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\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_{x*rt\} - (p-q)\rho^2 t \right. \cr
&\quad \left.-\sum_{x\ge th} \eta(x) 1\{X^x_t |