\magnification=1200 \scrollmode \baselineskip 15pt \overfullrule=0pt \def\Gax{{\Gamma}_{x,x+1}} \def\La{\Lambda} \def\la{\lambda} \def\ro{\rho} \def\sx{(\si,\xi)} \def\sxl{\sx_\La} \def\hmu{{\mu'}} \def\mtp{\mu_2'} \def\mdp{\mu_2''} \def\mttp{\mu_2'''} \def\m4p{\mu_2''''} \def\a{\alpha} \def\e{\eta} \def\el{\eta'} \def\ex{\eta(x)} \def\et{\eta_t} \def\etl{\eta'_t} \def\eo{\eta(0)} \def\elx{\eta'(x)} \def\elo{\eta'(0)} \def\st{\sigma_t} \def\tu{\tau_1} \def\th{\theta} \def\xt{X_t} \def\vep{{\varepsilon}} \def\de{{\delta}} \def\la{{\lambda}} \def\ga{{\gamma}} \def\Ga{{\Gamma}} \def\si{{\sigma}} \def\ze{{\zeta}} \def\om{{\omega}} \def\Om{\Omega} \def\up{\Upsilon} \def\nul{{\nu_\la}} \def\nur{{\nu_\rho}} \def\nurl{{\nu_{\rho,\la}}} \def\hnurl{{\nu'_{\rho,\la}}} \def\limt{\lim_{t\to\infty}} \def\today{\rightline{\ifcase\month\or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi \space\number\day,\space\number\year}} %%%%%%%%%%%%%%%%% EQUAZIONI CON NOMI SIMBOLICI %%% %%% per assegnare un nome simbolico ad una equazione basta %%% scrivere \Eq(...) o, in \eqalignno, \eq(...) o, %%% nelle appendici, \Eqa(...) o \eqa(...): %%% dentro le parentesi e al posto dei ... %%% si puo' scrivere qualsiasi commento; %%% per assegnare un nome simbolico ad una figura, basta scrivere %%% \geq(...); per avere i nomi %%% simbolici segnati a sinistra delle formule e delle figure si deve %%% dichiarare il documento come bozza, iniziando il testo con %%% \BOZZA. Sinonimi \Eq,\EQ,\EQS; \eq,\eqs; \Eqa,\Eqas;\eqa,\eqas. %%% All' inizio di ogni paragrafo si devono definire il %%% numero del paragrafo e della prima formula dichiarando %%% \numsec=... \numfor=... (brevetto Eckmannn); all'inizio del lavoro %%% bisogna porre \numfig=1 (il numero delle figure non contiene la sezione. %%% Si possono citare formule o figure seguenti; le corrispondenze fra nomi %%% simbolici e numeri effettivi sono memorizzate nel file \jobname.aux, che %%% viene letto all'inizio, se gia' presente. E' possibile citare anche %%% formule o figure che appaiono in altri file, purche' sia presente il %%% corrispondente file .aux; basta includere all'inizio l'istruzione %%% \include{nomefile} %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \global\newcount\numsec\global\newcount\numfor \global\newcount\numfig \gdef\profonditastruttura{\dp\strutbox} \def\senondefinito#1{\expandafter\ifx\csname#1\endcsname\relax} \def\SIA #1,#2,#3 {\senondefinito{#1#2} \expandafter\xdef\csname #1#2\endcsname{#3} \else \write16{???? ma #1,#2 e' gia' stato definito !!!!} \fi} \def\etichetta(#1){(\veroparagrafo.\veraformula) \SIA e,#1,(\veroparagrafo.\veraformula) \global\advance\numfor by 1 % \write15{\string\FU (#1){\equ(#1)}} \write16{ EQ \equ(#1) == #1 }} \def \FU(#1)#2{\SIA fu,#1,#2 } \def\etichettaa(#1){(A\veroparagrafo.\veraformula) \SIA e,#1,(A\veroparagrafo.