\magnification=1200 \scrollmode \input acentos \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\th{\theta} \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\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\cY{{\cal Y}} \def\cX{{\cal X}} \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\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}} \rightline {\today} \centerline {\bf Current fluctuations for the asymmetric simple exclusion process} %\centerline {\bf the one dimensional S.O.S. stochastic model.} \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 compute the diffusion coefficient of the current of particles through a fixed point in the one dimensional nearest neighbors asymmetric simple exclusion process in equilibrium. We find $D= \vert p-q \vert \rho(1-\rho) \vert 1-2\rho \vert$, where $p$ is the rate at which the particles jump to the right, $q$ is the jump rate to the left and $\rho$ is the density of particles. Notice that $D$ cancels if $p=q$ or $\rho = 1/2$. A law of large numbers and central limit theorems are also proven. Analogous results are obtained for the current of particles through a position travelling at a deterministic velocity $r$. As a corollary we get that the equilibrium density fluctuations at time $t$ are a translation of the fluctuations at time $0$. We also show that the current fluctuations at time $t$ are given, in the scale $t^{1/2}$, by the initial density of particles in an interval of length $\vert (p-q)(1-2\rho) \vert t$. The process is isomorphic to a growth interface process. Our result means that the growth fluctuations depend on the general inclination of the surface. In particular they vanish for interfaces roughly perpendicular to the observed growth direction. \vskip 3truemm \noindent {\it Keywords.} Asymmetric simple exclusion. Current fluctuations. Driven interface. \vskip 2truemm \noindent {\it AMS 1980 Classification.} 60K35, 82C22, 82C24, 82C41. \vskip 3truemm \noindent {\bf 1. Introduction.} \vskip 2truemm The nearest neighbor one dimensional simple exclusion process is the Markov process $\eta_t \in \{0,1\}^\bbz$ with generator given by $$ Lf(\eta)=\sum_{x\in \bbz} \sum_{y = x\pm 1} p(x,y) \eta(x) (1-\eta(y)) [f(\eta^{x,y}) - f(\eta)], $$ where $f$ is a continuous function, $$ p(x,y) = \cases { p & if $y=x+1$ \cr q & if $y=x-1$ \cr 0 &otherwise \cr }, $$ $ p+q = 1$ and $$ \eta^{x,y}(z) = \cases { \eta(z) &if $z\ne x,y$ \cr \eta(x) &if $z=y$ \cr \eta(y) & if $z=x$ \cr }. $$ A convenient way to describe the process is the so called graphical construction. At most one particle is admitted at each site $x\in \bbz$. Each pair of sites $(x,x+1)$ has associated two Poisson process with rates $p$ and $q$ respectively. An arrow pointing from $x$ to $x+1$ is attached to each event of the process with parameter $p$ and arrows pointing from $x+1$ to $x$ are attached to events of the process with parameter $q$. All these Poisson processes are independent and the null event ``two arrows occur at the same time'' is neglected. When an arrow appears pointing from $x$ to $y$, if there is a particle at $x$ and no particle at $y$, then at that time the particle jumps to the empty site. For any other configuration nothing happens. This process was introduced by Spitzer (1970) and has received a great deal of attention. The existence of the process and the ergodic properties were studied by Liggett (1976, 1985). The set of invariant measures