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CENTRAL LIMIT THEOREM FOR A TAGGED PARTICLE IN ASYMMETRIC SIMPLE EXCLUSION 7 0 0 PATRÍCIAGONÇALVES 2 n Abstract. We prove a Functional Central Limit Theorem for the position a J of a Tagged Particle in the one-dimensional Asymmetric Simple Exclusion Process in the hyperbolic scaling, starting from a Bernoulli product measure 4 conditioned to have a particle at the origin. We also prove that the position 2 of the Tagged Particle at time t depends on the initial configuration, by the numberofemptysitesintheinterval[0,(p−q)αt]dividedbyαinthehyperbolic R] andinalongertimescale,namelyN4/3. P . h t a 1. Introduction m The Exclusion process on Zd has been extensively studied. In this process, [ particlesevolveonZd accordingtointeractingrandomwalkswithanexclusionrule 2 which prevents more than one particle per site. The dynamics can be informally v described as follows. Fix a probability p() on Zd. Each particle, independently 5 · from the others, waits a mean one exponential time, at the end of which being at 0 5 x it jumps to x+y at rate p(y). If the site is occupied the jump is suppressed to 1 respecttheexclusionrule. Inbothcases,theparticlewaitsanewexponentialtime. 1 Zd The space state of the process is 0,1 and we denote the configurations by 6 { } the Greek letter η, so that η(x) =0 if the site x is vacant and η(x)=1 otherwise. 0 / Thecaseinwhichp(y)=0 y >1isreferredastheSimple Exclusionprocessand h ∀| | inthe Asymmetric Simple Exclusionprocess(ASEP)the probabilitypis suchthat t a p(1) = p, p( 1) = 1 p with p = 1/2 while in the Symmetric Simple Exclusion m − − 6 process (SSEP) p=1/2. : For 0 α 1, denote by ν the Bernoulli product measure on 0,1 Zd with v ≤ ≤ α { } i density α. It is known that να is an invariant measure for the Exclusion process X and that all invariant and translation invariant measures are convex combinations r of ν if p(.) is such that p (x,y)+p (y,x) > 0, x,y Zd and p(x,y) = 1, a α t t ∀ ∈ x y Zd, see [10]. P ∀ ∈ Assumethattheoriginisoccupiedattime0. TagthisparticleanddenotebyX t itspositionattimet. ApplyinganinvarianceprincipleduetoNewmanandWright [11], Kipnis in [8] proved a C.L.T. for the position of the Tagged Particle in the one-dimensional ASEP, provided the initial configuration is distributed according to ν , the Bernoulli product measure conditioned to have a particle at the origin. α∗ Date:February2,2008. 2000 Mathematics Subject Classification. 60K35. Key words and phrases. Asymmetric Exclusion, Equilibrium Fluctuations, Boltzmann-Gibbs Principle,TaggedParticle. The author wants to express her gratitude to F.C.T.(Portugal) forsupporting her Phdwith thegrant/SFRH/BD/11406/2002. 1 2 PATRÍCIAGONÇALVES Transforming the exclusion process into a series of queues, an asymmetric Zero- Range process with constant rate, the position of the TaggedParticle becomes the currentthroughthebond[ 1,0]. Kipnis[8],wasabletoapplyNewmanandWright − resultstotheZero-RangeprocessandderivetheL.L.N.andC.L.T.fortheposition of the Tagged Particle. Few years later, Ferrari and Fontes [6] proved that the position at time t of the Tagged Particle, X , can be approximated by a Poisson Process. More precisely, t they proved that for all t 0, if the initial distribution is ν and p > q, X = ≥ α∗ t N B + B , where N is a Poisson Process with rate (p q)(1 α) and B t t 0 t t − − − is a stationary process with bounded exponential moments. As a corollary they obtained the weak convergence of Xtǫ−1 −(p−q)(1−α)tǫ−1 (p q)(1 α)tǫ 1 − − − p to a Brownian motion. The argument is divided in