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%
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\null\vskip1.5truecm
\centerline {\titlefont Invariant Measures for a }
\vskip1\jot
\centerline{\titlefont Two Species Asymmetric Process}
\bigskip
\centerline {\authorfont P.\ A.\ Ferrari, L.\ R.\ G.\ Fontes}
\centerline {\authorfont Y.\ Kohayakawa}
\medskip
\centerline {\addressfont Universidade de S\~ao Paulo}
\bigskip
\noindent {\bf Summary.} We consider a process of two classes of
particles jumping on a one dimensional lattice. The marginal system of
the first class of particles is the one dimensional totally asymmetric
simple exclusion process. When classes are disregarded the process is
also the totally asymmetric simple exclusion process. The existence of
a unique invariant measure with product marginals with density~$\rho$
and~$\lambda$ for the first and first plus second class particles,
respectively, was shown by Ferrari, Kipnis and Saada~(1991). Recently
Derrida, Janowsky, Lebowitz and Speer~(1993) and Speer (1994) have
computed this invariant measure for finite boxes and performed the
infinite volume limit. Based on this computation we give a complete
description of the measure and derive some of its properties. In
particular we show that the invariant measure for the simple exclusion
process as seen from a second class particle with asymptotic
densities~$\rho$ and~$\la$ is equivalent to the product measure with
densities~$\rho$ to the left of the origin and~$\la$ to the right of
the origin.
\vskip 3truemm
\noindent {\it Keywords and phrases.} Two species process.
Asymmetric simple exclusion. Second class particles.
\vskip 2truemm
\noindent {\it AMS 1991 Classification.} 60K35, 82C22, 82C24,
82C41.
\vskip 2truemm
\noindent {\it Short title:} Two species asymmetric process.
\vskip 3truemm
\numsec=1\numfor=1
\beginsection 1. Introduction
The simplest way of defining the two species system is by using the
basic coupling of the totally asymmetric simple exclusion process
(\sep). We define the {\it simple exclusion process\/} $\eta_t\in
\{0,1\}^\Z$
($t\ge0$) as follows. At each site $x\in\Z$ we attach a random clock
that rings according to a Poisson process of parameter~1. The clocks
are mutually independent. When the clock of an occupied site $x$
rings, if $x+1$ is empty, the particle at~$x$ jumps to~$x+1$. If~$x+1$
is occupied, nothing happens. Thus, in this process, the particles are
generally drifting to the right. If one considers two initial
configurations~$\eta^1$ and~$\eta^2\in\{0,1\}^\Z$ such that
$\eta^1(x)\le \eta^2(x)$ for all~$x$ and uses the same clocks for both
realizations, then one has a coupled process~$(\eta^1_t,\eta^2_t)$
with the property that~$\eta^1_t(x) \le \eta^2_t(x)$ for all $x$ and
all $t\ge 0$ (cf.~Liggett~(1976, 1985)). The two species
process~$(\sigma_t,\xi_t)$ ($t\ge0$) is defined by putting~$\si_t(x) =
\eta^1_t(x)$ and $\xi_t(x) = \eta^2_t(x) - \eta^1_t(x)$. The $\si$
particles are the {\it first class\/} particles and the $\xi$
particles are the {\it second class\/} particles. The reason for these
terms is that, when a clock rings for a first class particle at
site~$x$ and a second class particle is at site~$x+1$, the particles
interchange positions, whereas if a second class particle is at~$x$
and a first class particle is at~$x+1$, they do {\it not\/} move. It
is easy to see that the two species process is Markovian.
In this paper we are concerned with the invariant measures for the two
species process. Since the marginal processes $\si_t$ and
$\si_t+\xi_t$ are simple exclusion processes, the corresponding
marginal measures of any invariant measure for the two species process
must be invariant for the \sep. Now, the invariant measures for the
{\sep} are convex combinations of the product measures~$\nur$ with
density $\rho\in[0,1]$ and the blocking measures concentrated on the
configuration $\dots000111\dots$ and its translates. Let us say that
a distribution of $(\si,\xi)$ has {\it good marginals\/} if, for
some~$\rho\leq\lambda$, its $\si$ marginal is $\nur$ and its $\si+\xi$
marginal is $\nul$. It is easy to construct a product measure $\pi_2$
for~$(\sigma,\xi)$ with good marginals. Ferrari, Kipnis and
Saada~(1991) proved that, for the two species
process~$(\sigma_t,\xi_t)$, there exists a unique invariant measure
$\mu_2$ with good marginals, and that the process started with the
product measure $\pi_2$ converges to $\mu_2$ as $t\to\infty$. Derrida,
Janowsky, Lebowitz and Speer~(1993) have recently computed the
invariant measure $\mu_2$ in finite boxes and, performing the infinite
volume limit, they have investigated~$\mu_2$. In a sequel,
Speer~(1994) makes this approach rigorous.
One important fact discovered by Derrida, Janowsky, Lebowitz and
Speer~(1993) is that, under the invariant measure $\mu_2$, the
distribution to the right of a second class particle is {\it
independent\/} of the distribution to its left. This suggests
studying the process `as seen from a second class particle'. To
make this precise, assume that at time~$t=0$ there is a second class
particle at the origin and let~$X_t$ be its position at time~$t$. The
process {\it as seen from this second class particle\/}
is~$(\tau_{X_t}\si_t,\tau_{X_t}\xi_t)$, where $\tau_x$ denotes
translation by~$x$. Thus, $(\tau_{X_t}\si_t(x),\tau_{X_t}\xi_t(x))=
(\sigma_t(x+X_t),\xi_t(x+X_t))$ for all~$x\in\Z$. Now assume that the
initial distribution~$\mu$ of the two species process is translation
invariant, and that it has a positive density of second class
particles. Then, at time~$t$, the process as seen from a second class
particle, started with the measure~$\mu$ conditioned to having a
second class particle at the origin, has the same distribution as the
two species process started with the unconditioned measure~$\mu$, but
itself conditioned to having a second class particle at the origin at
time~$t$. This means that, when the density of second class particles
is positive, the invariant measures for the two species process have a
corresponding invariant measure for the process as seen from a second
class particle (see Ferrari, Kipnis and Saada~(1991)). But in fact the
process as seen from a second class particle is richer: it has
invariant measures with only a finite number of second class particles
with no corresponding measure in the two species process.
Our main contribution is a complete description of the invariant
measures for the two species process as seen from a fixed second class
particle. This description is based on computations in Derrida,
Janowsky, Lebowitz and Speer~(1993). We consider two densities
$0<\rho\le\la< 1$ and construct a measure
$\mu'_2=\mu'_2(\rho,\lambda)$ that is invariant for our process (see
Theorem~1). The parameters correspond to the asymptotic densities (as
$x\to\pm\infty$) of the first and the first plus second class
particles. The cases $\rho=0$ or $\la=1$ are easier and were considered
before.
In the particular case in which~$\rho=0$ or~$\la=1$, either
the $\sigma$ marginal or the $\si+\xi$ marginal is trivial. Moreover, in
this case, the other marginal is trivial if $\rho=\la$ and it is the
{\sep} if $\rho<\la$. In the latter case, the process corresponds to
the {\sep} as seen from a tagged particle, as studied by Ferrari
(1986) and De Masi, Kipnis, Presutti and Saada~(1990). An important
general property of the measure $\mu'_2$ is `translation invariance',
in the sense that it is the same seen from {\it any\/} second class
particle. When $\rho <\la$ this `translation invariance' implies that
there exists a unique translation invariant $\mu_2$ such that $\mu'_2$
is $\mu_2$ conditioned to having a second class particle at the
origin. When $\la = \rho$, the average distance between two
successive second class particles is infinite. This implies that there
is no translation invariant measure $\mu_2$ such that $\mu'_2$ is
$\mu_2$ conditioned to having a second class particle at the origin.