is the set of convex combinations of the product measures $\nu_\rho$ and blocking measures. In the case $p>q$ the blocking measures concentrate on a denumerable set of configurations and have asymptotic density $0$ and $1$ to the left and right of the origin, respectively. When $p=q$ there are no blocking invariant measures. The hydrodynamical limit was studied by Andjel and Vares (1987) and extended by Benassi et al (1991) for monotone initial density profiles. Rezakhanlou (1990) proposed a general approach to prove a law of large numbers for the density fields of attractive particle systems that works for general initial density profiles. Landim (1992) uses this law of large numbers to prove local equilibrium. The current through $rt$ at time $t$ is defined by $\jrt = $ number of particles to the left of the origin at time zero and to the right of $rt$ at time $t$ minus number of particles to the right of the origin at time zero and to the left of $rt$ at time $t$. Let $X^x_t$ be the position of a tagged particle initially located at $x$. Then we define formally the current as the random process depending on the initial configuration $\eta$ given by $$ J_{rt,t}(\eta) = \sum_{x\le 0} \eta(x) 1\{X^x_t > rt\} - - \sum_{x>0} \eta(x) 1\{X^x_t \le rt\}. $$ We assume that the distribution of the initial configuration is the stationary measure $\nur$, the product measure with density $\rho$. Under this initial distribution, $$ E \jrt = ((p-q)\rho(1-\rho)-r\rho) t. \leqno (1.1) $$ Our main result is the following. It holds for any $p$, $q$, $p+q = 1$. \proclaim Theorem 1. Law of large numbers: $$ \limt {\jrt\over t} = ((p-q)\rho (1-\rho)-r\rho). \leqno (1.2) $$ Central limit theorem: Let $G(0,D)$ be a centered normal random variable with variance $D$. Then $$ \limt {\jrt - E\jrt \over \sqrt t } = G(0,D_J), \leqno (1.3) $$ in distribution, where $D_J = \limt (V\jrt/t)$, where $V$ is the variance. Furthermore $$ \limt {V\jrt\over t} = \rho (1-\rho)\vert (p-q)(1-2\rho)-r \vert. \leqno (1.4) $$ Dependence on the initial configuration. $$ \limt {E(\jrt - N_{th(r,\rho)} - (p-q) \rho^2 t)^2 \over t} = 0, \leqno (1.5) $$ where $h(r,\rho) = r-(1-2\rho)(p-q)$, $N_{r}(\eta) = -\sum_{x=0}^{r} \eta(x)$ for $r>0$ and $N_{r}(\eta) = \sum_{x=r}^{0} \eta(x)$ for $r\le 0$. $N_{th}(\eta)$ depends only on the initial configuration $\eta$. \noindent {\bf Remark.} Notice that for $p=q$ and $r=0$ or for $r=(p-q)(1-2\rho)$, $D_J = 0$. The first fact can be proven using Arratia (1983) or a formula given in De Masi and Ferrari (1985). Indeed, De Masi and Ferrari (1985b) showed that for $p=1/2$ and all $\rho$, $$ \limt {V\jrt \over t^{1/2}} = \sqrt {2/\pi} \rho (1-\rho) \leqno (1.6) $$ and that $$ \limt t^{-1/4} \jrt = N(0, \sqrt {2/\pi} \rho (1-\rho)). $$ The fact that $D_J = 0$ for $r=(p-q)(1-2\rho)$ is more surprising. For $p=1$ and $r=(1-2\rho)$ we show that $$ VJ_{(1-2\rho)t,t} = \rho(1-\rho) E \vert R^0_t - (1-2\rho)t \vert, \leqno (1.7) $$ where $R^0_t$ is the position of a second class particle initially located at the origin. For $p=1$, a second class particle interacts with the other particles in the following way: it jumps to empty sites to the right at rate $1$ and interchange positions with (``first class'') particles to its left at rate $1$. Spohn (1991) gives heuristic arguments suggesting that $VR^0_t$ behave as $t^{4/3}$. This would imply that the variance of the current through $(1-2\rho)t$ behaves as $t^{2/3}$. An