two steps. The convergence of the finite-dimensional distributions [4] is consequence of the fact that in the scale t12, the position Xt can be read from the initial configuration: Xt is given by the initial number of empty sites in the interval [0,(p q)αt] divided by α. Tightness − followsfromthesharpapproximationofXtǫ−1 bythePoissonprocessandtheweak convergence of the Poisson process to Brownian motion. Using the approximation of X by a Poisson process and Kipnis results for the Tagged Particle, the same t authors prove equilibrium density fluctuations for the ASEP in [5]. The density fluctuations for the Totally Asymmetric Simple Exclusion process (the case p=1) have also been obtained by Rezakhanlou in [18] in a more general setting than for the process starting from an equilibrium state. Recently, Jara and Landim in [7] showed that the asymptotic behavior of the Tagged Particle in the one-dimensional nearest neighbor exclusion process, can be recoveredfromajointasymptoticbehavioroftheempiricalmeasureandthecurrent through a bond. From this observation they proved a non-equilibrium C.L.T. for the position of the Tagged Particle in the SSEP, under diffusive scaling. In this paper, besides using this general method to reprove Ferrari and Fontes resultontheconvergenceoftherescaledpositionofthe TaggedParticletoaBrow- nianmotion in the hyperbolic time scale,we extended their resultby showingthat inalongertimescalethepositionoftheTaggedParticlestilldependsontheinitial configuration. The advantage of our approach is that it relates the C.L.T. for the position of the Tagged Particle to the C.L.T. for the empirical measure, a problem which is relatively well understood, see [9]. In particular, we can expect to apply this approachfor a one-dimensional system in contact with reservoirs. ItwasshownbyRezakhanlouin[17],thatinthe ASEPthemacroscopicparticle density profile in the hyperbolic scaling, evolves according to the inviscid Burgers equation, namely: ∂ ρ(t,u)+(p q) (ρ(t,u)(1 ρ(t,u))) = 0. To establish the t − ∇ − C.L.T. for the empirical measure we need to consider the density fluctuation field as defined in (2.2) below. We show that, in this time scale, the time evolution of the limit density fluctuation field is deterministic, in the sense that at any given time t, the density field is a translation of the initial one. As mentioned above, this result was previously obtained in [5]. In order to observe fluctuation from the dynamics one has to change to the diffusive scaling (see section six). C.L.T. FOR A TAGGED PARTICLE IN ASYMMETRIC SIMPLE EXCLUSION 3 The translation or velocity of the system is given by v = (p q)(1 2α) and − − for α = 1/2, the field does not evolve in time, and one is forced to go beyond the hydrodynamic scaling. We can consider the density fluctuation field in the longer timescaleasdefinedin(2.4),wherewesubtractthe velocityofthe systemandany value of α can be considered in this setting. It is conjecturedthat untilthe time scale N3/2 the density fluctuationfield does not evolve in time, see chap.5 of [19] and references therein. The result we obtain is a contribution in this direction, since we can accomplish the result just up to the time scale N4/3. The main difficulty in proving the C.L.T. for the empirical measureistheBoltzmann-GibbsPrinciple,whichweareabletoproveforthistime scale using a multi-scale argument. As a consequence of this translation behavior, we show the dependence on the initial configuration of the current through a bond and the position of the Tagged Particle in the longer time scale. This