\magnification=1200 %\voffset -15mm \scrollmode \baselineskip 15pt \overfullrule=0pt \def\bbz{{Z\kern-0.45emZ}} \def\bbr{{I\kern-0.3emR}} \def\bx{{\bf X}} \def\hbx{{\hat{\bf X}}} \def\by{{\bf Y}} \def\bbzd{{\bbz^d}} \def\bbrd{{\bbr^d}} \def\bbn{{I\kern-0.3emN}} \def\a{\alpha} \def\e{\eta} \def\el{\eta'} \def\ex{\eta(x)} \def\et{\eta_t} \def\etl{\eta'_t} \def\eo{\eta(0)} \def\elx{\eta'(x)} \def\elo{\eta'(0)} \def\st{\sigma_t} \def\tu{\tau_1} \def\tn{{1\over {2n+1}}} \def\ha{{1\over 2}} \def\en{E} \def\tq{t^{1/4}} \def\tl{\tau_{x'}} \def\tv{\tau_{vt}} \def\tvt{\tau_{vt+a\sqrt t}} \def\tva{\tau_{vt+a\sqrt t}} \def\tx{\tau_{(2k+1)x}} \def\ty{\tau_{(2k+1)y}} \def\th{\theta} \def\xt{X_t} \def\nh{{1\over \sqrt n}} \def\sn{\sum_{x=-n}^n} \def\be{\beta} \def\vep{{\varepsilon}} \def\de{{\delta}} \def\la{{\lambda}} \def\ga{{\gamma}} \def\Ga{{\Gamma}} \def\si{{\sigma}} \def\de{{\delta}} \def\ze{{\zeta}} \def\om{{\omega}} \def\Om{\Omega} \def\up{\Upsilon} \def\heta{{\hat\xi}} \def\hmu{{\mu'}} \def\nul{{\nu_\la}} \def\nur{{\nu_\rho}} \def\nurl{{\nu_{\rho,\la}}} \def\hnurl{{\nu'_{\rho,\la}}} \def\12{{\hbox{${1\over 2}$}}} \def\ofp{(\Omega, $ $ {\cal F}, $ $ P)} \def\lar{{\leftarrow}} \def\w{{\wedge}} \def\cX{{\cal X}} \def\cY{{\cal Y}} \def\cL{{\cal L}} \def\qed{$\clubsuit$\vskip 2truemm} \def\xil{{\it et al. }} \def \ie{{\it i.e.}} \def\limt{\lim_{t\to\infty}} \def\limn{\lim_{N\to\infty}} \def\lime{\lim_{\vep\to 0}} \def\jrt{{J_{rt,t}}} \def\jzt{{J_{0,t}}} \def\jt{{J_{1,t}}} \def\j2t{{J_{2,t}}} \def\nzt{{N_{0,t}}} \def\nt{{N_{1,t}}} \def\n2t{{N_{2,t}}} \def\today{\rightline{\ifcase\month\or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi \space\number\day,\space\number\year}} %%%%%%%%%%%%%%%%% EQUAZIONI CON NOMI SIMBOLICI %%% %%% per assegnare un nome simbolico ad una equazione basta %%% scrivere \Eq(...) o, in \eqalignno, \eq(...) o, %%% nelle appendici, \Eqa(...) o \eqa(...): %%% dentro le parentesi e al posto dei ... %%% si puo' scrivere qualsiasi commento; %%% per assegnare un nome simbolico ad una figura, basta scrivere %%% \geq(...); per avere i nomi %%% simbolici segnati a sinistra delle formule e delle figure si deve %%% dichiarare il documento come bozza, iniziando il testo con %%% \BOZZA. Sinonimi \Eq,\EQ,\EQS; \eq,\eqs; \Eqa,\Eqas;\eqa,\eqas. %%% All' inizio di ogni paragrafo si devono definire il %%% numero del paragrafo e della prima formula dichiarando %%% \numsec=... \numfor=... (brevetto Eckmannn); all'inizio del lavoro %%% bisogna porre \numfig=1 (il numero delle figure non contiene la sezione. %%% Si possono citare formule o figure seguenti; le corrispondenze fra nomi %%% simbolici e numeri effettivi sono memorizzate nel file \jobname.aux, che %%% viene letto all'inizio, se gia' presente. E' possibile citare anche %%% formule o figure che appaiono in altri file, purche' sia presente il %%% corrispondente file .aux; basta includere all'inizio l'istruzione %%% \include{nomefile} %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \global\newcount\numsec\global\newcount\numfor \global\newcount\numfig \gdef\profonditastruttura{\dp\strutbox} \def\senondefinito#1{\expandafter\ifx\csname#1\endcsname\relax} \def\SIA #1,#2,#3 {\senondefinito{#1#2} \expandafter\xdef\csname #1#2\endcsname{#3} \else \write16{???? ma #1,#2 e' gia' stato definito !!!!} \fi} \def\etichetta(#1){(\veroparagrafo.\veraformula) \SIA e,#1,(\veroparagrafo.\veraformula) \global\advance\numfor by 1 % \write15{\string\FU (#1){\equ(#1)}} \write16{ EQ \equ(#1) == #1 }} \def \FU(#1)#2{\SIA fu,#1,#2 } \def\etichettaa(#1){(A\veroparagrafo.\veraformula) \SIA e,#1,(A\veroparagrafo.\veraformula) \global\advance\numfor by 1 % \write15{\string\FU (#1){\equ(#1)}} \write16{ EQ \equ(#1) == #1 }} \def\getichetta(#1){Fig. \verafigura \SIA e,#1,{\verafigura} \global\advance\numfig by 1 % \write15{\string\FU (#1){\equ(#1)}} \write16{ Fig. \equ(#1) ha simbolo #1 }} \newdimen\gwidth \def\BOZZA{ \def\alato(##1){ {\vtop to \profonditastruttura{\baselineskip \profonditastruttura\vss \rlap{\kern-\hsize\kern-1.2truecm{$\scriptstyle##1$}}}}} \def\galato(##1){ \gwidth=\hsize \divide\gwidth by 2 {\vtop to \profonditastruttura{\baselineskip \profonditastruttura\vss \rlap{\kern-\gwidth\kern-1.2truecm{$\scriptstyle##1$}}}}} } \def\alato(#1){} \def\galato(#1){} \def\veroparagrafo{\number\numsec}\def\veraformula{\number\numfor} \def\verafigura{\number\numfig} \def\geq(#1){\getichetta(#1)\galato(#1)} \def\Eq(#1){\eqno{\etichetta(#1)\alato(#1)}} \def\eq(#1){\etichetta(#1)\alato(#1)} \def\Eqa(#1){\eqno{\etichettaa(#1)\alato(#1)}} \def\eqa(#1){\etichettaa(#1)\alato(#1)} \def\eqv(#1){\senondefinito{fu#1}$\clubsuit$#1\else\csname fu#1\endcsname\fi} \def\equ(#1){\senondefinito{e#1}\eqv(#1)\else\csname e#1\endcsname\fi} \let\EQS=\Eq\let\EQ=\Eq \let\eqs=\eq \let\Eqas=\Eqa \let\eqas=\eqa %%%%%%%%%%%%%%%%%% Numerazione verso il futuro ed eventuali paragrafi %%%%%%% precedenti non inseriti nel file da compilare \def\include#1{ \openin13=#1.aux \ifeof13 \relax \else \input #1.aux \closein13 \fi} \openin14=\jobname.aux \ifeof14 \relax \else \input \jobname.aux \closein14 \fi %\openout15=\jobname.aux %%%%%%%%%%%%%%%%%%%%%%%%%%%% %\BOZZA \rightline {\today} \centerline {\bf Shock fluctuations in the asymmetric simple exclusion process} \vskip 2truemm \centerline { P.