f)≤D(f)andthereforegap(L′)≤gap(L).

Example4.AssumethatthegraphGhasnverticesandcontainsaHamiltonpathΓ={x1,x2,...,xn},i.e.d(xi+1,xi)=1forall1≤i≤n−1andxi=xjforalli=j.ConsidertheFA-1fmodelonGwithonespecialvertex,e.g.xn,unconstrained(cxn≡1).Then,ifwereplaceGbyΓequippedwithitsnaturalgraphstructureandwedenotebyLandL′therespectivegenerators,wegetthatgap(L)≥gap(L′).ClearlyL′describestheFA-1fmodelonthefiniteinterval[1,...,n]⊂Zwiththelastvertexfreetoflip.ThisinturnisdominatedbyLEast,thegeneratoroftheEastmodelon[1,...,n],whichisknowntohaveapositive[3,12]spectralgapuniformlyinn.Thereforethelatterresultholdsalsoforgap(L′)andgap(L).

Example5.AlongthelinesofthepreviousexamplewecouldlowerboundthespectralgapoftheFA-2fmodelinZd,d≥2,withthatinZ2,byrestrictingthesetsA∈C0toe.g.the(e1,e2)-plane.

Facilitatedspinmodels:recentandnewresults11

Foralastandmoredetailedexampleofthecomparisontechniquewereferthereadertosection6.

Althoughthecomparisontechniquecanbequiteeffectiveinprovingposi-tivityofthespectralgap,oneshouldkeepinmindthat,ingeneral,itprovidesquitepoorbounds,particularlyinthelimitingcaseq↓qc.

Thesecondobservationwemakeconsistsinrelatinggap(L)whentheun-derlyinggraphisinfinitetoitsfinitegraphanalogue.Fixr∈VandletGn,r⊂Gbetheconnectedballcenteredatrofradiusn.Supposethatinfngap(LmaxGn,r)>0.Itistheneasytoconcludethatgap(L)>0.

Indeed,followingLiggettCh.4[27],foranyf∈Dom(L)withVar(f)>0pickfn∈L2(Ω,µ)dependingonlyonfinitelymanyspinssothatfn→fandLfn→LfinL2.ThenVar(fn)→Var(f)andD(fn)→D(f).Butsincefndependsonfinitelymanyspins

Var(fn)=VarGm,r(fn)andD(fn)=DGm,r(fn)

providedthatmisalargeenoughsquare(dependingonfn).Therefore

D(f)

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N.Cancrini,F.Martinelli,C.Roberto†,,andC.Toninelli†,

Withtheabovenotationthefirstmainresultof[12]canbeformulatedasfollows.

Theorem4.1Thereexistsauniversalconstantǫ0∈(0,1)suchthat,ifthereexistsℓandaǫ0-goodsetGℓonscaleℓ,theninfΛ∈Fgap(LmaxΛ)>0.Inpar-ticulargap(L)>0.

󰀐{y:y=x+di=1αiei,αi≥0}.IfthisisnotthecaseoneshouldinsteaduseanonrectangulargeometryforthetilesofthepartitionofZd,adaptedtothechoiceoftheinfluenceclasses.ForexamplefortheSpiralModelthebasictileatlenghtscaleℓisaquadrangularregionR0withonesideparalleltoe1andtwosidesparalleltoe1+e2,R0:=∪ℓ1S0+(i−1)(e1+e2)with

2

S0:={x∈Z:0≤x1≤ℓ−1,x2=0}.Inthiscasecondition(b)shouldalso

∗󰀆∗Λ0:=e1,e1−e2,−e2.bemodifiedbysubstitutingeverywhere∂+Λ0with∂+

Inseveralexamples,e.g.theFA-jfmodels,thenaturalcandidatefortheevent

GℓistheeventthatthetileΛ0is“internallyspanned”,anotionborrowedfrombootstrappercolation[1,34,14,24,15]:

Definition5.WesaythatafinitesetΓ⊂Zdisinternallyspannedbyaconfigurationη∈Ωif,startingfromtheconfigurationηΓequaltooneoutsideΓandequaltoηinsideΓ,thereexistsasequenceoflegalmovesinsideΓwhichconnectsηΓtotheconfigurationidenticallyequaltozeroinsideΓandidenticallyequaltooneoutsideΓ.

