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PHASE TRANSITION IN THE KMP MODEL WITH SLOW/FAST BOUNDARIES 6 1 TERTULIANOFRANCO 0 2 ABSTRACT. TheKipnis-Marchioro-Presutti(KMP)isaknownmodelconsist- c ingonaone-dimensionalchainofmechanicallyuncoupledoscillators, whose e interactionsoccurviaindependentPoissonclocks:whenaPoissonclockrings, D the total energy at two neighbors is redistributed uniformly at random be- tweenthem.Moreover,attheboundaries,energyisexchangedwithreservoirs 3 offixedtemperatures. WestudyhereageneralizationoftheKMPmodel by consideringdifferentratesatenergyisexchangedwiththereservoirs,andwe ] R thenprovetheexistenceofaphasetransitionfortheheatflow. P . h t a m 1. INTRODUCTION [ How microscopic interactions determine the macroscopic behavior of a giv- 4 en system is a question that guides a vast research in Statistical Mechanics v and Probability. In this context, since the seventies, a rigorousmathematical 2 4 theory has been developed in order to give a precise sense to the limit from 0 microscopical systems with stochastic time evolution towards its macroscopic 4 pointofview. Asaclassicalreferenceinthesubjectwecitethebook[13]. 0 AparticularimportantmicroscopicsystemistheKipnis-Marchioro-Presutti . 1 (KMP) model, see [15]. Such modelconsists of a one-dimensionalfinite chain 0 of oscillators, being each oscillator described by its velocity and position. The 6 oscillators interact in the following way. Associated to each pair of neighbors 1 there is a Poisson clock; when a certain clock rings, the total energy at the : v pair of neighbors is redistributed between them uniformly at random. The i X respective new positions and velocities are then chosen uniformly (according totheLebesguemeasure)amongallthepossibleconfigurationsonitssurfaces r a of constant energy. Besides, at the right (resp. left) boundary, also at arrival times of a Poisson clock, the energy is replaced according to an exponential distributionofparameterβ >0(resp. β >0). Thisisequivalenttosaythat + − thesystemisin contactwithreservoirsoftemperatureT = 1/(k β ),where ± B ± k standsfortheBoltzmannconstant. B Asexplainedin[15],attheinvariantstate,conditionallyontheenergy,posi- tionandvelocityareuniformlydistributed. Wethereforerestrictourattention hereonlytotheenergyprofile. SupposingT 6= T ,afluxofenergyisobservedinKMPmodel. This isthe + − content of [15], i.e., a rigorous proof of the Fourier Law. Due to its peculiar 2010MathematicsSubjectClassification. 60K35. Keywordsandphrases. Harmonicoscillators,weakconvergence,heatflow. 1 2 TERTULIANOFRANCO structure, which gives rise to an interesting duality and consequent manage- ability,theKMPmodelisaninterestingobjectofstudybothinProbabilityand StatisticalMechanics. Seeforinstance[2,11]andreferencestherein. Recently, several works have investigated how a slowed defect can utterly modify the scaling limit of a given microscopic system. See, for instance, [6, 7, 8, 9, 5]. By a slowed defect we mean that some specific site (or bond, or boundary), is rescaled differently from the rest of the system. This is pre- cisely what we investigate in this paper. Rescaling