\ A.\ Ferrari, L.\ R.\ G.\ Fontes} \vskip 1truemm \centerline {\it Universidade de S\~ao Paulo} \vskip 1truemm \noindent {\bf Summary.} We consider the one dimensional nearest neighbors asymmetric simple exclusion process with rates $q$ and $p$ for left and right jumps respectively; $qq\ge 0$. Let $S(t)$ denote the corresponding semigroup. In words, the process describes the evolution of particles in $\bbz$ under the constriction that at most one particle is allowed at each site. Particles jump to left and right nearest neighbor empty sites at rate $q$ and $p$ respectively. No jumps are allowed to occupied sites. We consider as initial measure $\nurl$, the product measure with densities $\rho$ and $\la$ to the left and right of the origin respectively. We fix $\rho<\la$. We say that a random position $X_t$ is a microscopic shock if the distribution of $\tau_{X_t}\eta_t$ has asymptotic distributions $\nu_\rho$ and $\nu_\la$ to the left and right of the origin respectively, uniformly in $t$. The operator $\tau_x$ is translation by $x$; $\nur$ and $\nul$ stand for product measures with density $\rho$ and $\la$ respectively. Ferrari, Kipnis and Saada (1990) showed that there exists a shock for this system. Ferrari (1992) showed that $Z_t$, the position at time $t$ of a second class particle with respect to $\eta_t$ is a shock. The motion of the second class particle can be described as follows. It jumps to empty left and right sites at rate $p$ and $q$ respectively and interchanges positions with left and right particles at rate $q$ and $p$ respectively. The process $\tau_{Z_t}\eta_t$ is Markovian and under initial distribution $\nurl$ converges weakly to an invariant measure with asymptotic product distributions with densities $\rho$ and $\la$. Our main result is the following \proclaim Theorem 1.1. Assume that the process $\eta_t$ has initial distribution $\nurl$. Let $Z_t$ be the position of the shock given by a second class particle initially put at the origin. Then $$ EZ_t = (p-q)(1-\la-\rho)t. \Eq(1.10) $$ Diffusion coefficient $$ D := \limt {E(Z_t)^2 - (EZ_t)^2 \over t} = (p-q) { \rho (1-\rho) + \la (1-\la) \over \la - \rho } \Eq(1.20) $$ Dependence on the initial configuration. \hfill\break Let $N_t(\eta) = \sum_{x=0}^{(p-q)(\la-\rho)t} (1-\eta(x))-\sum_{x=-(p-q)(\la-\rho)t}^0 \eta(x)$. Then $$ \limt {1\over t} E[Z_t -(\la-\rho)^{-1} N_t(\eta_0)]^2 = 0. \Eq(1.30) $$ In chapter 5 of Spohn (1991) \equ(1.10) was proven and (1.2) conjectured. Boldrighini et al.\ (1989) performed computer simulations confirming \equ(1.20). G\"artner and Presutti (1989) showed \equ(1.30) for $\rho=0$ and $p=1$. Ferrari (1992) showed that \equ(1.20) and \equ(1.30) are equivalent and that the right hand side of \equ(1.20) is a lowerbound for $D$ and \equ(1.30) for $\rho=0$ and all $p>q$. We show \equ(1.10) and \equ(1.20) using recent results relating the expected value and the variance of a tagged particle with the variance of the current of particles through a fixed or travelling position (Ferrari and Fontes (1993)). The following Theorem is a corollary to \equ(1.30). \proclaim Theorem 1.2. Convergence to the finite dimensional distributions of Brownian motion. Let $W(t)$ be Brownian motion with diffusion coefficient $D$. Then under the conditions of Theorem 1.1 $$ \lime \vep^{1/2} ( Z_{\vep^{-1}.} - EZ_{\vep^{-1}.}) = W(.) $$ weakly, in the sense of the finite dimensional distributions. It is well known that this process is related to the unviscous Burgers equation. This is the non linear equation for $u(r,t) \in [0,1]$, $$ {\partial u \over \partial t} = -\theta {\partial [u(1-u)]\over \partial r} \Eq(11) $$ The hydrodynamic limit plus local equilibrium give the following: Let $u_0(r)$ be a piecewise continuous function, and let $\nu^\vep_{u_0}$ be a family of product measures with marginals $\nu^\vep_{u_0}(\eta(\vep^{-1}r))=u_0(r)$. Then $$ \lime \nu^\vep_{u_0} S(\vep^{-1}t) \tau_{\vep^{-1}r} = \nu_{u(r,t)} \Eq(12) $$ in the continuity points of $u(r,t)$, the solution of \equ(11) with initial condition $u(r,0) = u_0(r)$. For general initial conditions this theorem is a consequence of the law of large numbers of Rezakhanlou (1990) and the proof of local equilibrium of Landim (1992). Before them, Liggett (1975, 1977) has shown this result for the case $r=0$. Rost (1982) and Benassi and Fouque (1987) showed \equ(12) for decreasing initial profiles. Andjel and Vares (1987) proved the hydrodynamical limit for the shock case. Then Benassi, Fouque, Saada and Vares (1991) computed the limit for monotone initial profiles. In the shock case \equ(12) means that under initial distribution $\nurl$, a traveller moving at deterministic velocity $r$ observes asymptotically that the particles are distributed as $\nur$ for $r>v$ and $\nul$ for $rvt\}$ is the solution of the Burgers equation when $u_0(r) = \la$ for $r>0$ and $\rho$ for $r\le 0$. It was conjectured that when $r=v$ the system converges to a fair mixture of $\nur$ and $\nul$. This was proven by Wick (1985) and De Masi et al.