OfcoursewhetherornotthesetΛ0isinternallyspannedforηdependsonlyontherestrictionofηtoΛ0.Oneofthemajorresultsinbootstrappercolationproblemshasbeentheexactevaluationoftheµ-probabilitythattheboxΛ0isinternallyspannedasafunctionofthelengthscaleℓandtheparameterq[24,34,14,15,1].Fornon-cooperativemodelsitisobviousthatforanyq>0suchprobabilitytendsveryrapidly(exponentiallyfast)tooneasℓ→∞,sincetheexistenceofatleastonecompletelyemptyfinitesetB+x⊂Λ0(seedefinition2),allowstoemptyallΛ0.Forsomecooperativesystemslikee.g.theFA-2finZ2,ithasbeenshownthatforanyq>0suchprobabilitytendsveryrapidly(exponentiallyfast)tooneasℓ→∞andthatitabruptlyjumpsfrombeingverysmalltobeingclosetooneasℓcrossesacriticalscaleℓc(q).Inmostcasesthecriticallengthℓc(q)divergesveryrapidlyasq↓0.Therefore,forsuchmodelsandℓ>ℓc(q),onecouldsafelytakeGℓasthecollectionofconfigurationsηsuchthatΛ0isinternallyspannedforη.Wenowformalizewhatwejustsaid.

Corollary4.2Assumethatlimℓ→∞µ(Λ0isinternallyspanned)=1andthattheMarkovchaininΛ0withzeroboundaryconditionson∪x∈K∗Λℓxis0ergodic.Thengap(L)>0.

Westressthatforsomemodelsanotionofgoodeventwhichdiffersfromrequiringinternalspanningisneeded.ThisisthecasefortheN-EandSpiralmodels,ascanbeimmediatelyseenbynoticingthatatanylengthscaleit

Facilitatedspinmodels:recentandnewresults13

ispossibletoconstructsmallclustersofparticlesinpropercornersofthetilesthatcanneverbeerasedbyinternalmoves.Thechoiceoftheproperǫ-goodsetofconfugurationsforN-Ehasalreadybeendiscussedin[12].FortheSpiralModelthedefinitionwhichnaturallyarisesfromtheresultsin[9]

󰀆0betheregionobtainedfromR0bysubtractingtwoisthefollowing.LetR

properquadrangularregionsatthebottomleftandtoprightcorners,namely󰀆0:=R0\\(Rbl∪Rtr)whereRbl(Rtr)havethesameshapeofR0shrinkedR

atlengthscaleℓ/4andhavethebottomleft(topright)cornerwhichcoincideswiththeoneofR0.Theǫ-goodsetofconfigurationsonscaleℓ,Gℓ,includesallconfigurationsηsuchthatthereexistsasequenceoflegalmovesinsideR0whichconnectsηR0(theconfigurationwhichhasallonesoutsideR0and

󰀆0.Lemmaequalsηinside)toaconfigurationidenticallyequaltozeroinsideR

4.7andProposition4.9of[9]prove,respectively,property(a)and(b)of

∗

Definition4(with∂+Λ0substitutedwithe1,e1−e2,−e2,seeremark8)whenthedensityisbelowthecriticaldensityoforientedpercolation.Thus,usingthisdefinitionforthegoodeventandTheorem4.1weconcludethatTheorem4.3gap(Lspiral)>0atanyρThesecondmainresultconcernsthelongtimebehaviorofthepersistencefunctionF(t)definedin(6).

Theorem4.4Assumethatgap(L)>0.ThenF(t)≤e−qgapt+e−pgapt.Remark9.Theabovetheoremsdisprovesomeconjectureswhichappearedinthephysicsliterature[21,23,5,6],basedonnumericalsimulationsandapproximateanalyticaltreatments,ontheexistenceofasecondcriticalpoint′qc>qcatwhichthespectralgapvanishesand/orbelowwhichF(t)woulddecayinastretchedexponentialform≃exp(−t/τ)βwithβ<1.

Theorem4.4alsoindicatesthatonecanobtainupperboundsonthespec-tralgapbyprovinglowerboundsonthepersistencefunction.Concretelyalowerboundonthepersistencefunctioncanbeobtainedbyrestrictingtheµ-averagetothoseinitialconfigurationsηforwhichtheoriginisblockedwithhighprobabilityforalltimess≤t.UnfortunatelyinmostmodelssuchastrategyleadstolowerboundonF(t)whichareusuallyquitefarfromtheaboveupperboundanditisaninterestingopenproblemtofindanexactasymptoticast→∞ofF(t).

FinallyweobservethatfortheNorth-EastmodelonZ2atthecriticalvalueq=√qcthespectralgapvanishesandthepersistencefunctionsatisfies󰀒∞

dtF(0

14

N.Cancrini,F.Martinelli,C.Roberto†,,andC.Toninelli†,

d

spaceS={0,1}ℓ,goodeventG:=Gℓ,singlesitemeasuretherestrictionofµtoSandrenormalizedconstraints{crenx}x∈Zd(ℓ)whichareastrengtheningoftheNorth-Eastonesnamely

crenx(η)=1

∗

iffηy∈Gforally∈Kx.