rates at which energy is exchanged with the reservoirs, we arrive at a phase transition for the steady state. In fact, taking rates as AL−a and BL−b, whereL is the scaling parameter, the invariant profile of temperatures disconnects from the reservoirs if some of the parameters a or b is equal to one. This result, which is the so- called localequilibrium(see[14]) is stated in the Theorem2.1, whereexplicit formulasforthelimitingprofilesarealsoprovided. Accordingto theseminalpaper[15],theKMPmodelhasthestriking prop- ertythat its dualprocessofparticles remainsinvariant(in somesense) when particles are added to the system. This is a fundamental ingredient in order to obtain the steady temperature profile. Since the original proof of this fact given in [15, Proposition 3.1] is somewhat unclear, we present here a simple, easilycomprehensibleproofofthisfactintheProposition4.1,basedonacom- binatorialidentity. Ourproofsuits forthe slow/fastboundariescase, covering theoriginalmodel[15]asaparticularcase. Since the KMP modelis non-gradient(see [13] fora precise definition),the hydrodynamiclimitofthemodelpresentedhereturnstobeachallengingprob- lem. In view of Theorem 2.1, we conjecture that its hydrodynamic behavior should be described by a non-linear heat equation with boundary conditions relatedtotheregimesdescribedinTheorem2.1. The outline of the paper is: Section 2 presents statements. Section 3 deals with the duality of KMP process with slow/fast boundaries. In Section 4, the Label Processis defined,which allows explicitcomputations,eventuallylead- ing to the proof of Theorem 2.1. In Section 5, further extensions and open problemsareconsidered. 2. STATEMENT OF RESULTS Notations: The cardinality of a finite set A will be denoted by |A|. We clarifythathereN={0,1,2,...}andR ={x∈R; x>0}. + ForapositiveintegerL,considerthestatespaceΩ =R(2L+1),whichrepre- L + sentsthesetofenergyconfigurationsof2L+1oscillatorsinaone-dimensional chain. Wedenoteanenergyconfigurationby ξ = (ξ ,...,ξ )∈Ω . (2.1) −L L L The KMP modelwithslow/fastboundarieswedefinehereis the Markov process {ξ ; t ≥ 0} on Ω characterized by its generator G acting on smooth t L L KMPMODELWITHSLOW/FASTBOUNDARIES 3 boundedfunctionsf :Ω →Ras L L 1 G f (ξ)= f(ξ ,...,p(ξ +ξ ),(1−p)(ξ +ξ ),...,ξ )−f(ξ) dp L −L x x+1 x x+1 L (cid:0) (cid:1) xX=−LZ0 h i A ∞ + f(y,ξ ,...,ξ )−f(ξ) β e−β−ydy La −L+1 L − B Z0∞h i + Lb f(ξ−L,...,ξL−1,y)−f(ξ) β+e−β+ydy, Z0 h i whereβ ,β ,A,B >0anda,b∈R. WedefineT ,thetemperaturesintheleft + − ± andrightreservoirs,respectively,bytheequalities 1 β = , ± k T B ± wherek standsfortheBoltzmannconstant. Someauthorsassumek =1for B B simplicity. Definition 1. For given T > 0, let ν be the Gibbs measure (for the energy) T of independent oscillators on Z. In other words, ν is the following product T measureonRZ: + 1 ξ x dν = exp − dξ . T k T k T x xY∈Zh B n B o i Notice that the marginal of the measure ν at any site is an exponential T distributionofparameter1/k T. B Definition2. LetusdivideRZ inblocksofsize(2L+1),whereoneoftheblocks + iscenteredattheorigin. Inotherwords,wewriteRZ asthefollowingCartesian + product: RZ = R{−L,...,0,...,L}+j(2L+1). + + j∈Z Y Let µ be the (unique) invariant measure of the KMP process with slow/fast L reservoirs. Denote by µ˜ its extension to RZ obtained by taking the product of L copiesofµ oneachoftheblocksaboveofsize(2L+1). L Theparticularchoicefortheextensionofµ hasnorelevancehereandany L other extension would suit our purpose as well. We have constructed µ˜ only L togivesensetothestatementbelow. Theorem 2.1. Denote by µ the invariant measure of the KMP process with L slow/fastboundaries,andletτ betheshiftof⌊uL⌋,withu∈(−1,1). ⌊uL⌋ Then, as L → ∞, the probability measure τ µ˜ converges weakly to ν , ⌊uL⌋ L T where: (i) Ifa,b<1, 1−u 1+u T = T(u)= T + T . − + 2 2 (ii) Ifa=1andb<1, (cid:16) (cid:17) (cid:16) (cid:17) A(1−u) A(1+u)+1 T(u)= T + T . − + 2A+1 2A+1 (cid:16) (cid:17) (cid:16) (cid:17) 4 TERTULIANOFRANCO (iii) Ifa<1andb=1, B(1−u)+1 B(1+u) T(u)= T + T . − + 2B+1 2B+1 (cid:16) (cid:17) (cid:16) (cid:17) (iv) Ifa=b=1, AB(1−u)+A AB(1+u)+B T(u)= T + T . − + 2AB+A+B 2AB+A+B (cid:16) (cid:17) (cid:16) (cid:17) (v) Ifa=b>1, A B T(u)= T + T . − + A+B A+B (cid:16) (cid:17) (cid:16) (cid:17) (vi) Ifa>max{1,b}, T(u)=T . + (vii) Ifb>max{1,a}, T(u)=T . − Some remarks: The regime (i) includes the seminal result of [15], which correspondstothecasea =b=0andA=B =1. Intheregimes(ii),(iii)and (iv), the temperature varies linearly for u ∈ (−1,1), but does not interpolate T and T . In fact, when some of the parameters a and b is equal to one, the − + temperature close to the boundary does not reach the temperature T of the ± corresponding reservoir. In the regimes (v), (vi) and (vii), the temperature onthe chain of oscillators is completelyhomogenized. See the Figure 1for an illustrationoftheregimeasafunctionoftheparametersaandb. b 1 1 a Regime(i) Regime(v) Regime(ii) Regime(vi) Regime(iii) Regime(vii) Regime(iv) FIGURE 1. Regimesfortheheatflow KMPMODELWITHSLOW/FASTBOUNDARIES 5 3. DUAL PROCESS OF WALKERS Weconstructinthissectionadiscretesystemofwalkerswhichisdual(ina sense to be defined)to the KMP process with slow/fast boundaries. We adapt hereideasfrom[15]. Definition3. LetΛ ={−L,...,L}∪{δ(−), δ(+)}anddenote L n=(nδ(−),n−L,...,nL,nδ(+))∈NΛL (3.1) ConsidertheMarkovprocesstakingvaluesonNΛL characterizedbythefollow- inggenerator A A f (n)= f(n +n ,0,n ,...,n )−f(n) L La δ(−) −L −L+1 δ(+) (cid:0) B(cid:1) h i + f(n ,...,n ,0,n +n )−f(n) Lb δ(−) L−1 δ(+) L Lh nj+nj+1 i + 1 f(n ,n ,...,n ,q,n +n −q,...,n )−f(n) . nj+nj+1+1 δ(−) −L i−1 i i+1 δ(+) jX=−L Xq=0 h i This particle system can be described in words as follows. We associate to each pair of neighbour sites a Poisson clock of parameter one. When a Pois- son clock rings, the particles in the corresponding sites are uniformly redistributed. Moreover,associatedtothesite−LthereisaPoissonclockofparam- eterA/La. WhenthisPoissonclockrings,alltheparticlesatthesite−Lmove tothesiteδ(−)andthenstay thereforever. Analogousdescriptionforthesite L. Moreover,allthePoissonclocksaretakenasindependent. Recall(2.1)and(3.1). Letusdefine 1 L ξnx F(n,ξ)= x . (3.2) β+nδ(+)β−nδ(−) x=−L nx! Y Theorem3.1(Duality). Fixζ ∈Ω and L k =(0,k ,...,k ,0)∈NΛL. (3.3) −L L Denote by E