\veraformula) \global\advance\numfor by 1 % \write15{\string\FU (#1){\equ(#1)}} \write16{ EQ \equ(#1) == #1 }} \def\getichetta(#1){Fig. \verafigura \SIA e,#1,{\verafigura} \global\advance\numfig by 1 % \write15{\string\FU (#1){\equ(#1)}} \write16{ Fig. \equ(#1) ha simbolo #1 }} \newdimen\gwidth \def\BOZZA{ \def\alato(##1){ {\vtop to \profonditastruttura{\baselineskip \profonditastruttura\vss \rlap{\kern-\hsize\kern-1.2truecm{$\scriptstyle##1$}}}}} \def\galato(##1){ \gwidth=\hsize \divide\gwidth by 2 {\vtop to \profonditastruttura{\baselineskip \profonditastruttura\vss \rlap{\kern-\gwidth\kern-1.2truecm{$\scriptstyle##1$}}}}} } \def\alato(#1){} \def\galato(#1){} \def\veroparagrafo{\number\numsec}\def\veraformula{\number\numfor} \def\verafigura{\number\numfig} \def\geq(#1){\getichetta(#1)\galato(#1)} \def\Eq(#1){\eqno{\etichetta(#1)\alato(#1)}} \def\eq(#1){\etichetta(#1)\alato(#1)} \def\Eqa(#1){\eqno{\etichettaa(#1)\alato(#1)}} \def\eqa(#1){\etichettaa(#1)\alato(#1)} \def\eqv(#1){\senondefinito{fu#1}$\clubsuit$#1\else\csname fu#1\endcsname\fi} \def\equ(#1){\senondefinito{e#1}\eqv(#1)\else\csname e#1\endcsname\fi} \let\EQS=\Eq\let\EQ=\Eq \let\eqs=\eq \let\Eqas=\Eqa \let\eqas=\eqa %%%%%%%%%%%%%%%%%% Numerazione verso il futuro ed eventuali paragrafi %%%%%%% precedenti non inseriti nel file da compilare \def\include#1{ \openin13=#1.aux \ifeof13 \relax \else \input #1.aux \closein13 \fi} \openin14=\jobname.aux \ifeof14 \relax \else \input \jobname.aux \closein14 \fi %\openout15=\jobname.aux %%%%%%%%%%%%%%%%%%%%%%%%%%%% %\BOZZA % % macros extras \let\@=@ %this is necessary because in mssymb @ is an undefined control sequence \input mssymb \font\csc=cmcsc10 \def\sep{{\csc sep}} \def\E{{\Bbb E}} \def\P{{\Bbb P}} \def\Z{{\Bbb Z}} \def\M{{\cal M}} \def\({\left(} \def\){\right)} \def\square{\ifmmode\sqr\else{$\sqr$}\fi} \def\sqr{\vcenter{ \hrule height.1mm \hbox{\vrule width.1mm height2.2mm\kern2.18mm\vrule width.1mm} \hrule height.1mm}} % This is a slimmer sqr. %\rightline {\today} \def\beginsection#1\par{\vskip0pt plus.08\vsize\penalty-250 \vskip0pt plus-.08\vsize\bigskip\vskip\parskip \leftline{\bf#1}\nobreak\smallskip\noindent} \font\titlefont=cmr17 \font\authorfont=cmr12 \font\addressfont=cmti12 % % % \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. $$ An immediate corollary of Lemma 3.1 is the following. Let $\Phi_\infty = \Psi_{-\infty} = I$, the identity operator. \proclaim Theorem 2. For any $0<\rho\le \la< 1$ and any $0x+1\}.$$ 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 E.D.~Andjel (1982). Invariant measures for the zero range process. {\sl Ann. Probab. \bf 10} 525-547. L.~Breiman (1968). {\sl Probability}, Addison-Wesley. A.~De Masi, C.~Kipnis, E.~Presutti, E.~Saada (1990). Microscopic structure at the shock in the asymmetric simple exclusion. {\sl Stochastics \bf 27} 151-165. B.~Derrida, S.~Janowsky, J.L.~Lebowitz, E.~Speer (1993). Exact solution of the totally asymmetric simple exclusion process: shock profiles. To appear in {\sl J. Stat. Phys.} W.~Feller (1968). {\sl An Introduction to Probability Theory and its Applications\/}, John Wiley \& Sons, Inc. P.A.~Ferrari (1986). 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