Suchamodelisreferredtoin[12]asthe*-generalmodel.Byassumption

theprobabilityofGcanbemadearbitrarilyclosetoonebytakingℓlargeenoughandtherefore,bythesocalledBisection-ConstrainedapproachwhichisdetailedinthenextsectionforthecasewhenµisahightemperatureGibbsmeasure,thespectralgapofthe*-generalmodelispositive.Nextoneobservesthatassumption(b)ofthetheoremisthereexactlytoallowonetoreconstructanylegalmoveofthe*-generalmodel,i.e.afullupdateofanentireblockofspins,bymeansofafinite(dependingonlyonℓ)sequenceoflegalmovesfortheoriginal0-1KCMS.Itisthenaneasystep,usingstandardpathtechniquesforcomparingtwodifferentMarkovchains(seee.g.[33]),togofromthePoincar´einequalityforthe*-generalmodeltothePoincar´einequalityfortheoriginalmodel.

Theproofof(aslightlylesspreciseversionof)Theorem4.4givenin[12]isbasedontheFeynman-Kacformulaandstandardlargedeviationconsider-ations.Howeveritispossibletoprovideasimplerandmorepreciseargumentasfollows.OnefirstobservethatF(t)=F1(t)+F0(t)where

󰀏

η

(s)=1foralls≤t)F1(t)=dµ(η)P(σ0andsimilarlyforF0(t).ConsidernowF1(t),thecaseofF0(t)beingsimilar,

anddefineTA(η)asthehittingtimeofthesetA:={η:η0=0}startingfromtheconfigurationη.Then(seee.g.Theorem2in[4])

󰀇󰀅

F1(t)=PµTA>t≤e−tλA

NoticethatforanyfasaboveVar(f)≥µ(A)=q.ThereforeλA≥qgapandtheproofiscomplete.

4.2Asymptoticsofthespectralgapneartheergodicitythreshold.Animportantquestion,particularlyinconnectionwithnumericalsimulationsornon-rigorousapproaches,isthebehaviorneartheergodicitythresholdqcofthespectralgapforeachspecificmodel.Hereisasetofresultsprovenin[12].

wherePµdenotestheprobabilityovertheprocessstartedfromtheequilibriumdistributionµandλAisgivenbythevariationalformulafortheDiricheltproblem󰀓󰀔

2

λA:=infD(f):µ(f)=1,f≡0onA(7)

Facilitatedspinmodels:recentandnewresults15

EastModel.

q→0

limlog(1/gap)/(log(1/q))2=(2log2)

−1

(8)

FA-1f.Foranyd≥1,thereexistsaconstantC=C(d)suchthatforanyq∈(0,1),thespectralgaponZdsatisfies:

C−1q3≤gap(L)≤Cq3

C−1q2/log(1/q)≤gap(L)≤Cq2

ford=1,ford=2,

C−1q2≤gap(L)≤Cq1+

2

q

󰀂(λ1󰀎−1

exp(−c/q5)≤gap(L)≤exp−

−ǫ)

󰀁d≥3

16

N.Cancrini,F.Martinelli,C.Roberto†,,andC.Toninelli†,

˜(t)isthefinitevolumepersistencefunction.IntegratingovertwhereF

andusingthemonoticityofthegap(see[12,Lemma2.11])giveE(T)≤e2c(qgap(LΛq)))−1≤e2c(qgap(L))−1.This,inviewofTheorem4.1,isin-compatiblewiththeassumedscalingqlog2(q).

MoreoveronecanobtainalowerboundonE(T)asfollows.LetλbesuchthatP(T≥λ)=e−1thenclearlyP(T≥t)≤e−⌊t/λ⌋andE(T)≥e−1λ.Wecanalwayscoupleinthenaturalwaytwocopiesoftheprocess,onestartedfromallonesandtheotherfromanyotherconfigurationη,andconcludethat

P(thetwocopieshavenotcoupledattimet)≤P(T≥t)≤e1−λt.Standardargumentsgiveimmediatelythatgap−1≤λi.e.E(T)≥e−1gap−1.Inconclusion

󰀄−1󰀄−1󰀃󰀃

≤E(T)≤e2cqgap(LΛq)e−1gap(LΛq)

5Extensiontointeractingmodels

Inthissectionweshowhowtoextendtheresultsonthepositivityofthe

spectralgapfor0-1KCSMonaregularlatticeZdtothecaseinwhichaweakinteractionispresentamongthespins.Webeginbydefiningwhatwemeanbyaninteraction.