\ (1988) for $\rho=0$ and by Andjel, Bramson and Liggett (1988) for $\la+\rho =1$. Our next result shows the conjecture for all cases. The proof is based on the central limit theorem for $Z_t$ established in Theorem 1.2. Let $w(r,t) = P(W(t)\le r) = (1/\sqrt{2\pi Dt})\int_{-\infty}^r \exp(-s^2/2Dt)ds$, the normal distribution with variance $Dt$. \proclaim Theorem 1.3. Dynamical phase transition. Let $v = (p-q)(1-\la-\rho)$. Then $$ \lime \nurl S(t) \tau_{vt+a t^{1/2}} = (1-w(a,1)) \nur + w(a,1) \nul \Eq(1.40) $$ Let $\up^\vep_{t}$ be the fluctuations fields defined by $$ \up^\vep_{t}(\Phi) = \vep^{1/2} \sum_{x\in \bbz} \Phi(\vep x)[\eta_{\vep^{-1}t}(x) - E\eta_{\vep^{-1}t}(x) ], \Eq (168) $$ for smooth integrable test functions $\Phi$. For $t=0$, if $\eta_0$ is distributed according to $\nurl$, $$ \lime \up^\vep(\Phi) = \up(\Phi), \Eq (162) $$ where $\up(\Phi)$ is Gaussian white noise with mean zero and covariance $$ E(\up(\Psi)\up(\Phi)) = \int u_0(r)(1-u_0(r)) \Psi(r) \Phi(r) dr. \Eq (163) $$ where $u_0(r) = \la 1\{r\ge 0\} + \rho 1\{r<0\}$. Our next result proves a conjecture of Spohn (1991) when the initial condition is a shock. \proclaim Theorem 1.4. Convergence of the fluctuation fields. Assume that the initial distribution of the process is $\nurl$. Let $v=(p-q)(1-\rho-\la)$. Let $u(r,t) = \la 1\{r> vt\} + \rho 1\{r < vt\}+\ha (\la+\rho)1\{r = vt\}$. As $\vep \to 0$, the fluctuation fields $\up^\vep_{t}$ defined in \equ(168) converge in a weak sense to the conservative solution $\up_{t}$ of the nonhomogeneous linear equation $$ {\partial \over \partial t} \up_t(r) = {\partial \over \partial r} (1-2u(r,t))\up_t(r), \Eq(6.5) $$ with initial condition $\up$, the Gaussian field with zero mean and covariance given by \equ(163). For $p=1$ and $\rho=0$ the convergence away from the shock have been obtained by Benassi and Fouque (1992). Theorem 1.4 is a consequence of the $L_2$ convergence of the fluctuation fields established in the next theorem. The weak solutions of \equ(6.5) present a singularity at the point $(vt,t)$ due to the discontinuity of $u(r,t)$ at $r=vt$. For this reason there is no unique solution. However there is only one conservative solution. To better describe it let us introduce some notation. Assume that $\Phi$ is the indicator of the interval $(a_1,a_2)$. For $i=1,2$ let $$ b_i(t) = \cases { a_i - (p-q)(1-2\rho) t &if $a_i < vt$ \cr a_i - (p-q)(1-2\la) t &if $a_i > vt$ \cr} $$ Then $\up_{t}$, the solution of \equ(6.5) is given by the following. $$ \int \Upsilon_{t}(r) \Phi(r)dr = \int_{a_1}^{a_2} \Upsilon_{t}(r) dr = \int_{b_1(t)}^{b_2(t)} \Upsilon_{0}(r) dr $$ We can interpret this by saying that if $vt\in (a_1,a_2)$ then, the fluctuations present in the interval $(-(p-q)(\la-\rho)t,(p-q)(\la-\rho)t)$ at time zero concentrate in the point $vt$ at time $t$. Formula (1.12) below says that these fluctuations are present in the scale $\sqrt t$. Indeed they reflect the shock fluctuations that occur in this scale. \proclaim Theorem 1.5. Let $E$ be the expected value with respect to the process with initial measure $\nurl$. Let $A_\vep = \bbz \cap (\vep^{-1}a_1,\vep^{-1}a_2)$, $B_\vep(t) = \bbz \cap (\vep^{-1}b_1(t),\vep^{-1}b_2(t))$. Then $$ \lime \vep E\left( \sum_{x\in A_\vep} [\eta_{\vep^{-1}t}(x) - E\eta_{\vep^{-1}t}(x) ] - \sum_{x\in B_\vep(t)} (\eta_0(x) - E\eta_0(x)) \right)^2=0. \Eq (68) $$ Let $c>0$, $C_\vep(t) = \bbz \cap (\vep^{-1}vt-\vep^{-1/2}c,\vep^{-1}vt+\vep^{-1/2}c)$ and $K_\vep(t) = \bbz \cap (-\vep^{-1}t(p-q)(\la-\rho), \vep^{-1}t(p-q)(\la-\rho))$. Then $$ \lime \vep E\left( \sum_{x\in C_\vep(t)} [\eta_{\vep^{-1}t}(x) - E\eta_{\vep^{-1}t}(x) ] - T_{\vep^{-1/2}c}\sum_{x\in K_\vep(t)} (\eta_0(x) - E\eta_0(x)) \right)^2 = 0, \Eq (67) $$ where $T_{c}$ is truncation by $c$: $$ T_{c}F(.) = \cases { F(.) &if $\vert F(.) \vert \le c$ \cr c & if $F(.)>c $ \cr - c & if $F(.)<-c. $ \cr } $$ Note that $C_\vep(t)$ is an interval of length proportional to $\vep^{-1/2}$ around the macroscopic point $vt$. When $c\to\infty$, \equ(67) says that the fluctuations at time $t$ in a region of length proportional to $\sqrt t$ around $vt$ are given by the fluctuations at time $0$ in a region of length proportional to $t$. \vskip 3truemm \noindent {\bf 2. Graphical construction and coupling.