important corollary of (1.4) is that it allows one to show that the equilibrium fluctuations translate rigidly in time. More precisely, let $\xi^\vep_t$ be the fluctuations fields defined by $$ \xi^\vep_t(\Phi) = \vep^{1/2} \sum_x \Phi(\vep x)[\eta_{\vep^{-1}t}(x) - E\eta_{\vep^{-1}t}(x) ], \leqno (1.8) $$ for smooth integrable functions $\Phi$. We prove in Section 6 that calling $\bar r = (p-q)(1-2\rho)$, $$ \lime E(\xi^\vep_t - \tau_{\vep^{-1}\bar rt} \xi^\vep_0)^2 = 0, \leqno (1.9) $$ where the translation $\tau$ is defined by $\tau_y \xi^\vep_t(\Phi) = \xi^\vep_t(\tau_y\Phi)$ and $\tau_y\Phi(x) = \Phi(x+y)$. In Section 2 we give some results on the behavior of tagged and second class particles. In Section 3 we compute the current fluctuations (1.4). The law of large numbers (1.2) is shown in Section 4. The dependence of the initial configuration (1.5) and the central limit theorem (1.3) are shown in Section 5. In Section 7 we discuss consequences of our results on the motion of an interface model related to the simple exclusion process. \vskip 3truemm \noindent {\bf 2. The motion of tagged and second class particles.} \vskip 2truemm We recall briefly some results concerning the motion of a tagged particle and show a lemma relating the tagged particle with a second class particle. We assume that the initial distribution of $\eta_t$ is the equilibrium measure $\nur$. At time $0$, a particle is put at a fixed site $x$, regardless of the value of the configuration $\eta_0(x)$. This particle is tagged and followed. It interacts by exclusion with the other particles. Its position is denoted $X^x_t$. The joint process $(\eta_t,X^x_t)$ is Markov and the process $\tau_{X^x_t}\eta_t$ has as extremal invariant measure $\nu'_\rho = \nur(.\vert \eta(0) = 1)$. Under this distribution, $$ EX^0_t = (1-\rho)(p-q)t. \leqno (2.1) $$ Kipnis (1986) proved the following law of large numbers $$ \limt {X^0_t \over t} = (1-\rho)(p-q) \leqno (2.2) $$ and central limit theorem: $$ \limt {X^0_t - (1-\rho)(p-q) t \over \sqrt t } = G(0, D_X), \leqno (2.3) $$ in distribution. The variance $D_X$ is given by $$ D_X = \limt {VX^0_t \over t} = (1-\rho)(p-q). \leqno (2.4) $$ The limit was computed by De Masi and Ferrari (1985). These results also follow from a recent extension of Burke's theorem due to Ferrari and Fontes (1992) that states the following. Assume that the initial distribution of $\eta_t$ is given by $\nu'_\rho$. Then there exist random variables $K\ge 0$ with a finite exponential moment (\ie for some positive $\theta$, $E\exp (\theta K) < \infty$) and $K_t$ satisfying $P(\vert K_t\vert \ge k) \le P(K\ge k)$ for all $k\ge 0$ (\ie $\vert K_t\vert \le K$ stochastically), such that $$ X^0_t = N_t + K_t, $$ for all $t\ge 0$, where $N_t$ is a Poisson process of parameter $(1-\rho)(p-q)$. In particular this implies that if $r<(1-\rho)(p-q)$, then $$ \limt {E ((X^0_t- rt)^2 1\{X^0_t \le rt\}) \over t } = 0. \leqno (2.6) $$ Now we recall the definition of the so called ``second class particle'' and some results concerning its asymptotic behavior. Let $\eta^x$ be the configuration $\eta$ modified at $x$, \ie $\eta^x(x) = 1-\eta(x)$, $\eta^x(y) = \eta (y)$ for $y\ne x$. Let $\eta_t^x$ be the process with initial configuration $\eta^x$. Then, using the graphical construction, the processes $\eta_t$ and $\eta^x_t$ can be realized simultaneously with the same arrows. In this way the number of sites where the two configurations