Recall that our particles are drifting to~$+\infty$. The definition
of the two species process is such that a first class particle can
overtake a second class one, but the other way around is
prohibited. Let us start the process with a second class particle at
the origin, and, along the evolution of the process, we refer to this
particle as the {\it $0$-th second class particle\/}. We consider the
second class particles ordered from left to right, so that we may
speak of the $i$-th second class particle for any~$i\in\Z$. If one
identifies the two classes of particles starting from the $i$-th
second class particle ($i>0$), one has an operator~$\Phi_i$ acting on
the configurations~$(\sigma,\xi)$, which commutes with the semigroup
corresponding to the evolution. Similarly, for any fixed~$j<0$,
one can identify {\it holes\/}, i.e.~empty sites, with second class
particles starting from, and to the left of, the $j$-th second class
particle to obtain an
operator~$\Psi_j$ that also commutes with the semigroup. Hence,
applying any (or both) of these operators to the `translation
invariant' stationary measure $\mu'_2$, we obtain another invariant
measure (see Theorem~2). Incidentally, these new measures are clearly
not `translation invariant'. Now, identifying first and second class
particles to the right of the particle at the origin and holes and
second class particles to the left of it, we obtain the invariant
measure for the process as seen from a single, isolated second class
particle. When $\rho<\la$ this corresponds to a shock in the {\sep}
(Ferrari, Kipnis and Saada (1991), Ferrari (1992)). When $\rho=\la$
there is a reminiscence of the shock, as the density to the right of
the second class particle is bigger than the density to the left of it
and the approach to the asymptotic density $\la$, which equals~$\rho$
here, is slow (Derrida, Janowsky, Lebowitz and Speer (1993)). If one
makes the identification for all but two second class particles, one
gets that the distance~$d$ between the two second class particles is,
following the terminology of Derrida, Janowsky, Lebowitz and
Speer~(1993), a `bounded state' even when $\la=\rho$. Indeed, the
distribution of~$d$ is the same as the distribution of the distance
between two successive second class particles under $\mu'_2$. It turns
out that this distance has the same distribution as the hitting time
of~$1$ for a nearest neighbor random walk with jumps in~$\{-1,0,1\}$
with probabilities $\rho(1-\la)$, $1-\la(1-\rho)-\rho(1-\la)$ and
$\la(1-\rho)$ respectively (see Lemma~2.5).
Our approach relies on the work of Derrida, Janowsky, Lebowitz and
Speer (1993) and Speer (1994) but we work directly in the infinite
volume. In Section 3 we describe completely the measure $\mu'_2$ and
show that it is invariant for the process as seen from a fixed second
class particle. Derrida, Janowsky, Lebowitz and Speer~(1993) state the
following remarkable property of the measure~$\mu'_2$: the
distribution of first class particles to the right of the tagged
second class particle is the product measure~$\nu_\rho$ with density
$\rho$ while the distribution of empty sites to the left of the tagged
second class particle is the product measure~$\nu_{1-\lambda}$ with
density $1-\la$. Speer (1994) proves this statement and here we give
an alternative proof of this fact by showing that one may
construct~$\mu'_2$ as follows. We first put a second class particle at
the origin and distribute the first class particles to the right of
the origin according to the product measure $\nur$. Then we give a
recipe for deciding where to put the second class particles among the
unnoccupied sites.
%
%In order to do this one must prove that a certain sum depending on
%$\eta$ and $\la$ must be one. We are able to prove this only for
%$\nur$ almost all $\eta$. [yk: aqui da a impressao que ficou faltando
%algo. Na verdade, podemos simplesmente eliminar este pedaco? Acho que
%nao e tao relevante assim.]
%
To the left of the second class particle at the origin the positions
of the empty sites are chosen according to the product measure
$\nu_{1-\la}$ and a similar recipe is used to decide where to put the
second class particles. (See Proposition~1.)
When $\la>\rho$ there exists a unique translation invariant measure
$\mu_2$ with the property that it coincides with~$\mu'_2$ when it is
conditioned to having a second class particle at the origin. As explained
above, the invariance of~$\mu'_2$ for the process as seen from the second
class particle implies that the
measure~$\mu_2$ is invariant for the two species process. Using the
property that the first class particles to the right of the tagged
second class particle are distributed according to a product measure,
we show that $\mu_2$ has good marginals (cf.~Theorem~3).
This already followed from
the infinite volume limit of Derrida, Janowsky, Lebowitz and Speer
(1993) but in a somewhat indirect way. We also show that it is
possible to construct a coupling $\tilde\mu$ with marginals $\mu'_2$
and $\mu_2$ in such a way that the number of sites where the two
marginals differ is a random variable with a finite exponential
moment. (See Theorem~4.)
%
%This is done in Section 3.
Let~$\nurl$ be the product measure with density~$\rho$ to the left of
the origin and density~$\lambda$ to the right of the origin. Using
the results of Ferrari, Kipnis and Saada (1991), Ferrari (1992) proved
that the {\sep} as seen from a second class particle starting with the
product measure $\nurl$ presents a shock: uniformly in time the
asymptotic densities are $\rho$ and $\la$ to the left and right of the
origin, respectively. Indeed the process with a unique second class
particle, started at the origin,
with initial product distribution $\nurl$ can be coupled to
the two species process with initial product distribution $\pi_2$
(with marginals $\nur$ and $\nul$) in such a way that at all times the
single second class particle of the first process has the same
position as the tagged second class particle in the second process. As
mentioned above, this can be done by identifying first and second
class particles to the right of the origin and empty sites and second
class particles to the left of the origin. Applying the results for
the two species process to the shock in the {\sep}, Derrida, Janowsky,
Lebowitz and Speer (1993) have computed the rate of convergence of the
density of the shock to the asymptotic densities $\rho$ and $\la$. We
make a further step proving that the invariant measure $\mu'$ for the
process as seen from a single second class particle has the following
property. One may construct a coupling between $\mu'$ and $\nurl$ in
such a way the the number of sites where the two marginals differ is a
random variable with a finite exponential moment. This implies in
particular that $\mu'$ is equivalent to $\nurl$. (See Theorem~5 and
its corollary.)
%
%This is done in Section 4.
Let us now mention some related results. Speer~(1994) have described
the set of all invariant measures for the two species process and has
shown that the invariant measure $\mu_2$ is not Gibbsian. Ferrari and
Fontes~(1993) have computed the asymptotic variance of the position of
the second class particle for the process with initial distribution
$\mu'_2$, and they have studied the density fluctuation fields for the
exclusion process with a shock initial condition.
This article is organised as follows. In the next section we prove
three basic lemmas (Lemmas~2.1, 2.2, and~2.3) that are used in later
sections. In Section~3 we give our construction of
the measure~$\mu'_2$, and we prove Theorem~1, which asserts
that~$\mu'_2$ is invariant for the
process~$(\tau_{X_t}\sigma_t,\tau_{X_t}\xi_t)$. Also in this section
are Theorem~2, concerning other invariant measures constructed
from~$\mu'_2$ with the aid of the operators~$\Phi_i$ and~$\Psi_j$, and
Proposition~1. In Section~4 we deal with the invariant
measure~$\mu_2$ for the process~$(\sigma_t,\xi_t)$, and prove that it
has good marginals (cf.~Theorem~3). In that section we also prove
Theorem~4, concerning the coupling~$\tilde\mu$ between~$\mu_2$
and~$\mu'_2$ mentioned above. The last section is devoted to proving
Theorem~5, on the coupling between~$\mu'$ and~$\nurl$.
%
% nova Secao 2.
%
\def\Zt{{\widetilde Z}}
%
\beginsection 2. A distribution on the set of finite configurations
\numsec=2\numfor=1
Let ${\bf Y}$ be the space of {\it finite configurations\/} of $0$s
and $1$s, i.e.
$$
{\bf Y}=\bigcup_{n\ge0}\{0,1\}^n
=\{\emptyset,0,1,00,01,10,11,000,\dots\}.
$$
Usually, we think of a sequence~$\zeta$ in~${\bf Y}$ of length~$n$ as
indexed by~$\{1,\ldots,n\}$. In this section we define and study a
certain probability distribution~$p$ on the space~${\bf Y}$. This
distribution will be used in the next section to construct the
invariant measure for the system as seen from a second class particle.
Let~$\zeta\in{\bf Y}$ be given. We write~$N(\zeta)$ for the length
of~$\zeta$, and~$K(\zeta)$ for the number of $1$s
in~$\zeta$. Formally, we have~$N(\zeta)=n$ if and only
if~$\zeta\in\{0,1\}^n$, and~$K(\zeta)=\sum_{x=1}^{N(\zeta)}\zeta(x)$.
An important definition that we shall need is the following.
For~$\zeta\in{\bf Y}$, let~$M(\zeta)$ be the number of {\sl
distinct configurations that can be obtained from~$\zeta$ by
shifting ones to the right, including $\zeta$ itself\/}. Thus, for
example, we have~$M(100)=3$, $M(0011)=1$, and~$M(1010)=5$.