work is organized as follows. In the second section we introduce some notation and we state the results. The sketch of the proof of the C.L.T. for the empirical measure associated to the ASEP in the hyperbolic scaling is exposed in the third section. In section four, we use the same strategy as in [7] to obtain L.L.N.andtheconvergenceoffinite-dimensionaldistributionsofthepositionofthe TaggedParticle to those of Brownianmotion. Tightness is provedby means of the Zero-Range representation as Kipnis in [8]. In this time scale we show that the current through a fixed bond and the position of the Tagged Particle at time tN, can be read from the initial configuration, in section five. In the following sections we study the same problem up to the time scale N1+γ withγ <1/3. WestartbyshowingtheC.L.T.fortheempiricalmeasureassociated to this process, in section six. Since a Boltzmann-Gibbs Principle is needed, its proof is the content of the seventh section and in the subsequent section we treat the problemoftightness. Inthe lastsectionwe provethe dependence onthe initial configurationforthecurrentthroughabondthatdependsontimeandtheposition of the Tagged Particle, in this longer time scale. 2. Statement of Results Theone-dimensionalasymmetricsimpleexclusionprocessistheMarkovProcess Z η 0,1 with generator given on local functions by t ∈{ } Lf(η)= c(x,y,η)[f(ηx,y) f(η)], (2.1) − Xx Zy=Xx 1 ∈ ± wherec(x,y,η)=p(x,y)η(x)(1 η(y)), p(x,x+1)=p, p(x,x 1)=q =1 pand − − − η(z), if z =x,y 6 ηx,y(z)= η(y), if z =x .  η(x), if z =y  Its description is the following. At most one particle is allowed at each site. If there is a particle at site x, it jumps at rate p to site x+1 if there is no particle at that site. If the site x 1 is empty, the particle at x jumps to x 1 at rate q. − − Z Initially,place the particlesaccordingto a Bernoulliproductmeasurein 0,1 , of { } parameter α (0,1), denoted by ν . α ∈ 4 PATRÍCIAGONÇALVES For each configuration η, denote by πN(η,du) the empirical measure given by 1 πN(η,du)= η(x)δx(du) N N Xx Z ∈ and let πN(η,du) = πN(η ,du). First, we state the C.L.T. for the empirical mea- t t sure, for which we need to introduce some notation. For eachintegerz 0,letH (x)=( 1)zex2dze x2 be the Hermite polynomial, ≥ z − dx − and h (x) = 1 H (x)e x2 the Hermite function, where c = z!√2π . The set z cz z − z h ,z 0 is an orthonormal basis of L2(R). Consider in L2(R) the operator z { ≥ } K =x2 ∆. A simple computation shows that K h =γ h where γ =2z+1. 0 0 z z z z Foran−integerk 0,denoteby theHilbertspaceinducedbyS(R)(thespace k ≥ H of smooth rapidly decreasing functions) and the scalar product < , > defined k by < f,g > =<f,Kkg >, where < , > denotes the inner product·o·f L2(R) and k 0 · · denote by the dual of , relatively to this inner product. k k Fix α H(0−,1) and an intHeger k. Denote by YN the density fluctuation field, a linear fun∈ctional acting on functions H S(R) a.s ∈ 1 x YN(H)= H (η (x) α). (2.2) t √N xXZ (cid:16)N(cid:17) tN − ∈ Denote by D(R+, ) (resp. C(R+, )) the space of H -valued functions, k k k H− H− − right continuous with left limits (resp. continuous), endowed with the uniform weak topology, by Q the probability measure on D(R+, ) induced by the N k density fluctuation field YN and ν . Consider PN = P thHe −probability measure on D(R+, 0,1 Z) induced. by ν aαnd the Markovναproceνsαs η speeded up by N and α t denote by{E }expectation with respect to P . να να Theorem 2.1. Fix an integer k > 2. Denote by Q be the probability measure on C(R+, ) corresponding to a stationary Gaussian process with mean 0 and k H− covariance