disagree is exactly one for all $t$. This is the basic coupling of Liggett (1976, 1985). Calling $R^x_t$ the site where the configurations disagree by time $t$, one can show that the process $(\eta_t, R^x_t)$ is Markovian and that $R^x_t$ can be described as a second class particle: it jumps over nearest neighbor empty sites at rates $p$ and $q$ to the right and left respectively and exchange positions with (first class) nearest neighbor particles at rates $q$ and $p$ to the right and left respectively. Details can be found in Ferrari (1992), as well as the following law of large numbers: $$ \limt {R^0_t \over t} = (p-q)(1-2\rho), \ \ \ a.s. \leqno (2.7) $$ Since the absolute value of the position of a second class particle is dominated above by a Poisson process of rate $1$, $R^0_t/t$ is uniformly integrable. Then for $\rho \ge 1/2$ $$ \limt {E(R^0_t-rt)^+ \over t} = \cases { 0 &if $r >(p-q)(1-2\rho)$ \cr (p-q)(1-2\rho)-r &otherwise. \cr } \leqno (2.8) $$ We also have, for all $\rho$ and $p\ge q$, $$ \limt {E(R^0_t - X^0_t)^+ \over t} = 0. \leqno (2.9) $$ Next we show a technical identity needed in the computation of the current fluctuations. Fix a configuration $\eta$ with infinitely many particles to the right and left of the origin and with a particle at the origin. Let $U^y_t$ be the position at time $t$ of a tagged particle initially at $y$ for the configuration $\eta$. Let $Z^y_t$ be the position at time $t$ of a tagged particle initially at $y$ for $\eta^0$, the configuration $\eta$ without the particle at the origin. \proclaim Lemma 2.10. For all $r\in \bbr$ it holds $$ \sum_{y<0} \eta(y) 1\{Z^y_t >r, U^y_t \le r\} = 1\{R^0_t \le r, X^0_t >r\} \ \ \hbox {a.s.} \leqno (2.11) $$ \noindent {\bf Proof.} Let $\{y_i: i \in \bbz\}$ be the ordered occupied sites of $\eta$, such that $y_0 =0$. Let $\{z_i: i \in \bbz \setminus \{0\}\}$ be the ordered occupied sites of $\eta^0$, in such a way that $y_i = z_i$ for all $i\ne 0$. Let $\Pi^i_t$ denote the label of the $\eta^0_t$ particle that at time $t$ is in the position $y_i(t)=U^{y_i}_t$, if there is such a particle. Assign to $\Pi^i_t$ the symbol $\emptyset$ otherwise. In this way, $\{(i,\Pi^i_t): i \in \bbz\}$ tells us how the particles of the processes $\eta_t $ and $\eta^0_t $ are coupled. Assuming $v_0 = 0$, $T_0 = 0$, define for $n \ge 1$ $$ T_n = \inf \{ t > T_{n-1} : \Pi^{v_{n-1}}_t \ne \emptyset \}, \ \ \ v_n = i, \hbox { for $i$ satisfying } \Pi^{v_{n-1}}_{T_n} = i. $$ There is always a discrepancy of particles between $\eta_t $ and $\eta^0_t $, and $\eta_t $ has one more particle. Initially the (only) discrepancy is located at $0$ and $\Pi^0_0=\emptyset$, but this location changes in time. $T_n$ is the time of the $n$-th change while $v_n$ is the index of the new location. At time $t$ the discrepancy is located at $y_i(t)$ if $\Pi^i_t=\emptyset$. It holds by induction on $n$ that if $t \in [T_n, T_{n+1})$, then $$ \leqalignno { & \Pi^{v_n}_t = \emptyset & (2.12.1) \cr & \hbox{If }v_n \ge 0, \hbox{ then }\Pi^i_t = \cases {i& if $i\in [0,v_n]^c$ \cr {i+1} &if $i\in [0,v_n)$} & (2.12.2)\cr & \hbox {If }v_n \le 0,\hbox{ then }\Pi^i_t = \cases { i & if $i\in [v_n,0]^c$ \cr {i-1} &if $i\in (v_n,0].$} & (2.12.3)\cr} $$ Now, we have $$ X^0_t = U^{y_0}_t, \leqno (2.13) $$ and all $(2.12)$ is saying is that for $t \in [T_n, T_{n+1})$, $$ \leqalignno { & R^0_t = U^{y_{v_n}}_t & (2.14.1) \cr & \hbox{If }v_n \ge 0, \hbox{ then }U^{y_i}_t = \cases { Z^{y_i}_t & if $i\in [0,v_n]^c$ \cr Z^{y_{i+1}}_t &if $i\in [0,v_n)$} & (2.14.2)\cr & \hbox {If }v_n \le 0,\hbox{ then }U^{y_i}_t = \cases {Z^{y_i}_t & if $i\in [v_n,0]^c$ \cr Z^{y_{i-1}}_t &if $i\in (v_n,0].