We may now define the distribution~$p$ on~${\bf Y}$. In fact, we shall
define a distribution~$p=p_{\rho,\lambda}$ for
each~$0<\rho\leq\lambda<1$. Let~$\rho$ and~$\lambda$ as above be
fixed. Given~$\zeta\in{\bf Y}$, we put
$$
p(\zeta)=p_{\rho,\lambda}(\zeta)=\lambda(1-\rho)M(\zeta)
(\lambda\rho)^{K(\zeta)}
\big((1-\lambda)(1-\rho)\big)^{N(\zeta)-K(\zeta)}. \Eq(1)
$$
We show in Lemma~2.1({\it i\/}) below that~$p$ does indeed define
a probability distribution over~$\bf Y$. It is with the aid
of~$p=p_{\rho,\lambda}$ that we shall construct the invariant measure
for the two species process as seen from a second class particle, when
the asymptotic densities of the first class particles and the first
plus second class particles are respectively~$\rho$ and~$\lambda$.
The rest of this section is devoted to proving that~$p$ gives a probability
measure over~{\bf Y} and to the study of some simple properties of the
space~$({\bf Y},p)$ and of the function~$M(\zeta)$ ($\zeta\in{\bf Y}$). In
particular, we shall consider the random variable~$N=N(\zeta)$, that is, the
random length of a sequence~$\zeta$ drawn from~${\bf Y}$ according to~$p$.
The main results of this section are given in Lemmas~2.1, 2.2 and~2.3, which
we now state.
\proclaim Lemma 2.1. Let~$\rho$, $\lambda\in(0,1)$ be fixed.
Then {\rm({\it i\/})}~$\sum_{\zeta\in{\bf Y}}p(\zeta)=1$ if and only
if~$\rho\le\la$. Assuming that~$\rho\leq\lambda$ and,
writing~$\E=\E_{\rho,\lambda}$ for
the expectation in~$({\bf Y},p)$, we have:
{\rm({\it ii\/})}~if~$\rho<\lambda$ then~$\E(N+1)=1/(\lambda-\rho)$, and
{\rm({\it iii\/})}~if~$\rho=\lambda$ then $\E(N+1)=\infty$.
Finally, {\rm({\it iv\/})}~if $\rho<\la$, then $N$~has a finite
exponential moment. In other words, there exists~$\theta>0$ such that
$$
\E e^{\theta N} < \infty. \Eq(exp)
$$
The distribution of the random variable~$N$ is given in Lemma 2.5 below.
The generating function of~$N$ is given in the following lemma.
\proclaim Lemma 2.2. Let~$\rho\le\lambda$. The generating function of
$N$ is given by
$$
\E s^N={1\over 2as^2}
\left\{1-cs-\sqrt{(1-cs)^2-4abs^2}\right\},\Eq(fg)
$$
where $a=\rho(1-\lambda)$, $b=\lambda(1-\rho)$ and $c=1-a-b$.
The closed disc $|s|\le\left\{1-(\sqrt{b}-\sqrt{a})^2\right\}^{-1}$ is
its domain of convergence.
Our next lemma, Lemma~2.3, is inspired by Derrida, Janowsky, Lebowitz
and Speer (1993).
\proclaim Lemma 2.3. For all~$\zeta$, $\gamma\in{\bf Y}$, we
have~$M(\ze10\ga)=M(\ze1\ga)+M(\ze0\ga)$.
We now define a random walk on the integers~$\Z$ that will be
important in the sequel. Let~$X_1,X_2,\ldots\,$, $Y_1,Y_2,\ldots$ be
independent $0$--$1$~random variables with~$\E(X_i)=\lambda$
and~$\E(Y_i)=\rho$ ($i\ge1$). Put~$Z_i=X_i-Y_i$ ($i\ge1$), and
let~$\Zt_n=\sum_{1\leq i\leq n}Z_i$ ($n\ge0$). Note that
then~$(\Zt_n)_0^\infty$ is a random walk on~$\Z$, and
let~$T=\inf\{n\ge0:\Zt_n=1\}$ be the hitting time of the
event~$\{\Zt_n=1\}$.
The rest of this section is devoted to prove the lemmas above. The other
sections of this note may be read independently from what follows. Our first
auxiliary lemma is the following
\proclaim Lemma 2.4. For all integers~$n$, $k\ge0$, we have
$$
\sum M(\zeta)={1\over k+1}{n\choose k}{n+1\choose k}, \Eq(0)
$$
where the sum ranges over all $\zeta\in{\bf Y}$ with
$N(\zeta)=n$ and $K(\zeta)=k$.
We defer the proof of Lemma~2.4 until later, and pass on to a result
that is crucial in the proof of Lemma~2.1({\it i\/}).
For a finite configuration~$\zeta\in{\bf Y}$, recall that~$N(\zeta)$
denotes its length and~$K(\zeta)$ its number of~$1$s. For
integers~$n$ and~$k$, set~$p_{n,k}=\sum_\zeta p(\zeta)$, where the sum
ranges over all~$\zeta\in{\bf Y}$ with~$N(\zeta)=n$ and~$K(\zeta)=k$.
Note that once we know that~$p$ is a probability measure on~$\bf Y$,
the quantity~$p_{n,k}$ is simply the probability that a random
configuration~$\zeta\in{\bf Y}$ has length~$n$ and~$k$ elements equal
to~$1$. In particular, the lemma below in this case simply states
that~$P\{N=n\}=P\{T=n+1\}$.
\proclaim Lemma 2.5. Let~$\la$, $\rho\in(0,1)$ be fixed. Then, for
any~$n\ge 0$, we have
$$
\sum_kp_{n,k}=\P\{T=n+1\}. \Eq(t=n)
$$
\noindent{\bf Proof.} By Lemma~2.4, we have
$$
\eqalign{
p_{n,k}&={1\over n+1}{n+1\choose k+1}\lambda^{k+1}(1-\lambda)^{n-k}
{n+1\choose k}\rho^k(1-\rho)^{n+1-k} \cr
&={1\over n+1}\P(W_1=k+1)\P(W_2=k),\cr
}
$$
where~$W_1$ and~$W_2$ are two independent binomial random variables with
parameters~$n+1$ and~$\lambda$ and~$n+1$ and~$\rho$, respectively. Now,
summing over all~$k$, for $n\ge 0$ we have
$$
\sum_kp_{n,k}={1\over n+1}\P\{W_1-W_2=1\}
={1\over n+1}\P\{\Zt_{n+1}=1\}, \Eq(Nn)
$$
where~$(\Zt_n)_0^\infty$ is the random walk introduced above. On the
other hand, recalling that $T$ is the hitting time of~$1$ for that
walk, we have
$$
\P\{T=n+1\}={1\over n+1}\P\{\Zt_{n+1}=1\} \Eq(raney)
$$
for all integers~$n\ge0$. Identity~\equ(raney) is Exercise~(IV.12) of
Spitzer~(1976), but for completeness we give a combinatorial proof for
it in Lemma~2.6 below. Lemma~2.5 follows from \equ(Nn) and
\equ(raney).\quad\square
\bigskip
\noindent {\bf Remark 2.1.} The Local Central Limit Theorem (or direct
calculations) and \equ(raney) imply that, when $0<\rho=\lambda<1$, we
have~$\P(T=n)=(c+o(1))n^{-3/2}$ as~$n\to\infty$, where~$c=c(\rho)>0$
depends only on~$\rho$. For the case in which~$\rho<\lambda$, see
Remark~2.2.
\def\Vt{{\widetilde V}}
Let us now prove~\equ(raney). The proof below is entirely
combinatorial and more elementary than the one suggested in
Spitzer~(1976), which is based on Lagrange's inversion formula.
\bigskip
\proclaim Lemma 2.6. Let~$(V_i)_1^\infty$ be a family of {i.i.d.}
$\{\pm1,0\}$-random variables and let~$\Vt_n=\sum_{1\leq i\leq n}V_i$
$(n\ge0)$ be the associated random walk on~$\Z$.
Let~$T=\inf\{n:\Vt_n=1\}$ be the hitting time
of~$\{\Vt_n=1\}$. Then~$\P\{T=n\}=n^{-1}\P\{W_n=1\}$ for all
integers~$n\ge1$.