} \vskip 2truemm \numsec=2 \numfor=1 The main tool to deal with this process is coupling, the joint realization of two versions of the process with different initial configurations. One way to define a coupling is via the joint generator (Liggett (1976), (1985)). Another way is by graphical constructing the process. This is something like to use the same random numbers for different initial configurations. To describe the graphical construction attach two Poisson processes to each pair of sites $(x,x+1)$. One of rate $p$ and the other of rate $q$. A Poisson process is a sequence of random times. To each time of the Poisson process of rate $p$ an arrow going from $x$ to $x+1$ is drawn and for the times of the process of rate $q$ an arrow is drawn from $x+1$ to $x$. The product of these Poisson processes induces a probability space $\ofp$. We discard the null event ``two arrows occur at the same time''. Given an initial configuration $\eta$, the configuration at time $t$ for the set of arrows $\om$, starting from $\eta$ is denoted $\eta^{\eta,\om}_t$ and is constructed in the following way. When an arrow appears from site $x$ to $y$, if there is a particle at $x$ and no particle at $y$ then, after the arrow the particle will be at $y$ and $x$ will be empty. We denote $\eta^\eta_t$ the random process defined on $\ofp$ with initial configuration $\eta$. Consider now two initial configurations $\eta^0$ and $\eta^1$ and write $\eta^i_t=\eta^{\eta^i}_t$, for the configurations at time $t$. Use the same structure of arrows for $\eta^0_t$ and $\eta^1_t$. In this case $(\eta^0_t,\eta^1_t)$ is the ``basic coupling''(Liggett (1985)). If $\eta^0(x) \le \eta^1(x)$ for all $x\in\bbz$ (in this case we say $\eta^0 \le \eta^1$) then for all times $\eta^0_t \le \eta^1_t$. This property is called attractivity. Let $\nur$ be the product measure with density $\rho$. Take $\rho <\la$ and realize jointly the measures $\nur$ and $\nul$ in the following way. Let $U(x)\in [0,1]$ be i.i.d.\/ uniformly distributed random variables. Then define $\eta^0(x) = 1\{U(x)\le \rho\}$, $\eta^1(x) = 1\{U(x)\le \la\}$. Hence, $\eta^0$ is distributed according to $\nur$, $\eta^1$ is distributed according to $\nul$ and $\eta^0\le \eta^1$. Define $\si(x) = \eta^0(x)$ and $\xi(x) = \eta^1(x) - \eta^0(x)$. We say that the distribution of $(\si,\xi)$ has the good marginals if the $\si$ marginal is $\nur$ and the $\si+\xi$ marginal is $\nul$. Calling $\pi_2$ the distribution of $(\si,\xi)$, we have that $$ \hbox{$\pi_2$ is a product measure with the good marginals.} \Eq (2.20) $$ Define $\si_t(x) = \eta^0_t(x)$ and $\xi_t(x) = \eta^1_t(x) - \eta^0_t(x)$. The motion of $(\si_t,\xi_t)$ obeys the following rule. The $\si$ particles have priority over the $\xi$ particles: when an arrow from a $\si$ particle to a $\xi$ particle appears, then after the arrow the particles interchange positions. Otherwise the particles interact by exclusion. We say that the $\xi$ particles behave as ``second class particles''. If the distribution of $(\si_0,\xi_0)$ has the good marginals, the same is true for the distribution of $(\si_t,\xi_t)$. We call $S_2(t)$ the corresponding semigroup. Let $\nu_2$ be a translation invariant measure with the good marginals and $\nu'_2 = \nu_2(.\vert \xi(0)=1)$. Let $X_t$ be the position of the $\xi$ particle initially at the origin. Let $S'_2(t)$ be the semigroup of the process as seen from the second class particle $(\tau_{X_t}\si_t,\tau_{X_t}\xi_t)$. The key tool in Ferrari, Kipnis and Saada (1991) to show that $X_t$ is a microscopic shock is the following. If $\nu_2$ is translation invariant and has the good marginals, then $$ (\nu_2S_2(t))'=\nu'_2S'_2(t) \Eq(2.19) $$ In words, the law of the process as seen from the tagged second class particle looks as the law of the process seen from the origin conditioned to have a second class particle at the origin. Ferrari (1992) showed the following law of large numbers. Let $\nu_2$ have the good marginals, then under initial measure $\nu'_2$, $$ \limt {X_t \over t} = v \ \ \ \hbox{ almost surely.} \Eq(3.0) $$ Let $X^1$ denote the position of the first $\xi$ particle to the right of the origin. \proclaim Lemma 2.1. There exist positive constants $c', c''$ such that $$ \sup_{\nu_2}\nu'_2 (X^1 > n) \le c' \exp(-c'' n) $$ where the sup is taken over $\{\nu_2: \nu_2$ is translation invariant and has the good marginals$\}$. \noindent {\bf Proof.} Write $$ \eqalign { \nu'_2 (X^1 &> n) (\la-\rho)\le \nu_2\left(\sum_{x=1}^n \xi(x)=0\right) \cr &\le \nu_2\left(\sum_{x=1}^n \xi(x)=0,\ \left\vert\sum_{x=1}^n \si(x)-n\rho\right\vert \le \vep n ,\right. \left.