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 \def\title{Exact Solution of the Totally
Asymmetric Simple Exclusion Process: Shock Profiles}

 \def\author{B.\ Derrida\thanks{\small Service de Physique Th\'eorique,
CEN Saclay, F91191 Gif-sur-Yvette, France.},
 S.\ A.\ Janowsky\thanks{\small Department of Mathematics, Rutgers
University, New Brunswick, NJ 08903.},
 J.\ L.\ Lebowitz\thanks{\small Departments of Mathematics and Physics,
Rutgers University, New Brunswick, NJ 08903.},
 E.\ R.\ Speer\footnotemark[2]}

\def\date{\today}

\begin{titlepage}
\begin{center}{\LARGE \title\par}
\vskip1em
{ \large \author\par}
\vskip3em
 \end{center}
 \vskip5em 

 \begin{abstract} The microscopic structure of macroscopic shocks in the
one dimensional totally asymmetric simple exclusion process is obtained
exactly from the complete solution of the stationary state of a model
system containing two types of particles---``first'' and ``second''
class.  This nonequilibrium steady state factorizes about any second
class particle, which implies factorization in the one component system
about the (random) shock position.  It also exhibits several other
interesting features, including long range correlations in the limit of
zero density of the second class particles.  The solution also shows
that a finite number of second class particles in a uniform background
of first class particles form a weakly bound state. 

  \end{abstract}

\vskip5em\noindent
{\bf Date:} March 2, 1993
\vskip1em\noindent
{\bf Submitted to:} Journal of Statistical Physics
\vskip1em\noindent
{\bf Key words:} asymmetric simple exclusion process, shock profiles,
second class particles, Burgers equation
\renewcommand{\thepage}{}
\end{titlepage}
\baselineskip=18pt plus 3pt minus 3pt

\section{Introduction}

Much attention has focused recently on nonequilibrium model systems
consisting of particles on a lattice, evolving microscopically under
conservative dynamical rules, and described on the macroscopic level by a
continuous density field which satisfies some equations of hydrodynamic
type~\cite{Spohn,LPS}.  Of particular interest are cases in which the
macroscopic equations produce shocks.  Such models usually have microscopic
transition rates which do not satisfy any detailed balance condition; that
is, the dynamics is not reversible with respect to any measure.  In this
paper we provide an analytic description of the shock structure of one such
model, the much studied one dimensional totally asymmetric simple exclusion
process, whose macroscopic behavior is described by the Burgers equation,
by obtaining an exact solution for a related model system.  Before
explaining these models in detail we describe briefly their context within
nonequilibrium statistical mechanics.

For systems with dynamics not satisfying detailed balance, there is no
general formula of the Gibbs type for the steady states.  As a result, we do
not in general have even qualitative information about the nature of these
states or their dependence on the parameters entering the dynamics.  Almost
all of our information about such driven diffusive systems comes from
computer simulations or from some approximate calculations using
renormalization group ideas or field theoretic methods~\cite{BS}.  These
indicate the existence of generic long range correlations in dimensions
greater than one~\cite{BS,GLMS,Grinstein}, even when the dynamics is short
ranged.  This situation is in strong contrast to the behavior when the
dynamics satisfies detailed balance; in this case the steady states are
essentially Gibbs states for some short ranged Hamiltonian and therefore
have exponential decay of correlations in the weak coupling or high
temperature regime.

One of the simplest models of driven diffusive systems is the {\it
asymmetric simple exclusion process}~\cite{L}, in which particles on
the one dimensional lattice $\bbz$ attempt at random times to jump to an
adjacent site, choosing the site to their right with some fixed probability
$p$ and that to their left with probability $1-p$, with the attempt
succeeding if the target site is not already occupied.  Here the Bernoulli
measure $\nu_\rho$, in which each lattice site is occupied independently
with probability $\rho$ ($0\le\rho\le1$), is a (translation invariant)
stationary state for any value of $p$; in this state there is a current
$J=(2p-1)\rho(1-\rho)$.  The dynamics satisfies detailed balance with
respect $\nu_{\rho}$ if and only if $p=1/2$.  (In fact, the model may be
formulated in any dimension and always has product measures as invariant
states; in this regard it is not typical of driven diffusive systems.)

For the asymmetric simple exclusion process the mass density $u(x,t)$ on
the macroscopic level---which is an appropriately scaled continuum limit
of the particle density in the microscopic model---is
described~\cite{L,R,BF,AV} by the inviscid Burgers' equation
 %
 \begin{equation}
  u_t
    +(2p-1)(u[1-u])_x=0.\label{Burg}
 \end{equation}
 %
It is well known that solutions of this equation can exhibit shocks (for
$p\ne1/2$).  In the simplest such situation the macroscopic density
$u(x,t)$ satisfies $u(x,t)=u_+$ for $x>x_0(t)$ and $u(x,t)=u_-$ for
$x<x_0(t)$, where $u_+>u_-$ (when $p>1/2$) and the shock position $x_0(t)$
moves with the constant velocity $(2p-1)(1-u_+-u_-)$.

An important question to ask of any model which produces shocks is whether
the sharpness of the density transition persists on the microscopic scale.
In the asymmetric simple exclusion process this problem has been
extensively studied both theoretically and
numerically~\cite{W,F,ABL,DKPS,FKS,F2,BCFN,JL}.  In either case one must
first locate the shock---that is, define precisely its position---on the
microscopic level.  This may be done by introducing into the system a
special {\it second class particle}~\cite{ABL}, which evolves by a modified
dynamical rule, chosen to keep the second class particle near the shock;
the position of this particle may then conveniently be taken as the
definition of the (microscopic) shock location.

With this definition of shock location it has been
established~\cite{F,DKPS,FKS,F2} that the shock remains sharp on the
microscopic level---i.e., that the (ensemble averaged) particle density as
viewed from the second class particle has a time invariant distribution
which approaches the densities $\rho_\pm=u_\pm$ at $\pm\infty$.  Computer
studies of this shock profile in the case $p=1$ are described
in~\cite{BCFN} (see also~\cite{JL}).  These indicate that in general the
density approaches its asymptotic value exponentially fast.  In the
limiting case $\rho_+=\rho_-=\rho$, however, when the invariant measure on
the original particles (now called {\it first class particles}) is
Bernoulli and the density in a fixed frame is thus constant and equal to
$\rho$, the choice to view the system from the second class particle
produces a ``shock'' profile for which the approach to $\rho$ is governed
by a power law.

In this paper (see also~\cite{DJLS}) we study these questions for the {\it
totally} asymmetric model (TASEP) in which all jumps are to the right, that
is, in which $p=1$; for this system we derive exact formulae for the shock
density profiles discussed above.  (In the conclusion we comment on
extensions to more general models.) Our method is to study a model
containing arbitrary numbers of first and second class
particles~\cite{FKS}, which we call the {\it two species} TASEP.  In
particular, we first obtain an explicit expression for the invariant
measure of the two species TASEP on a ring of $N$ sites.  This solution,
which is based on an earlier exact solution~\cite{DEHP} (see
also~\cite{DDM}) derived for a finite system with open boundaries and only
first class particles, expresses the probability of any configuration as
the trace of a product of matrices or operators.

 From this solution on the ring we then study shock profiles and other
phenomena by taking the infinite volume limit $N\to\infty$ in various
ensembles. We consider three main cases.  The shock profile between
distinct densities $\rho_+$ and $\rho_-$ is studied---via an
artifice~\cite{FKS}---through the infinite volume limit of systems with
fixed, non-zero densities of both first and second class particles.  On the
other hand, the ``shock'' profile for the constant density case is obtained
from the infinite volume limit of systems containing one second class
particle and a fixed density $\rho$ of first class particles.  Finally, we
study also a pair of second class particles in the presence of a finite
density of first class particles, and find that they form a bound state.

In the next section we describe more explicitly the two species TASEP and
the method we use to obtain shock profiles on the infinite line from the
exact solution of the model on the ring.  We derive there a simple symmetry
property of the shock profiles and discuss as well several general
properties of the invariant measures.  We also describe the organization of
the remainder of the paper, where the exact solution is obtained and its
consequences in the infinite volume limit are explored.

\section{Description of the model and summary of results} 

We begin by describing the two species TASEP on a ring of $N$
lattice sites, with $K_1$  first class particles, $K_2$  second
class particles, and $K_0=N-K_1-K_2$ empty sites.  No two particles are
allowed to occupy the same site, so that we may specify a configuration of
the system by an $N$-tuple $(\tau_1,\tau_2,\ldots,\tau_N)$, where
$\tau_i=1$ if there is a first class particle at site $i$, $\tau_i=2$ if
there is a second class particle at site $i$, and $\tau_i=0$ if site $i$ is
empty.  We treat the site index as cyclic, so that $\tau_{N+i}=\tau_i$.

The system evolves under a stochastic dynamical rule.  During a time interval
$dt$ each adjacent pair of sites in the system---say sites $i$ and
$i+1$---has probability $dt$ of being selected for a possible exchange of
their current states.  The exchange takes place if (a) site $i$ is occupied
by a first class particle and site $i+1$ is either occupied by a second class
particle or is empty, or (b) site $i$ is occupied by a second class particle
and site $i+1$ is empty; that is, the possible transitions are
 %
 \begin{equation}
\ldots1\;0\to0\;1\;,\qquad 1\;2\to2\;1\;,
   \qquad\hbox{and}\qquad 2\;0\to0\;2\;.
 \end{equation}
 %
 Note that the number of particles of each type is conserved.  It is easy
to see that any configuration can evolve into any other, so that the system
is ergodic and there is a unique stationary probability measure.

In Section 3 we derive an explicit expression for the probability of any
configuration in this invariant measure; the remainder of the paper
explores the implications of this representation for the behavior of the
system in the infinite volume limit.  Although it is possible
to carry out the analysis using the canonical ensembles introduced above,
it simplifies the computations to introduce superpositions of these---that
is, grand canonical ensembles---in which the populations of particles can
fluctuate, adjusting the fugacities of the first and second class
particles, and of the holes, to produce some fixed densities of the various
species.

In Section 4 we consider a system which contains a single second class
particle and a fixed density $\rho$ of first class particles: $K_2=1$,
$\E{K_1}/N=\rho$; the quantity which has a nontrivial infinite volume limit
is the density of first class particles as seen from the second class
particle.  In this relative frame the location of the second class particle
is taken to be $i=0$ and we let $d_i$ denote the average density of first
class particles at sites $i\ne0$.  The average drift velocity of the second
class particle in a fixed frame is $1-2\rho$, as we show at the end of this
section (see also~\cite{F2}).  Because this drift velocity decreases with
$\rho$ the effect of density fluctuations is that the second class particle
is attracted toward regions in which the local density has a positive
gradient.  Thus the density profile has $d_i>\rho$ for $i>0$ and $d_i<\rho$
for $i<0$.  We find that $d_i$ converges monotonically to $\rho$ as $|i|$
increases and, as indicated in the introduction,
$|d_i-\rho|\simeq C|i|^{-1/2}$ for $|i|\gg1$.