\noindent{\bf Proof.}
We deduce this result from a lemma of Raney~(1960) (see also Example~4
in Section~7.5 of Graham, Knuth, and Patashnik~(1989)):
{\sl if~${\bf x}=(x_1,\ldots,x_n)$ is a sequence of integers
with~$\sum_{1\leq i\leq n}x_i=1$, then there is a unique cyclic
permutation of~${\bf x}$,
say~$(x_j,x_{j+1},\ldots,x_n,x_1,\ldots,x_{j-1})$, all of whose proper
initial partial sums are non-positive, {\it i.e.\/}~such that~$x_j$,
$x_j+x_{j+1},\ldots,x_j+\cdots+x_{j-2}\leq0$.\/}
Let~${\bf x}=(x_1,\ldots,x_n)$ be a $\{\pm1,0\}$-sequence
with~$\sum_{1\leq i\leq n}x_i=1$, and let~$E_{\bf x}$ be the event
that~$(V_1,\ldots,V_n)$ is a cyclic permutation of~${\bf x}$. It is
simple to check, and in fact it follows from Raney's lemma, that all
the~$n$ cyclic permutations of~${\bf x}$ are distinct. Also, clearly,
the probability that~$(V_i)_1^n$ is any of these~$n$ permutations
is~$(1/n)\P(E_{\bf x})$. Now, by Raney's lemma, exactly one of these
permutations corresponds to the event~$\{T=n\}$, and hence Lemma~2.6
follows.\quad\square
\bigskip
We may now prove Lemma~2.1, the first main result of this section.
\bigskip
\noindent {\bf Proof of Lemma 2.1.} ({\it i\/})~We need to prove
that~$\sum_{n,k}p_{n,k}=1$ if and only if~$\rho\leq\lambda$. In view
of~\equ(t=n), we have~$\sum_{n,k}p_{n,k}=\sum_n\P(T=n+1)=\P(T<\infty)$,
where~$T$ is the hitting time of~$1$ for the walk~$(\Zt_n)_1^\infty$
defined just after Lemma~2.3. It now suffices to notice
that~$T<\infty$ almost surely if and only if the
walk~$(\Zt_n)_0^\infty$ has non-negative drift. This proves~({\it
i\/}).
We assume from now on that~$\rho\leq\lambda$, and rewrite~\equ(t=n)
as~$\P\{N=n\}=\P\{T=n+1\}$ ($n\in\Z$). Let us now prove~({\it
ii\/}). Suppose that~$\rho<\lambda$. Then, again considering the
random walk~$\Zt_n=\sum_{1\leq i\leq n}Z_i$ and the hitting time~$T$,
by Wald's identity we
obtain~$1=\E(\Zt_T)=\E(Z_i)\E(T)=(\lambda-\rho)\E(T)$, and
hence~$\E(N+1)=\E(T)=1/(\lambda-\rho)$, as required.
To see~({\it iii\/}), note that for~$\la=\rho$ the expected hitting
time~$T$ is infinite. Finally, to prove~({\it iv\/}), we prove
that~$\P\{N=n\}$ decays exponentially with~$n$. By~\equ(t=n)
and~\equ(raney), we
have~$\P\{N=n\}=\P\{T=n+1\}=(n+1)^{-1}\P\{\Zt_{n+1}=1\}
\leq\P\{\Zt_{n+1}=1\}$
for all integers~$n\ge0$. But then it suffices to notice that this
last probability is exponentially small,
since~$\E(Z_i)=\lambda-\rho>0$. Indeed, if~$n$ is large enough with
respect to~$\lambda-\rho$, we have that
$$
\P(\Zt_{n+1}=1)
\leq\exp\left\{-{(\lambda-\rho)^2\over5(\lambda-\rho+1)}n\right\}
\Eq(Hoeff)
$$
by Hoeffding's inequality. (See Hoeffding~(1963) or McDiarmid~(1989).)
\quad\square
\bigskip
\noindent {\bf Proof of Lemma 2.2.} The result is obtained by a standard
application of Wald's identity to the stopping time $T$
(see Breiman (1968)) and standard analytic continuation arguments.
\quad\square
\bigskip
\noindent {\bf Remark 2.2.}
It follows from Lemma~2.2 that~$\P\{N=n\}$ decays a little faster than
it is suggested in~\equ(Hoeff) in the proof of Lemma~2.1({\it iv\/}).
The rate of exponential decay of the distribution
of $N$ when $\lambda>\rho$ is given by
$$\limsup_{n\rightarrow\infty}\P\{N=n\}^{1/n}=
1-\left\{\sqrt{\lambda(1-\rho)}-\sqrt{\rho(1-\lambda)}\right\}^2.
$$
\bigskip
We now turn to the proof of Lemma~2.3.
Let~$\zeta\in{\bf Y}$ be given. Write~$\M(\zeta)$ for the set of
configurations that can be obtained from $\ze$ by translating ones to
the right. Thus~$M(\zeta)$ is simply the
cardinality~$|\M(\zeta)|$ of~$\M(\zeta)$. If~$\eta\in{\bf Y}$
then~$\eta\zeta$ will denote the sequence in~$\bf Y$ obtained by the
concatenation of~$\eta$ and~$\zeta$. Finally, if~${\bf X}\subset{\bf
Y}$, we let~${\bf X}\zeta=\{\eta\zeta:\eta\in{\bf X}\}$.
\bigskip
\noindent {\bf Proof of Lemma 2.3.} We fix~$\zeta\in{\bf Y}$, and use
induction on~$N(\gamma)$. If~$N(\gamma)=0$, that is, if~$\gamma$ is
the empty sequence, then it suffices to notice
that~$\M(\zeta10)=\M(\zeta1)0\cup\M(\zeta0)1$, where the union is
clearly disjoint. Thus the result follows in this case. Assume now
that~$N(\gamma)\ge 1$, and that the result holds for smaller values
of~$N(\gamma)$. We now analyse two cases.
\medskip
\noindent
{\sl Case\/}~1. The sequence~$\gamma$ does not contain the segment~$10$.
\smallskip\noindent
In this case we clearly have that~$\gamma=0^k1^\ell$ for some~$k$,
$\ell\ge0$. If~$\ell\ge1$, using the fact that~$\M(\eta1)=\M(\eta)1$ for
any~$\eta\in{\bf Y}$ and the induction hypothesis, we are home. Thus we may
assume that~$\gamma=0^k$ for some~$k\ge1$. Now note that
$$
\M(\zeta100^k)
=\M(\zeta)10^{k+1}\cup\M(\zeta0)10^k\cup\cdots\cup\M(\zeta0^{k+1})1,
\Eq(yu1)
$$
where clearly the sets on the right-hand side are pairwise disjoint.
Similarly, we have
$$
\M(\zeta10^k)
=\M(\zeta)10^k\cup\M(\zeta0)10^{k-1}\cup\cdots\cup\M(\zeta0^k)1,
\Eq(yu2)
$$
with all the unions disjoint. We now observe that, by~\equ(yu2),
the elements in all
but the last set on the right-hand side of~{\equ(yu1)} are in natural
one-to-one correspondence with the elements in~$\M(\zeta10^k)$.
Moreover, since the elements in the last set on the right-hand side
of~{\equ(yu1)} correspond to the elements
in~$\M(\zeta0^{k+1})=\M(\zeta0\gamma)$ in an obvious way, we have that
$$
M(\zeta10\gamma)=|\M(\zeta10^{k+1})|=|\M(\zeta10^k)|+|\M(\zeta0^{k+1})|
=M(\zeta1\gamma)+M(\zeta0\gamma),
$$
as required.
\medskip
\noindent
{\sl Case\/}~2. The sequence~$\gamma$ contains the segment~$10$.
\smallskip\noindent
In this case let us write~$\gamma=\gamma_110\gamma_2$. Using the induction
hypothesis, we have that
$$
\eqalignno{
M(\zeta10\gamma)&=M(\zeta10\gamma_110\gamma_2)
=M(\zeta10\gamma_11\gamma_2)+M(\zeta10\gamma_10\gamma_2)\cr
&\qquad=M(\zeta1\gamma_11\gamma_2)+M(\zeta0\gamma_11\gamma_2)
+M(\zeta1\gamma_10\gamma_2)+M(\zeta0\gamma_10\gamma_2)\cr
&\qquad\qquad=M(\zeta1\gamma_110\gamma_2)+M(\zeta0\gamma_110\gamma_2)
=M(\zeta1\gamma)+M(\zeta0\gamma),
}
$$
completing the induction step, and hence the proof.