given by E [Y (H)Y (G)]=χ(α) H(u+v(t s))G(u)du (2.3) Q t s ZR − for every 0 s t and H, G in . Here χ(α) = Var(ν ,η(0)) and v = k α ≤ ≤ H (p q)χ(α). Then, the sequence (Q ) converges weakly to the probability ′ N N 1 − ≥ measure Q. We remark that last theorem holds for the ASEP evolving in any Zd, with the appropriate changes. In this case, the limit density fluctuation field at time t is a translationoftheinitialdensityfield,sinceforeveryH S(R):Y (H)=Y (T H), t 0 t ∈ where T H(u)=H(u+vt). t Havingestablishedtheequilibriumdensityfluctuations,wecanobtaintheL.L.N. andthe C.L.T.for the currentoverabond, as in[7]. Denote byν the measureν α∗ α conditionedtohaveaparticleattheorigin. BycouplingtheASEPstartingfromν α with the ASEP starting from ν , in such a way that both processes differ at most α∗ in one site at any given time, the L.L.N. and the C.L.T. for the empirical measure and for the current starting from ν , follows from the L.L.N. and the C.L.T. for α∗ the empirical measure and for the current starting from ν . α meAassusuremoennDow(Rt+h,at0t,h1eZi)niitnidalucmedeabsuyreν isaνnα∗d,tlheet MPNνaα∗rk=ovPpνrα∗ocbeessthηe sppreoebdaebdiliutpy by N and denote by{Eν∗}expectation witαh∗ respect to Pν∗. t α α C.L.T. FOR A TAGGED PARTICLE IN ASYMMETRIC SIMPLE EXCLUSION 5 Denote by X the position at time tN 0 of the tagged particle initially at tN ≥ the origin. We reprove the L.L.N. for the position of the Tagged Particle, which was previously obtained by Saada in [14]: Theorem 2.2. Fix t 0. Then, ≥ X tN v =(p q)(1 α)t t N −N−−−+−→ − − → ∞ in Pν∗-probability. α and the convergence to the Brownian motion, which was already obtained by Ferrari and Fontes in [6]: Theorem 2.3. Under Pν∗, α X v N tN t − B t N p q (1 α) −N−−−+−→ | − | − → ∞ p weakly, where B denotes the standard Brownian motion. t Another interesting property is the dependence on the initial configuration for the position of the Tagged Particle, which was previously obtained by Ferrari in [4]. Suppose p>q. Corollary 2.4. Fix t 0. Then for every ǫ>0, ≥ Nlim+ Eνα∗hX√tNN − Px(p=−0q)ααtN√(N1−η0(x))i2−ǫ =0. → ∞ Inthe hyperbolicscaling,wehaveseenabovethatforthecaseα=1/2the limit density fluctuation field at time t is the same as the initial one. This forced us to consider a longer time scale in order to observe other fluctuations than the shifted version of the initial ones. Henceforth, consider the ASEP evolving in the time scale N1+γ, with γ >0. In the sequel, we point out the restrictions needed in γ in order to obtain the results. Let α (0,1) and redefine the density fluctuation field on H S(R) by: ∈ ∈ 1 x vtN1+γ YtN,γ(H)= √N xXZH(cid:16) − N (cid:17)(ηtN1+γ(x)−α). (2.4) ∈ Weremarkhere,thanonecandefineinthehyperbolicscalingoftimethedensity fluctuationfieldasabove. Butinthatcasethecurrentcouldnotbedefinedthrough a fixed bond, instead it would have to be defined through a bond that depends on time(seesection9). AswewewanttoshowtheC.L.T.forthepositionofaTagged Particleusingthe relationbetweenthe densityofparticlesandthe currentthrough a fixed bond (4.3), we have the need to defined the density fluctuation field as in (2.2). As above, let Qγ be the probability measure on D(R+, ) induced by the density fluctuationNfield YN,γ and ν , let PN,γ = Pγ be theHp−rkobability measure on D(R+, 0,1 Z) induced. by ν andα the Mνaαrkov prνoαcess η speeded up by N1+γ α t anddenote{by}Eγ expectationwith respect to Pγ . Now, we state Theorem2.1 in να να this longer scaling: 6 PATRÍCIAGONÇALVES Theorem 2.5. Fix an integerk >1 and γ <1/3. Let Q bethe probability measure on C(R+, ) corresponding