We could, of course, take a infinite volume limit with any constant number
of second class particles. In Section 5 we consider the case $K_2=2$,
$\E{K_1}/N=\rho$.  Somewhat surprisingly, the two second class particles
form a bound state, with a power law decay $ r^{-3/2}$ of the probability
of finding them a distance $r$ apart; the distance between them thus has
infinite expectation.

A time invariant state in the ring geometry cannot support two regions of
distinct densities of first class particles.  It is possible, however, to
compute the density profile in the infinite system for the shock between
asymptotic densities $\rho_+$ and $\rho_-$ (with $\rho_+>\rho_-$) by taking
the infinite volume limit from the ring geometry at fixed densities
$\rho_1=\rho_-$ of first class particles and $\rho_2=\rho_+-\rho_-$ of
second class particles.  Again we suppose that we look at the system
relative to one second class particle, located at the origin; let us denote
by $\delta_i$ the average density of first class particles at site $i$, for
$i<0$, and the average density of all particles---both first and second
class---at site $i$, for $i>0$.  Because the second class particles ahead
of the distinguished one behave toward it as if they were first class
particles, it is easy to see that $\delta_i$ is exactly the profile we
seek.  (This is a special case of a method of~\cite{FKS} for identifying a
shock profile in a system of first class particles from the knowledge of a
stationary measure for a system with uniform densities of first and second
class particles, viewed from a second class particles.  The general method
works for any asymmetry in the ASEP---that is, for any value of $p$, not
just $p=1$.)  The calculations for this situation are carried out in
Section~6, and the shock profile is discussed in Section~7.  We find that
$\delta_i$ converges exponentially to $\rho_\pm$ as $|i|\to\pm\infty$ and
that the characteristic length of this decay diverges as
$(\rho_+-\rho_-)^{-2}$ when $(\rho_+-\rho_-)\searrow0$.

There are several simple general properties of the invariant states of
the two species model.  One of these, derived immediately below, is a
symmetry property---symmetry around a second class particle, in the
two component system, or symmetry of the shock, in the one component
system:
 %
 \begin{equation}
\delta_{-i} + \delta_i = \rho_- + \rho_+.\label{symm}
 \end{equation}
 %
Another, derived at the end of Section~3 as a simple consequence of our
representation of the invariant measure, is that in the infinite system a
second class particle has the effect of decoupling the regions on each side
of it; thus if we condition (as in the discussions above) on the presence
of such a particle at the origin, and let $f(\tau)$ and $g(\tau)$ be
functions which depend only on the occupation variables $\tau_i$ for $i<0$
and $i>0$, respectively, then
 %
 \begin{equation}
   \E{fg\mid\tau_0=2}=\E{f\mid\tau_0=2}\E{g\mid\tau_0=2}.\label{indep}
 \end{equation}
 %
 (A similar independence holds for finite systems in the grand canonical
ensemble, for expectations conditioned on the presence of at least two
second class particles at specified sites.)  A third property depends for
its derivation on the somewhat elaborate computations from the explicit
formula for the invariant measure which will be given later (see Sections~6
and 7): if we again condition on the presence of a second class particle at
the origin in the infinite system with densities $\rho_1$ and $\rho_2>0$ of
first and second class particles, then the distributions of first class
particles to the right of the origin, and of holes to the left of the
origin, are Bernoulli, with densities $\rho_1$ and $1-\rho_1-\rho_2$,
respectively.

The derivation of the symmetry (\ref{symm}) is based on the fact~\cite{L}
that the invariant measure for the TASEP at density $\rho$, with only first
class particles, is Bernoulli.  Let us introduce new occupation variables
$\xi_i$ and $\zeta_i$ taking values 0 and 1, where $\xi_i=1$ (respectively
$\zeta_i=1$) if and only if there is a first class (respectively second
class) particle at site $i$.  Now the second class particles appear to the
first class particles to be indistinguishable from holes; thus the first
class particles considered separately have a Bernoulli distribution at
density $\rho_1$.  On the other hand, we may ignore the distinction between
the two types of particle as well as interchanges ($1\;2\to2\;1$) between
them; these new particles themselves form a TASEP system which will have an
invariant measure which is Bernoulli with density $\rho_1+\rho_2$.  Thus
for all $j\ne k$,
 %
 \begin{equation}
\E{\xi_j\xi_k}=\rho_1^2 \qquad \hbox{and}\qquad
\E{(\xi_j+\zeta_j)(\xi_k+\zeta_k)}=(\rho_1+\rho_2)^2.\label{ind}
 \end{equation}
 %
 If we write $\E{(\xi_j+\zeta_j)(\xi_k+\zeta_k)}
=\E{\xi_j\xi_k}+\E{\xi_j\zeta_k}+\E{\zeta_j(\xi_k+\zeta_k)}$,  
then (\ref{ind}) implies that 
 %
 \begin{equation}
\rho_2^{-1}\E{\xi_j\zeta_k}+\rho_2^{-1}\E{\zeta_j(\xi_k+\zeta_k)}
 = 2\rho_1+\rho_2.
 \end{equation}
 %
  Since $\rho_2^{-1}\E{\xi_j\zeta_k}$ is the probability of a first class
particle at site $j$, conditioned on there being a second class particle at
site $k$, and $\rho_2^{-1}\E{\zeta_j(\xi_k+\zeta_k)}$ is the probability of
a first or second class particle at site $k$, conditioned on there being a
second class particle at site $j$, for $j<k$ this is just (\ref{symm}) with
$i=k-j$.

 The fact that we can view the two species TASEP as a one species TASEP in
two different ways also yields a simple argument that in a uniform one
component TASEP at density $\rho$ the velocity of a single second class
particle is $1-2\rho$. For if a small density $d\rho$ of second class
particles is added to such a system, then the new system may be regarded as
a one component TASEP at density $\rho+d\rho$ and hence the current must
increase from $\rho(1-\rho)$ to $(\rho+d\rho)(1-\rho-d\rho)$. On the other
hand, since the first class particles alone form a TASEP, they still
contribute a current $\rho(1-\rho)$, and the additional current due to the
second class particles is $(1-2\rho)d\rho$.

\section{The stationary measure}

Our goal in this section is to derive an explicit formula for the
probabilities of all configurations, in the stationary measure, for a
system of $K_1$ first class and $K_2$ second class particles on a ring of
$N$ sites.  In an earlier study~\cite{DEHP}, of a system with open
boundaries and with only first class particles, such formulas were given in
terms of certain matrix products.  Here we extend this method to the case
of periodic boundary conditions for a system with an arbitrary number of
second class particles.

If the system contains no second class particles then the stationary
measure is well known: all configurations are equally likely.  For the
rest of the section we will therefore assume that the system contains at
least one second class particle.  We will show that the stationary
measure may be constructed with the aid of two matrices or operators
$D$ and $E$ which satisfy (\cite{DEHP})
 %
 \begin{equation}
      DE=D+E.\label{DE}
 \end{equation}
 %
 If we define
 %
 \begin{equation}
A = DE-ED,\label{A}
 \end{equation}
 %
 then it follows from (\ref{DE}) that $A$ is a projection ($A^2=A$) and
satisfies
 %
 \begin{equation}
DA=A\qquad\hbox{and}\qquad AE=A;\label{ADAE}
 \end{equation}
 %
 we suppose also that $\Tr A<\infty$.  (An example of such matrices will be
given below.) Then we claim that for fixed ${\K}=(K_0,K_1,K_2)$ the
probability in the stationary  measure of the configuration
$\tau$ is given, up to an overall normalizing factor dependent on $\K$, by
the weight $w_\K(\tau)$ which is defined by
 %
 \begin{equation}
w_\K(\tau_1,\ldots,\tau_N)\equiv\Tr[X_1\cdots X_N],\label{fund}
 \end{equation}
 %
 where $X_i=E$ if $\tau_i=0$, $X_i=D$ if $\tau_i=1$, and $X_i=A$ if
$\tau_i=2$.  Note that the invariance of the system under rotation of
the ring of sites, that is, under cyclic permutation of the $\tau_i$, is
preserved by the corresponding invariance of the trace. 

We next write down the stationarity conditions on the measure which
follow from the dynamics described in Section~2.  In doing so we will
use the following standard notation: if $\tau=(\tau_1,\ldots,\tau_N)$ is
a system configuration then $\tau^{ij}$ denotes the configuration
obtained from $\tau$ by interchanging $\tau_i$ and $\tau_j$.  Now
consider the possible change in the probability of a specific
configuration $\tau$ during some time interval $dt$.  For each pair of
indices $i,i+1$ with $\tau_i\,\tau_{i+1}$ of the form $1\,0$, $1\,2$, or
$2\,0$ there is a probability $dt$ that the states of these sites will
be interchanged during the time interval; on the other hand, for each
such pair $i,i+1$ with $\tau_i\,\tau_{i+1}$ of the form $0\,1$, $2\,1$,
or $0\,2$ there is a probability $dt$ that an exchange will occur in the
configuration $\tau^{i,i+1}$, leading to the configuration $\tau$.  Thus
the condition for stationarity of the measure with weights $w_\K$ is
that for all $\tau$,
 %
 \begin{equation}
 \sum_{\{\,i\mid\tau_i\tau_{i+1}=10,\,12,\;{\rm or}\;20\,\}}w_\K(\tau)
  =\sum_{\{\,i\mid\tau_i\tau_{i+1}=01,\,21,\;{\rm or}\;02\,\}}
        w_\K(\tau^{i,i+1})\;.
  \label{bal}
  \end{equation}
 %