\quad\square
\bigskip
To close this section, we need to prove Lemma~2.4. To this end, we
consider a function~$R(\zeta)$ ($\zeta\in{\bf Y}$), implicit in
Derrida, Janowsky, Lebowitz and Speer (1993), which will turn out to
give an alternative combinatorial description of the
quantity~$M(\zeta)$. It is using this description that we shall prove
Lemma~2.4.
Let~$W=(W_i)_1^n$ be a $\{\pm1,0\}$-sequence and~$L=(L_i)_1^n$ a
$0$--$1$~sequence. We say that~$(W,L)$ is a {\it labelled closed
walk\/} of length~$n$ if ({\it i\/})~$W$ is a {\it closed walk
on~$\Z_+$ starting at~$0$\/}, that is, if all initial partial
sums~$\sum_{1\leq i\leq j}W_i$ ($0\leq j\leq n$) are non-negative
and~$\sum_{1\leq i\leq n}W_i=0$, and ({\it ii\/})~$L$ is such that,
for all~$1\leq i\leq n$, if~$W_i=1$ then~$L_i=1$, if~$W_i=-1$
then~$L_i=0$, and if~$W_i=0$ then~$L_i\in\{0,1\}$. For brevity, we
refer to a closed walk on~$\Z_+$ starting at~$0$ simply as a {\it
closed walk\/}. Given~$\zeta\in{\bf Y}$, let~${\cal R}(\zeta)$ be the
set of all labelled closed walks~$(W,L)$ with~$L=\zeta$, and
put~$R(\zeta)=|{\cal R}(\zeta)|$.
\bigskip
\noindent{\bf Proof of Lemma~2.4.}
We start by proving the following claim.
\proclaim Claim. For all~$\zeta$, $\gamma\in{\bf Y}$, we
have~$R(\zeta10\gamma)=R(\zeta1\gamma)+R(\zeta0\gamma)$.
\noindent{\bf Proof of the Claim.} Let~$\zeta$, $\gamma\in{\bf Y}$ be
fixed. Suppose~$W=(W_i)_1^n$ is a closed walk for
which~$(W,\zeta10\gamma)$ is a labelled closed walk.
Assume~$W=W^{(1)}w_1w_2W^{(1)}$, where~$W^{(1)}$ and~$W^{(2)}$ are
$\{\pm1,0\}$-sequences of length~$N(\zeta)$ and~$N(\gamma)$
respectively, and~$w_1$, $w_2\in\{\pm1,0\}$. We put
$$
\varphi(W,\zeta10\gamma)
=\cases{(W^{(1)}0W^{(2)},\zeta1\gamma) &if~$(w_1,w_2)=(0,0)$\cr
(W^{(1)}0W^{(2)},\zeta0\gamma) &if~$(w_1,w_2)=(1,-1)$\cr
(W^{(1)}1W^{(2)},\zeta1\gamma) &if~$(w_1,w_2)=(1,0)$\cr
(W^{(1)}(-1)W^{(2)},\zeta0\gamma) &if~$(w_1,w_2)=(0,-1)$.\cr
}
$$
Then it is straightforward to check that~$\varphi$ defines a bijection
between~${\cal R}(\zeta10\gamma)$ and~${\cal R}(\zeta1\gamma)\cup{\cal
R}(\zeta0\gamma)$, proving the claim.\quad\square
\bigskip
Putting together the claim above and Lemma~2.3, we deduce
that~$R(\zeta)=M(\zeta)$ for all~$\zeta\in{\bf Y}$,
since~$R(0^k1^\ell)=M(0^k1^\ell)=1$ for all~$k$, $\ell\ge0$.
We are now ready to start the proof of Lemma~2.4 proper. The
calculations below, which are included for completeness, appear in the
Appendix of Derrida, Janowsky, Lebowitz and Speer (1993) in a
slightly different form.
Let~$a$ and~$b\ge0$ be integers. For convenience, let us say that a
$0$--$1$~sequence~$L$ is of {\it type~$(a,b)$\/} if~$L$ has~$a$
elements equal to~$1$ and~$b$ elements equal to~$0$. Let~$r_{a,b}$ be
the number of labelled closed walks~$(W,L)$ with~$L$ of
type~$(a,b)$. Thus~$r_{a,b}=\sum_\zeta R(\zeta)=\sum_\zeta M(\zeta)$,
where the sum ranges over all~$\zeta$ with~$N(\zeta)=a+b$
and~$K(\zeta)=a$. Moreover, if~$W_0$ is a given closed walk,
let~$r_{W_0,a,b}$ be the number of labelled closed walks~$(W,L)$
with~$W=W_0$ and~$L$ a sequence of type~$(a,b)$.
Clearly~$r_{a,b}=\sum_Wr_{W,a,b}$, where the sum ranges over all
closed walks~$W$ of length~$a+b$.
The easiest way of handling the numbers~$r_{a,b}$ and $r_{W,a,b}$ is
by using generating functions. In the sequel, we shall consider
bivariate formal power series with formal variables~$x$ and~$y$.
Let~$n\ge0$ be an integer. We
put~$\psi_n(x,y)=\sum_{a,b}r_{a,b}x^ay^b$, where the sum ranges over
all pairs~$(a,b)$ with~$a$, $b\ge0$ and~$a+b=n$. Moreover, for a
closed walk~$W$, we put~$\psi_W(x,y)=\sum_{a,b\ge0}r_{W,a,b}x^ay^b$.
Then clearly~$\psi_n(x,y)=\sum_W\psi_W(x,y)$, where the sum is over
all closed walks~$W$ of length~$n$.
Now, if a closed walk~$W=(W_i)_1^n$ has~$2q$ non-zero entries,
it is immediate that we have~$\psi_W(x,y)=(x+y)^{n-2q}x^qy^q$. Now
note that the number of
closed walks~$W=(W_i)_1^n$ of length~$n$ with~$2q$ non-zero entries
is~$(q+1)^{-1}{2q\choose q}{n\choose2q}$. Indeed, to each such
walk~$W$, associate the walk~$W'=(W_i')_1^{n+1}$ with~$W_i'=W_i$
for~$1\leq i\leq n$ and~$W_{n+1}'=-1$. Then all proper partial initial
sums of~$W'$ are non-negative and~$\sum_{1\leq i\leq n+1}W_i'=-1$. The
number of such sequences~$W'$ is~$(n+1)^{-1}{2q+1\choose
q}{n+1\choose2q+1}$: choose where to have the~$\pm1$ in~$W'$ randomly,
and then Raney's lemma (cf.~the proof of Lemma 2.6) tells us that a
fraction of~$1/(n+1)$ of such choices will do for~$W'$. Thus the
number of closed walks of length~$n$ and~$2q$ non-zero entries is
$$
{1\over n+1}{2q+1\choose q}{n+1\choose2q+1}
={1\over 2q+1}{2q+1\choose q}{n\choose2q}
={1\over q+1}{2q\choose q}{n\choose2q},
$$
as claimed. (Here and in the sequel, the reader is referred to
Chapter~5 of Graham, Knuth, and Patashnik~(1989) for identities
involving binomial coefficients.) Therefore
$$
\eqalignno{
\psi_n(x,y)&=\sum_q{1\over q+1}{2q\choose q}{n\choose2q}
(x+y)^{n-2q}x^qy^q\cr
&=\sum_{q,j}{1\over q+1}{2q\choose q}{n\choose2q}
{n-2q\choose j}x^{q+j}y^{n-(q+j)}\cr
&=\sum_{q,k}{1\over q+1}{2q\choose q}{n\choose2q}
{n-2q\choose k-q}x^ky^{n-k}\cr
&=\sum_{q,k}{1\over q+1}{n\choose k}{k\choose q}
{n-k\choose q}x^ky^{n-k}\cr
&=\sum_k{1\over k+1}{n\choose k}x^ky^{n-k}
\sum_q{k+1\choose q+1}{n-k\choose q}\cr
&=\sum_k{1\over k+1}{n\choose k}{n+1\choose k}x^ky^{n-k}.\cr
}
$$
Therefore we
have that~$r_{k,n-k}=(k+1)^{-1}{n\choose k}{n+1\choose k}$, and hence
Lemma~2.4 follows.\quad\square
\bigskip
We close this section with the following remark. Recall
that~$M(\zeta)$ appears in the definition of the probability
measure~$p=p_{\rho,\lambda}$ on~$\bf Y$, and that we shall use~$p$ to
construct invariant measures for our two species asymmetric processes.
