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\begin{document}
\draft
\preprint{UCL-IPT-01-09}
\title{Scaling Fields in the Two-Dimensional Abelian Sandpile Model}
\author{St\'ephane Mahieu and Philippe Ruelle\thanks{Chercheur qualifi\'e FNRS}}
\address{Universit\'e catholique de Louvain\\
Institut de Physique Th\'eorique\\
B--1348 \hskip 0.5truecm Louvain-la-Neuve, Belgium
}
\date{\today}
\maketitle
\widetext
\begin{abstract}
We consider the isotropic two-dimensional abelian sandpile model from a perspective
based on two--dimensional (conformal) field theory. We compute lattice correlation
functions for various cluster variables (at and off criticality), from which we infer
the field--theoretic description in the scaling limit. We find a perfect agreement
with the predictions of a $c=-2$ conformal field theory and its massive
perturbation, thereby providing direct evidence for conformal invariance and more
generally for a description in terms of a local field theory. The question of the
height 2 variable is also addressed, with however no definite conclusion yet.
\end{abstract}

%\pacs{PACS numbers:\ 11.25.Hf,\ 05.50.+q,\ 75.10.Hk}

%\narrowtext

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}
\label{sec:intro}

Sandpile models have been invented by Bak, Tang and Wiesenfeld \cite{btw} as
prototypical examples for a class of models which show self--organized
criticality. The main peculiarity of these models is that they possess a dynamics
which drives them to a critical regime, robust against various perturbations. The
ineluctable criticality as well as the robustness of these specific dynamics could
provide a universal explanation of the ubiquity of power laws in natural phenomena.
Various physical situations have been discussed along this idea, see the recent
books \cite{b,j}.

Sandpile models are among the simplest models showing self--organized criticality.
Although their physical relevance can be questioned, it is believed that they have
all the features that should be present in more complicated/physical models.
Therefore they constitute a useful playground where the most important features can
be understood.

One of the most interesting models is the two--dimensional unoriented abelian
sandpile model (ASM) \cite{btw}, which we first briefly recall (recent reviews are
\cite{dhar1,dhar2}). The model is defined on a $L \times M$ square lattice. At each
site $i$, we assign a random variable $h_i$, taking its values in the set
$\{1,2,3,4\}$. We think of $h_i$ as a height variable, which counts the number
of grains of sand at $i$. Thus a sand configuration is specified by a set of values
$\{h_i\}_i$ of the height variables. A configuration is stable if all $h_i \leq 4$,
and unstable if $h_i > 4$ for one or more sites. The number of stable configurations
is equal to $4^{LM}$. 

The discrete dynamics of the model takes a stable configuration ${\cal C}_t$ at time
$t$ to another stable configuration ${\cal C}_{t+1}$, and is defined in two steps. 
First step is the addition of sand: one grain of sand is dropped on a randomly chosen
site of ${\cal C}_t$, and this produces a new configuration ${\cal C}'_t$. The second step is the
relaxation to ${\cal C}_{t+1}$. If ${\cal C}'_t$ is stable, we simply set  ${\cal
C}_{t+1} = {\cal C}'_t$. If not, the site where $h_i > 4$ topples: it looses 4
grains of sand, and each of its neighbours receives one grain, something we write in
the form $h_j \rightarrow h_j-\Delta_{ij}$ for all sites $j$, with $\Delta$ the
discrete laplacian. In the process, one neighbour can have its height $h>4$, in
which case it too topples: it looses 4 grains of sand, each of its neighbours
receiving one grain. And so on for each site which has a height $h > 4$, until we
reach a stable configuration. ${\cal C}_{t+1}$ is then set equal to this new stable
configuration. The relaxation process is well--defined: it always stops (sand can
leave the system at the boundaries) and produces the same result ${\cal C}_{t+1}$
independently of the order in which the topplings are performed (the abelian
property). 

One can let an initial distribution over the stable configurations evolve in time
according to the dynamics, and examine its time limit. Under mild
assumptions, one shows \cite{dhar3} that all initial distributions converge to a
well-defined and unique distribution $P^*$, call the SOC (for self--organized
critical) state. The theory of Markov chains and the abelian property allow for a
complete characterization of it: $P^*$ is uniform on the set $\R$ of so--called
recurrent configurations, and is zero elsewhere (the transient configurations). The
number of recurrent configurations is $|\R| = {\rm det}\,\Delta \sim (3.21)^{LM}$,
with
$\Delta$ the discrete laplacian on the $L \times M$ lattice with open boundary
conditions. Although the counting of recurrent configurations is easy, the criterion
which actually decides whether a given stable configuration is recurrent or
transient is well--known \cite{dhar3,md1} but hard and non--local: in a generic
case, one has to scan the whole configuration in order to decide whether it is
recurrent or not. Explicit calculations are therefore difficult (and few). 