\left\vert\sum_{x=1}^n (\si(x)+\xi(x))-n\la\right\vert \le \vep n \right) \cr &\quad + \nu_2 \left(\left\vert \sum_{x=1}^n \si(x)- n\rho\right\vert > \vep n \right) + \nu_2 \left(\left\vert \sum_{x=1}^n (\si(x)+\xi(x))- n\la\right\vert > \vep n \right)}\Eq(20) $$ For $\vep<(\la-\rho)/2$, the first term in the right hand side of \equ(20) vanishes. The second and third term depend only on the marginals $\nur$ and $\nul$ respectively. The result follows from large deviations of Bernoulli measures. \qed Using the same arrows there is a natural coupling between $(\si_t,\xi_t)$ with initial measure $\pi'_2$ and $\eta_t$ with initial measure $\nurl$. To describe it one let $(\si,\xi)$ to be a configuration taken from the distribution $\pi'_2$. Now mark independently the $i$-th $\xi$ particle as $\ga$ with probability $(p/q)^i/(1+(p/q)^i)$, otherwise as $\ze$. Then consider the process $(\si_t,\ga_t,\ze_t)$ with priorities $\si$ over $\ga$ over $\ze$. In this way $\si_t$ has distribution $\nur$ for all $t$, $\eta_t=\si_t+\ga_t$ has distribution (absolutely continuous with respect to) $\nurl S(t)$ and $\si_t+\ga_t+\ze_t$ has distribution $\nul$. See Ferrari, Kipnis and Saada (1991) and Ferrari (1992) for details. \vskip 3truemm \noindent {\bf 3. Tagged second class particles and currents.} \numsec=3\numfor=1 \vskip 2truemm Consider the joint process $(\si_t,\xi_t)$ described in the previous section. Define the current of $\xi$ particles as $\j2t :=$ number of $\xi$ particles to the left of the origin at time $0$ and to the right of the origin at time $t$ minus number of $\xi$ particles to the right of the origin at time $0$ and to the left of the origin at time $t$. Analogously define $\jzt$ for the current of $\si$ particles and and $\jt$ for the total current of $\si+\xi$ particles. Consider a configuration $(\si,\xi)$ taken from $\pi'_2$, the measure $\pi_2$ conditioned to have a $\xi$ particle at the origin. This configuration has $\xi(0)=1$ and $\si(0)=0$, \ie, it has a $\xi$ particle at the origin. Let $\si^*(x) = 1\{x\ne 0\} \si(x) + 1\{x=0\}(1-\si(x))$ and analogously $\xi^*$. Now, using the same arrows, couple $(\si_t, \xi_t)$ with $(\si_t,\xi^*_t)$. At time $t$ the two processes will differ at only one site whose position is called $R_t$. Similarly, coupling $(\si_t, \xi_t)$ with $(\si^*_t,\xi^*_t)$ we get only one discrepancy located at a position denoted $\bar R_t$. In words, $R_t$ is like a third class particle, while $\bar R_t$ is a second class particle with respect to $\si_t$ but has priority over $\xi_t$. %A perturbation of a configuration $\eta$ with a second class particle at %the origin is the discrepancy which exists in the %coupled simple exclusion process starting from $\eta$ and $\eta^\ast$ %respectively, where %$\eta^\ast$ is equal to $\eta$ everywhere except at the origin; there it can % have either no particle (first case) or a first class particle (second case). %Let $R_t$ (resp. $\bar R_t$) be the position of the perturbation in the first %case (resp. second case). \proclaim Theorem 3.1. Let $(\si_t,\xi_t)$ be the joint process of first and second class particles with initial product measure $\pi_2$ defined in \equ(2.20). Let $X_t$ be the position of the tagged second class particle put initially at the origin. Then it holds that $$ E \j2t = (\la - \rho) E X_t \Eq(3.9) $$ where the expected values are taken with respect to the process with initial distribution $\pi_2$. Furthermore, denoting the variance by $V$, $$ \eqalign { V\j2t &= (\la-\rho)^2 VX_t - (\la-\rho)(1-(\la-\rho)) E(X_t) \cr &\qquad+ 2(\la-\rho)(1-\la) (E(R_t)^+ - E(R_t - X_t)^+)\cr &\qquad+ 2(\la-\rho)\rho (E(\bar R_t)^+ - E(\bar R_t - X_t)^+).\cr } \Eq(3.10) $$ \noindent {\bf Proof. } The proof or \equ(3.9) is the same as the proof of (3.2) in Ferrari and Fontes (1993). The proof of \equ(3.10) is very similar to the proof of equation (3.10) in the same paper, where the variance of the current of (first class) particles in simple exclusion is written as a function of moments of a (first class) tagged particle and a discrepancy. We just sketch it, pointing out the main different point, referring the reader to the mentioned paper for details. Write $\j2t=(\j2t)^+-(\j2t)^-$, where $$ (\j2t(\si,\xi))^+ = \sum_{x\le 0} \xi(x) 1\{X^x_t(\si,\xi) >0\}, \ \ \ (\j2t(\si,\xi))^- = \sum_{x> 0} \xi(x) 1\{X^x_t(\si,\xi) \le 0\}. $$ Here $X^x_t(\si,\xi)$ is the position at time t of a tagged $\xi$ particle starting at $x$, when the initial condition is $(\si,\xi)$. The variance of $\j2t$ is then expressed in terms of variances and expectations of $(\j2t)^+$ and $(\j2t)^-$. The main calculation which follows is that of $E((\j2t)^+)^2$ and $E((\j2t)^-)^2$. The first one is expressed in various terms one of which is $$ 2(\la-\rho) \sum_{y0, \xi(y) = 1 \vert \xi(x) = 1) - P(X^y_t > 0, \xi(y) = 1)], \Eq(3.11) $$ The sum in \equ(3.11) can be rewritten as $$ \eqalign { (1-\la)&\sum_{y0, \xi(y) = 1 \vert \xi(x) = 1)- P(X^y_t > 0, \xi(y) = 1\vert \xi(x)=\sigma(x)=0)]\cr +\rho &\sum_{y0, \xi(y) = 1 \vert \xi(x) = 1)- P(X^y_t > 0, \xi(y) = 1\vert \sigma(x)=1)].