We feel that the definition of~$M(\zeta)$ makes it natural
that this quantity should be involved in our construction. The
alternative description of~$M(\zeta)$ as a certain number of labelled
walks on~$\Z_+$, given in the proof of Lemma~2.4 above, allows us to
perform some calculations, and in particular to prove Lemma~2.4.
\numsec=3\numfor=1
\beginsection 3. Invariant measures for the process as seen from a second
class particle
In the sequel, we shall always have~$\rho\le\lambda$.
Given $0<\rho\le \la <1$ we construct here a `translation
invariant' measure $\mu'_2$, in the sense this measure is invariant
under translations that leave a second class particle at the origin.
The parameter~$\rho$ corresponds to the asymptotic density of the
first class particles, and~$\lambda$ corresponds to the asymptotic
density of all the particles, with classes disregarded. In
Proposition~1 we show that under $\mu'_2$ the distribution of first
class particles to the right of the origin and the distribution of
empty sites to the left of it are product measures with densities
$\rho$ and $1-\la$ respectively. Another important and nice property
of the measure $\mu'_2$ is that the distribution of the distance
between two successive second class particles is the same as the
distribution of the hitting time of~$1$ for the random walk $\tilde
Z_n$ introduced after Lemma~2.3. This observation and Proposition~1
give an alternative way of computing the decay of densities found
by Derrida, Janowsky, Lebowitz and Speer (see Remark 3.2 below).
In Theorem~1 we show that $\mu'_2$ is invariant for the process. We
then construct other invariant measures for the process as seen from a
second class particle randomly drawing a configuration according
to~$\mu'_2$ and identifying first and second class particles to the
right of the origin and empty sites and second class particles to the
left of it. In particular, we get the shocks when $\la>\rho$: the
invariant measure as seen from a single, isolated second class
particle. We may also obtain an invariant measure with only two second
class particles. If $\la=\rho$ the distance between these two
particles is a non-degenerate random variable with an infinite
first moment. In this case the corresponding random walk~$\tilde Z_n$
is {\it symmetric} and the hitting time of~$1$ is finite with
probability one but has an infinite mean.
Let $\{\zeta_i\}_{i\in\Z} \subset {\bf Y}$ be a doubly
infinite {i.i.d.}~sequence of finite configurations with distribution
$\P(\zeta_i = \zeta) = p(\zeta)$, where $p(\ze)$ is given in \equ(1).
A configuration $(\si,\xi)$ with distribution $\mu'_2$ is obtained by
displaying the~$\zeta_i$ on the
integers separated by second class particles. More rigorously,
for~$i\ge0$, let $N_i
= N(\zeta_i)+1$ and $S_i = \sum_{j=0}^{i-1} N_j$. Let $I(x) = i$ if and only
if $S_i \le x < S_{i+1}$ ($i\in\Z$).
Set $\si(0)=0$, $\xi(0)=1$ and for $x>0$ put
$$
\si(x)=\cases{\zeta_{I(x)}(x-S_{I(x)})
&if $S_{I(x)}0$ and $j<0$, the operators
$\Phi_i$ and $\Psi_j$ commute
with the generator~$L'_2$ of the process as seen from the second class
particle:
$$
\Phi_i L'_2 = L'_2\Phi_i, \qquad \Psi_j L'_2 = L'_2\Psi_j.
$$
An immediate corollary of Lemma 3.1 is the following. Let $\Phi_\infty =
\Psi_{-\infty} = I$, the identity operator.
\proclaim Theorem 2. For any $0<\rho\le \la< 1$ and any
$0*x+1\}.$$
We first show $\mttp(1\Ga)=\m4p(\Ga^{-}12)$.
Notice that $$\mttp(1\Ga)=\mtp(\si(-1)=1)\mdp(\Ga)=\ro\la\mdp(\Ga).$$
On the other hand
$$
\m4p(\Ga^{-}12)=\prod_{i=-k}^{k-1}p(\ga_i)\times p(\ga_k1)=
\prod_{i=-k}^{k-1}p(\ga_i)\times p(\ga_k)\ro\la=\ro\la\mdp(\Ga),
$$
where $p$ is the probability measure given by $(2.1)$.
Similarly, $\m4p(\Ga0)=\mttp(20\Ga_{-})$, so we only need to show
that the two central sums in \equ(teo11) and \equ(teo12) are equal.
The first thing to notice is that $\mdp(\Ga)$ factors
in the following way.
$$
\mdp(\Ga)=m(\Ga)\times
\la(1-\ro)(\la\ro)^{k(\Ga)}((1-\la)(1-\ro))^{n(\Ga)-k(\Ga)},\Eq(teo13)
$$
where $m(\Ga)=\prod_{i=-k}^k M(\ga_i)$,
$k(\Ga)=\sum_{i=-k}^kK(\ga_i)$ and $n(\Ga)=\sum_{i=-k}^kN(\ga_i)$.
The measure
$\mdp(\Gax)$ factors in a similar way when
$x\in\La^{-}\backslash\{\sum_{i=-k}^{-1}(N(\ga_i)+1)\}$.
If moreover $x\in\La_2$,
then $k(\Ga)=k(\Gax)$ and $n(\Ga)=n(\Gax)$. So the last factor in the
product \equ(teo13) for $\mdp(\Ga)$ equals the corresponding one in
$\mdp(\Gax)$.
When $l_-\in\La_2$, we have
$$
\eqalign{\mdp(\Gax)&= m(\Gax)\times
\la(1-\ro)(\la\ro)^{k(\Gax)}((1-\la)(1-\ro))^{n(\Gax)-k(\Gax)}\cr
&\qquad\qquad\times\mtp(\si(-1)=1).}
$$
Since in this case $k(\Gax)=k(\Ga)-1$, $n(\Gax)=n(\Ga)-1$ and
$\mtp(\si(-1)=1)=\la\ro$, the product of the last two terms in the above
expression
equals the last term in $\equ(teo13)$. A similar thing happens when
$l_+\in\La_2$. Hence the factors dependent on
$\la$ and $\ro$ in the terms of both central sums in $(3.2)$ and $(3.3)$
are the same and so it is sufficient to verify
$$
\sum_{x\in\La_1}m(\Ga)=\sum_{x\in\La_2}m(\Gax).\Eq(teo14)
$$
This is proven in exactly the same way as ($3.5$) in
Derrida, Janowsky, Lebowitz and Speer
(indeed, $m$ here is the same object as $w$ in that paper),
by observing the following properties of $m$, which are inherited from
$M$:
$$
m(\Ga'10\Ga'')=m(\Ga'1\Ga'')+m(\Ga'0\Ga''),
$$
$$
m(\Ga'12\Ga'')=m(\Ga'2\Ga'')\quad{\rm and}\quad
m(\Ga'20\Ga'')=m(\Ga'2\Ga'').
$$
The conclusion is that both sides in $(3.5)$ equal
$$
\sum_{j:y_j=1,0}m(y_1^{k_1}\ldots y_j^{k_j-1}\ldots y_m^{k_m}),
$$
where $y_i^{k_i}$ ($i=1,\ldots,m$) are the maximal blocks of $y_i$
($y_i\in\{0,1,2\}$) constituting $\Ga$, that is,
$\Ga=y_1^{k_1}\ldots y_j^{k_j}\ldots y_m^{k_m}$.\quad\square
\bigskip
The next result was announced by Derrida, Janowsky, Lebowitz and
Speer~(1993) and proved by Speer~(1994) through direct
computations. Our proof is based on constructing the measure~$\mu'_2$
by first displaying the first class particles to the right of the
origin according to a product measure and then specifying the
positions of the second class particles. The same is done to the left
of the origin with the empty sites.
\proclaim Proposition 1. Under $\mu'_2$, the distribution of first
class particles to the right of the origin is the product
measure~$\nu_\rho$ of parameter~$\rho$. Similarly, the distribution
of holes to the left of the origin is the product
measure~$\nu_{1-\lambda}$ with parameter $1-\la$.