to a stationary Gaussian process with mean 0 and k H− covariance given by E [Y (H)Y (G)]=χ(α) H(u)G(u)du (2.5) Q t s ZR for every s,t 0 and H, G in . Then, the sequence (Qγ ) converges weakly ≥ Hk N N≥1 to the probability measure Q. Aswefollowthemartingaleapproach,themaindifficultyinprovingthistheorem is the Boltzmann-Gibbs Principle, which we canprove for γ <1/3and in this case is stated in the following way: Theorem 2.6. (Boltzmann-Gibbs Principle) Fix γ <1/3. For every t>0 and H S(R), ∈ t Nγ x 2 lim Eγ H η¯ (x)η¯ (x+1)ds =0. N→∞ ναhZ0 √N xXZ (cid:16)N(cid:17) s s i ∈ In order to keep notation simple, here and after we denote by X¯ the centered random variable X. Let PNνα∗,γ =Pγνα∗ be the probability measure on D(R+,{0,1}Z) induced by ν and the Markov process η speeded up by N1+γ. α∗ t By the results juststated, inthis longertime scalethe systemtranslatesintime at a certain velocity v. This allows us to deduce from the previous results the asymptotic behavior of the position of the Tagged Particle even in the longer time scale: Corollary 2.7. Fix t 0, suppose that p>q and γ <1/3. Then, ≥ XtN1+γ x(p=−0q)αtN1+γ(1−η0(x)) 0 √N − P α√N −N−−−+−→ → ∞ in Pγν∗-probability. α 3. Density Fluctuations in the Hyperbolic Scaling The aim of this section is to prove Theorem 2.1. We just give a sketch of the proofofthis result, since we aregoingto use similar techniques to the ones usedin chap. 11 of [9] when describing the equilibrium fluctuation field of the Symmetric Zero-Range process under diffusive scaling. Fix a positive integer k and recall the definition of the density fluctuation field in (2.2). The purpose is to show that YN converges to a process Y whose time- . . evolution is deterministic. Denote by A the operator v defined on a domain of L2(R) and by T ,t 0 t thesemigroupassociatedtoA.∇Fort 0,let betheσ-algebraonD([0{,T], ≥ }) t k generated by Y (H) for s t and H ≥in S(R)Fand set =σ( ). H− s ≤ F t 0Ft Toprovethetheoremweneedtoverifythat(QN)N 1istigSht≥andtocharacterize ≥ the limit field. To check the last assertion, we consider a collection of martingales associated to the empirical measure. Fix a function H S(R). Then: ∈ t 1 x MN,H =YN(H) YN(H) NH W (η )ds t t − 0 −Z0 √N Xx Z∇ (cid:16)N(cid:17) x,x+1 s ∈ C.L.T. FOR A TAGGED PARTICLE IN ASYMMETRIC SIMPLE EXCLUSION 7 is a martingale with respect to the filtration ˜ = σ(η ,s t), whose quadratic t s F ≤ variation is given by: t 1 x 2 NH c(x,x+1,η )+c(x+1,x,η ) ds, Z0 N2 xXZ(cid:16)∇ (cid:16)N(cid:17)(cid:17) h s s i ∈ where W (η) denotes the instantaneous current between the sites x and x+1: x,x+1 W (η)=c(x,x+1,η) c(x+1,x,η) x,x+1 − and x x+1 x NH =N H H . ∇ (cid:16)N(cid:17) (cid:16) (cid:16) N (cid:17)− (cid:16)N(cid:17)(cid:17) Using the fact that NH(x) = 0, the integral part of the martingale is x Z∇ N equal to: P ∈ t 1 x NH W¯ (η ) ds. x,x+1 s Z0 √N xXZ∇ (cid:16)N(cid:17)h i ∈ As we need to write the expression inside last integral in terms of the fluctuation fieldYN,weareabletoreplacethe functionW¯ (η )by(p q)χ(α)[η (x) α], s x,x+1 s − ′ s − with the use of the: Theorem 3.1. (Boltzmann-Gibbs Principle) For every local function g, for every H S(R) and every t>0, ∈ t 1 x 2 Nl→im∞Eναh(cid:16)Z0 √N xXZH(cid:16)N(cid:17)nτxg(ηs)−g˜(α)−g˜′(α)[ηs(x)−α]ods(cid:17) i=0, ∈ where g˜(α)=E [g(η)]. να In spite of considering the ASEP in the hyperbolic scaling, the proof of last result is very close to the one presentedfor the Zero-Rangeprocess in the diffusive scaling, and for that reason we have omitted it. Assume now, that (Q ) is tight and let Q be one of its limiting points. By N N 