Definition6.AfiniterangeinteractionΦisacollectionΦ:={ΦΛ}Λ∈Fwhere

WewillsaythatΦ∈BM,rifr(Φ)≤rand󰀰Φ󰀰≤M.

i)ΦΛ:ΩΛ→RforeveryΛ∈F;

ii)ΦΛ=0ifdiam(Λ)≥rforsomefiniter=r(Φ)calledtherangeoftheinteraction;󰀐

iii)󰀰Φ󰀰≡supx∈ZdΛ∋x󰀰ΦΛ󰀰∞<∞;

GivenaninteractionΦ∈Br,MandΛ∈F,wedefinetheenergyinΛofaspinconfigurationσ∈Ωby

󰀖

ΦA(σ)HΛ(σ)=

A∩Λ=∅

τ

(σ):=HΛ(σ·τ)whereσ·τdenotesForσ∈ΩΛandτ∈ΩΛcwealsoletHΛ

theconfigurationequaltoσinsideΛandtoτoutsideit.Finally,foranyΛ∈Fandτ∈ΩΛc,wedefinethefinitevolumeGibbsmeasureonΩΛwithboundaryconditionsτandapriorisinglespinmeasureνbytheformula

µΦ,τΛ(σ):=

1

Facilitatedspinmodels:recentandnewresults17

ThekeypropertyofGibbsmeasuresisthat,foranyV⊂ΛandanyξinΛ\\V,theconditionalGibbsmeasureinΛwithboundaryconditionsτgivenξcoincideswiththeGibbsmeasureinVwithboundaryconditionτΛc·ξ.Moreformally

Φ,τΛc·ξ

µΦ,τ(·)Λ(·|σVc=ξ)=µVClearlyaveragesw.r.t.µΦ,τΛ(·|σVc=ξ)arefunctionofξand,whenevercon-fusiondoesnotarise,wewillsystematicallydropξfromournotation.

Asitiswellknown(seee.g.[35]),foranyr<∞thereexistsM0>0suchthatforany0Λ↑Zd

Φ

limµΦ,τΛ=µ

wherethelimitistobeunderstoodasaweaklimit.Moreoverthelimitis

reached“exponentiallyfast”inthestrongestpossiblesense.Namely,foranyd⊂Λ∈Fandanytwoboundaryconditionsτ,τ′,

󰀋µΦ,τ′(σ)󰀋dmax󰀋Λ

σd

18

N.Cancrini,F.Martinelli,C.Roberto†,,andC.Toninelli†,

concretecaseoftheNorth-Eastmodelintroducedinsection2.4.Moreover,inordernottoobscurethediscussionwithrenormalizationorblockconstruc-tions,wewillmaketheunnecessaryassumptionthatthebasicparameterqofthereferencemeasureνisveryclosetoone.

Theorem5.1Let{cx}x∈Z2bethoseoftheNorth-Eastmodel.Thereexistsq0∈(0,1)andforanyr<∞thereexistsM1suchthat,foranyMΦ∈Br,M

inf

gap(LΦ)>0

Remark12.Aswewillseeintheproofofthetheorem,therestrictiononstrengthoftheinteractioncomesfromtwodifferentrequirements.ThefirstoneisthatthefinitevolumeGibbsmeasurehastheverystrongmixingprop-ertyuniformlyintheboundaryconditionsgivenin(10).That,aswepointedoutpreviously,isguaranteedaslongasMTheorem5.2Let{cx}x∈ZbethoseoftheEastmodel.Foranyfinitepair(r,M)

infgap(LΦ)>0

Φ∈Br,M

Proof(ofTheorem5.1).Wewillfollowthepatternoftheproofforthenoninteractingcasegivenin[12]andwewillestablishthestrongerresult

supγ(Λ)<+∞,

whereγ(Λ):=

󰀅

Φ∈Br,M

Λ∈F

inf

infgap(LΦ,τΛ)τ∈MaxΛ

󰀇−1

(11)

providedthatq>q0islargeandMistakensufficientlysmall.AboveMaxΛ

denotesthesetofconfigurationsinΩΛcwhichareidenticallyequaltozeroon∗∂+Λ.Inwhatfollowsinordertosimplifythenotationwewillnotwritethedependenceontheboundaryconditionofthetransitionrates.

Asin[12]thefirststepconsistsinprovingacertainmonotonicitypropertyofγ(Λ).