\noindent{\bf Proof.} Let a configuration~$\eta\in\{0,1\}^\Z$ be
given. For~$x\ge1$, let~$\eta|_x\in{\bf Y}$ be the finite
configuration~$\eta(1)\eta(2)\dots\eta(x-1)$ of length~$x-1$
determined by~$\eta$. For all~$x\ge1$, put~$M(\eta,x)=M(\eta|_x)$ and
similarly put~$K(\eta,x)=K(\eta|_x)$. For~$x\ge1$, let
$$
p(x\,|\,\eta) = \cases { 0 &if $\eta(x) = 1$ \cr\noalign{\vskip1\jot}
\la M(\eta,x) \la^{K(\eta,x)} (1-\la)^{x-K(\eta,x)-1} &if $\eta(x) =0$. \cr}
\Eq (pp)
$$
Notice that $p(x\,|\,\eta)$ depends on $\eta$ only through sites
$1,\dots,x$. To prove our result it suffices to prove that
$$
\sum_{x> 0} p(x\,|\, \eta) =1 \Eq(22)
$$
$\nur$ almost surely, which is proven in Lemma 3.2 below.
The reason is that we can interpret $p(x\,|\, \eta)$ as the probability of
the leftmost second class particle to the right of the origin be at site $x$ given
that the first class particles are at the sites occupied by $\eta$. To see
this compute for instance the probability that the
configuration~$\zeta=11010$ appears between the second class particle
at the origin and the next second
class particle (at site 6). According to our construction, first
ditribute three first class particles and two holes at
sites~$\{1,\dots,5\}$ with probability~$\rho^3(1-\rho)^2$. Then put a
hole at site~$6$ with probability~$1-\rho$. Finally, the conditional
probability of putting a second class particle at site~$6$ given the
configuration~$110100\dots$ is
$$
p(6\,|\, 110100\dots)=\la M(11010)\la^3(1-\la)^2.
$$
%
%(notice that~$p(x\,|\,\eta)$
%depends on $\eta$ only through
%$\eta(1),\dots,\eta(x-1)$.)
%
The resulting distribution is exactly the one given by \equ(1). This
argument can be applied to an arbitrary configuration but the
the notation is too heavy, and hence we omit the details.
\quad\square
\proclaim Lemma 3.2. Let $p(x\,|\, \eta)$ be defined as in \equ(pp).
Then for all $\rho\in [0,\la]$
$$
\sum_{x>0} p(x\,|\, \eta)=1 \Eq(sp)
$$
$\nur$ almost surely. Furthermore \equ(sp) holds for all
configurations~$\eta\in\{0,1\}^\Z$ with a finite number of particles.
\noindent {\bf Proof.} We first prove the identity \equ (sp)
for configurations~$\eta\in\{0,1\}^\Z$ with a
finite number of particles. Observe that if $\eta(x)=0$ for
all~$x>0$, then
$$
\sum_{x>0} p(x\,|\, \eta)= \la \sum_{x>0} (1-\la)^{x-1} = 1.
$$
Assume that the identity holds for any configuration with $n$ particles.
Let $\eta$ be a configuration with $n+1$ particles, whose rightmost particle
is located at $z>0$. Let $\eta^z$ be the configuration $\eta$ modified only
at site $z$. Hence $\eta^z$ has $n$ particles. From Lemma 2.3, for
$x\ge z+2$, we have
$$
M(\eta,x) = M(\eta^z,x-1) + M(\eta,x-1). \Eq (mm)
$$
Divide the sum in~\equ(sp) in two parts:
$$
\sum_{x>0} p(x\,|\, \eta) = \sum_{x=1}^{z+1} p(x\,|\, \eta) +
\sum_{x=z+2}^\infty p(x\,|\, \eta). \Eq(twoparts)
$$
Apply identity \equ(mm) to
all terms of the second sum of the right-hand side of~\equ(twoparts)
to obtain
$$
\eqalign{
\sum_{x=z+2}^\infty p(x\,|\, \eta)
&= \sum_{x=z+2}^\infty (1-\eta(x))M(\eta^z,x-1)
\la^{K(\eta,x)+1} (1-\la)^{x-1-K(\eta,x)} \cr
&\quad + \sum_{x=z+2}^\infty (1-\eta(x)) M(\eta,x-1)
\la^{K(\eta,x)+1} (1-\la)^{x-1-K(\eta,x)}. \cr
}
$$
Since for $x\ge z+2$, $K(\eta,x) = K(\eta,x-1) =
K(\eta^z,x-1)+1$ and $1-\eta(x)=1-\eta(x-1)=1$, we obtain
$$
\eqalign{
&= \la \sum_{x=z+2}^\infty (1-\eta(x-1)) (\eta^z,x-1)
\la^{K(\eta^z,x-1)+1} (1-\la)^{(x-1)-1-K(\eta^z,x-1)} \cr
& \quad\quad+ (1-\la)\sum_{x=z+2}^\infty (1-\eta(x-1)) M(\eta,x-1)
\la^{K(\eta,x-1)+1} (1-\la)^{(x-1)-1-K(\eta,x-1)}. \cr}
$$
Hence
$$
\sum_{x=z+2}^\infty p(x\,|\, \eta)
= \la \sum_{x=z+1}^\infty p(x\,|\, \eta^z)
+ (1-\la)\sum_{x=z+1}^\infty p(x\,|\, \eta). \Eq(>z)
$$
Observe that for $x\le z$, $M(\eta,x) = M(\eta^z,x)$,
$K(\eta,x)=K(\eta^z,x)$ while $1-\eta(x)=1-\eta^z(x)$ for
$xz) and \equ(0} p(x\,|\, \eta) = (1-\la) \sum_{x>0} p(x\,|\, \eta) + \la
\sum_{x>0} p(x\,|\, \eta^z). \Eq(puzzle)
$$
Since the second sum in the right-hand side of~\equ(puzzle) above
is one by inductive hypothesis, this complete the induction step.
Thus the result holds for finite~$\eta$.
Since the sum of the first $n$ terms in \equ(sp) depends only on
$\eta(1),\dots,\eta(n)$, the validity of \equ(sp) for finite $\eta$
implies that for any $\eta$ and $n\ge 1$, we have
$\sum_{x=1}^n p(x\,|\, \eta) \le 1$,
which in turns implies that for any $\eta$
$$
\sum_{x=1}^\infty p(x\,|\, \eta) = c(\eta) \le 1. \Eq(le1)
$$
Assume that there exists a set ${\bf X_0}$ with positive $\nur$ probability
such that if $\eta\in {\bf X_0}$, then $c(\eta)\le c<1$. This and \equ(le1)
imply that
$$
1>\int d\nur(\eta) \sum_{x=1}^\infty p(x\,|\, \eta) = \P(N<\infty) = 1,
\qquad\hbox{for } \rho\le \la,
$$
where $N$ is the random variable whose distribution is given by
\equ(1) and \equ(fg). The
contradiction above proves that \equ(sp) holds $\nur$ almost
surely for any $\rho \in [0,\la]$.
\quad\square
\bigskip
\noindent{\bf Remark 3.2.} Using the generating function of~$N$ given
in~\equ(fg), one can estimate precisely the rate of convergence of the
densities of the particles computed by Derrida, Janowsky, Lebowitz and
Speer~(1993). Let $(\Zt_n)_1^\infty$ be the random walk
on~$\Z$ defined after Lemma~2.3. We say that there is a {\it
record\/} at time~$n\ge1$ if $\Zt_n>\Zt_j$ for all $0\le j \rho$ and like $n^{-1/2}$ when $\la =
\rho$. Since the density of first class particles to the right of the
origin is a product measure with constant density $\rho$, this gives
also the asymptotic density of holes to the right of the
origin. Analogous arguments work to the left of the origin by
observing that the density of holes is~$1-\la$.
\numsec=4\numfor=1
\beginsection 4. The invariant measure for the translation invariant process
In this section we assume $0<\rho<\la< 1$. Let $\mu_2$ be the unique
translation invariant measure satisfying $\mu_2(\;\cdot\;|\,\xi(0)=1)=
\mu'_2(\;\cdot\;)$. As mentioned in the introduction, $\mu_2$ must be
invariant for the two species process. We show next that the measure
$\mu_2$ has good marginals.
\proclaim Theorem 3. The $\si$ marginal of $\mu_2$ is $\nur$ while the
$\si+\xi$ marginal of $\mu_2$ is $\nul$.