$} & (2.14.3)\cr} $$ The exclusion interaction implies that, for $j r$, $U^{y_i}_t \le r$, implies by (2.14.2-3) that $t\in [T_n, T_{n+1})$ for which $v_n < 0$. This, (2.14.3) and (2.15) imply that for all $j\ne i$ either $Z^{y_j}_t \le r$ and $U^{y_j}_t \le r$ or $Z^{y_j}_t > r$ and $U^{y_j}_t > r$. Hence $$ \eqalign { \sum_{i<0} 1\{Z^{y_i}_t >r, U^{y_i}_t \le r\} & = 1\{\bigcup_{i<0} \{ Z^{y_i}_t >r, U^{y_i}_t \le r\}\} \cr & \le 1\{R^0_t \le r, X^0_t >r\},\cr } $$ where the inequality holds by (2.13-15). For the reverse inequality observe that if $t\in[T_n, T_{n+1})$, then $$ U^{y_0}_t \ge r, U^{y_{v_n}}_t < r \hbox { implies } Z^{y_i}_t > r, U^{y_i}_t \le r $$ for some $i<0$, by taking $i = \min \{k\le 0: U^{y_k}_t \ge r\}-1$. This proves the lemma. \qed \vskip 3truemm \noindent {\bf 3. Current fluctuations.} \vskip 2truemm In this section we prove (1.4). Recall that $X^x_t$ denotes the position of a tagged particle that at time $0$ is put at $x$. For a fixed initial configuration $\eta$ we write $J_{rt,t}(\eta) = (J_{rt,t}(\eta))^+ -(J_{rt,t}(\eta))^-$, where $$ (J_{rt,t}(\eta))^+ = \sum_{x\le 0} \eta(x) 1\{X^x_t >rt\}, \ \ \ (J_{rt,t}(\eta))^- = \sum_{x> 0} \eta(x) 1\{X^x_t \le rt\}. \leqno (3.1) $$ By translation invariance, $$ \eqalign { E(\jrt)^+ & =E\left(\sum_{x\le 0} \eta(x) 1\{X^x_t >rt\}\right) = \rho \sum_{x\le 0} P\{X^x_t >rt \} = \rho E(X^0_t - rt)^+, \cr E(\jrt)^- & =E\left(\sum_{x> 0} \eta(x) 1\{X^x_t \le rt\}\right) = \rho \sum_{x\le 0} P\{X^x_t \le rt\} = \rho E(X^0_t - rt)^-. \cr } \leqno (3.2) $$ Since $J^+J^- \equiv 0$, $$ V\jrt = V(\jrt)^+ + V (\jrt)^- + 2E(\jrt)^+ E(\jrt)^-. \leqno (3.3) $$ We compute now $V(\jrt)^+ = E((\jrt)^+)^2 - (E(\jrt)^+)^2$. We have $$ \eqalign { E((\jrt)^+)^2 &= \rho E(X^0_t - rt)^+ + 2 \sum_{yrt \}1\{X^y_t >rt\}) \cr & = \rho E(X^0_t - rt)^+ + 2 \rho^2 \sum_{y rt) \cr &\qquad + 2 \sum_{y rt\}) - \rho^2 P(X^y_t > rt) \right) \cr & = A_1(t) + A_2(t) + A_3(t). \cr} \leqno (3.4) $$ Reordering the sum in the second term of (3.4), $$ A_2(t)= \rho^2 E((X^0_t-rt)^+)^2 - \rho^2 E(X^0_t-rt)^+. $$ The third term in (3.4) is $$ A_3(t) = 2\rho \sum_{yrt, \eta(y) = 1 \vert \eta(x) = 1) - P(X^y_t > rt, \eta(y) = 1)]. $$ Let $A$, $B$ and $B^c$, the complementary of $B$, be events with positive probability. Then $P(A\vert B) - P(A) = P(B^c) (P(A\vert B) - P(A \vert B^c))$. Hence we write $$ \eqalign { A_3(t) &= -2\rho(1-\rho) \sum_{yrt, \eta(y) = 1 \vert \eta(x) = 0) - P(X^y_t > rt, \eta(y) = 1 \vert \eta(x) = 1)] \cr &= -2\rho(1-\rho) \sum_{y rt, U^{y,x}_t \le rt\}\right), \cr } \leqno (3.5) $$ where $U^{y,x}_t$ (respectively $Z^{y,x}_t$) is the position of the tagged particle starting at $y$ for the system where a particle is present at $x$ (respectively, is not present at $x$). In order to compute the last line of (3.5) we couple two processes that start with a configuration chosen according to $\nu_\rho$ but one of them has a particle at site $x$ while the other has a hole at $x$. We choose the basic coupling for which the number of discrepancies is always one (see the discussion before (2.7)). Denote $R_t^x$ the position at time $t$ of the discrepancy initially at $x$. By (2.11), $$ \eqalign { A_3(t) &= -2\rho(1-\rho) \sum_{x\le 0} P(X^x_t >rt, R^x_t \le rt) \cr &= -2\rho(1-\rho) \sum_{x\le 0} P(X^x_t > rt) + 2\rho(1-\rho)\sum_{x\le 0} P(R^x_t>rt, X^x_t >rt) \cr &= -2\rho(1-\rho) E(X^0_t - rt)^+ + 2\rho(1-\rho) \sum_{x\le 0} P(R^0_t-rt > x, X^0_t-rt > x) \cr &= -2\rho(1-\rho) E(X^0_t - rt)^+ + 2\rho(1-\rho) (E(R^0_t-rt)^+ - L^+_{rt}), \cr } \leqno (3.6) $$ where $L^+_{rt} = \sum_{x\ge 0} P(R^0_t-rt > x, X^0_t-rt \le x)$. The identity of the first terms in the second and the third line of (3.6) holds if $rt$ is integer, which we assume without loss of generality (if not, the difference is $O(1)$). For the identity of the second terms of these lines we used translation invariance. From (3.4), $$ \eqalign { E((\jrt)^+)^2 &= \rho E(X^0_t-rt)^+ + \rho^2 E((X^0_t-rt)^+)^2 - \rho^2 E(X^0_t-rt)^+ \cr &\quad - 2\rho(1-\rho) E(X^0_t-rt)^+ + 2\rho(1-\rho) (E(R^0_t-rt)^+ -L^+_{rt}) } $$ and using (3.2), $$ V(\jrt)^+ = \rho^2 V(X^0_t-rt)^+ - \rho(1-\rho) E(X^0_t-rt)^+ + 2\rho(1-\rho) (E(R^0_t-rt)^+-L^+_{rt}). \leqno (3.7) $$ Now we compute the variance of $(\jrt)^-$. $$ \eqalign { E((\jrt)^-)^2 &= \rho E(X^0_t - rt)^- + 2 \sum_{0 rt) \cr &\qquad + 2 \sum_{0x, X^0_t - rt \le x) = - -2\rho(1-\rho) L^-_{rt}, $$ from where $$ V(\jrt)^- = \rho^2 V(X^0_t-rt)^- + \rho(1-\rho) E(X^0_t-rt)^- - 2\rho(1-\rho) L^-_{rt}. \leqno (3.9) $$ Now $$ L^+_{rt} + L^-_{rt} = \sum_x P(R^0_t- rt >x, X^0_t - rt \le x) = E(R^0_t - X^0_t)^+. $$ We can now put all together and compute the variance of the current. Substitute (3.2), (3.7) and (3.9) in (3.3) to obtain $$ \eqalign { V\jrt &= \rho^2 (V(X^0_t-rt)^+ +V(X^0_t-rt)^- +2E (X^0_t-rt)^+ E(X^0_t-rt)^-) \cr &\quad - \rho(1-\rho) (E (X^0_t-rt)^+ - E(X^0_t-rt)^-) \cr &\quad + 2\rho(1-\rho) (E(R^0_t-rt)^+ - E(R^0_t - X^0_t)^+) \cr & = \rho^2 VX^0_t - \rho(1-\rho) E(X^0_t-rt) + 2\rho(1-\rho) (E(R^0_t-rt)^+ - E(R^0_t - X^0_t)^+). } \leqno (3.10) $$ Taking the limit as $t\to\infty$ and using (2.1), (2.4), (2.8) and (2.9), $$ \limt {V\jrt \over t} = \rho(1-\rho) \vert (p-q)(1-2\rho) - r\vert $$ This shows (1.4). In order to show (1.7) we assume $p=1$. In this case it is known that $X^0_t$ is a Poisson process of rate $(1-\rho)$ (Spitzer (1970), Liggett (1985)) for which $$ E(X^0_t)^+ = E(X^0_t) = V(X^0_t)^+= V(X^0_t) = (1-\rho) t \ \ \hbox{ and }\ \ (R^0_t - X^0_t)^+ \equiv 0. $$ Using the fact that the current through $-rt$ when the density is $1-\rho$ has the same law as $\jrt$, (3.10) reads $$ VJ_{(1-2\rho)t} = \rho(1-\rho) E\vert R^0_t - (1-2\rho)t \vert. $$ Observe that (3.10) works also for $p=1/2$: from (3.10) and $VX^0_t = \sqrt {2t/\pi}(1-\rho)/\rho)+ o(\sqrt t)$ (Arratia (1983)) one can deduce (1.6). The key point is that a second class particle in symmetric exclusion behaves just as a simple symmetric random walk. \vskip 3truemm \noindent {\bf 4. Law of large numbers.} We prove now the law of large numbers. It holds $$ \limt {J_{rt,t} \over t} = (p-q)\rho (1-\rho) - r \rho. \leqno (4.1) $$ The proof of (4.1) would be a consequence of the ergodic theorem if one knew that the product measures $\nur$ are extremal invariant for the process $\tau_{[rt]} \eta_t$, where $[.]