\cr} \Eq(3.12) $$ These terms are reexpressed after a coupling argument as $$ (1-\la)(1-\la) \sum_{x\le 0} P(X^x_t >0, R^x_t \le 0) +\rho (1-\la) \sum_{x\le 0} P(X^x_t >0, \bar R^x_t \le 0) \Eq(3.123) $$ The expresion \equ (3.123), when combined with expressions obtained similarly in the calculation of $E((J_{2,t})^-)^2$, lead to the desired result in a straightforward manner.\qed \proclaim Theorem 3.2. Under the conditions of Theorem 3.1, it holds that $$ \limt {1 \over t} E(\j2t - \n2t(\si_0,\xi_0)- (p-q)(\la^2 - \rho^2)t)^2 = 0, \Eq(3.19) $$ where $\n2t(\si,\xi)$ is a random variable that does not depend on $\om$. It depends only on the initial configurations $\si$ and $\xi$ and it is given below by (3.13). \noindent {\bf Proof.} By mass conservation: $$ \jt = \jzt + \j2t. \Eq(3.199) $$ The current $\jzt$ depends only on the $\si$ marginal of the process, while $\jt$ depends on the $\si+\xi$ marginal. Hence, writing $E$ for the expectation of the process with initial distribution $\pi_2$ and noting that the distribution of $(\si_t,\xi_t)$ has the good marginals, $$ EJ_{2,t} = (p-q)(\la(1-\la)-\rho(1-\rho)). \Eq(3.18) $$ On the other hand, (1.5) of Ferrari and Fontes (1993) implies that $$ \eqalign { \limt { E(\jzt - \nzt(\si_0,\xi_0) - (p-q)\rho^2t)^2 \over t} &= 0, \cr \limt { E(\jt - \nt(\si_0,\xi_0) - (p-q)\la^2t)^2 \over t} &= 0,\cr} \Eq(3.20) $$ where $$ \nzt(\si,\xi) = \cases{\displaystyle {\sum_{x=-(p-q)(1-2\rho)t}^0} \si(x), & when $1-2\rho>0$, \cr \displaystyle { -\sum^{(p-q)(2\rho-1)t}_{x=0}}\si(x),\ \ & when $1-2\rho\le0$, \cr}\Eq(3.22) $$ $$ \nt(\si,\xi) = \cases{\displaystyle {\sum_{x=-(p-q)(1-2\la)t}^0} (\si(x)+\xi(x)), & when $1-2\la>0$,\cr \displaystyle { -\sum^{(p-q)(2\la-1)t}_{x=0}} (\si(x)+\xi(x)), & when $1-2\la\le0$.\cr }\Eq(3.24) $$ Define $$ \n2t (\si,\xi) = \nt(\si,\xi) - \nzt(\si,\xi).\Eq(3.25) $$ The result follows from \equ(3.199) \equ(3.20) and \equ(3.25).\qed \noindent {\bf Proof of Theorem 1.1.} We first show \equ(1.10) and \equ (1.20) for $X_t$ instead of $Z_t$. It follows from \equ(3.9) and \equ(3.18), $$ EX_t =(p-q) (1-\la-\rho) t \Eq(3.35) $$ >From \equ(3.22), \equ(3.24) and \equ(3.25), we have that $\n2t(\si,\xi) $ equals $$ \eqalign{\sum_{x=-(p-q)(1-2\rho)t}^{-(p-q)(1-2\la)t}\si(x) &+\sum_{x=-(p-q)(1-2\la)t}^0\xi(x),\ \ \hbox{ when }\ \ \la\le1/2,\cr \sum_{x=-(p-q)(1-2\rho)t}^0\si(x)&+\sum_{x=0}^{(p-q)(2\la-1)t}(\xi(x) +\si(x)), \ \ \hbox{ when }\ \ \rho\le1/2<\la,\cr \sum_{x=0}^{(p-q)(2\rho-1)t}\xi(x) &+\sum_{x=(p-q)(2\rho-1)t}^{(p-q)(2\la-1)t}(\xi(x) +\si(x)), \ \ \hbox{ when }\ \ \rho>1/2.\cr} $$ Hence $\limt (V\j2t / t) = \limt (V\n2t/t)$ equals $$ \eqalign{&2(p-q)\rho(1-\rho) (\la-\rho) + (p-q)(\la -\rho)(1-\la+\rho) (1-2\la), \ \hbox{ when } \ \la\le1/2,\cr &(p-q)(1-2\rho)\rho(1-\rho)+(p-q)(1-2\la)\la(1-\la), \ \hbox{ when } \ \rho\le1/2<\la,\cr &(p-q)(1-2\rho)(\la-\rho)(1-\la+\rho)+2(p-q)(\la-\rho)\la(1-\la), \ \hbox{ when } \ \rho>1/2. \cr} \Eq(3.33) $$ On the other hand, it is proven by Ferrari and Fontes (1993) that $$ \limt {E(R_t)^+ \over t} = \cases { (p-q)(1-2\la) &if $\la < 1/2$ \cr 0 & otherwise, \cr} \Eq(3.40) $$ $$ \limt {E(\bar R_t)^+ \over t} = \cases { (p-q)(1-2\rho) &if $\rho < 1/2$ \cr 0 & otherwise, \cr} \Eq(3.45) $$ $$ \limt {E(R_t - X_t)^+ \over t} = 0 \Eq(3.50) $$ and $$ \limt {E(\bar R_t - X_t)^+ \over t} = (p-q)(\la-\rho).\Eq(4.55) $$ Substituting \equ(3.35), \equ(3.33), \equ(3.40), \equ(3.45), \equ(3.50) and \equ(4.55) in \equ(3.10) we get $$ \limt {E(X_t)^2 - (EX_t)^2 \over t} = (p-q) { \rho (1-\rho) + \la (1-\la) \over \la - \rho } \Eq(3.34) $$ Now we show the theorem for $Z_t$. We consider the process $(\eta_t, Z_t)$, where $Z_t$ is a second class particle with respect to $\eta_t$. Ferrari (1992) has shown that it is possible to realize the processes $(\eta_t,Z_t)$ and $(\si_t,\xi_t,X_t)$ with initial distribution $\pi'_2$, in such a way that if one calls $X^i_t$ the $i$-th $\xi$ particle ($X^0_t = X_t$), and let ${\cal F}_{2,t}$ be the sigma algebra generated by $\{(\si_s,\xi_s):s\le t\}$, then for all times $$ \eqalign{ P(Z_t &= X^i_t\vert {\cal F}_{2,t} ) = m(i), \hbox{ where } \cr m(i) &= M\left(\left( 1+(p/q)^{i-\ha}\right)\left( 1+(q/p)^{i+\ha}\right)\right)^{-1} \cr} \Eq(3.46) $$ and $M$ is a normalizing constant making $\sum_{i\in\bbz} m(i) = 1$. The symmetry of $m(i)$, \equ(3.35) and \equ(3.46) show \equ (1.10). Since $m(i)$ is a probability with exponential decay, to show \equ(1.20) it suffices to prove that $$ \limt {E(X_t - X^i_t)^2 \over t} = 0 ,\ \ \hbox{ for all } i \in \bbz\Eq(3.55) $$ But \equ(3.55) follows from Lemma 2.1 and the fact that, by translation invariance, the law of $X^{i-1}_t - X^i_t$ is independent of $i$. Ferrari (1992) showed that \equ(1.20) and \equ(1.30) are equivalent.