Lemma5.3ForanyV⊂Λ∈F,

0<γ(V)≤γ(Λ)<∞

Proof(ProofoftheLemma).FixΦ∈Br,Mand,foranyξ∈MaxV,definethenewinteractionΦξasfollows:

Facilitatedspinmodels:recentandnewresults19

ΦξA(σA)

=

󰀉

Noticethat,byconstruction,

r(Φ)≤r(Φ)

ξ

0

󰀐

A′:A′∩V=A

ΦA′(σA·ξA′\\A)

󰀖

ifA∩Vc=∅ifA⊂V

andsup

x

A∋x

󰀰ΦξA󰀰∞≤󰀰Φ󰀰∞

sothatΦξ∈Br,M.NextobservethattheGibbsmeasureonΛwithinteraction

Φξissimplytheproductmeasure

Φ,ξ

µΦΛ(σΛ):=µV(σV)⊗νΛ\\V(σΛ\\V)onΩΛ=ΩV⊗ΩΛ\\V

Φ,τ

Thus,foranyf∈L2(ΩV,µΦ,ξ≡V)andτ∈MaxΛ,wecanwrite(VarΛVarµΦ,τ)

Λ

ξ

Φ

VarΦ,ξV(f)=VarΛ

ξ

,τ

Φ

(f)≤γ(Λ)DΛ

ξ

,τ

Φ,ξ

(f)≤γ(Λ)DV(f)

where,inthelastinequality,weusedthefactthat,foranyx∈Vandany

ω∈ΩΛ,cx,Λ(ω)≤cx,V(ω)becauseξ∈MaxV,togetherwith

VarΦΛ

⊓⊔

ThankstoLemma5.3weneedtoprove(11)onlywhenΛrunsthroughall

possiblerectangles.Forthispurposeourmainingredientwillbethebisectiontechniqueof[28]which,initsessence,consistsinprovingasuitablerecursionrelationbetweenspectralgaponscale2LwiththatonscaleL,combinedwiththenovelideaofconsideringanacceleratedblockdynamicswhichisitselfconstrained.Suchanapproachisreferredtoin[12]astheBisection-ConstrainedorB-Capproach.

Inordertopresentitwefirstneedtorecallsomesimplefactsfromtwodimensionalpercolation.

Apathisacollection{x0,x1,...,xn}ofdistinctpointsinZ2suchthatd(xi,xi+1)=1foralli.A∗-pathisacollection{x0,x1,...,xn}ofdistinct

∗

foralli.GivenarectangleΛandadirectionpointsinZ2suchthatxi+1∈Nxi

ei,wewillsaythatapath{x0,...,xn}traversesΛintheith-directionif{x0,...,xn}⊂Λandx0,xnlayonthetwooppositesidesofΛorthogonaltoei.

Definition7.GivenarectangleΛandaconfigurationω∈ΩΛ,apath{x0,...,xn}iscalledatop-bottomcrossing(left-rightcrossing)ifittra-versesΛinthevertical(horizontal)directionandωxi=0foralli=0,...,n.Therightmost(lower-most)suchcrossings(see[22]page317)willbedenotedbyΠω

ξ

,τ

(f|{σy}y=x)=VarΦ,ξV(f|{σy}y=x).

20

N.Cancrini,F.Martinelli,C.Roberto†,,andC.Toninelli†,

Remark13.GivenarectangleΛandapathΓtraversingΛine.g.theverticaldirection,letΛΓconsistsofallthesitesinΛwhichareinΓortotherightofit.Then,asremarkedin[22],theevent{ω:Πω=Γ}dependsonlyonthevariablesωxwithx∈ΛΓ.

Wearenowreadytostarttheactualproofofthetheorem.Atthebeginningthemethodrequiresasimplegeometricresult(see[7])whichwenowdescribe.

Letlk:=(3/2)k/2,andletFkbethesetofallrectanglesΛ⊂Z2which,modulotranslationsandpermutationsofthecoordinates,arecontainedin[0,lk+1]×[0,lk+2].ThemainpropertyofFkisthateachrectangleinFk\\Fk−1canbeobtainedasa“slightlyoverlappingunion”oftworectanglesinFk−1.Lemma5.4Forallk∈Z+,forallΛ∈Fk\\Fk−1thereexistsafinitesequence

1/3(i)(i)k1

lk−2,{Λ1,Λ2}si=1inFk−1,wheresk:=⌊lk⌋,suchthat,lettingδk:=(i)Λ=Λ1∪Λ2,

(i)(i)

(ii)󰀅d(Λ\\Λ1,Λ\\Λ2)≥δk,󰀇󰀇󰀅

(j)(j)(i)(i)

=∅,ifi=j.(iii)Λ1∩Λ2∩Λ1∩Λ2

(i)

(i)

TheB-Capproachthenestablishesasimplerecursiveinequalitybetweenthe

quantityγk:=supΛ∈Fkγ(Λ)onscalekandthesamequantityonscalek−1asfollows.