$ is integer part (see Kipnis (1986)). It is not clear to us how to show this extremality. To overcome the difficulty consider a Poisson process $U(t)$ at rate $\lambda$, independent of $\eta_t$. It is not hard to show that the invariant measures for the process $\tau_{U(t)}\eta_t$ are translation invariant. Then use Liggett's (1976, 1985) techniques to show that the the set of extremal invariant measures for $\tau_{U(t)}\eta_t$ is $\{\nur: 0\le\rho\le 1\}$. Hence $J_{U(t),t}$, the current through $U(t)$ satisfies a law of large numbers: $$ \limt {J_{U(t),t} \over t} = (p-q)\rho (1-\rho) - \lambda \rho. \leqno (4.2) $$ Now use that $U(t)/t$ converges to $\lambda$ almost surely and the fact that the current is a decreasing function of $r$ to conclude the proof of (4.1). This argument was used by Ferrari (1992) to show a law of large numbers for a second class particle. \vskip 3truemm \noindent {\bf 5. Dependence on the initial configuration.} \vskip 2truemm Since $N_{th}$ is a sum of independent random variables, (1.5) implies the central limit theorem (1.3). To show (1.5) for $r< (p-q)(1-\rho)$ write $$ \eqalign { & \jrt - N_{th(r,\rho)} - (p-q)\rho^2 t \cr &= \sum_{x<0} \eta(x)1\{X^x_t > rt \} - - \sum_{x\ge 0 } \eta(x) 1\{X^x_t \le rt \} - - \sum_{x=th}^0 \eta(x)- (p-q)\rho^2 t \cr &= \sum_{x=th}^{0} \eta(x)(1\{X^x_t > rt\} -1) + \left(\sum_{xrt\} - (p-q)\rho^2 t \right. \cr &\quad \left.-\sum_{x\ge th} \eta(x) 1\{X^x_t 0\} [f(\xi^{x,+}) - f(\xi)] \cr &\qquad + p 1\{\xi(x-1) + \xi(x+1) - 2 \xi(x) <0\} [f(\xi^{x,-}) - f(\xi)]\big), \cr } $$ where $\xi^{x,\pm}(x) = \xi (x) \pm 2$ and $\xi^{x,\pm}(y) = \xi(y)$ otherwise. In words, interpreting $\xi(x)$ as the height of a surface at $x$, the process can be described by saying that at rate $q$ the surface at $x$ increases two units if both heights at $x-1$ and $x+1$ are bigger than the height at $x$. Analogously, at rate $p$ the surface decreases two units if both neighbor heights are smaller than the height at $x$. For a given configuration $\eta \in \{0,1\}^\bbz$ define $\xi = T\eta\in \bbz^\bbz$ by $$ \xi(x) = \sum_{y=0}^x (2\eta(y) - 1). $$ Denoting $\xi^\xi_t$ and $\eta^\eta_t$ the interface and the simple exclusion processes with initial configuration $\xi$ and $\eta$ respectively, it holds $$ \xi^{T\eta}_t(x) = (T\eta^\eta_t)(x) + J_{0,t}. $$ Hence $$ \xi_t(0) \equiv J_{0,t}. $$ The density in the simple exclusion process gives the general inclination of the surface. Density $1/2$ gives a surface parallel to the $x$ axis (flat). Our results on the current mean that the diffusion coefficient for a flat surface scales in a different way than the diffusion coefficient for a inclined surface, no matter for which inclination. For the flat surface the correct normalization would be $t^{2/3}$. Our interpretation is that a flat surface has ``more memory'' than an inclined surface. In this last one sees a flux of particles falling down the hill and pick the space fluctuations of the initial configuration. This does not happen in the flat case. Alexander et al (1992) studied a two dimensional asymmetric simple exclusion process. For this process the transition function is given by $p((x,y),(x,y+1)) = 1/2$, $p((x,y),(x\pm 1,y))=1/4$ and $p((x,y),(z,w))= 0$ otherwise (flat initial surface). The process starts with a product measure with density $0$ and $\rho>0$ in the semiplanes $\{y<0\}$ and $\{y\ge 0\}$ respectively. Defining $Y(t)$ as the first coordinate of the leftmost particle on the $y$ axis, they found via simulations that the variance of $Y(t)$ behaves as $t^{1/4}$. Is this normalization correct for inclined surfaces? For which reason the normalization factor in the flat case depends on the model? \vskip 3truemm \noindent {\bf Acknowledgments.} We thank K. 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