\qed \noindent {\bf Remark 3.1.} Note that \equ(3.33) implies that $\limt {V\j2t \over t} = 0$ if $\la+\rho =1$. In this case $v=0$. We do not use this. We finish this section with a lemma to be used in Section 5. Let $J_t^{b,a}$ be the number of $\eta$ particles to the left of $b$ at time zero and to the right of $a$ at time $t$ minus number of $\eta$ particles to the right of $b$ at time zero and to the left of $a$ at time $t$. Let $J_{i,t}^{b,a}$ be the analogous current for particles $\si$, $\si+\xi$ and $\xi$ for $i=0,1,2$ respectively. \proclaim Lemma 3.1. Consider the process $\eta_t$ with initial distribution $\nurl$ and the process $(\si_t,\xi_t)$ under initial distribution $\pi_2$ coupled as described at the end of Section 2. If $b>0$ and $a>v$ then $$ \limt {E(J_t^{bt,at}-J^{bt,at}_{1,t})^2 \over t} = 0. \Eq (322) $$ If $b<0$ and $a0$ and $a>v$, $J_t^{bt,at}-J^{bt,at}_{1,t}= (J^{0,at}_{2,t})^+ \le (X_t - at)^+ \le (X_t - vt)^+$. By \equ(3.0) $\limt P(((X_t -at)^+)^2/t >s) = 0$ for all $s\ge 0$. By Theorem 1.2, $\limt P((X_t -vt)^2/t >s) = 2(1-w(\sqrt s,1))$. Write $$ {E( (X_t - at)^+)^2 \over t} = \int_0^\infty P((X_t - at)^+/\sqrt t>\sqrt s)ds $$ Now $ P((X_t - at)^+ /\sqrt t>\sqrt s)\le P((X_t - vt)/\sqrt t>\sqrt s)$ and $\limt \int_0^\infty P((X_t - vt/\sqrt t)>\sqrt s) ds = \int_0^\infty 2(1-w(\sqrt s,1)) ds = D \le \infty$ by Theorem 1. By dominated convergence we get \equ(322). Analogously we get \equ(323). If $1>p>q$ one repeates the argument using $D_t$, the position of the rightmost $\ze$ particle and the fact that $\limt E(D_t - vt)^2 /t<\infty$. For \equ(323) one uses $G_t$, the position of the leftmost $\ga$ particle.\qed \noindent {\bf Remark 3.2.} Since for the process $(\si_t,\xi_t)$ the $\si$ marginal is $\nur$ and the $\si+\xi$ marginal is $\nul$, it follows from Ferrari and Fontes (1993) that $$ \limt {E(J^{bt,at}_{i,t}-J^{bt,at}_{i,t})^2 \over t} = \cases { (p-q)\rho(1-\rho)\vert1-2\rho-(a-b)\vert &if $i=0$ \cr (p-q)\la(1-\la)\vert1-2\la-(a-b)\vert &if $i=1$. \cr} \Eq(324) $$ \vskip 3truemm \noindent {\bf 4. Dynamical phase transition.} \numsec=4\numfor=1 \vskip 2truemm In order to prove Theorem 1.3 we need the following result. \proclaim Lemma 4.1. Weak limits as $t\to\infty$ of $\nurl S(t)\tvt$ are translation invariant. \noindent {\bf Proof.} Since $\tu\nurl S(t)=\nurl S(t)\tu$, it suffices to show $$ \limt \vert \tu\nurl S(t)\tvt f-\nurl S(t)\tvt f \vert = 0 \Eq(4.10) $$ for any cylinder $f$. Since the measures $\nurl$ and $\tu\nurl$ are product, we can construct a product measure $\tilde \nu$ on $\{0,1\}^\bbz \times \{0,1\}^\bbz$ with marginals $\nurl$ and $\tau_1\nurl$ in such a way that if $(\eta,\eta^*)$ is distributed according to $\tilde\nu$, then $\ex=\eta^*(x)$ $\forall x\ne 0$ and $$ \tilde\nu[\eo=\eta^*(0)]= 1-(\la-\rho),\ \ \ \tilde\nu[\eo=1,\eta^*(0)=0]=\la-\rho. \Eq(40) $$ Then we construct the coupled process $(\eta_t,\eta^*_t)$ with initial distribution $\tilde\nu$, using the same arrows. If $\eta(0) \ne \eta^*(0)$, then the processes $\et$ and $\eta^*_t$ differ at most in one site. The position of this discrepancy behaves as a second class particle with respect to $\eta$. We call it $Z_t$. At time $0$, $Z_0=0$. If $f$ depends on sites $\{-k,\ldots,k\}$, the expresion inside the limit in \equ(4.10) is bounded above by $$ (\la-\rho) \vert\vert f\vert\vert_\infty P(\vert Z_t-vt+a \sqrt t \vert\le k). \Eq(41) $$ The probability in \equ(41) converges to zero as $t\rightarrow\infty$, by the convergence of $(Z_t-vt)/t^{(1/2)}$ to a normal random variable with nonzero diffusion coefficient $D$, proven in Theorem 1.2.\qed \noindent {\bf Proof of Theorem 1.3.} We first show the result for $p=1$. In this case $Z_t\equiv X_t$. To avoid heavy notation we prove the theorem for $a=0$. The extension is straightforward. Assume $f$ depends on the sites $\{-k,\ldots,k\}$. Then by Lemma 4.1 we have (along convergent subsequences) $$ \limt \nurl S(t)\tv f=\tn\limt \nurl S(t)\tv\sn\tx f. \Eq(4.20) $$ for all $n\ge 0$. We choose to translate by $(2k+1)x$ because in that way the support of $\tx f$ is disjoint of the support of $\ty f$ if $x\ne y$. To compute the second limit in \equ(4.20) write $x'=(2k+1)x$ and $$ \eqalign{ \tn\nurl S(t)\tv\sn\tx f &=\tn\en\left[\tv\sn\tl f(\et)\right] \cr &=\tn\en\left[\tv\sn\tl f(\et) 1\{\xt-vt>\tq\}\right]\cr &\quad+\tn\en\left[\tv\sn\tl f(\et) 1\{\xt-vt<-\tq\}\right]\cr &\quad+\tn\en\left[\tv\sn\tl f(\et) 1\{|\xt-vt|<\tq\}\right]\cr &=I_1(t)+I_2(t)+I_3(t).