FixΛ∈Fk\\Fk−1andwriteitasΛ=Λ1∪Λ2withΛ1,Λ2∈Fk−1satisfyingthepropertiesdescribedinLemma5.4above.WithoutlossofgeneralitywecanassumethatallthehorizontalfacesofΛ1andofΛ2layonthehorizontalfacesofΛexceptforthefaceorthogonaltothefirstdirectione1andthat,alongthatdirection,Λ1comesbeforeΛ2.Setd≡Λ1∩Λ2andwrite,fordefiniteness,d=[a1,b1]×[a2,b2].Lemma5.4impliesthatthewidthofdinthefirstdirection,b1−a1,isatleastδk.Setalso

I≡[a1+(b1−a1)/2,b1]×[a2,b2]

andlet∂rI={b1}×[a2,b2]betherightfaceofIalongthefirstdirection.Definition8.Givenaconfigurationω∈ΩwewillsaythatωisI-goodiffthereexistsatop-bottomcrossingofI.

Givenτ∈MaxΛ,werunthefollowingconstrained“blockdynamics”onΩΛ(inwhatfollows,forsimplicity,wesuppresstheindexi)withboundaryconditionsτandblocksB1:=Λ1\\I,B2:=Λ2.TheblockB2waitsameanoneexponentialrandomtimeandthenthecurrentconfigurationinsideitisrefreshedwithanewonesampledfromtheGibbsmeasureoftheblockgiventhepreviousconfigurationoutsideit(andτoutsideΛ).TheblockB1doesthesamebutnowtheconfigurationisrefreshedonlyifthecurrentconfigurationωinBisI-good(seeFigure5.1).

Thegeneratoroftheblockdynamicsappliedtofcanbewrittenas

Facilitatedspinmodels:recentandnewresults21

22

N.Cancrini,F.Martinelli,C.Roberto†,,andC.Toninelli†,

Proof.Itfollowsimmediatelyfromstandardpercolationargumentstogetherwith

2M

supsupµΦ,τ{x}(σx=1)≤(1−q)e

Φ∈Br,M

τ

WecannowstatethemainconsequenceofLemma5.5,5.6.

Proposition5.7Thereexistsq0∈(0,1)andforanyr<∞thereexistsM1

suchthat,foranyM󰀅󰀇−1󰀅󰀕(k)Φ,τgap(Lblock)≤1−8supγblock:=sup

Φ∈Br,Mτ∈MaxΛ

⊓⊔

λksinceotherwisethereisnothingtobeproved.

ByapplyingµB1tobothsidesof(15)andusing(13)weobtain

󰀕󰀁󰀂

⇒󰀰µB1(f)󰀰∞≤(1+λ)µB1f=µB1µB2(f)

1+λ+c1

λk,weget

󰀰µB2(f)󰀰∞≤󰀰µB2(f)󰀰∞󰀰µB2(1+λ+c1󰀈

≤󰀰µB2(f)󰀰∞󰀰µB2(

whichispossibleonlyif

󰀰µB2(

i.e.

λ≤−1+8

1

󰀕

λk1

µB2(f)+

c1

󰀰∞󰀰µB1(f)󰀰∞

1

λk

󰀊

(17)

Facilitatedspinmodels:recentandnewresults23

Thesecondterminther.h.s.of(18),usingthedefinitionofγkandthefactthatB2=Λ2∈Fk−1isboundedfromaboveby

󰀇󰀅󰀖Φ,τ󰀂󰀁Φ,τΦ

µΛVarB2(f)≤γk−1µΛcx,B2VarΦ(f)(19)x

x∈B2

BywritingdownthestandardPoincar´einequalityfortheblockauxiliary

chain,wegetthatforanyf

󰀇󰀅

(k)Φ,τΦ,τΦΦ

VarΛ(f)≤γblockµΛc1VarB1(f)+VarB2(f)(18)

NextweexaminethemorecomplicatetermµΦ,τc1VarΦB1(f).ForanyΛ

ωsuchthatthereexistsarightmostcrossingΠωinIdenotebyΛωthesetofallsitesinΛwhicharetotheleftofΠω.SinceVarΦB1(f)dependsonlyon

I{Πω=Γ}doesnotdependonωΛ\\B1and,foranytop-bottomcrossingΓofI,1

thevariablesω’stotheleftofΓ,wecanwrite

󰀅󰀇󰀅󰀁󰀇󰀂Φ,τΦ,τΦΦΦ

(20)I{∃ΠωinI}µΛωVarB1(f)µΛc1VarB1(f)=µΛ1Theconvexityofthevarianceimpliesthat

󰀁󰀂Φ(f)≤VarΦVarµΦΛω(f)B1Λω

Noticethat,byconstruction,forallx∈B2andallω,cx,B2(ω)=cx,Λ(ω).