 We now verify that (\ref{fund}) provides the weights of a stationary
measure, that is, according to (\ref{bal}), that for all $\tau$,
 %
 \begin{eqnarray}
 \sum_{\{\,i\mid\tau_i\tau_{i+1}=10,\,12,\;{\rm or}\;20\,\}}
   \Tr[X_1\cdots X_iX_{i+1}\cdots X_N]\hskip100pt\nonumber\\
  =\sum_{\{\,i\mid\tau_i\tau_{i+1}=01,\,21,\;{\rm or}\;02\,\}}
  \Tr[X_1\cdots X_{i+1}X_i\cdots X_N].
  \label{balx}
  \end{eqnarray}
 %
Let us write $X_1\cdots X_N$, possibly after a cyclic permutation which
will not affect the trace, in terms of blocks of consecutive identical
matrices: $X_1\cdots X_N=Y_1^{k_1}\cdots Y_m^{k_m}$, where each $Y_j$ is
$D$, $A$, or $E$, and $Y_j\ne Y_{j+1}$, $Y_m\ne Y_1$.  Now in each term
on the left hand side of (\ref{balx}) we use the algebraic relations
(\ref{DE}) and (\ref{ADAE}) to replace the product $X_iX_{i+1}$ by a
sum of two operators or a single operator---that is, $DE$ is replaced by
$D+E$ while $DA$ and $AE$ are replaced by $A$.  The net effect of these
replacements is that each boundary $DE$, $DA$, or $AE$ between blocks in
the product gives rise to one or two terms:
% 
\begin{equation}
\matrix{\cdots DE\cdots&\to
    &\Tr[\cdots \hat DE\cdots]+\Tr[\cdots D\hat E\cdots]\;,\cr
\cdots DA\cdots&\to&\Tr[\cdots \hat DA\cdots]\;,\cr
\cdots AE\cdots&\to&\Tr[\cdots A\hat E\cdots]\;,\cr}
 \end{equation}
 %
where the caret denotes an omitted factor.  Since each $D$ block is
followed by an $E$ or an $A$, and each $E$ block is preceded by a $D$ or an
$A$, each $D$ block (respectively $E$ block) gives rise to a term in which
the rightmost (respectively leftmost) factor in the block has disappeared;
notice that in the case of a DE boundary we get a term for each block.
Thus we find
 %
 \begin{equation}
 \sum_{\{\,i\mid\tau_i\tau_{i+1}=10,\,12,\;{\rm or}\;20\,\}}
     \Tr[X_1\cdots X_iX_{i+1}\cdots X_N]
  = \sum_{\{\,j\,|\, Y_j=D,E\,\}}
   \Tr[Y_1^{k_1}\cdots Y_j^{k_j-1}\cdots Y_m^{k_m}].
  \label{left}
  \end{equation}
 %
 Similarly, each block boundary $X_iX_{i+1}$ of the form $ED$, $AD$, or
$EA$ contributes a term to the right hand side of (\ref{balx}) in
which $X_iX_{i+1}$ has become $X_{i+1}X_i$; if we make the same
replacements as above for the latter product, then the net
effect on block boundaries is
 % 
\begin{equation}
\matrix{\cdots ED\cdots&\to&\cdots DE\cdots&\to
    &\Tr[\cdots E\hat D\cdots]+\Tr[\cdots\hat ED\cdots]\;,\cr
\cdots AD\cdots&\to&\cdots DA\cdots&\to
   &\Tr[\cdots A\hat D\cdots]\;,\cr
\cdots EA\cdots&\to&\cdots AE\cdots&\to
   &\Tr[\cdots \hat EA\cdots]\;.\cr}
 \end{equation}
 %
 Now the $D$ blocks lose their leftmost and the $E$ blocks their rightmost
factors,  and we find that 
 %
 \begin{equation}
 \sum_{\{\,i\mid\tau_i\tau_{i+1}=01,\,21,\;{\rm or}\;02\,\}}
   \Tr[X_1\cdots X_{i+1}X_i\cdots X_N]
  = \sum_{\{\,j\,|\,Y_j=D,E\,\}}
   \Tr[Y_1^{k_1}\cdots Y_j^{k_j-1}\cdots Y_m^{k_m}].
  \label{right}
  \end{equation}
 %
Comparison of (\ref{left}) and (\ref{right}) verifies (\ref{balx}). 

We turn finally to the question of existence of matrices $D$ and $E$
satisfying (\ref{DE}).  If we write $D=I+d$ and $E=I+e$ then the algebra
(\ref{DE}) satisfied by $D$ and $E$ reduces to $de=I$, so that if $D$ and
$E$ are finite dimensional then $d=e^{-1}$, $D$ and $E$ commute, and $A=0$.
This trivial solution may in fact be regarded as giving the correct weights
for the system with no second class particles.  To treat second class
particles, however, we must take $D$ and $E$ to be infinite dimensional; a
simple solution~\cite{DEHP} of $de=I$ with noncommuting $d,e$ takes $e$ to
be the right shift operator on semi-infinite vectors and $d$ to be its
adjoint, leading to
 %
  \begin{eqnarray}
{D} = \left( \begin{array}{cccccc}
		  1&1&0&0&. &.\\
		  0&1&1&0&& \\
		  0&0&1&1&&\\
		  0&0&0&1&.  &\\
		  . &&&&. &. \\
		  . &&&&&.
		   \end{array}
		   \hspace{0.2in} \right)
\;,\hspace{0.3in}
{E} = \left( \begin{array}{cccccc}
		  1&0&0&0&. &.\\
		  1&1&0&0&& \\
		  0 &1&1&0&&\\
		  0 &0&1&1&&\\
		  . &&&.&. & \\
		  .&&&&.&.
		   \end{array}
		   \hspace{0.2in} \right)\;.
\label{DEdef}
 \end{eqnarray}
 %
$A$ is then the projection on the first component of the vector and has
trace 1:
 %
  \begin{eqnarray}
A = | 1 \rangle \langle 1 | = \left( \begin{array}{cccccc}
		  1&0&0&0&. &.\\
		  0&0&0&0&& \\
		  0 &0&0&0&&\\
		  0 &0&0&0&&\\
		  . &&&.&. & \\
		  .&&&&.&.
		   \end{array}
		   \hspace{0.2in} \right)\;,
\label{DEA}
 \end{eqnarray}
 %
where $\langle1|=[1,0,0,\ldots]$.  Of course, our results are independent
of any explicit representation of the matrices, depending only on
(\ref{DE}), (\ref{A}), and the fact that $A$ has finite trace.  For notational
convenience in the balance of the paper, however, we will assume that
$A$ is a one dimensional projector and write $A=|1\rangle\langle1|$ as
in (\ref{DEA}).  We will also use the specific representation
(\ref{DEdef}), (\ref{DEA}) in the appendix. 

There is, however, one important property of the stationary state which is
easy to see in this representation. Let us consider configurations
$(\tau_1,\ldots,\tau_N)$ which have second class particles at two specified
sites, say $j$ and $N$.  Then the weights of such configurations factorize:
 %
 \begin{eqnarray}
w_{\K}(\tau_1,\ldots,\tau_N)
= \langle1|X_{1}\cdots X_{j-1}|1\rangle\;
 \langle1|X_{j+1}\cdots X_{N-1}|1\rangle\label{factor}
 \end{eqnarray}
 %
 when $\tau_j=\tau_N=2$. In later sections we will consider the infinite
volume limit of these systems, conditioning on the presence of a second
class particle at site $N$, or equivalently at the origin in the limit.
Because of the factorization (\ref{factor}), there is in the limit
independence between the portions of the system to the left and right of
the origin, as expressed by (\ref{indep}).

In the remainder of the paper we will consider several different infinite
volume limits of the system described here, in which we either fix a finite
number of second class particles or consider a finite density of second
class particles.  In each case we will concentrate on calculating the
limiting values of the particle density profiles; however, it is easy to
extend the computations to show that the (weak) limit of the measure
defined here---that is, the limit of all correlation functions---exists in
each case.

 \section{The profile seen from a single second class particle}

In this section we consider systems which contain a single second class
particle, and ultimately derive an exact expression for the
stationary density profile of first class particles, seen from
the second class particle, in an infinite system in which the first
class particles have average density $\rho$.  In the process we also find
formulae for this profile in the canonical and grand canonical ensembles
for finite systems. 

We begin by considering a system of $N$ sites, with a single second
class particle ($K_2=1$) and $K_1=K$ first class particles.  Since the
cyclic symmetry of the model implies that the weights of configurations
in the steady state depend only on the positions of the first class
particles relative to the second class particle we may choose a frame in
which the second class particle is always at position $N$.  Equation
(\ref{fund}), with the notation $A=|1\rangle\langle1|$, then becomes
 %
 \begin{eqnarray}
w_{\K}(\tau_1,\tau_2, \ldots, \tau_{N-1},2)  
   = \langle 1 | \prod_{i=1}^{N-1} X_i | 1 \rangle,\label{f12}
 \end{eqnarray}
 %
 where $X_i = D$ if $\tau_i =1$ and $X_i = E $ if $ \tau_i =0$.  From this
one can in principle calculate the conditional density
$\langle \tau_i\,|\,\tau_N=2 \rangle_{\K}$ at any site $i\ne N$ in the
system---that is, the probability that site $i$ is occupied---by
 %
 \begin{eqnarray}
\langle \tau_i\,|\,\tau_N=2 \rangle_{\K} =
\frac { \sum_{{\rm configurations}} w_{\K}(\tau_1,\tau_2, \ldots,
\tau_{N-1},2)\, \tau_i }
 { \sum_{{\rm configurations}} w_{\K}(\tau_1,\tau_2, \ldots, \tau_{N-1},2)}\;,
 \label{ratio}
 \end{eqnarray}
 %
where the sums run over all the configurations with $K$ first
class particles and $K_0=N-1-K$ empty sites.

The quantities $\langle \tau_i\,|\,\tau_N=2 \rangle_{\K}$ can in fact be
computed from (\ref{ratio}) (see Remark~4.1 at the end of this section),
but the constraint of a fixed number of first class particles makes the
calculation rather complicated.  It is more convenient to work in the grand
canonical ensemble, in which the number of first class particles can
fluctuate.  For this purpose we introduce two parameters which play the
role of fugacities: $x$ for first class particles and $z$ for empty sites.
(It would be sufficient to introduce only one of these, since the total
number of available sites is fixed at $N-1$, but the use of both makes the
symmetry between first class particles and holes more apparent).