From the point of view of critical systems and conformal field theory, one is
interested in the thermodynamic limit $\lim_{L,M \to \infty} P^*$. The result
should be a probability measure on the space of spatially unbounded configurations,
or equivalently on the infinite collection of random variables $h_i$. Despite the
fact that these variables are strongly coupled ---the couplings are even non--local
because of the recurrence condition---, their correlation functions seem to be of the
usual, local form. In the scaling limit, one could therefore hope to recover a local
field theory. 

There are indications that indeed a conformal field theory emerges, like in
ordinary critical, equilibrium lattice models. In \cite{md1}, a connection with
spanning trees was established, which suggests a relationship with the $q=0$
limit of the $q$--state Potts model, hence with a $c=-2$ conformal field theory,
a value confirmed by the calculation of the universal finite size correction to the
free energy on a finite strip \cite{md1}. The 2--site probability $\prob[h_i=h_j=1]$
was shown in \cite{md2} to decay algebraically, with an exponent that can be easily
accomodated in a $c=-2$ free grassmanian scalar field theory \cite{ip}. Also the
2--site probabilities for height variables on the boundary of a half--plane domain
have been computed in \cite{bip,ivash}, and show the same algebraic fall--off as the
height 1 variables in the bulk.

Beyond these concordant elements, no systematic investigation in the sandpile model
has been made, to our knowledge, which can solidly confirm the connection with a
$c=-2$ conformal field theory. It is our purpose to provide a more explicit link
between the two. We do this by computing multi--site probabilities of various height
variables, and by comparing them with the conformal predictions. More specifically,
we compute the scaling limit of the 2--, 3-- and 4--site correlations of height 1
variables, but also of other lattice variables, namely finite subconfigurations that
can be handled by the technique developed in \cite{md2}. 

In fact, we compute these correlations in an off--critical extension of the
abelian sandpile model. We evaluate them in the scaling regime, extract the
scaling limit, and then establish a correspondence with a field theory.
In this way we strengthen the field--theoretic connection away from
criticality, by relating a massive perturbation of the ASM to the massive extension
of the $c=-2$ fermionic field theory. One can therefore probe more deeply the
structure of both pictures, leaving little doubt about the identifications
that are to be made. 

The conclusion these calculations allow us to draw is that the $c=-2$ theory, and its
massive extension, seems to provide a field--theoretic description of the height
profile of the sandpile model. At least for the cluster variables examined in this
paper, this is a statement that we could verify explicitly. Other important
spatial, non--dynamical features of the SOC state must be studied. These include
boundary features and avalanche distributions. The latters are undoubtedly
much more difficult to handle, because they lie at a higher level of non--locality
than the height variables, since they depend on height values in unbounded regions.  
Whether they can be accounted for by the non--local sectors of the $c=-2$ conformal
theory remains a largely open question. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Lattice calculations in the sandpile model}
\label{sec:sec2}

As recalled above, explicit calculations in the bulk of the lattice are notoriously
hard, because of the non--local nature of the SOC state (probability measure) $P^*$. 

All four 1--site probabilities $\prob[h_i = a]$, for $a=1,2,3,4$, have been
computed exactly in the thermodynamic limit, but already the calculation for $a \geq
2$ \cite{priezz} is formidably more complicated than for $a=1$ \cite{md2}. The only
2--site probability that has been computed is again for the unit height variables
\cite{md2}. 

The technique used to compute the correlation of two unit height variables is a
particular case of a beautiful idea put forward by Majumdar and Dhar \cite{md2}. It
is based on the important notion of forbidden subconfigurations (FSC), and its
relation to recurrent configurations. A cluster $F$ of sites, with its heights
$h_i$, is a FSC if, for each site $j \in F$, the number of sites in $F$ and
connected to $j$ is bigger or equal to $h_j$. Simple examples of FSCs are two
adjacent 1's (11), a linear arrangement (121), or a cross--shape arrangement with
four 1's surrounding a central site with any height value. A configuration is then
recurrent if and only if it contains no FSC \cite{dhar3,md1}.

The idea used in \cite{md2} allows to compute the probability of occurrence, in the
SOC state, of any cluster that becomes an FSC if any of its heights is decreased
by one unit. A simple case is a height 2 next to a height 1, but more examples are
given below in Figure 1. Following \cite{md3}, let us call them weakly allowed
cluster variables. 