1 the result just stated and sin≥ce lim E [(MN,H)2]=0, under Q N→+∞ να t t Y (H)=Y (H)+ Y (AH)ds. (3.1) t 0 s Z 0 So, dY (H) = Y (AH). Take r < t, and note that d < Y ,T H >= 0. As a dt t t dr r t−r consequence, Y (H)=Y (T H) where T H(u)=H(u+vt). t 0 t t It is easy to show that Q restricted to , is a Gaussian field with covariance 0 F given by E (Y (G)Y (H)) = χ(α) < G,H > and it is immediate that the limit Q 0 0 field has covariance given by (2.3). Tofinishtheproof,itremainstoshowthat(Q ) istightwhoseprooffollows N N 1 ≥ closely the same arguments as the ones for the Zero-Range process in the diffusive scaling. Lastly,wenotethatoncetheprocessevolvesonZandthehyperbolicscale is considered, we must take an integer k>2 in order have the density fluctuations field well defined in . k H− 8 PATRÍCIAGONÇALVES 4. Law of Large Numbers and Central Limit Theorem for the Position of the Tagged Particle In this section we prove Theorems 2.2 and 2.3 following the same arguments as Jara and Landim in [7]. For that reason we give an outline of the proofs. First we state the C.L.T. for the current through a fixed bond. For a site x, denote the currentthroughthe bond [x,x+1] by JN (t), as the total number of x,x+1 jumps from the site x to the site x+1 minus the total number of jumps from the site x+1 to the site x during the time interval [0,tN]. Since JN (t)= η (x) η (x) , −1,0 xX0(cid:16) t − 0 (cid:17) ≥ the current can be written in terms of the density fluctuation field as 1 JN (t) E [JN (t)] =YN(T H ) YN(H ), √Nn −1,0 − να −1,0 o t t 0 − 0 0 where H is the Heaviside function, H (u)=1 (u). By approximatingH by a 0 0 [0, ) 0 sequence(G ) ,definedforeachu RbyG (∞u)=(1 u)+1 (u),weobtain: n n≥1 ∈ n −n [0,∞) Proposition 4.1. For every t 0, ≥ J¯N (t) 2 nlim+ Eναh −√1,N0 −(YtN(TtGn)−Y0N(Gn))i =0 → ∞ uniformly in N. Proof. For a site x, consider the martingale MN (t) equal to x,x+1 t JN (t) NW (η ) vNη (x)ds (4.1) x,x+1 −Z x,x+1 s − s 0 whose quadratic variation is given by t <MN > =N c(x,x+1,η )+c(x+1,x,η ) ds. x,x+1 t Z0 n s s o Since the number of particles is preserved,it holds that: JN (t) JN (t)=η (x) η (x) x−1,x − x,x+1 t − 0 for all x Z, t 0, and we have that ∈ ≥ 1 x YN(T G ) YN(G )= G J¯N (t) J¯N (t) . t t n − 0 n √N xXZ n(cid:16)N(cid:17)n x−1,x − x,x+1 o ∈ Making a summation by parts and using the explicit knowledgeof G , last expres- n sion can be written as J¯N (t) 1 Nn 1 −√1,N0 −hYtN(TtGn)−Y0N(Gn)i= √N xX=1NnJ¯xN−1,x(t). Representingthe currentJN (t)intermsofthe martingalesMN (t), the right x 1,x x 1,x hand side of the last express−ion becomes equal to − 1 Nn 1 1 t 1 Nn MN (t)+ [W¯ (η ) v(η (x 1) α)]ds. (4.2) √N Xx=1Nn x−1,x √N Z0 nxX=1 x−1,x s − s − − C.L.T. FOR A TAGGED PARTICLE IN ASYMMETRIC SIMPLE EXCLUSION 9 The martingale term convergesto 0 in L2(P ) as n + , since we can estimate να → ∞ their quadratic variation by Nt, use the fact that they are orthogonal to obtain that its L2(P )-norm is bounded above by Ct. να Nn Making an elementary computation it is easy to show that the L2(P )-norm of να the integral term is bounded above by C. Taking the limit as n , the proof is concluded. n →∞ (cid:3) Putting together, last result and the C.L.T. for the empirical measure, it holds: Theorem 4.2. Fix x Z and let ∈ 1 ZN = JN (t) E [JN (t)] . t √Nn x,x+1 − να x,x+1 o Then, for every k 1 and every 0 t < t < .. < t , (ZN,..,ZN) converges in ≥ ≤ 1 2 k t1 tk law to a Gaussian vector (Z ,..,Z ) with mean zero and covariance given by t1 tk E [Z Z ]=χ(α)v s Q t s | | provided s t. ≤ Assumenow,theinitialmeasuretobeν . LetX bethepositionoftheTagged