In the grand canonical ensemble the unnormalized weight 
$W_N(\tau_1,\tau_2, \ldots, \tau_{N-1},2)$ of the
configuration $(\tau_1,\ldots,\tau_{N-1},2)$ with $K_1$ first class and
$K_0=N-K_1-1$ second class particles is given by
 %
 \begin{eqnarray}
W_N(\tau_1,\tau_2, \ldots, \tau_{N-1},2) 
 &=&x^{K_1}z^{K_0}
    w_{\K}(\tau_1,\tau_2, \ldots, \tau_{N-1},2)\nonumber\\
 &=& \langle 1 | \prod_{i=1}^{N-1}
  [ x D \tau_i + z E (1 - \tau_i) ] | 1 \rangle.\label{gN}
 \end{eqnarray}
 %
 The density $d_i(N)$ at site $i\ne N$ is then
 %
 \begin{eqnarray}
d_i(N) &\equiv&
  \frac { \sum_{\tau} W_N(\tau_1,\tau_2, \ldots,\tau_{N-1},2)\,\tau_i}
 {\sum_{\tau}W_N(\tau_1,\tau_2, \ldots, \tau_{N-1},2)};
\label{taui0}
 \end{eqnarray}
 %
 where the sums, no longer  restricted to fixed $K_0$ and $K_1$, run
over $\tau_j=0,1$ for $j=1,\ldots,N-1$. 
Introducing 
 %
 \begin{eqnarray}
C = x D + z E,
\label{Cdef}
 \end{eqnarray}
 %
 we see that these densities are easily expressed in terms of the matrix
$C$:
 %
 \begin{eqnarray}
d_i(N)  &=&  x  \  \frac{ \langle 1 | C^{i-1} D C^{N-i-1}
| 1 \rangle  }
 {  \langle 1 |  C^{N-1}
| 1 \rangle}.
\label{taui}
 \end{eqnarray}
 %
 From (\ref{A}) and (\ref{ADAE}) it follows that 
$D|1\rangle=|1\rangle$, $\langle1|E=\langle1|$, and 
$D C - C D = z | 1 \rangle \langle 1 |$, so that
 %
 \begin{eqnarray}
d_1(N) = 1 -  z \ \frac{ \langle 1 | C^{N-2} | 1 \rangle }
   { \langle 1 | C^{N-1} | 1 \rangle},
\label{tau1}
 \end{eqnarray}
 %
 \begin{eqnarray}
d_i(N) - d_{i+1}(N)
 =   x z  \ \frac{ \langle 1 | C^{i-1} | 1 \rangle
 \langle 1 | C^{N-i-2} | 1 \rangle }
   {\langle 1 | C^{N-1} | 1 \rangle},
\label{tauiip1}
 \end{eqnarray}
 %
 and
 %
 \begin{eqnarray}
 d_{N-1}(N) =   x  \ \frac{\langle 1 | C^{N-2} | 1 \rangle  }
    {\langle 1 | C^{N-1} | 1 \rangle}.
\label{tauNm1}
 \end{eqnarray}
 %
 Thus the computation of the $d_{i}(N)$ is reduced to the evaluation
of the matrix elements $ \langle 1 | C^{n} | 1 \rangle $.  These are
calculated in the Appendix, both exactly (see (\ref{Cna})),
 %
 \begin{eqnarray}
   \langle 1 | C^{n} | 1 \rangle  =
\sum_{p=0}^n  x^p z^{n-p}  \frac{1}{p+1}
\left( \begin{array}{c}
n + 1 \\ p
 \end{array} \right)
\left( \begin{array}{c}
n  \\ p
 \end{array} \right),
\label{Cn}
  \end{eqnarray}
 %
 and in the large $n$ limit (see (\ref{Casympta})),
 %
 \begin{eqnarray}
   \langle 1 | C^{n} | 1 \rangle  \simeq
   \frac{ ( \sqrt{x} + \sqrt{z})^{2 n+3}}
   { 2 \sqrt{\pi} n^{3/2} ( x z ) ^{3/4} }\;.
\label{Casympt}
    \end{eqnarray}
 %
  Note that these results are symmetric in $x$ and $z$.

   As always in the grand canonical ensemble approach, one must choose
the values of the fugacities $x$ and $z$ in order to fix the
density of the first class particles to be $\rho$.  It follows from
(\ref{gN}) and (\ref{taui}) that the average number $\Ex{K_1}$ of first
class particles in a system of N sites (including the second class
particle, i.e., with $N-1$ sites available for the first class
particles) is
 %
 \begin{eqnarray}
  \Ex{K_1} = x \  \frac{ d}{dx} \log  \ \langle 1 | C^{N-1} | 1 \rangle.
    \end{eqnarray}
 %
Then for large $N$, from (\ref{Casympt}), 
 %
 \begin{eqnarray}
    \Ex{K_1} \simeq N \  \frac{ \sqrt{x}}{ \sqrt{x} + \sqrt{z}},
    \end{eqnarray}
 %
 so that, in the infinite volume limit, a density $\rho$ of first class
particles in the system is obtained when $x$ and $z$ satisfy
 %
 \begin{eqnarray}
    \frac{ \sqrt{x}}{ \sqrt{x} + \sqrt{z}}=\rho;
\label{rhodef}
     \end{eqnarray}
 %
 note that $x$ and $z$ are determined only up to an overall factor.
Note also that the fluctuations in the density go to zero in this limit:
from $\Ex{K_1^2}-\Ex{K_1}^2=x(d/dx)\Ex{K_1}$ we have that for large $N$,
 %
 \begin{equation}
 \left\langle\left(K_1\over N\right)^2
   -\left(\Ex{K_1}\over N\right)^2\right\rangle^{1/2}
      \simeq{1\over\sqrt{N}},
 \end{equation}
 %
 so that the results of the canonical and grand canonical ensembles
agree in the large $N$ limit. 
 

We now turn to the evaluation of the infinite volume limit of the profile.
Let us define, for all $i>0$, $d_i=\lim_{N\to\infty}d_i(N)$ and
$d_{-i}=\lim_{N\to\infty}d_{N-i}(N)$.  We may use the asymptotic formula
(\ref{Casympt}) for $\langle1|C^n|1\rangle$ to evaluate the $N\to\infty$
limits of (\ref{tauNm1}) and (\ref{tauiip1}):
 %
 \begin{eqnarray}
     d_{-1} =  \frac{x}{(\sqrt{x} + \sqrt{z})^2 } = \rho^2 ,
\label{taum1}
     \end{eqnarray}
 %
and for all $i>0$, from the exact expression (\ref{Cn}) for the matrix
elements of $C$,
 %
 \begin{eqnarray}
 d_{-i-1}-d_{-i} &=&
      x z  \ \langle 1 | C^{i-1} | 1 \rangle   
\  ( \sqrt{x} + \sqrt{z} ) ^{-2 i - 2 } \nonumber \\
&=& \sum_{p=0}^{i-1} \frac{1}{p+1} \left(\begin{array}{c}  i \\ p
  \end{array}  \right) \left( \begin{array}{c} i-1 \\ p \end{array} \right)
  (1 - \rho)^{2 i - 2 p} \rho^{2 p + 2 }.
\label{tauia}
   \end{eqnarray}
 %
   These formulae determine the profile to the left of the second class
particle in the infinite volume limit: for $i>0$,
 %
 \begin{equation}
 d_{-i}
 =\rho^2 + \sum_{n=1}^{i-1}\sum_{p=0}^{n-1} 
 \frac{1}{p+1}   {n \choose p} {n-1 \choose p}
  (1 - \rho)^{2 n - 2 p} \rho^{2 p + 2 }.\label{dmif}
 \end{equation}
 %
   Similarly, the $ N \rightarrow \infty$ limits of (\ref{tau1}) and
(\ref{tauiip1}) determine the profile to the right of the second class
particle:
 %
 \begin{eqnarray}
  d_1 &=& 1 - \frac{z}{(\sqrt{x} + \sqrt{z})^2 } =
    2 \rho - \rho^2,
\label{tau1a}
   \end{eqnarray}
 %
 and  for $i>0$,
 %
 \begin{eqnarray}
 d_{i } -d_{i+1 } =
    x z  \langle 1 | C^{i-1} | 1 \rangle  ( \sqrt{x} + \sqrt{z} ) ^{ -2 i - 2 }
    = d_{-i-1} - d_{-i} ;
\label{taumi}
     \end{eqnarray}
 %
then
 %
 \begin{equation}
 d_{i}
 =2\rho-\rho^2 - \sum_{n=1}^{i-1}\sum_{p=0}^{n-1} 
 \frac{1}{p+1}   {n \choose p} {n-1 \choose p}
  (1 - \rho)^{2 n - 2 p} \rho^{2 p + 2 }.\label{dif}
 \end{equation}
 %
 For example, considering (\ref{dmif}) and (\ref{dif}) for $i=2$ and $3$
yields
 %
 \begin{eqnarray}
d_{-2}= 2 \rho^2 - 2 \rho^3 + \rho^4, \label{ex1}&&
d_{-3}=  3 \rho^2 - 6 \rho^3 + 9  \rho^4 - 6 \rho^5 + 2 \rho^6 ,\\
d_2=  2 \rho - 2 \rho^2 + 2 \rho^3 - \rho^4 ,&&
d_3= 2 \rho - 3 \rho^2 + 6 \rho^3 - 9  \rho^4 +6 \rho^5 -2 \rho^6,
 \end{eqnarray}
 %
 and so on.  

 With the aid of the formula (see (\ref{Csuma})) 
 %
 \begin{eqnarray}
 \sum_{i=1}^{\infty} \langle 1 | C^{i-1} | 1 \rangle  
   ( \sqrt{x} + \sqrt{z})^{-2 i + 2}  = \frac{1}{\rho ( 1 - \rho)},
\label{Csum}
 \end{eqnarray}
 %
 we see that
 %
 \begin{eqnarray}
  \lim_{i\rightarrow \infty} d_{-i}
  =  d_{-1} + x z \sum_{i=1}^{\infty}  \langle 1 | C^{i-1}| 1 \rangle  
     ( \sqrt{x} + \sqrt{z})^{-2 i - 2}
  = \rho \ .  \label{asym}
 \end{eqnarray}
 %
 This is of course expected, since at large distance we should recover
the uniform density $\rho$.  Similarly,
 $\lim_{ i \rightarrow \infty} d_{i} = \rho$. We
may also calculate the approach to this limiting density:
 %
 \begin{eqnarray}
 d_{-i-1} -d_{-i} 
\simeq \frac { x z } {  2 \sqrt{\pi} (\sqrt{x } + \sqrt{z}) 
    ( x z )^{3 / 4}} \ \frac{1}{ i^{3/2} }
= \sqrt{ \frac{ \rho(1 - \rho)  }{ 4 \pi}}  \ \frac{1}{ i^{3/2} }
\label{power}
 \end{eqnarray}
 %
 for large $i$, which implies that 
 %
 \begin{eqnarray}
 \rho -d_{-i} \simeq
 \sqrt{ \frac{ \rho(1 - \rho) } {  \pi}} \ \frac{1}{ i^{1/2} } \ .
  \end{eqnarray}
 %
 We see that the profile decays as the square root of the distance; the
mean field theory~\cite{BCFN} predicts an exponent $1$.  

Note that from (\ref{dmif}) and (\ref{dif}) we recover that fact that
 $d_i+d_{-i} = 2 \rho$ for all $i$, as already known from
(\ref{symm}). 
 \vskip1em\noindent
 {\bf Remark 4.1}: The results obtained within the grand canonical
ensemble can be recovered from a direct calculation in the canonical
ensemble with $K$ first class particles, that is, from (\ref{ratio}). 
For this purpose we consider the denominator $Z_N(K)$ of (\ref{ratio}):
 %
 \begin{equation}
 Z_N(K) \equiv
   \sum_{ {\rm configurations} } w_{\K}(\tau_1,\tau_2,\ldots,
   \tau_{N-1},2),\label{cpf}
 \end{equation}
 %
 where the sum is restricted to the configurations with one second class
particle on site $N$ and $K$ first class particles distributed on the
remaining $N-1$ sites.  If we multiply $Z_N(K)$ by $x^K z^{N-1-K}$ and sum
over $K$ we obtain (see (\ref{gN})) the grand canonical partition function:
 %
 \begin{eqnarray}
 \sum_{K=0}^{N-1} x^K z^{N-1-K} Z_N(K) = \langle 1 | C^{N-1} | 1 \rangle.
\label{ZNa}
 \end{eqnarray}
 %
If we now compare the coefficients of $x$ and $z$ in (\ref{Cn}) and
(\ref{ZNa}) we see that
 %
 \begin{eqnarray}
 Z_N(K) = \frac{1}{K+1}
{ N \choose K}
{ N \choose K-1}\;.
\label{ZNb}
 \end{eqnarray}
 %
We can then make use of the commutation relations and some algebra to
find formulae for the average occupations in this ensemble. For example,
 %
 \begin{eqnarray}
 \langle \tau_{N-1} \rangle_{\K}\label{ce1}
 &=& \frac{ Z_{N-1}(K-1)}{Z_N(K)} = \frac{K(K+1)}{N(N-1)},\\
\noalign{\vskip0.6em}
 \langle \tau_{N-2} \rangle_{\K}\label{ce2}
 &=& \frac{ Z_{N-1}(K-1) + Z_{N-2}(K-1)}{Z_N(K)} \nonumber \\
 &=& \frac{K(K+1)}{N(N-1)} + \frac{ K (K+1)(N-K)(N-K-1)}{N (N-1)^2(N-2)},
 \end{eqnarray}
 %
and so on.  In the limit $K,N\to\infty$, with $K/N\to\rho$, we find that 
$\langle\tau_{N-i} \rangle_{\K}$ converges to $d_{-i}$, for example,
(\ref{ce1}) and (\ref{ce2}) recover (\ref{taum1}) and (\ref{ex1}).