\cr} \Eq(4.30) $$ By Theorem 1.2 $\limt I_3(t) = 0$. Couple $\eta_t$ with initial distribution $\nurl$ and $(\si_t,\xi_t)$ with initial distribution $\pi_2$ as described at the end of Section 2. For $\tq>n(2k+1)$, since $p=1$, $$ I_1(t) = \tn E[\tv\sn\tl f(\st) 1\{\xt-vt>\tq\}]. $$ Now, $$ \eqalign{ &\left\vert I_1(t) - E[\nur f \ \ 1\{\xt-vt>\tq\}] \right\vert^2 \cr &\quad\le\left\vert E\left[\left(\tv\tn\sn\tl f(\st)-\nur f\right)1\{\xt-vt>\tq\}\right]\right\vert ^2\cr &\quad\le E\left[\tv\tn\sn(\tl f(\st)-\nur f)\right]^2\cr} \Eq(4.4) $$ But $\{\tx f(\st)\}_x$ are i.i.d.\/ with distribution induced by $\nur$, hence the r.h.s.\/ of \equ(4.4) does not depend on translations by $vt$ and equals $\nur(f-\nur f)^2/(2n+1)$. By Theorem 1.2 (central limit theorem for $X_t$) $\limt E[1\{\xt-vt>\tq\}]=1/2$. Hence $$ \left\vert\limt I_1(t) - \ha\nur f\right\vert \le O({1\over\sqrt n}). $$ Analogously, $$ \left\vert\limt I_2(t)-\ha\nul f\right\vert \le O(\nh). $$ We get \equ(1.40) for $p=1$ and $a=0$ by taking $n$ to infinity. To obtain the result for $a\ne 0$ it suffices to make a partition inside the expectation in \equ(4.30) according to $\{\xt-vt>a t^{1/2}+\tq\}$, $\{\xt-vtFrom this it is not difficult to construct an argument similar to the case $p=1$ to show \equ(1.40) for all cases. \qed \vskip 2truemm \noindent {\bf 5. Fluctuation fields.} \vskip 2truemm \numsec=5 \numfor=1 In this section we show the convergence of the density fluctuation fields in the case of a shock. We only need to show Theorem 1.5 as the passage from \equ(68) to \equ(6.5) is standard. The proof of \equ(68) is based on the fact that the variance of the current through certain lines parallel to $(t(1-2\rho), t)$ and $(t(1-2\la), t)$ vanishes. The proof of \equ(67) is based in Theorems 1.1, 1.2, 1.3. \noindent {\bf Proof of Theorem 1.5.} We first show \equ(68). Since the number of particles can change only on the boundaries, $$ \sum_{x\in A_\vep} \eta_{\vep^{-1}t}(x) -\sum_{x\in B_\vep} \eta_0(x) = J^{\vep^{-1}b_1(t),\vep^{-1}a_1}_t -J^{\vep^{-1}b_2(t),\vep^{-1}a_2}_t, \Eq(5.0) $$ where $J^{\vep^{-1}b_i(t),\vep^{-1}a_i}_t$ has been defined before Lemma 3.1. By \equ(324) and Lemma 3.1, $$ \lime \vep E(J^{\vep^{-1}b_i(t),\vep^{-1}a_i}_t-EJ^{\vep^{-1}b_i(t),\vep^{-1}a_i}_t)^2 = 0,\ \ \ i=1,2. \Eq(70) $$ Then \equ(68) is a consequence of \equ(5.0) and \equ(70). Now we show \equ(67). We prove below that $$ \lime E\left(\vep^{1/2}\sum_{x\in C_\vep(t)} \eta_{\vep^{-1}t}(x) - (\la (c-T_cW_\vep(t))+\rho (T_cW_\vep(t)+c))\right)^2 = 0, \Eq(556) $$ where $W_\vep(t) = \vep^{1/2}(Z_{\vep^{-1}t} - \vep^{-1}vt)$ and $T_c$ is truncation by $c$ defined in Theorem 1.5. Since $\sum_{x\in K_\vep(t)}E\eta_0(x) = \vep^{-1}vt$, \equ(1.30) implies $$ \lime E \left(\vep^{1/2} T_{\vep^{-1/2}c}\sum_{x\in K_\vep(t)} (\eta_0(x) - E\eta_0(x)) - (\la - \rho) T_c W_\vep(t) \right)^2 = 0. \Eq(557) $$ By Theorem 1.3, $$ \lime \vep^{1/2}\sum_{x\in C_\vep(t)} E\eta_{\vep^{-1}t}(x) = \int_{-c}^c (\rho (1-w(r,t)) +\la w(r,t)) dr, \Eq(555) $$ where $w(r,t)$ is defined in Theorem 1.3. Finally, by symmetry of $(r,w(r,t))$ with respect to $(0,1/2)$, $$ \int_{-c}^c (\rho (1-w(r,t)) +\la w(r,t))dr -(\la (c-T_cW_\vep(t))+\rho (T_cW_\vep(t)+c)) = (\la - \rho) T_c W_\vep(t). \Eq(558) $$ Then \equ(67) follows from \equ(556), \equ(557), \equ(555) and \equ(558). \qed \noindent {\bf Proof of \equ(556).} We first show it for $p=1$. Let $C^-_\vep(t) = [-\vep^{-1/2}(T_cW_\vep(t)+c),0]\cap \bbz$, $C^+_\vep(t) = [0,\vep^{-1/2}(c-T_cW_\vep(t))] \cap \bbz$. Use the coupling described at the end of Section 2. Let $\eta'_t = \tau_{X_t}\eta_t$, $\si'_t = \tau_{X_t}\si_t$, $\xi'_t = \tau_{X_t}\xi_t$. Then, $$ \eqalign { \sum_{x\in C_\vep(t)} \eta_{\vep^{-1}t}(x) &= \sum_{x\in C^-_\vep(t)} \eta'_{\vep^{-1}t}(x) + \sum_{x\in C^+_\vep(t)} \eta'_{\vep^{-1}t}(x)\cr &= \sum_{x\in C^-_\vep(t)} \si'_{\vep^{-1}t}(x) + \sum_{x\in C^+_\vep(t)} (\si'_{\vep^{-1}t}(x)+\xi'_{\vep^{-1}t}(x))\cr }\Eq(656) $$ The first marginal of $(\si_t,\xi_t)$ is $\nur$ for all $t$ and $\vert C^-_\vep(t) \vert \le 2c\vep^{-1}$. Hence, by \equ(2.19), $$ \lime \vep^{1/2}\sum_{x\in C^-_\vep(t)} \si'_{\vep^{-1}t}(x)- \rho(T_cW_\vep(t)+c)) = 0 \ \ \hbox { a.s.} \Eq(5.10) $$ Then \equ(5.10) and dominated convergence imply $$ \lime E\left(\vep^{1/2}\sum_{x\in C^-_\vep(t)} \si'_{\vep^{-1}t}(x)- \rho(T_cW_\vep(t)+c) \right)^2 = 0 $$ Analogously $$ \lime E\left(\vep^{1/2}\sum_{x\in C^+_\vep(t)} (\si'_{\vep^{-1}t}(x)+\xi'_{\vep^{-1}t}(x))- \la(c-T_cW_\vep(t)) \right)^2 = 0 $$ We leave to the reader the proof for $p\in(1/2,1)$.\qed \bigskip \noindent {\bf Acknowledgements.} We thank Henrique von Dreifus for useful discussions. This research is part of FAPESP ``Projeto Tem\'atico'' Grant number 90/3918-5. Partially supported by CNPq. \vfill\eject \noindent{\bf References} \bigskip \item {[abl]} E.\ D.\ Andjel, M.\ Bramson, T.\ M.\ Liggett (1988). Shocks in the asymmetric simple exclusion process. {\sl Probab.\ Theor.\ Rel.\ Fields \bf 78 } 231-247. \item{[av]} E.\ D.\ Andjel, M.\ E.\ Vares (1987). 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