the expectation induced by the Markov process of generator A k L startingattheconfigurationk anddenotebyE theexpectationinducedbythe ζ Markov process of generator G starting at the initial configuration ζ. Then, L forallt≥0, E F(k,ξ ) = E F(η ,ζ) . (3.4) ζ t k t h i h i The proof of above is very similar to the one in [15, Thm 2.1], and consists on checking that A F = G F. We left this to the reader. By letting t → ∞ in L L (3.4)weobtain: Corollary3.2. Givenkasin(3.1),denotekkk= L k . For0≤j ≤kkklet x=−L x q (k,j) = P j particleshitδ(+)andkkkP−j particleshitδ(−) . L k Then h(cid:8) (cid:9)i kkk 1 F(k,ξ)µ (dξ) = q (k,j). L βjβkkk−j L Z j=0 + − X 6 TERTULIANOFRANCO 4. LABEL PROCESS We will call by Label Process the Markov process constructed by labelling particles in the process presented in Definition 3 in the following way. First, weputa labelto distinguish each particle. Consider atime whenthe Poisson clockassociatedtoapairofsitesk,k+1rings. Inthatmoment,letussaythat thetotalquantityofparticlesisn +n . Makeabijectionbetweenthesetof k k+1 labelsofthoseparticlesandthesetofintegers{0,...,n +n }. Then,choose k k+1 an integer U uniformly between 0 and n + n and independently choose k k+1 uniformly a permutation ζ of the integers {0,...,n + n }. The particles k k+1 correspondingtothefirstU positionsofthepermutationζ willbeaddressedto thesitekandtheremainingtothesitek+1. Attheboundaries,theprocedure is the following: when the Poisson clock associated to the site L rings, all the particlesin the site L moveto the site δ(+) andstay thereforever. Analogous descriptionforthesite−L. Notice that particles are not independent in this dynamics. Moreover, by counting how many particles there is at each site we can recover the process givenintheDefinition3. Definition4. LetPL betheprobabilityinducedbytheLabelProcessstart- x1,...,xn ingfromndistinctparticleslocatedatthesitesx ,...,x ∈{−L,...,L}. More- 1 n over,forx ,...,x ∈{−L,...,L}andε ,...,ε ∈{δ(+), δ(−)},denote 1 n 1 n p (x ,...,x ;ε ,...,ε ):=PL fori=1,...,n,theparticlestarting L 1 n 1 n x1,...,xn h(cid:8) atx hitsthesiteε . i i Fromthedefinitionabove,weget (cid:9)i q (k,j) = p (x ,...,x ;ε ,...,ε ), L L 1 n 1 n wherethesumaboveistakenoXverallsequencesε ,...,ε suchthatthecardi- 1 n nalityoftheset{i; ε =δ(+)}isequaltoj. i The next proposition tell us that if some of the particles initially located at {x ,...,x } are removed(or some particles are added), the behavior of the 1 n remainingparticlesisnotmodified. Proposition 4.1. Recall that PL denotes the probability induced by the x1,...,xn Label Process starting from particles initially located at the sites x ...,x ∈ 1 n {−L,...,L}. Letm<nand{i ,...,i }⊂{1,...,n}. Definey =x . Then,the 1 m k ik probabilityPL restrictedto the classofevents that dependonlyonthe set x1,...,xn ofparticlesinitiallylocatedat{x ,...,x }coincideswithPL . i1 im y1,...,ym Inparticular,forany1≤i≤n, p (x ,...,x ;ε ,...,ε ) L 1 n 1 n εi∈{δ(X−),δ(+)} = p (x ,...,x ,x ,...,x ;ε ,...,ε ,ε ,...,ε ). L 1 i−1 i+1 n 1 i−1 i+1 n We point out that the original proof of this result presented in [15, Prop 3.1],is of difficultreading1. For this