\section{The bound state between two second class particles}

In this section we consider the case of two second class particles in a
system of first class particles at density $\rho$.  We will show that in
the infinite volume limit the two second class particles form a bound
state, and that the probability of finding them a distance $r$ apart has
power law behavior, decaying as $ r^{-3/2}$ for large $r$.

 From our formula (\ref{fund}) for the weights $w_k$ as traces we know that
in the canonical ensemble, with two second class particles and a fixed
number $K=K_1$ of first class particles, the weight for the configuration
with the second class particles $r$ units apart (with
$1\le r\le(N-1)/2$)---say at positions $N$ and $r$---is easily computable
if we specify the occupation variables for the other sites to be
$\tau_i=0,1$.  This weight (if we take $N$ odd to simplify counting) is
given by
 %
 \begin{eqnarray}
w_{\K}( \tau_1,\ldots, \tau_{r-1},2,\tau_{r+1},\ldots,\tau_{N-1},2) =
\langle 1 | \prod_{i=1}^{r-1} X_i |1 \rangle \
\langle 1 | \prod_{i=r+1}^{N-1} X_i |1 \rangle,\label{fnr}
 \end{eqnarray}
 %
where $X_i = D $ if $ \tau_i =1 $ and $X_i = E$ if $ \tau_i =0 $.  As in
the previous section it is convenient to introduce the grand canonical
ensemble with fugacities $x$ for first class particles and $z$ for holes.
Then from (\ref{fnr}) the probability $p_N(r)$ of finding the two second
class particles a distance $r$ apart is
 %
 \begin{eqnarray}
p_N(r) =
\langle 1 | C^{r-1}  |1 \rangle \
\langle 1 | C^{N-r-1}  |1 \rangle
\left[ \sum_{s=1}^{(N-1)/2}
\langle 1 | C^{s-1}  |1 \rangle \
\langle 1 | C^{N-s-1}  |1 \rangle   \right]^{-1}\;,
 \end{eqnarray}
 %
where $C = x D + z E$ as above.  In the infinite volume limit we must again
choose $x$ and $z$ to satisfy (\ref{rhodef}) to ensure that the density of
first class particles is $\rho$; then the probability
$p_\infty(r)=\lim_{N\to\infty}p_N(r)$ that the particles are a distance $r$
apart is
 %
 \begin{eqnarray}
 p_\infty(r) \label{pinf0}
 &=& \langle 1 | C^{r-1} |1 \rangle \
   ( \sqrt{x} + \sqrt{z})^{-2(r-1)}
  \left[ \sum_{s=0}^{\infty}
  \langle 1 | C^{s}  |1 \rangle \  ( \sqrt{x} + \sqrt{z})^{-2s}
  \right]^{-1}\\
 &=&\rho ( 1 - \rho) \sum_{p=0}^{r-1 } \rho^{2p}
 (1- \rho)^{2 r -2 p -2}\frac{1}{p+1}
\left( \begin{array}{c} r \\ p \end{array} \right)
\left( \begin{array}{c} r - 1 \\ p \end{array} \right).
\label{pinf}
 \end{eqnarray}
 %

 For small $r$ we may use (\ref{pinf}) to find
$p_\infty(r)$ explicitly; for example, $p_{\infty}(1) = \rho (1 - \rho)$
and $p_{\infty}(2)= \rho ( 1 - \rho) [ \rho^2 + (1 - \rho)^2 ]$.  For
large $r$ we may either estimate $p_\infty(r)$ directly from (\ref{pinf})
or use (\ref{pinf0}) together with the asymptotic results (\ref{Csum}) and
(\ref{Casympt}) for the matrix elements of $C$; in either case, we find
 %
 \begin{eqnarray}
 p_\infty( r )
 \simeq \frac{1}{ 2 \sqrt{ \pi \ \rho (1 - \rho) } } \  \frac {1}{ r^{3/2}} .
 \end{eqnarray}
 %
 Thus the two second class particles form a rather unusual bound state:
the decay of $p_\infty(r)$ is a power law and the average distance
between the two second class particles is infinite. 

It is easy to extend this analysis to a system with any fixed number $m$ of
second class particles.  We find that the probability that the particles
are located at positions $x,x+r_1,\ldots,x+r_{m-1}$ for some $x$, with all
$r_i>0$, is $\prod_{i=1}^{m-1}p_\infty(r_i)$.  This
factorization is an example of the decoupling of the system across second
class particles, discussed at the end of Section~3.

 \vskip1em\noindent
 {\bf Remark 5.1}: All these results can be easily recovered with the
canonical ensemble.  Here
 %
 \begin{eqnarray}
p_N(r) = \sum_{k=0}^{r-1} Z_r(k) Z_{N-r}(K-k)
\left[  \sum_{s=1}^{(N-1)/2}\ 
   \sum_{q=0}^{s-1} Z_s(q) Z_{N-s}(K-q) \right]^{-1},
 \label{canpnr}
 \end{eqnarray}
 %
where $Z_N(k)$ is given by (\ref{ZNb}).  Now one can show that if
$N,K\to\infty$ with $K/N \rightarrow \rho$ (and $r$ and $k$ are fixed),
 $ Z_{N-r}(K-k) / Z_N(K) \simeq \rho^{2k} ( 1 - \rho)^{2(r - k)} $, so
that (\ref{canpnr}) becomes
 %
 \begin{eqnarray}
p_\infty(r) = \sum_{k=0}^{r-1} Z_r(k) \rho^{2k} ( 1 - \rho)^{2( r-k-1)}
\left[ \sum_{s=1}^\infty \sum_{q=0}^{s-1} Z_s(q) \rho^{2 q}
 (1 -\rho)^{2 ( s -q -1 )} \right]^{-1}.\label{canpinfr}
 \end{eqnarray}
 %
The denominator in (\ref{canpinfr}) is, from (\ref{ZNa}) and (\ref{Csum}),
 %
 \begin{eqnarray}
\sum_{N=0}^{\infty} \langle 1 | C^{N-1} | 1 
\rangle ( \sqrt{x} + \sqrt{z})^{-2(N-1)} = \frac{1}{\rho(1-\rho)}\;,
 \end{eqnarray}
 %
 and therefore with (\ref{ZNb}) one recovers (\ref{pinf}).  We note that
the argument in fact gives additional information: the probability that
the two second class particles are a distance $r$ apart with exactly $k$
first class particles between them.  In the infinite system this
probability is
 %
 \begin{eqnarray}
 Z_r(k) ( 1 - \rho)^{2 r - 2 k - 1} \rho^{2 k + 1} 
 = \frac{1}{k+1}
{ N \choose k}
{ N \choose k-1}( 1 - \rho)^{2 r - 2 k - 1} \rho^{2 k + 1}\;.
 \end{eqnarray}
 %


\section{A finite density of second class particles}

In this section we study the infinite volume limit in the case in which the
first and second class particles have positive densities $\rho_1$ and
$\rho_2$, respectively.  We begin with a finite system---a ring of $N$
sites---and since we want to calculate the particle densities as seen from
a second class particle, we choose the frame in which site $N$ is occupied
by such a particle.  One approach is to work in the canonical ensemble,
with $K_1$ first class and $K_2$ second class particles, and then take the
limit $N,K_1,K_2 \rightarrow \infty$ with $K_1 /N \to \rho_1$ and
$K_2 /N\to\rho_2$.  As in the previous sections, however, it is more
convenient to work in the grand canonical ensemble for the sites
$1,\ldots,N-1$.

We therefore introduce three
fugacities (two would be sufficient), $x$ for the first class particles,
$y$ for the second class particles and $z$ for the holes, and 
define a matrix $G$ as the obvious generalization of the matrix $C$
introduced previously:
 %
 \begin{eqnarray}
G = x D + y A + z E.\label{Gdef}
 \end{eqnarray}
 %
Then the densities $d_i(N)$, $a_i(N)$, and $e_i(N)$ of first class
particles, second class particles, and holes respectively are given by
 %
 \begin{eqnarray}
d_i(N) = x\;\frac {\langle 1 | G^{i-1} D G^{N-i-1} | 1 \rangle }{
\langle 1 | G^{N-1} | 1 \rangle }\;,
\label{di}
 \end{eqnarray}
 %
 \begin{eqnarray}
a_i(N) = y\;\frac { \langle 1 | G^{i-1} A G^{N-i-1} | 1 \rangle }{
\langle 1 | G^{N-1} | 1 \rangle   } \;,
\label{ai}
 \end{eqnarray}
 %
and
 %
 \begin{eqnarray}
e_i(N) = z\;\frac { \langle 1 | G^{i-1} E G^{N-i-1} | 1 \rangle }{
\langle 1 | G^{N-1} | 1 \rangle    }\;.
\label{ei}
 \end{eqnarray}
 %
 In the Appendix we give a recursion relation (\ref{Gna}) and from that an
explicit formula (see (\ref{GGG})) for the matrix elements
$\langle1|G^n|1\rangle$; to complete the determination of the densities, we
will express the numerators in (\ref{di})--(\ref{ei}) in terms of the known
quantities $\langle1|G^n|1\rangle$ and $\langle1|C^n|1\rangle$.  