α∗ tN ParticleattimetN 0initiallyattheorigin. Fix,apositiveintegern. Sinceweare ≥ consideringtheone-dimensionalsetting,particlescannotjumpoverotherparticles, and therefore it holds the following relation: n 1 − X n = JN (t) η (x) (4.3) { tN ≥ } n −1,0 ≥ xX=0 t o which allows, together with the previous results, to obtain L.L.N. and the C.L.T. for the position of the Tagged Particle. Now, we give a sketch of the proof of this results. Proof of Theorem 2.2. Inorderto showL.L.N. for the TaggedParticle,denote by a the smallestinteger ⌈ ⌉ larger or equal to a, fix u > 0 and take n = uN in (4.3). Using the martingale ⌈ ⌉ decomposition of the current (4.1) and Theorem 4.2 it is easy to show that JN (t) 1,0 − (p q)χ(α)t N −N−−−+−→ − → ∞ in P -probability. Since < πN,1 > converges in probability to αu, we obtain να t [0,u] that N→lim+∞Pνα∗hXNtN ≥ui=(cid:26) 01,, iiff ((pp−−qq))χχ((αα))tt <≥ααuu . For u<0 we obtain a similar result, which concludes the proof. ⊔⊓ We proceedbyprovingthe convergenceofthe TaggedParticleprocess,properly centered and rescaled, to the standard Brownianmotion. 10 PATRÍCIAGONÇALVES Proof of Theorem 2.3. Let W = 1 (X v N). The result follows from showing the convergence of tN √N tN − t finite dimensionaldistributions of W to those of Brownianmotion together with tN tightness. Using (4.3), Theorems 2.1 and 4.2 above, it is not hardto show that under Pν∗, α k 1, 0 t <..<t , (W ,..,W ) converges in law to a Gaussian vector ∀ ≥ ∀ ≤ 1 k t1N tkN (W ,...,W ) with mean zero and covariance given by t1 tk E W W = p q (1 α)s Q t s h i | − | − for 0 s t. ≤ ≤ Toendtheproofitremainstoshowtightness. Forthatweusearelationbetween the ASEP and a Zero-Range process, as Kipnis in [8]. For the latter, the product measures µ with marginals given by µ η(x) = k = α(1 α)k are extremal α α { } − invariant. This process has space state = NZ and generator defined on local functions X by Ωf(η)= 1 [pf(ηx,x 1)+qf(ηx,x+1) f(η)], η(x) 1 − Xx Z { ≥ } − ∈ where p+q =1 and η(z), if z =x,y 6 ηx,y(z)=η(x) 1, if z =x .  − η(y)+1, if z =y  The process can also be reversed with respect to any µ , and the reversed process α is denoted by ηˆ, whose generator Ωˆ is the same as Ω, except that p is replaced by q and vise-versa. The position of the Tagged Particle in the Zero-Range representation becomes the current through the bond [−1,0]: Xt = −Nt++Nt−, where Nt+ (resp. Nt−) is the number ofparticlesthat jumped fromsite 1to site 0during the time interval − [0,t] (resp. from site 0 to 1). − As a consequence, the proof ends if we show tightness of the distributions of v1√(tNN) and v2√(tNN), where v1(t) = Nt+ −qt(1−α) and v2(t) = Nt− −pt(1−α). With this purpose, we use Theorem 2.1 of [15], with a slightly different definition for weakly positively associated increments given in [16], namely: Definition 1. A process v(t):t 0 has weakly positive associated increments if for all coordinatewise incr{easing fu≥nct}ions f :R R, g :Rn R → → E [f(v(t+s) v(s))g(v(s ),..,v(s ))] E [f(v(t))]E [g(v(s ),..,v(s ))], µα − 1 n ≥ µα µα 1 n for all s,t 0 and 0 s <..<s =s (weakly negative associated in the sense of 1 n ≥ ≤ the reversed inequality). Following the same argumentsas in Theorem2 of[8]we note that the processes Nt+ and Nt−, have weakly positive associated increments. In the sake of complete- ness, we give a sketch of the proof of this result for the process N+. t Let s,t 0 and 0 s < ... < s = s, and f,g coordinatewise increasing 1 n functions f≥:R R, g :≤Rn R. We have to show that → → E [f(N+ N+)g(N+,..,N+)] E (f(N+))E (g(N+,..,N+)). µα t+s− s s1 sn ≥ µα t µα s1 sn