\noindent {\bf Proof.} To construct the measure $\mu'_2$ we started by
assigning the positions of the second class particles and then we gave
the distribution of the first class particles, given the position of
the second class particles. The positions of the second class
particles form a (discrete time) renewal process with finite
interarrival time, being the first renewal at time~$0$. When
$\rho<\la$ the average distance between two renewals is
$(\la-\rho)^{-1}<\infty$. Hence we can use the key renewal theorem to
construct $\mu_2$ in the following way:
$$
\mu_2 = \lim_{x\to-\infty} \mu'_2\tau_x = \lim_{x\to +\infty}
\mu'_2\tau_x \Eq(mp)
$$
where $\tau_x$ is the translation by $x$ operator. To show the theorem take a
cylinder function $f(\si,\xi)$ depending only on $\si$. Take a negative $z$
such that the support of $f$ is contained in $(z,\infty)$. By Proposition 1,
$$
\mu'_2 \tau_z f = \nur f.
$$
This and \equ(mp) implies that $\mu_2 f = \nur f$. To show that the $\si+\xi$
marginal is $\nul$ apply the same reasoning for a positive $z$ and show that
$\mu_2 f = \nul$ if $f$ depends only on $\si+\xi$.\quad\square
\bigskip
Our next result exploits the embedded renewal process in both $\mu_2$ and
$\mu'_2$.
\proclaim Theorem 4. It is possible to construct a coupling $\tilde\mu_2$ with
marginals $\mu_2$ and $\mu'_2$ such that if $(\si,\xi,\si',\xi')$ has
distribution $\tilde\mu_2$ and $H(\si,\xi,\si',\xi')= \sum_x \vert \si(x)
-\si'(x) \vert + \vert \xi(x)
-\xi'(x) \vert$ is the number of sites where $(\si,\xi)$ is different from
$(\si',\xi')$ then, under $\tilde\mu_2$, the random variable~$H$ has a
finite exponential moment. In other words, there exists $\theta>0$
such that
$$
\int d\tilde\mu_2 e^{\theta H} < \infty. \Eq(h)
$$
\noindent {\bf Proof.} Let $(\si,\xi)$ be a realization of
the translation invariant point
process related to the point process with distribution $\mu'_2$
and $T_i$ the stationary process
related to $\xi$. Thus~$T_i$ denotes the position of the $i$-th $\xi$
particle,
where~$T_0\le 0$ is the position of the rightmost $\xi$
particle to the left of the origin. Let
$(\bar\si',\bar\xi')$ be a realization of the process with distribution
$\mu'_2$ and let $S_i$ be
the renewal process associated to $\bar\xi'$, with $S_0=0$ and $S_i$ denoting
the position of the $i$-th $\bar\xi'$ particle.
The random variables $T_i-T_{i-1}$ are
independent and have the same distribution as $S_i-S_{i-1}$ for $i\ne 1$,
while $(T_0,T_1)$ has the limiting distribution
$$
P(T_1> u,- T_{0} > v) =
\limt P(S_{I(t)+1}>u, S_{I(t)} >v).
$$
Similarly, for all cylinder $f$, we have~$\mu_2 f = \lim_{x\to\infty}
\mu'_2 \tau_x f$.
Now we construct a coupling $(\si,\xi,\si',\xi')$ with the property that the
two first marginals have distribution $\mu_2$ and the two last marginals have
distribution $\mu'_2$.
If $T_0=0$, put~$\xi'(x)\equiv \xi(x)$ and $\si'(x) \equiv \si(x)$.
If $T_0\ne 0$, let~$J^+
= \min\{i>0: S_i \in \xi\}$ and $J^- = \max\{i\le 0: S_i \in \xi\}$,
and let
$$
(\si'(x),\xi'(x)) =
\cases { ( \si(x),\xi(x)) & if $x\ge J^+$ or $x\le J^-$ \cr
(\bar \si'(x),\bar\xi'(x)) &if $J^- < x < J^+$. \cr }
$$
It is clear that the resulting distribution of $(\si,\xi,\si',\xi')$ has
marginals $\mu_2$ and $\mu'_2$. To show \equ (h) notice that under
$\tilde\mu_2$, we have
$H \le J^+ - J^-$. Since $T_0$ and $T'_0$ have a finite exponential
moment, it follows from Lindvall~(1992), pp.~30--31, that both $J^+$
and $\vert J^-\vert$ have a finite exponential moment.
\quad\square
\numsec=5\numfor=1
\beginsection 5. Shocks in the simple exclusion process
If we take the process as seen from a second class particle and
identify particles of both classes to the right of the origin and
second class particles with holes to the left of it, we get $\eta'_t$,
the simple exclusion process as seen from an isolated second class
particle. Rigorously, Lemma~3.1 says that the process
$\eta'_t:=\tau_{X_t}\eta_t = \Phi_1\Psi_{-1}\tau_{X_t}(\si_t + \xi_t)$
is the {\sep} as seen from a second class particle. We consider the
shock measure constructed in the remark after Theorem 2 of Section
3. Let $\mu'=\Phi_1\Psi_{-1}\mu'_2$. For $0<\rho \le \la <1$, it
follows from Theorem 2 that $\mu'$ is invariant for the process
$\eta'_t$. Notice that $X_t$ can be seen as either a tagged second
class particle for the $(\si_t,\xi_t)$ process or as an isolated
second class particle for the $\eta_t$ process. Our next result
implies in particular that, for $0<\rho<\la<1$, the measure~$\mu'$ is
equivalent to $\nurl$ the product measure with densities $\rho$ and
$\la $ to the left and right of the origin respectively.
\proclaim Theorem~5. If $0<\rho < \la <1$, it is possible to
construct jointly the invariant measure $\mu$ and the product measure
$\nurl$ in such a way that the number of sites where the
configurations differ has a finite exponential moment.
\noindent {\bf Proof.}
First we construct a configuration with distribution $\nurl$ using two
independent configurations with distribution $\mu_2$. Let $(\si^+,\xi^+)$ and
$(\si^-,\xi^-)$ be two independent realizations of $\mu_2$ and $(\si',\xi')$ a
realization of $\mu'_2$ independent of the other two.
Define~$\eta\in\{0,1\}^\Z$ by letting
$$
\eta(x) = \cases { \si^+(x) + \xi^+(x) & if $x\ge 0$ \cr
\si^-(x) &if $x<0$ \cr}
$$
for all~$x\in\Z$.
Then, by the marginal properties of $\mu_2$ given by Theorem~3, it is
easy to
see that~$\eta$ constructed above has distribution $\nurl$. Here it is
important that we take independent realizations of $\mu_2$ to the right and
left of the origin. Now couple $(\si^+,\xi^+)$ with $(\si',\xi')$ as in the
proof of Theorem~4, letting~$J_+$ be the leftmost positive site where
$(\si^+,\xi^+)$ is different from $(\si',\xi')$. Similarly, couple
to the left of the origin letting $J^-$ be the rightmost negative site
where
$(\si^-,\xi^-)$ differs from $(\si',\xi')$. By the same argument as before
$J^+$ and $J^-$ have a finite exponential moment. Hence, setting~$\eta' =
\Phi_1\Psi_{-1} (\si',\xi')$, we get
$$
\sum_x \vert \eta'(x) - \eta(x) \vert \le J^+ - J^-.
$$
The result now follows again from Lindvall~(1992).\quad\square
An immediate consequence of the result above is the following.
\proclaim Corollary. The measures $\mu'$ and $\nurl$ are equivalent,
{i.e.}~one
is absolutely continuous with respect to the other.
\noindent {\bf Proof.} Since under both measures all non-empty
cylinder sets have positive probability, and non-empty sets of measure
zero depend on infinitely many coordinates, the corollary follows from
Theorem~4.\quad\square
\bigskip
\noindent {\bf Acknowledgments.} We thank E.~Speer for insightful
comments and very helpful discussions. The final version of this paper
was written while {P.A.F.} was a participant of the programme {\it
Random Spatial Processes\/} held at the Isaac Newton Institute for
Mathematical Sciences, University of Cambridge, whose warm
hospitality is gratefully acknowledged.
This paper is partially supported by FAPESP ``Projeto Tem\'atico''
Grant number 90/3918-5, CNPq and the SERC under Grant GR~G59981.
{Y.K.}~was partially supported by FAPESP under grant~93/0603-1, and by
CNPq under grant~300334/93-1.
\beginsection References
\bgroup
\frenchspacing
\parindent=0pt\parskip=1\jot
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\egroup
\vskip 6truemm
\+Instituto de Matem\'atica e Estat\'\i stica --- %
Universidade de S\~ao Paulo \cr
\+Cx.\ Postal 20570 --- 01452-990 S\~ao Paulo SP --- Brasil \cr
\+{\tt}, {\tt}, %
{\tt} \cr
\bye
*