First, it is simple to obtain the explicit form for the density of second
class  particles: since $A = |1 \rangle \langle 1 | $, we have from
(\ref{ai}) that 
  %
 \begin{eqnarray}
 a_i(N) = y \ \frac{ \langle 1 | G^{i-1}  | 1 \rangle \
  \langle 1 | G^{N-i-1}  | 1 \rangle }
  { \langle 1 | G^{N-1}  | 1 \rangle }\;.\label{ai1}
  \end{eqnarray}
 %

 To treat  the density $d_i(N)$ of first class
particles we first note that, from (\ref{di}) and the fact that
$D|1\rangle=|1\rangle$ (see (\ref{ADAE})),
 %
 \begin{eqnarray}
  d_{N-1}(N) = x\;\frac{\langle 1 | G^{N-2} |1 \rangle }
  {\langle 1 | G^{N -1} |1 \rangle }\;.
\label{dm1}
  \end{eqnarray}
 %
 Next we note a simple consequence of the matrix algebra
(\ref{DE})--(\ref{ADAE}):
   %
 \begin{eqnarray}
  \lefteqn{ x(DG-GD) = xy(DA-AD) + xz(DE-ED)}  \nonumber \\
&& = (xy+xz)A -xyAD
 =(x+y)(z+y)A -yAG.
  \label{DG}
  \end{eqnarray}
 %
  Clearly (\ref{di}) and (\ref{DG}) imply that for $i>1$,
  %
 \begin{eqnarray}
 d_i(N) - d_{i+1}(N) = \frac{ (x+y)(z+y) \langle 1 | G^{i-1} |1 \rangle
  \langle 1 | G^{N - i-2} |1 \rangle
  - y \langle 1 | G^{ i-1} |1 \rangle
  \langle 1 | G^{N - i-1} |1 \rangle }
  {\langle 1 | G^{N -1} |1 \rangle }\;,\nonumber\\
  \end{eqnarray}
 %
which can be rewritten using the recursion (\ref{Gna}) as
 %
 \begin{eqnarray}
 d_i(N) - d_{i+1}(N) = xz\;\frac{ \langle 1 | G^{i-1} |1 \rangle
  \langle 1 | C^{N - i-2} |1 \rangle }
  {\langle 1 | G^{N -1} |1 \rangle }\;.
\label{di1}
  \end{eqnarray}
 %
 The formulae (\ref{dm1}) and (\ref{di1}) determine all the  $d_i(N)$. 

The determination of the $e_i(N)$ is similar.  First,
 %
 \begin{eqnarray}
  e_{1}(N) = z\;\frac{ \langle 1 | G^{N-2} |1 \rangle }
  {\langle 1 | G^{N -1} |1 \rangle }\;;
\label{e1}
  \end{eqnarray}
 %
then from 
   %
 \begin{eqnarray}
  \lefteqn{z(EG-GE) =  zx(ED-DE) + zy(EA-AE)} \nonumber \\
&& = - (zx+zy)A + zy EA
= -(x+y)(z+y)A +yGA
  \label{EG}
  \end{eqnarray}
 %
 and (\ref{Gna}) we find
 %
 \begin{eqnarray}
 e_i(N) - e_{i+1}(N) = -xz\;\frac{ \langle 1 | C^{i-1} |1 \rangle
  \langle 1 | G^{N - i-2} |1 \rangle }
  {\langle 1 | G^{N -1} |1 \rangle }\;.
\label{ei1}
  \end{eqnarray}
 %
Equations  (\ref{e1}) and (\ref{ei1}) determine all the $e_i(N)$.

We now consider the infinite volume limit of the these densities.  The
large $n$ behavior of $ \langle 1 | G^n | 1 \rangle $ for $y^2>xz$ is
determined in (\ref{Glarge}):
 %
 \begin{eqnarray}
\langle 1 | G^n | 1 \rangle 
  \simeq  \left( 1 -\frac{xz}{y^2} \right) 
     \left[ \frac{(y+x)(y+z)}{y} \right]^n.
\label{Gasympt}
 \end{eqnarray}
 %
 From this we may determine the values of $x$, $y$ and
$z$ which in the large $N$ limit give densities $\rho_1$ of first class
particles, $\rho_2$ of second class particles and $1 -\rho_1 - \rho_2$ of
holes:
 %
 \begin{eqnarray}
\rho_1 = \lim_{N \rightarrow \infty}  \frac{x}{N} \  \frac{d}{dx}
 \log \langle 1 | G^N |1 \rangle = \frac{x}{y+x};
\label{rho1}
 \end{eqnarray}
 %
 \begin{eqnarray}
\rho_2 = \lim_{N \rightarrow \infty}  \frac{y}{N}  \ \frac{d}{dy}
 \log \langle 1 | G^N |1 \rangle = \frac{y^2 -xz}{(y+x)(y+z)};
\label{rho2}
 \end{eqnarray}
 %
 \begin{eqnarray}
 1 - \rho_1 - \rho_2 = \lim_{N \rightarrow \infty}  \frac{z}{N}  \ \frac{d}{dz}
 \log \langle 1 | G^N |1 \rangle = \frac{z}{y+z}.
\label{rho3}
 \end{eqnarray}
 %
It is clear from (\ref{rho2}) that the assumption $y^2>xz$ leads to a
consistent determination of $x$, $y$, and $z$, unique up to an overall
factor; for $y\searrow\sqrt{xz}$ we recover
$\rho_1=\sqrt{x}/(\sqrt{x}+\sqrt{z})$ as in the case (\ref{rhodef}) with a
finite number of second class particles.  (For $y^2<xz$ a calculation
similar to that yielding (\ref{Glarge}) shows that
$\langle1|G^n|1\rangle\simeq(1-y/\sqrt{xz})^{-2}\langle1|C^n|1\rangle$, so
that the number of second class particles in the infinite system is finite
and diverges when $y\nearrow\sqrt{xz}$.)

We can now compute the infinite volume limit of the densities $d_i(N)$,
$a_i(N)$, and $e_i(N)$ at site $i$ relative to the second class
particle; as in Section~3, we define $d_i=\lim_{N\to\infty}d_i(N)$ and
$d_{-i}=\lim_{N\to\infty}d_{N-i}(N)$, for $i>0$, with similar
definitions for $a_{\pm i}$ and $e_{\pm i}$.  We begin with a discussion
of the situation to the left of the origin.  From (\ref{dm1}),
(\ref{Gasympt}), and (\ref{rho1})--(\ref{rho2}),
 %
 \begin{eqnarray}
d_{-1} = \frac{x y}{(y+x)(y+z)}= \rho_1 ( \rho_1 + \rho_2),\label{dm1inf}
 \end{eqnarray}
 %
 while the difference equation (\ref{di1}) yields
 %
 \begin{eqnarray}
d_{-i} &=& d_{-1} +
 xz\sum_{n=1}^{i-1} \left(\frac{y}{(y+x)(y+z)}\right)^{n+1}
\langle1|C^{n-1}|1\rangle\label{fmia}\nonumber\\
 &=& \rho_1(\rho_1+\rho_2) \\
&&\quad +  \sum_{n=1}^{i-1} \sum_{p=0}^{n-1}   \frac{1}{p+1}
{n \choose p}
{n-1 \choose p}
[ \rho_1(\rho_1 + \rho_2) ]^{p+1}
[(1 - \rho_1 )( 1 - \rho_1   -  \rho_2) ]^{n-p},
\nonumber
 \end{eqnarray}
 %
where we have used the asymptotic behavior (\ref{Cn}) of
$\langle1|C^n|1\rangle$ in the last step.  Moreover, because the condition
$y^2>xz$ guarantees that the asymptotic growth of $\langle1|C^n|1\rangle$
is slower than that of $\langle1|G^n|1\rangle$ (compare (\ref{Gasympt}) and
(\ref{Casympt})), the difference equation (\ref{ei1}) for the $e_i$'s
implies that $e_{-i}=e_{-(i+1)}=e_{-1}$ for all $i>0$. This constant
density is easily determined from  $e_{-1}=1-a_{-1}-d_{-1}$; since 
(\ref{ai1}) and (\ref{Gasympt}) imply that
$a_{-1}=y^2/(y+x)(y+z)=\rho_2+\rho_1(1-\rho_1-\rho_2)$,
using the value (\ref{dm1inf}) for $d_{-1}$ yields
 %
 \begin{eqnarray}
e_{-i}=e_{-1}=1-a_{-1}-d_{-1}=1-\rho_1-\rho_2.\label{elft}
 \end{eqnarray}
 %
 Finally, the density of second class particles is determined simply from
the above expressions for $d_{-1}$ and $e_{-1}$:
 %
 \begin{equation}
a_{-i}= 1-e_{-i}-d_{-i}=\rho_1+\rho_2-d_{-i}.\label{alft}
 \end{equation}
 %

We obtain in a very similar way expressions for the densities to the right
of the second class particle.  From (\ref{e1}), 
 %
 \begin{eqnarray}
e_1 = ( 1 - \rho_1)( 1 - \rho_1 - \rho_2) \ ,\label{erght1}
 \end{eqnarray}
 %
 and then  from the difference equation (\ref{ei1}), for all $i>0$,
 %
 \begin{eqnarray}
\lefteqn{e_{i} = ( 1 - \rho_1)(1 - \rho_1-\rho_2)} \label{erght}\\
&&+ \sum_{n=1}^{i-1} \sum_{p=0}^{n-1}   \frac{1}{p+1}
{n \choose p} {n-1 \choose p}
[ \rho_1(\rho_1 + \rho_2) ]^{p+1}
[(1 - \rho_1 )( 1 - \rho_1   -  \rho_2) ]^{n-p}.
\nonumber
 \end{eqnarray}
 %
Finally,  for all $i>0$, from (\ref{di1}) (which implies $d_i=d_{i+1}$),
(\ref{erght1}), and the fact that $a_1=\rho_2+\rho_1(1-\rho_1-\rho_2)$,
 %
 \begin{eqnarray}
d_{i} =   \rho_1 \label{drght}
 \end{eqnarray}
 %
and
 %
 \begin{eqnarray}
 a_i = 1 - d_i -e_i  = 1 - \rho_1 -e_i .\label{arght}
 \end{eqnarray}
 %

 In the remainder of this section we make several comments on these
results; in the next we discuss their relation to shock profiles in a
system consisting only of first class particles. 

Notice from these formulae that for the measure in the infinite volume
limit, conditioned on the presence of a second class particle at the
origin, the density $e_{-i}$ of holes to the left of the origin, and the
density $d_{i}$ of first class particles to the right of the origin (where
$i>0$ in each case), is constant.  It is easy to extend these results and
show that, under this conditioning, the holes to the left and particles to
the right of the origin have Bernoulli distributions with densities
$1-\rho_1-\rho_2$ and $\rho_1$, respectively.  We have no simple
explanation for this extremely simple property of the invariant measure.

 Next we observe that, using (\ref{GC}), it can be shown from (\ref{fmia})
that
 %
 \begin{eqnarray}
\lim_{i \rightarrow \infty} d_{-i} = \rho_1\label{das}
 \end{eqnarray}
 %
and that the approach is exponential, since from the difference equation
(\ref{di1}) and the asymptotic formulae (\ref{Casympt}) and
(\ref{Gasympt}) one has
 %
 \begin{eqnarray}
d_{-i-1} - d_{-i} &=&  \frac{1}{2 \sqrt{\pi} i^{3/2}}
\bigl[\rho_1(\rho_1 + \rho_2)(1 -\rho_1)
 (1 - \rho_1 - \rho_2)\bigr]^{\frac{1}{4}} \nonumber\\
 &&\qquad
\times\;\left[\sqrt{\rho_1(\rho_1+\rho_2)}
   +\sqrt{(1-\rho_1)(1-\rho_1-\rho_2)}\right]^{2i-1}.
\label{dasym}
 \end{eqnarray}
 %
Similarly
 %
 \begin{eqnarray}
\lim_{i \rightarrow \infty} e_i = 1-\rho_1-\rho_2,\label{eas}
 \end{eqnarray}
 %
 with the same exponential approach.