reason,it isprovidedherean alternative 1Theproofof[15,Prop3.1]isconcernedwiththecasea=b=0andA=B=1forourmodel. Anyway,sincetheboundariesdonotplayanyroleintheproof,thestatementsareessentiallythe same. KMPMODELWITHSLOW/FASTBOUNDARIES 7 proofbased on the tricky combinatorialidentity which wedevelopin the next lemma. Recall the convention that n = 0 whenever n,m are integers such m thatn<m. (cid:0) (cid:1) Lemma4.2. Foranyq ∈{0,...,M}, N p N −p N +1 = . (4.1) q M −q M +1 p=0(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19) X Proof. We prove (4.1) by a combinatorial argument. That is, we are going to count, by two different procedures, how many ways we can choose a subset with M +1 objects from a set with N +1 objects. Obviously, a first answer is the righthand side of (4.1). Fix q ∈ {0,...,M}. Withoutloss of generality,let ussay that the set of objectsis a set ofreal numbersO = {a ,...,a } such 1 N+1 thata <a wheneveri<j. Define i j S := {A⊂O;|A|=M +1, a ∈A, thereareq elements p p+1 inAstrictlysmallerthana , andthereare p+1 M −q elementsinAstrictlybiggerthana }. p+1 It is easy to check that S ,...,S are disjoint sets, its union is the set of all 0 N subsetsofOofcardinalityM +1,and|S |= p N−p . Thisimplies(4.1). (cid:3) p q M−q ProofofProposition4.1. Wewillmakeuseof(cid:0)th(cid:1)e(cid:0)grap(cid:1)hicalconstructionofpar- ticlessystemsviaPoissonprocess. AssumethesamePoissonprocessareused to evolve both the Label Process with particles starting from sites x ,...,x 1 n and the Label Process with particles starting from y ,...,y . For short, we 1 m willcallbyx-LabelProcessthefirstprocessandbyy-LabelProcessthesecond one. If thePoisson clock associated tothe site δ(+)rings, allthe particlesin the site L movesto the site δ(+) and stay there forever,and analogousstatement holds for the site δ(−). Both situations do not interfere in the trajectory of particles before hitting {δ(+),δ(−)}. Therefore, consider the time when the Poissonclockassociatedtoapairofsitesk,k+1rings,withk ∈{−L,...,L−1}. As described in the Label Process definition, let n +n = N the total k k+1 number of particles at the sites k,k+1 in that instant, being M of those particles belonging to the y-Label Process. Fix a bijection between these set of particlesandthesetofintegersnumbers{1,...,N}. LetU,ζ areindependent, being U an uniform random variable in the set {1,...,N} and ζ is uniformly chosen on the set of permutations of {1,...,N}. The particles corresponding the first U positions of ζ will be sent to the site k and the remaining particles willbesenttothesitek+1. Since an uniform permutation ζ on N objects induces an uniform permutation on M of them, we only need to assure that the quantity of particles belonging to the y-Label Process that will be sent to the site k is uniformly distributedon{0,1,...,M}. DenotethisquantityofparticlesbyY,whichisa function of U and ζ. For short, denote by P the probability from to the space whereU,ζ havebeenconstructed. Inotherwords,ourgoalistoshowthat 1 P Y =q = , (4.2) M +1 (cid:2) (cid:3) 8 TERTULIANOFRANCO forany0≤q ≤M integer. Wehavethat N P Y =q = P Y =q U =p ·P U =p p=0 (cid:2) (cid:3) X (cid:2) (cid:12) (cid:3) (cid:2) (cid:3) N (cid:12) N p N−p 1 1 = P Y =q U =p = q M−q , N +1 N +1 N (cid:0) (cid:1)(cid:0) (cid:1) p=0 p=0 M X (cid:2) (cid:12) (cid:3) X wherein the last equality we haveuse(cid:12)d that ζ is picked uniform(cid:0)ly(cid:1)at random inthesetofpermutations. RecallingnowLemma4.2yields(4.2),finishingthe proof. (cid:3) Proposition4.3. FixN integer,x ,...,x andε ,...,ε ∈{δ(+),δ(−)}. Then, 1 N 1 N for any u ∈ (−1,1) and any permutation σ of the set {1,...,N}, the following limitholds: lim p (x +⌊uL⌋,...,x +⌊uL⌋;ε ,...,ε ) L 1 N 1 N L→∞ (4.3) n −p (x +⌊uL⌋,...,x +⌊uL⌋;ε ,...,ε ) = 0, L 1 N σ(1) σ(N) o wherewehavedenotedby⌊uL⌋theintegerpartofuL. Since the boundaries do not play any role in the result above, its proof can bestraightforwardlyadaptedfrom[10,Proposition3.5]andforthisreasonwe omitithere. Nextwecalculatethehittingprobabilitiesforasingleparticle. Proposition4.4. Considerthestoppingtimes τ = inf t>0; X =δ(+) and τ = inf t>0; X =δ(−) . δ(+) t δ(−) t Then,foranyx∈(cid:8){−L,...,L}, (cid:9) (cid:8) (cid:9) AB(x+L)+BLa p (x;δ(+)) = PL τ <τ = . L x δ(+) δ(−) 2ABL+ALb+BLa h i In particular, the limit below, which we denote by p(u), does not depend on x 1 regardlessthechosenvaluesofaandb: p(u) := lim p (x +⌊uL⌋,δ(+)) = lim p (⌊uL⌋,δ(+)) L 1 L L→∞ L→∞ AB(⌊uL⌋+L)+BLa = lim . (4.4) L→∞ 2ABL+ALb+BLa Proof. In order to not overload notation, we write down c = A/La, d = B/Lb, a =PL τ <τ ,a =PL τ <τ anda =PL τ < L+1 δ(+) δ(+) δ(−) −(L+1) δ(−) δ(+) δ(−) x x δ(+) τ ,forx∈h{−L,...,L}i. h i h δ(−) ApiplyingtheMarkov Property,onecan checkthat(ax)x isasolution ofthe followinglinearsystem: a = 1,a =0,a =(a +d)/(1+d),a = L+1 −(L+1) L L−1 −L a /(1+c),and,for−L+1≤x≤L−1,a =(a +a )/2. −L+1 x x−1 x+1 Ingeneral,itisnotaneasytasktoexhibitthesolutionofasystemasabove inasimpleway. However,thefactthata =(a +a )/2holdsfor−L+1≤ x x−1 x+1 x ≤ L−1 leads us to the guess that, except at the boundaries, the solution KMPMODELWITHSLOW/FASTBOUNDARIES 9 shouldbearestrictiontotheintegersofsomelinearfunction. Withthisguess inmind,onecandeducebysomelongalbeitelementarycalculationsthat 1, ifx=L+1, cdx+d(Lc+1) ax =  , if −L≤x≤L, c(Ld+1)+d(Lc+1) 0, ifx=−(L+1), finishingtheproofsincec=A/La andd=B/Lb. (cid:3) Proposition4.5. Foranyx ,...,x integers,foranyε ,...,ε ∈{δ(+),δ(−)} 1 N 1 N andforanyu∈(−1,1),thefollowinglimitholds: lim p (x +⌊uL⌋,...,x +⌊uL⌋;ε ,...,ε ) L 1 N i N L→∞ n N (4.5) − p (x +⌊uL⌋;ε ) = 0. L i i iY=1 o Proof. Recall that u ∈ (−1,1) is fixed. For N = 2, the result follows from the fact that, with high probability, particles starting at x +⌊uL⌋ and x +⌊uL⌋ 1 2 will meet before hit the boundaries. The concerning technical details can be easilyadaptedfromtheproofof[10,Lemma3.5]. InpossessofthecaseN =2, we proceed to prove the result for any positive integer. Let I = {−L,...,L} L anddenoteelementsof{−1,+1}IL byη. Given{x1,...,xN}⊂IL,wedefine αL {η ∈{−1,+1}IL ; η(xi)=εi fori=1,...,N} := p (x +⌊uL⌋,...,x +⌊uL⌋;ε ,...,ε ). (cid:0) L 1 N (cid:1) i N Proposition4.1guaranteesthatαL isaprobabilitymeasureon{−1,+1}IL. Now,weextendα tosomeprobabilitymeasureα˜ on{−1,+1}Z,beingthe L L particular