Let $S$ be such a cluster. The authors of \cite{md2} show how one
can define a new sandpile model, with its own toppling rules (and a new matrix
$\Delta'$), such that the number of its recurrent configurations is the number, in
the original model, of recurrent configurations which contain $S$. From this, a
simple determinantal formula follows, $\prob(S)=\det\Delta' /\det\Delta$. Because the
new sandpile model is obtained by modifying the original one in the region localized
around $S$, the ratio of the two  determinants reduces to a finite determinant, even
for an infinite lattice. 

This technique has been used to compute the probabilities of various
subconfigurations, like those in Figure 1 below. The simplest one is the cluster
reduced to one site, with height equal to 1. In this case, the new model is obtained
by changing the toppling rules at 4 sites (the height 1 and three neighbours).
A 4--by--4 determinant then yields $P(1) \equiv \prob[h_i=1] = {2 \over \pi^2} (1-{2
\over \pi}) \sim 0.074$. Allowing for disconnected clusters leads to multi--site
correlations such as the 2--site correlation of unit heights:
\be
\prob[h_i = h_j = 1] = P(1)^2 \, \big[1 - {1 \over 2r^4} + \ldots \big],
\qquad  r=|i-j| \gg 1.
\label{2pt}
\ee
It was also remarked in \cite{md2} that more general clusters ---for instance, a
single site with height equal to 2--- can be handled using the same ideas, but the
corresponding probabilities become infinite series, the terms of which involve
weakly allowed clusters, of increasing size. Unfortunately, these series seem to be
slowly convergent.

\begin{figure}[htb]
\leavevmode
\begin{center}
\begin{picture}(320,120)(0,-100)
\put(11,0){\circle{10}}
\put(11,0){\makebox(0,0)[c]{\footnotesize 1}}
\put(4,0){\line(1,0){2}}
\put(16,0){\line(1,0){2}}
\put(11,5){\line(0,1){2}}
\put(11,-5){\line(0,-1){2}}

\put(53,-3){\conf1}
\put(117,-3){\conf2}
\put(202,-8){\conf3}
\put(267,-3){\conf4}
\put(-5,-50){\conf5}
\put(70,-50){\conf6}
\put(135,-50){\conf7}
\put(205,-47){\conf8}
\put(285,-58){\conf9}
\put(20,-110){\conf{10}}
\put(105,-110){\conf{11}}
\put(178,-105){\conf{12}}
\put(255,-112){\conf{13}}

\end{picture}
\begin{minipage}{14cm}
\bigskip \bigskip \bigskip 
\caption{On the first two lines are shown the ten smallest weakly allowed cluster
variables, up to orientations, which contain no more than four sites. Taking the
different orientations into account makes a total of 57 clusters of weight smaller
or equal to 4. In addition, calculations involving the four clusters on the last line
will be considered in the text.  All these clusters will be numbered $S_0$ to
$S_{13}$ from left to right and top to bottom. The reason for including the last
two clusters is explained in Section \ref{sec:sec7}.}
\end{minipage}
\end{center}
\end{figure}


In general, the way the original sandpile model is modified is by removing some
of the bonds linking $S$ to its nearest neighbourhood, and at the same time, by
reducing the threshold at which the sites become unstable (4 in the original model),
so that the threshold at every site remains equal to its connectivity. These
modifications affect all the sites of $S$, plus a certain number of sites which are
nearest neighbours of $S$. All together they form a set we call $M_S$,
the cardinal of which depends on the shape of $S$. The new toppling matrix is then
given by $\Delta' = \Delta + B$, where the symmetric matrix $B$ has entries
$B_{ij}=1$ if the bond linking $i$ to $j$ has been removed, $B_{ii}=-n$ if $n$ bonds
off the site $i$ have been removed, and is zero otherwise. Then the probability of
$S$ (in the original model) is
\be
P(S) = {\det \Delta' \over \det \Delta} = \det \,({\Bbb I} + G B) = \det \,({\Bbb I}
+ G B)\big|_{M_S}.
\ee
Because $B$ is zero outside the finite set $M_S$, the determinant is finite, in
fact of size $|M_S|$, but requires the knowledge of the Green function $G
\equiv \Delta^{-1}$ of the laplacian, at all sites belonging to $M_S$.

In the above example where $S$ is just one site with a height equal to 1, the
modifications can be pictorially described as follows:
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\end{picture}
\end{center}
\end{figure}

\medskip \noindent
The dashed segments represent the removed bonds, and the numbers on the
right lattice indicate the thresholds at which the sites become unstable and topple. 