 If one sets $\rho_1=\rho$ and $\rho_2=0$ in (\ref{dasym}) then the
difference in neighboring densities reduces to (\ref{power}) and the decay
of the density to its asymptotic value is algebraic; conversely, it is
interesting to notice that the only case in which the decay of $d_{-i}$ is
not exponential is $ \rho_2 = 0$.  As $\rho_2\searrow0$ the characteristic
length of the exponential decay is asymptotically
$4\rho(1-\rho)\rho_2^{-2}$.  On the other hand, according to the remark in
the paragraph above, the density of first class particles to the right of
the origin here ($d_i$, with $i>0$) is constant for all values of
$\rho_1,\rho_2$, and in particular in the $\rho_1=\rho$, $\rho_2=0$ limit;
this is in apparent disagreement with the situation (see (\ref{dif})) when
there is only one second class particle in the system.  The resolution of
this paradox is that the slow decay of the densities to their asymptotic
values when $\rho_2=0$ (see (\ref{dasym})) implies that even in this limit
an infinite number of second class particles remain in the system.  These
particles form a bound state, so that the second class particle at the
origin is surrounded by infinitely many others and sees a very different
environment from that seen by a single second class particle in the
presence of a density $\rho$ of first class particles.

Finally, by combining the explicit expressions above for $d_i$ and $e_i$ we
obtain
 %
 \begin{eqnarray}
d_{-i} + a_i + d_i = 2 \rho_1 + \rho_2 
 \end{eqnarray}
 %
 for all $i>0$, in agreement with (\ref{symm}).

\section{Microscopic shapes of fronts}

As observed in Section~2, the results of the previous section allow us to
determine the microscopic shape of the shock front between densities
$\rho_-=\rho_1$ and $\rho_+=\rho_1+\rho_2$ in a system of first class
particles, as seen from the position of a single second class particle
which is introduced into the system to locate the shock front.  To do so we
define densities $\delta_i$ for all $i\ne0$ by
 %
 \begin{equation}
 \delta_i=\cases{d_i,& if $i<0$,\cr d_i+a_i,& if $i>0$.}
 \end{equation}
 %
 Thus $\delta_i$ is the density profile of a new species of particles,
consisting of all first class particles to the left of the origin and of
all first and second class particles to the right of the origin.
Consideration of the dynamics shows that in fact the particles of this new
species, together with the distinguished second class particle located at
the origin, do behave as a system of first class particles as seen from a
single second class particle (in the new system one ignores interchanges
between particles of the new species, and ignores also the old second class
particles to the left of the origin).

Since $\lim_{i\to\pm\infty}d_i=\rho_1$ and
$\lim_{i\to\pm\infty}a_i=\rho_2$, the densities $\delta_i$ satisfy
 %
 \begin{equation}
\lim_{i\to-\infty}\delta_i=\rho_-\qquad{\rm and}\qquad
\lim_{i\to\infty}\delta_i=\rho_+,
 \end{equation}
 %
 so that this is indeed a system with a shock between these two densities.
The exact shock profile to the left of the origin follows from
(\ref{fmia}): for $i>0$,
 %
 \begin{eqnarray}
\delta_{-i} = \rho_-\rho_+
 +  \sum_{n=1}^{i-1} \sum_{p=0}^{n-1}   \frac{1}{p+1}
{n \choose p}
{n-1 \choose p}
(\rho_-\rho_+)^{p+1}
[(1 - \rho_- )( 1 - \rho_+ ) ]^{n-p};\label{dellft}
 \end{eqnarray}
 %
  the shock profile to the right of the origin follows from (\ref{erght}),
since $\delta_i=1-e_i$ for $i>0$:
 %
 \begin{eqnarray}
\lefteqn{\delta_{i} = \rho_++\rho_-(1 - \rho_+)} 
\label{delrght}\\
&&\qquad - \sum_{n=1}^{i-1} \sum_{p=0}^{n-1}   \frac{1}{p+1}
{n \choose p} {n-1 \choose p}
(\rho_-\rho_+)^{p+1}
[(1 - \rho_- )( 1 - \rho_+ ) ]^{n-p}.
\nonumber
 \end{eqnarray}
 %
For example, we find
 %
 \begin{eqnarray}
\delta_{-1} = \rho_-\rho_+,\qquad
\delta_{-2} = \rho_-\rho_+ + 
   \rho_-\rho_+(1-\rho_-)(1-\rho_+),\label{del12}
%\delta_1&=& \rho_++\rho_-(1 - \rho_+),\\
%\delta_2&=&\rho_++\rho_-(1 - \rho_+)
%   -  \rho_-\rho_+(1-\rho_-)(1-\rho_+),
 \end{eqnarray}
 %
 and so forth; for $i$ large there is exponential convergence to $\rho_-$,
 %
 \begin{eqnarray}
\delta_{-i-1}-\delta_{-i}=
   \frac{[\rho_-\rho_+(1-\rho_-)(1-\rho_+)]^{1/4}}{2\sqrt{\pi}i^{3/2}}
  \left[\sqrt{\strut\rho_-\rho_+}+\sqrt{(1-\rho_-)(1-\rho_+)}\right]^{2i-1},
     \label{deliip1}
 \end{eqnarray}
 %
 with characteristic length diverging as $(\rho_+-\rho_-)^{-2}$ when
$(\rho_+-\rho_-)\searrow0$.  The corresponding results for $\delta_i$,
$i>0$, may be obtained from the symmetry
$\delta_i+\delta_{-i}=\rho_++\rho_-$.


In the limit $\rho_1=\rho$, $\rho_2=0$, i.e., $\rho_-=\rho_+=\rho$, these
results recover the ``shock'' profile found in Section~4.  But, as
discussed at the end of Section~6, the makeup of the density profiles in
the two situations is quite different.  In the $\rho_2\searrow0$ limit, the
excess density ahead of the origin is contributed by second class
particles; the density of first class particles is constant and equal to
$\rho$ in this region.  Similarly, behind the origin the density of holes
is constant and equal to $1-\rho$, and the lowered density of first class
particles reflects the presence of second class particles.  The density
profile $\delta_i$ found here coincides with the profile $d_i$ of
Section~4, however, because a second class particle interacts with the
second class particles ahead of it as if they were first class particles
and with those behind it as if they were holes.


\section{Conclusion}

In this paper we have found  an exact solution of the two species
totally asymmetric simple exclusion process, first on a ring and then in
the infinite volume limit on the lattice $\bbz$.  From this solution
we have calculated exact shock profiles in a one species TASEP as seen from
the location of a single second class particle, both for the true shock
between different asymptotic densities and for the artificial ``shock''
seen by the second class particle in a uniform density of first class
particles.  We have also observed that two second class particles in a
uniform density of first class particles form a weakly bound state.

The exact solution which we have introduced in a ring of sites can be
extended in two ways to models with other jump rates.  First, we may
relax the requirement that the process be totally asymmetric and
consider the case in which particles---both first and second
class---jump to the right with probability $p>1/2$ and to the left with
probability $q=1-p$.  In this case we may find the steady state with the
aid of matrices $D$ and $E$ which satisfy (\cite{DEHP})
 %
 \begin{equation}
      pDE-qED=(p-q)(D+E).\label{DEpq}
 \end{equation}
 %
  $A = DE-ED$ will then satisfy
 %
 \begin{equation}
pDA-qAD=(p-q)A\qquad\hbox{and}\qquad pAE-qEA=(p-q)A.\label{ADAEpq}
 \end{equation}
 %
 If we assume that $\Tr A$ is finite we may again take the weight
$w_\K(\tau)$ of a configuration $\tau=(\tau_1,\ldots,\tau_N)$ to be
 %
 \begin{equation}
w_\K(\tau_1,\ldots,\tau_N)=\Tr[X_1\cdots X_N],\label{fundpq}
 \end{equation}
 %
 where $X_i=E$ if $\tau_i=0$, $X_i=D$ if $\tau_i=1$, and $X_i=A$ if
$\tau_i=2$.  By a slight generalization of the proof given in Section~3
for the $p=1$ case, we may then show that the $w_{\K}$ are the weights
of a steady state for the dynamics.  Equations (\ref{DEpq}) and
(\ref{ADAEpq}) may be realized by the choice~\cite{DEHP}
 %
  \begin{eqnarray}
{D} = \left( \begin{array}{cccccc}
		  1&\alpha_1&0&0&. &.\\
		  0&1&\alpha_2&0&& \\
		  0&0&1&\alpha_3&&\\
		  0&0&0&1&.  &\\
		  . &&&&. &. \\
		  . &&&&&.
		   \end{array}
		   \hspace{0.2in} \right)
\;,\hspace{0.3in}
{E} = \left( \begin{array}{cccccc}
		  1&0&0&0&. &.\\
		  \alpha_1&1&0&0&& \\
		  0 &\alpha_2&1&0&&\\
		  0 &0&\alpha_3&1&&\\
		  . &&&.&. & \\
		  .&&&&.&.
		   \end{array}
		   \hspace{0.2in} \right)\;,
\label{DEdefpq}
 \end{eqnarray}
 %
where $\alpha_i$ satisfies $\alpha_i^2=1-(q/p)^i$. The matrix
$A=DE-ED$ is then given by
 %
  \begin{eqnarray}
A = \left( \begin{array}{cccccc}
		  \alpha_1&0&0&0&. &.\\
		  0&\alpha_2-\alpha_1&0&0&& \\
		  0 &0&\alpha_3-\alpha_2&0&&\\
		  0 &0&0&\alpha_4-\alpha_3&&\\
		  . &&&.&. & \\
		  .&&&&.&.
		   \end{array}
		   \hspace{0.2in} \right)\;.
\label{DEApq}
 \end{eqnarray}
 %
Because we have assumed that $q<p$, the matrix elements of $A$ decrease
exponentially and $\Tr A = \lim_{i\to\infty}\alpha_i=1$.

A different extension is to the case in which the asymmetry is still
total---only jumps to the right are permitted---but the three types of
jumps, $1\;0\to0\;1$, $1\;2\to2\;1$, and $2\;0\to0\;2$, occur at
different rates, say $1$, $r$, and $s$, respectively.  
Now the invariant weights are given by (\ref{fundpq}) if the matrices
$D$, $E$, and $A$ satisfy
 %
 \begin{equation}
      DE=(D+E), \qquad
   rDA=A, \qquad\hbox{and}\qquad 
  sAE=A;\label{DEArs}
 \end{equation}
 %
 the proof is as in Section 3.  To satisfy (\ref{DEArs}) we may, as in
the case $r=s=1$ considered in the body of the paper, take $A$ to be a
projection operator~\cite{DEHP}, but now $A=|V\rangle\langle W|$, with
$|V\rangle$ a right eigenvector of $D$ and $\langle W|$ a left
eigenvector of $E$: $D|V\rangle=r^{-1}|V\rangle$ and $\langle
W|E=s^{-1}\langle W|$.  Several explicit realizations of such $D$, $E$,
$|V\rangle$, and $\langle W|$ are given in~\cite{DEHP}. 