choice of the extension not relevant. Consider the weak conver- genceofprobabilitymeasures(see[3]). Since{−1,+1}Z isacompactspace,by Prohorov’s Theorem there exists a probability measure α which is a limit of τ α˜ alongsomesubsequence. ⌊uL⌋ L Weclaimnowthatαistheonlypossiblelimitalongsubsequencesofτ α˜ ⌊uL⌋ L and, moreover, it is a Bernoulli product measure of constant parameter. No- ticethatthisclaim puttogetherwithProposition4.4immediatelyimply(4.5), finishingtheproof. Proposition 4.3 implies that α is an exchangeable measure. Hence, by de Finetti’s Theorem, we conclude that α is a mixture of Bernoulli productmea- sures,thatis, 1 α = Θ m(dp), p Z0 whereΘ istheBernoulliproductmeasureon{−1,+1}Zofconstantparameter p p∈[0,1]andmisaprobabilitymeasureontheBoreliansetsof[0,1],calledthe law of the mixture. On de Finetti’s Theorem and exchangeability, we refer to thesurvey[12]. Providedby the caseN = 2,wealreadyknowthatthe marginalofα atthe sitesx,x+1∈ZisaBernoulliproductmeasurewithsameparameteratxand 10 TERTULIANOFRANCO x+1,becauseoftheProposition4.4. Forthisreason,onecandeducethat 1 1 2 p2m(dp) = pm(dp) . Z0 (cid:16)Z0 (cid:17) Since f(p) = p2 is a strictly convex function, then m must be a Delta of Dirac measure,whichimpliesthat α isa Bernoulliproductmeasureof constantpa- rameterp(u). Thisprovestheclaimandconcludestheproof. (cid:3) Weareinpositiontoproveourmainresult. Recallthedefinitionsofν and T τ µ˜ . ⌊uL⌋ L ProofofTheorem2.1. Fix u ∈ (−1,1). We start with two observations. First, foranexponentiallawwehavethat ∞ yj λe−λydy = λ−j, ∀j ∈N. j! Z0 Thus,sinceν isaproductmeasure,wehavethat F(k,ξ)ν (dξ) = k T kkk, T T B foranyk ∈NZ suchthatkkk<∞. Second,theclassofpolynomialsp:RZ →R R (cid:0) (cid:1) inafinitenumberofvariablesisaweakconvergence-determiningclassforthe setofprobabilitymeasuresconcentratedonRZ. + These two observations implies that, in order to prove that τ µ˜ con- ⌊uL⌋ L vergesweaklytotheprobabilitymeasureν ,itissufficienttoassurethat,for T anyk ∈NZ suchthatkkk<∞,thefollowinglimitholds: lim F(k,ξ)τ µ˜ (dξ)= k T kkk, (4.6) ⌊uL⌋ L B L→∞ Z whereT =T(u)istobeachievedaccordingtothe(cid:0)chos(cid:1)envaluesoftheparam- eters a and b. By the Corollary 3.2, the limit in the left side of above is equal to kkk 1 lim q (k+⌊uL⌋,j). (4.7) βjβkkk−j L→∞ L j=0 + − X BytheProposition4.4andtheProposition4.5,wehavethat kkk j kkk−j lim q (k+⌊uL⌋,j) = p(u) 1−p(u) . L L→∞ j (cid:18) (cid:19) (cid:0) (cid:1) (cid:0) (cid:1) Wethereforeconcludethatexpression(4.7)isequalto p(u) 1−p(u) kkk kkk + = k p(u)T +(1−p(u))T . B + − β β + − (cid:16) (cid:17) (cid:16) n o(cid:17) Then, evaluating (4.4) in each regime of the parameters a,b ∈ R implies (4.6) where T = T(u) is the one in the statement of the Theorem 2.1, finishing the proof. (cid:3) 5. FURTHER EXTENSIONS 5.1. AvariantoftheKMPmodel. Weconsidererherethemodelasdefined in Section 2.4 of [1], which is a slight variation of the original KMP model of [15]. Tobetterlinkthemodels,weadoptinthissectionipseliteristhenotation of[1]. DefineΛ = {1,...,N −1}and by ξ ∈ RΛN the energyconfigurationof N + oscillators,beingξ itsenergyatsitex∈Λ . Givenp∈[0,1],denotebyξ(x,y),p x N