󰀁󰀂󰀐Φ

Var(f)isnothingbutthecontributioncThereforethetermx∈B2µΦ,τx,B2xΛ

Φ,τ

carriedbythesetB2tothefullDirichletformDΛ(f󰀇󰀅).

whereitisunderstoodthatther.h.s.dependsonthevariablesinΠωandtotherightofit.Thekeyobservationatthisstage,whichexplainstheroleandtheneedoftheevent{∃ΠωinI},isthefollowing.ForanyωsuchthatΠωexiststhevarianceVarΦΛω(f)iscomputedwithboundaryconditions(τoutsideΛandωΛ\\Λω)whichbelongtoMaxΛω.ThereforewecanbounditfromaboveusingthePoincar´einequalityby

ΦΦ

VarΦΛω(f)≤γ(Λω)DΛω(f)≤γ(B1∪I)DΛω(f)

whereweusedLemma5.3togetherwiththeobservationthatΛω⊂B1∪I=

Λ1.Inconclusion

󰀅󰀁󰀇󰀂Φ,τΦΦ

I{∃ΠωinI}µΛωVarB1(f)µΛ1

󰀇󰀅

Φ,τΦ

≤γ(Λ1)µΛ1I{∃ΠωinI}DΛω(f)

󰀅󰀇󰀖Φ,τΦ

≤γ(Λ1)µΛ1I{∃ΠωinI}cx,ΛωVarx(f)

≤

γ(Λ1)µΦ,τΛ

󰀅󰀖

x∈Λω

cx,ΛVarΦx(f)

x∈Λ1

󰀇

24

N.Cancrini,F.Martinelli,C.Roberto†,,andC.Toninelli†,

because,byconstruction,foreveryωsuchthatthereexistsΠωinI

cx,Λω(ω)=cx,Λ(ω)∀x∈Λω.

(21)

Ifwefinallyplug(5.1)intother.h.s.of(20)andrecallthatΛ1∈Fk−1,weobtain

󰀇󰀅󰀖󰀁Φ,τ󰀂Φ,τΦ

µΛc1VarB1(f)≤γk−1µΛcx,ΛVarΦ(f)(22)x

x∈Λ1

Inconclusionwehaveshownthat

󰀅󰀖Φ,τ󰀂󰀁󰀇(k)Φ,τΦ,τ

µΛcx,ΛVarx(f)VarΛ(f)≤γblockγk−1DΛ(f)+

x∈d

1/3

(23)

Averagingoverthesk=⌊lk⌋possiblechoicesofthesetsΛ1,Λ2gives

VarΛ(f)≤γblockγk−1(1+

(k)

1

sk

(k)

)γblockγk−1

≤γk0

j=k0

k󰀍

(1+

1

sj)

isbounded.⊓⊔

6Onespinfacilitatedmodelonageneralgraph

Inthissectionweproveoursecondsetofnewresultsbyexaminingtheonespin

facilitatedmodel(FA-1finshort)onageneralconnectedgraphG=(V,E).OurmotivationcomesfromsomeunpublishedspeculationbyD.Aldous[2]that,inthisgeneralsetting,theFA-1fmayserveasanalgorithmforinforma-tionstorageindynamicgraphs.

Webeginbydiscussingthefinitesetting.LetrbeoneoftheverticesandTbearootedspanningtreeofGwithrootr.OnΩ={0,1}VconsidertheFA-1fconstraints:

󰀉

cx,G(ω)=1ifωy=0forsomeneighboryofx

(26)

0otherwiseˆbethecorrespondingMarkovandletcˆx,G=cx,Gifx=randcˆr,G≡1.LetL

generatorandnoticethatassociatedMarkovchainisergodicsincethevertex

ˆasthe(G,r,risunconstrained.ForshortnesswewillreferinthesequeltoL

FA-1f)model.Ourfirstresultreadsasfollows.

Facilitatedspinmodels:recentandnewresults25

Theorem6.1

gap(G,r,FA-1f)≥gap(Z,East)

Proof.Bymonotonicitycˆx,G(ω)≥cˆx,T(ω)andthereforegap(G,r,FA-1f)≥gap(T,r,FA-1f).Wecanpushthemonotonicityargumentabitfurtherandconsiderthefollowing(T,r,East)model:

󰀉

1ifeitherx=rorωy=0,whereyistheancestor(inT)ofx

c˜x,T(ω)=

0otherwise

(27)

Clearlycˆx,T(ω)≥c˜x,T(ω)andthereforegap(G,r,FA-1f)≥gap(T,r,East).Wewillnowproceedtoshowthat

gap(T,r,East)≥gap(Z,East)

(28)

IfalltheverticesofThavedegree2withtheexceptionoftherootandtheleaves,i.e.ifT⊂Z,then(28)followsfrom[12,Lemma2.11].Thusletusassumethatthereexistsx∈Twithdx≥3andletusordertheverticesofTbyfirstassigningsomearbitraryordertoallverticesbelongingtoanygivenlayer(≡samedistancefromtheroot)andthendeclaringx26

N.Cancrini,F.Martinelli,C.Roberto†,,andC.Toninelli†,

ByrecursivelyapplyingtheaboveresulttoAandBseparately,weimme-diatelyreduceourselvestothecaseofatreeT′⊂Zandtheproofofthetheoremiscomplete.