\vskip2em\noindent
{\bf Acknowledgments}
\vskip1em
This work was supported in part by NSF Grant DMR89-18903; SAJ was
supported in part by NSF Mathematical Sciences Postdoctoral Research
Fellowship DMS 90-07206.  BD thanks the Mathematics Department of
Rutgers University, where this work was initiated, for its
hospitality, and  SAJ, JLL, and ERS thank the IHES.


\appendix

In this appendix we collect various formulae involving the matrix elements
of powers of the fundamental matrices $C$ and $G$ used in computations
in the grand canonical ensembles. 

We begin with the calculation of $ \langle 1 | C^n | 1 \rangle$.  The
matrix $C = x D + z E $ defined in (\ref{Cdef}) is a tridiagonal matrix
in the representation defined in (\ref{DEdef}):
 %
 \begin{eqnarray} {C} = \left( \begin{array}{cccccc}
		  x + z&x&0&0&. &.\\
		  z&x+z&x&0&& \\
		  0&z&x+z&x&&\\
		  0&0&z&x+z&.  &\\
		  . &&&&. &. \\
		  . &&&&&.
		   \end{array}
		   \hspace{0.2in} \right)\;.
 \end{eqnarray}
 %
We may think of $C$ as the unnormalized transition matrix of a one
dimensional walk, in which the walker can at each step go to the right with
a weight $x$, go to the left with a weight $z$, or remain at its present
position with a weight $x+z$; because the matrix is semi-infinite the walk
cannot go to the left of  site $1$.  The quantity
$ \langle 1 | C^n | 1 \rangle$ is thus just the total weight of all the
walks of $n$ steps starting and ending at $1$.  But the number of such
walks which take a total of $p$ steps to the right (and also, of course,
$p$ steps to the left) is
 %
 \begin{eqnarray}
\left( \begin{array}{c}
n \\
2 p \end{array} \right)
\left[
\left( \begin{array}{c}
 2 p \\
 p \end{array} \right) \ - \
\left( \begin{array}{c}
 2 p \\
 p -1 \end{array} \right) \right]\;,
 \end{eqnarray}
 %
 where the first factor is the number of ways to choose $2p$ non-zero
steps among the $n$ steps and the second factor is the number of walks
of $ 2 p $ steps, with $p$ steps each to the right and left, which
go from $1$ to $1$ without visiting $0$; this number can be computed by
the method of images.  Thus
 %
 \begin{eqnarray}
   \langle 1 | C^{n} | 1 \rangle  =
\sum_{p=0}^{\left[ \frac{n}{2} \right]}
\left( \begin{array}{c}
n \\ 2p
 \end{array} \right)
( x + z )^{n-2p} (xz)^p \left[
\left( \begin{array}{c}
 2p \\ p
\end{array} \right) -
\left( \begin{array}{c}
2p \\ p-1
\end{array} \right)  \right].
\label{Cnb}
 \end{eqnarray}
 %

Equation (\ref{Cnb}) may be rewritten in a convenient alternate form
using the identity
 %
 \begin{equation}
\sum_j{a\choose j}{b\choose j+c}={a+b\choose a+c},
 \end{equation}
 %
 easily verified by induction on $a$ from the fundamental identity
${n+1\choose m}={n\choose m}+{n\choose m-1}$, as follows:
 %
 \begin{eqnarray}
   \langle 1 | C^n | 1 \rangle 
  &=& \sum_{p=0}^{\left[ \frac{n}{2} \right]}
       {n\choose 2p}(x+z)^{n-2p}(xz)^p{1\over p+1}{2p\choose p}\nonumber\\
  &=&\sum_{p=0}^{\left[ \frac{n}{2} \right]}\sum_{k=0}^{n-2p}
   {n\choose 2p}{n-2p\choose k}x^{k+p}z^{n-(k+p)}
       {1\over p+1}{2p\choose p}\nonumber\\
  &=&\sum_{q=0}^n{n!\over q!(n+1-q)!}x^qz^{n-q}
      \sum_p{q\choose p}{n+1-q\choose p+1}\nonumber\\
  &=&\sum_{q=0}^n{1\over q+1}{n+1\choose q}{n\choose q}x^qz^{n-q}.
\label{Cna}
 \end{eqnarray}
 %
 From this we may obtain the large $n$ behavior of
$\langle1|C^n|1\rangle$, for example by calculating the contribution of the
values of $q$ which dominate the sums:
 %
 \begin{eqnarray}
   \langle 1 | C^{n} | 1 \rangle  \simeq
   \frac{ ( \sqrt{x} + \sqrt{z})^{2 n+3}}
   { 2 \sqrt{\pi} n^{3/2} ( x z ) ^{3/4} }\;.
\label{Casympta}
    \end{eqnarray}
 %


It will also be useful to have an expression for the generating function
of the $\langle 1 | C^n | 1 \rangle$.  From (\ref{Cnb}), for
$|\lambda|\le(\sqrt{x}+\sqrt{z})^{-2}$,
 %
 \begin{eqnarray}
 \sum_{n=0}^{\infty}  \lambda^n  \langle 1 | C^n | 1 \rangle  
 &=& \sum_{n=0}^\infty \sum_{p=0}^{\left[ \frac{n}{2} \right] }
 {n \choose n-2p} \lambda^n (x + z )^{n-2p} ( xz)^p   \
\frac{ (2p)!}{ p! (p+1)!}\nonumber\\
  &= & \sum_{p=0}^\infty \sum_{q=0}^\infty
 {2 p + q \choose  q }
(x + z)^q \lambda^{2p+q} ( x z )^p   \
\frac{ (2p)!}{ p! (p+1)!}
\nonumber \\
&=&  \sum_{p=0}^\infty [ 1 - \lambda(x+z)]^{-2p-1}
( \lambda^2 x z)^p \frac{(2p)!  }{ p! (p+1)!}\nonumber\\
 &=& \frac{1 - \lambda(x+z) - \sqrt{ (1-\lambda(x+z))^2 - 4 \lambda^2 xz}}
{2 \lambda^2 x z},
\label{GC}
 \end{eqnarray}
 %
where  we have used the standard Taylor series
 %
 \begin{eqnarray}
 \frac{ 1 - \sqrt{ 1 - 4 u}}{ 2 u} =
  \sum_{p=0}^{\infty} \frac{ (2p)!}{ p! ( p+1)! } u^p.
 \end{eqnarray}
 %
The formula (\ref{GC}) simplifies at the radius of convergence: when $
\lambda = (\sqrt{x}+\sqrt{z})^{-2} $,
 %
 \begin{eqnarray}
\sum_{n=0}^{\infty} \langle 1 | C^n | 1 \rangle ( \sqrt{x}+\sqrt{z})^{-2 n}  =
\frac{ ( \sqrt{x} + \sqrt{z})^2}{ \sqrt{x} \sqrt{z}}
 = \frac{ 1}{ \rho(1 -\rho)},\label{Csuma}
 \end{eqnarray}
 %
where in the last step we have taken $\rho = \sqrt{x}/(\sqrt{x}+\sqrt{z})$,
as in Sections~3 and 4.

We finally turn to the discussion of the matrix elements of
$G=xD+yA+zE$ defined in (\ref{Gdef}).  It is convenient to begin with
the calculation of the generating function of the $\langle1|G^n|1\rangle$.
By writing $G=yA+C=y|1\rangle\langle1|+C$ we see that for $\lambda$
sufficiently small, 
 %
\begin{eqnarray}
\sum_{n=0}^\infty \lambda^n \langle 1 | G^n | 1 \rangle 
 &=& \sum_{k=0}^\infty ( \lambda y )^k \left[ \sum_{n=0}^\infty
 \lambda^n \langle 1 | C^n |1 \rangle \right]^{k+1}\\
  &=&  \sum_{n=0}^\infty
 \lambda^n \langle 1 | C^n |1 \rangle  \
 \left[  1 -  \lambda y \sum_{n=0}^\infty
 \lambda^n \langle 1 | C^n |1 \rangle  \right]^{-1}.
\label{GG1}
 \end{eqnarray}
 %

 We now claim that the $ \langle 1 | G^n | 1 \rangle$
satisfy the recursion
 %
 \begin{eqnarray}
\langle 1 | G^n | 1 \rangle = \frac{(y+x)(y+z)}{y}
\langle 1 | G^{n -1} | 1 \rangle  - \frac{xz}{y}
\langle 1 | C^{n -1} | 1 \rangle.
\label{Gna}
 \end{eqnarray}
 %
 To verify this claim we note that if (\ref{Gna}) is satisfied then
 %
 \begin{eqnarray}
\sum_{n=0}^\infty \lambda^n \langle 1 | G^n | 1 \rangle =
\left[ 1 -\frac{(y+x)(y+z)}{y} \lambda \right]^{-1} \left[
1 - \frac{x z}{y} \lambda \sum_{n=0}^\infty
 \lambda^n \langle 1 | C^n |1 \rangle \right] \  ,
\label{GG2}
 \end{eqnarray}
 %
 and it is easy to see from the explicit formula (\ref{GC}) for the
generating function of the $\langle1|C^n|1\rangle$ that (\ref{GG1}) and
(\ref{GG2}) are identical.  The recursion (\ref{Gna}) together with the
initial case $\langle1| G^0|1\rangle=1$ leads to
 %
 \begin{eqnarray}
 \langle 1 | G^n | 1 \rangle 
   =  \left[\frac{(y+x)(y+z)}{y} \right]^n 
  \left[1-\frac{xz}{(y+x)(y+z)} \sum_{q=0}^{n-1}
  \left(\frac{y}{(y+x)(y+z)}\right)^q\langle1|C^q|1\rangle\right],
\label{GGG}\nonumber\\
 \end{eqnarray}
 %
so that an the explicit expression for the $\langle 1 | G^n | 1 \rangle$
may be obtained with the aid of either of the expressions (\ref{Cnb}),
(\ref{Cna}) for $\langle1|C^q|1\rangle$.  Finally, we may find the
asymptotic form of $\langle1|G^n|1\rangle$ directly from (\ref{GGG}).  We
consider only the case $y^2 > xz$ needed in Section~6; then taking
$\lambda=y/[(y+x)(y+z)]$ in the generating function (\ref{GC}) we have
 %
\begin{equation}
 \sum_{q=0}^\infty
  \left(\frac{y}{(y+x)(y+z)}\right)^q\langle1|C^q|1\rangle
= \frac{(y+x)(y+z)}{y^2},
\end{equation}
 %
so that (\ref{GGG}) yields
 %
 \begin{eqnarray}
 \langle 1 | G^n | 1 \rangle \simeq \left( 1 - \frac{xz}{y^2} \right)
\label{Glarge}
\left[ \frac{(y+x)(y+z)}{y} \right]^n.
 \end{eqnarray}
 %

\clearpage

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