In fact, in the modified lattice shown on the right, the only site to which the
1 is connected has a height bigger or equal to 2. So one could as well decrease its
height and its threshold by 1, and remove the connection. In this way, the site with
a height originally equal to 1 is completely cut off from the rest of the lattice,
defining a different modification of the original ASM\footnote{Strictly speaking, in
this second modified ASM, the removal of the bond connecting the height 1 to its
western neighbour should be supplemented by the creation of a bond connecting the
height 1 to a sink site, so that sand brought by seeding can be evacuated. The part
of the modifications that affect the sink site plays no role whatsoever, so we may
ignore it completely. See the second appendix for a detailed argument.}. Either of
them can be used to compute correlations involving heights 1. 
 
Correspondingly the matrix $B$ that specifies the modifications is a 4--by--4 or a
5--by--5 matrix given by (in an obvious ordering)
\be
B = \pmatrix{-3 & 1 & 1 & 1 \cr 1 & -1 & 0 & 0 \cr 1 & 0 & -1 & 0 \cr
1 & 0 & 0 & -1} \qquad {\rm or\ } \qquad
\pmatrix{-3 & 1 & 1 & 1 & 1\cr 1 & -1 & 0 & 0 & 0\cr 1 & 0 & -1 & 0 & 0\cr
1 & 0 & 0 & -1 & 0\cr 1 & 0 & 0 & 0 & -1}.
\ee

For bigger clusters, there is a fair amount of ambiguity in the way the modifications
are made in order to freeze the cluster heights to what we want. These
modifications can affect regions of different sizes, and so can be more or less
computationally convenient. The least economical solution is the analog of the
second modification explained above for the unit height. It is also the easiest to
describe: one simply cuts the cluster off the rest of the lattice, removing all bonds
inside the cluster and all connections between the cluster and the outside lattice. 
There are many other choices of intermediate efficiency. 
For the second cluster in Figure 1 for instance, namely a 2 next to a 1, one may
consider the following three modifications (among others):
\begin{figure}[htb]
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\begin{center}
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\medskip \noindent
with corresponding $B$ matrices of dimension 8, 7 or 6. For bigger clusters, the
difference can be computationally noticeable, and so choosing the modifications
which affect the smallest possible region makes the calculation of determinants
easier\footnote{The reader familiar with the technique knows that these determinants
can be reduced by appropriate summations of rows (or columns). The gain in size is
equal to the size of the cluster one considers, but it has its prize,
because it renders the entries of the reduced determinant more complex. This gain
is the same no matter how the ASM is modified.}. So one should cut as few links as
possible, a prescription which makes sure that the modified ASM remains conservative
where the original one is: the removal of a bond off one site is always accompanied
by the lowering by 1 of the threshold at that site, or equivalently the $B$ matrix
has row and column sums equal to 0.

When the cluster $S = \cup_k \, S_k$ is disconnected, the matrix $B$ is the direct
sum of submatrices $B_k$. The probability $\prob (S)$ (the
correlation of the subparts $S_k$) involves the Green function $G(i,j)=G(0,i-j)$ at
all sites $i,j$ of $S$, and thus depends on the relative locations and orientations
of the various $S_k$'s, and in particular on their separation distances. As the
original sandpile model is invariant under lattice translation, the probabilities
retain the translation invariance. For $S$ containing two heights equal to
1, separated by a distance $r$, the evaluation of the 8--by--8 determinant yields
the dominant term $r^{-4}$ given in (\ref{2pt}), independently of the angular
distance of the two sites.

Precisely in the case $S$ is disconnected and contains different pieces separated by
large distances, a simple but important observation can be made. Because the
probability of $S$ is going to depend on the Green function $G(z_k,z_{k'}) \sim
\log{|z_k-z_{k'}|}$ evaluated at points where the subparts are located, one could
expect at first sight a logarithmic dependence in the separation distances. However
due to the property that sand in conserved in the modified ASM, the
probability in fact only depends on the derivatives (or finite differences) of the
Green function. This evacuates the logarithmic dependences and turn all correlations
into rational series in the various distances $(z_k-z_{k'})$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The massive sandpile model}
\label{sec:sec3}

The previous section summarized the calculation of correlations of cluster
variables in the standard ASM. Even though it is critical, and self--organized in
the dynamical sense, one can drive it off criticality by switching on relevant
perturbations. There are various ways of doing it, but one of the simplest is to add
dissipation, whose rate is controlled by a parameter $t$. In effect this introduces
a mass $m \sim \sqrt{t}$, or equivalently a non--zero reduced temperature
$t$. The resulting model can be described as a massive (or thermal) perturbation of
the massless, critical sandpile model. For the purpose of comparing the correlations
in the ASM with those of a local field theory, the inclusion of some neighbourhood of
the critical point is important as it strengthens the connection. 