⊓⊔Proof(ofLemma6.2).InL2(Ω,µ)considerthesetHBoffunctionsfthatdonotdependonωx,x∈Ta.Becauseofthechoiceoftheconstraintsc˜x,T(ω),HBisaninvariantsubspaceforthegeneratorofthe(T,r,East)modeland

f∈HBµ(f)=0

inf

˜(f)D

2

gap(Z,East)µ(Ω+)

Facilitatedspinmodels:recentandnewresults27

Proof.AsintheproofofTheorem6.1wecansafelyassumethatGisatreeT

˜onΩbysettingwithrootr∈V.Weextendanyf:Ω+→Rtoafunctionf

˜(ηy=1∀y)≡f(ηy=1∀y=r,ηr=0).UsingTheorem6.1,wethenwritef

󰀂󰀁

˜)≤µ(Ω+)−2Var(f˜)Var+(f)=Var+(f

󰀇󰀖󰀅󰀂󰀁+−2−1˜≤µ(Ω)gap(T,r,East)µcˆx,TVarx(f)

x

wheretheconstraints{cˆx,T}x∈Thaverightafter(26).󰀅beendefined󰀇

˜)withx=r.RememberLetusexamineagenerictermµcˆx,TVarx(f

thatcˆx,T=cx,Tandmoreovercx,T(η)=0ifηy=1forally=x.Fur-thermore,foranyηsuchthatthereexistsy=xwithηy=0,µ+(ηx=1|{ηy}y=x)=p.Inconclusionwehaveshownthat

󰀅󰀇󰀂󰀁

˜(f)∀x=r(31)µcˆx,TVarx(f)=µ(Ω+)µ+cx,TVar+x󰀅󰀇󰀅󰀇

˜˜Wenowexaminethedangeroustermµcˆr,TVarr(f)=µVarr(f).Be-˜wecansafelyrewriteitascauseofthedefinitionoff

󰀇󰀂󰀁

˜µVarr(f)=µχ{∃y=r:ηy=0}Varr(f)󰀅

LetusordertheverticesofthetreeTstartingfromthefurthermostones

byfirstassigningsomearbitraryordertoallverticesbelongingtoanygivenlayer(≡samedistancefromtheroot)andthendeclaringxd(y,r)ord(x,r)=d(y,r)andxcomesbeforeyintheorderassignedtotheirlayer.Next,foranyηsuchthatηy=0forsomey=r,defineξ=min{y:ηy=0}andletTξ:={z∈T:z>ξ}(seeFig3).

2328

N.Cancrini,F.Martinelli,C.Roberto†,,andC.Toninelli†,

InordertoboundfromaboveVarTξ(f)weapplythePoincar´einequalityinTξwithconstraints{cˆz,Tξ}androotvtogetherwithTheorem6.1:

VarTξ(f)≤gap(Z,East)−1

z∈Tξ

󰀂󰀁󰀂󰀁µχ{∃y=r:ηy=0}Varr(f)=µ(χξ=rµ(Varr(f)|ξ))≤µχξ=rVarTξ(f)

󰀖

󰀂󰀁ˆz,TξVarz(f)µTξc

Noticethat,byconstruction,cˆz,Tξ(η)=cz,T(η)foranyz∈Tξ,includingthe

rootvofTξwherecˆv,Tξ(η)=1bydefinitionandcv,T(η)=1becauseηξ=0.Puttingalltogetherweconcludethat

󰀖󰀂󰀂󰀁󰀁

µχ{∃y=r:ηy=0}Varr(f)≤gap(Z,East)−1µχ{∃y=r:ηy=0}cx,TVarx(f)

≤gap(Z,East)−1µ(Ω+)

x∈T

󰀖

x∈T

󰀂󰀁

µ+cx,TVar+(f)x

wherewehaveusedoncemoretheobservationbefore(31)towrite

cx,TVarx(f)=cx,TVar+x(f).

Ifwenowcombine(6),(31)and(6)togetherweget

󰀂󰀁−1󰀖+󰀂󰀁

Var+(f)≤2gap(Z,East)µ(Ω+)(f)µcx,TVar+x

x∈T

andtheproofiscomplete.⊓⊔

Acknowledgement.F.MartinelliwouldliketowarmlythankRomanKoteck´yfor

theveryniceinvitationtolectureatthePragueSummerSchoolonMathemati-calStatisticalMechanics.WealsoacknowledgeP.Sollichforusefuldiscussionsandcommentsonthetopicsofthispaper.

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