The way a mass can be introduced in the model is most straightforward, and
corresponds to a dissipation of sand each time a toppling takes place. We define the
perturbed ASM by its toppling matrix (we suppress the explicit dependence on $x$)
\be
\Delta_{i,j} = \cases{
x & if $i=j$, \cr
-1 & if $i$ and $j$ are nearest neighbours, \cr
0 & otherwise.}
\label{delta}
\ee
The external driving rate of the sandpile remains the same (one grain per unit of
time), but the threshold beyond which the sites become unstable is increased from 4
to $x$. As a consequence, the height variables now take the values between 1 and
$x$. Each time a site topples, $t=x-4 \geq 0 $ grains of sand are dissipated. 

In order to assess the robustness of their SOC features, perturbations of sandpile
models have been often discussed, with various and sometimes surprising conclusions,
see for instance \cite{pcdca} for a review. Among the many contributions on the
subject, the reference \cite{mkk} has been one of the first attempts to see how the
non--conservation of sand in the toppling rules can alter the critical properties of
the model. In particular, the massive perturbation defined above in (\ref{delta})
corresponds to the globally dissipative model studied in that paper, and for which
the authors have found that the avalanche distributions decay exponentially. More
recently, the same perturbation was reconsidered in \cite{tk}, in which the
exponential decay of the 2--site probability for unit height variables, our Eq.
(\ref{2s0}) below, was proved. 

The advantage of the perturbation (\ref{delta}) is that it allows the same
calculations as the non--perturbed model, in the way that has been recalled in
Section \ref{sec:sec2}. One can in particular compute the correlation functions
by the same formulae, with however two minor changes. The first one is of course
that one uses the massive Green function, with a mass fixed by $\sqrt{x-4}$. The
second one concerns the $B$ matrices, that define the modified ASM. Because the
height variables now take the values from 1 up to $x$, the diagonal entries of $B$
corresponding to sites of the cluster $S$ must be set equal to $1-x$, in order to
lock the heights into their minimal values. (As a consequence, note that sand is 
conserved at those sites, in the modified model.)

When doing concrete computations, one needs the value of the Green function at points
close to the origin (at sites belonging to the same connected subpart), and at
points far from the origin (at sites located in different connected parts). For
the former, one uses a development around $t=0$ (in powers of $t$ with $\log{t}$
terms), whereas for the latter, one performs a double expansion in inverse powers
of the distances, and in (half--integral) powers of the perturbing parameter
$t$. For arbitrary positions, this development is cumbersome as it depends also on
the angular positions. In the calculations to be presented in the following
sections, we have therefore restricted ourselves to configurations of clusters which
only require the knowledge of the Green function at points close to a principal or a
diagonal axis, for which all useful expressions are collected in an appendix. 

The field theory enters as a description of the long distance regime of the ASM
correlations (perturbed or not). As usual, this requires at the same time an
adjustment of the correlation length, or equivalently of the mass. So we are
interested in computing the scaling regime of correlations. To reach it, we take
simultaneously the long distance limit $R={r \over a}
\rightarrow \infty$ and the critical limit\footnote{We deliberately take the stand to
formally continue all the expressions from integer values of $x$ to arbitrary values
$x \geq 4$. Thus we do not define a family of well--defined sandpile models,
parametrized by a real number $x \geq 4$. For $x$ rational, this can easily be done,
however the limit for $x$ going to zero by rational values is not the usual,
original model defined for $x=4$. We suspect that the model one gets in this specific
limit is a model in which the height variables are completely decoupled. See
\cite{vazq} for a related discussion.} $x-4=a^2M^2 \rightarrow 0$ so that the
product $\sqrt{x-4} R \rightarrow Mr$ defines the effective mass $M$ and the
macroscopic distances $r$. The scale $a \to 0$ controls the way the limit is taken,
and can be thought of as a lattice spacing. 
 
In the actual calculations of correlation functions, large determinants are needed,
with entries given by Green function values, themselves expressed as power series.
In the scaling regime $x \sim 4$, it is convenient to expand all matrix entries and
the correlations as power series of $\sqrt{t}$. The first non--zero term in a
correlation should then be directly related to its scaling limit.

We will finish this section by commenting on the way the calculations have been
done, before presenting in the next section the results for the unit height
variables. 

Suppose that we want to compute the joined probability for having a certain cluster
$S$ at the origin say and an other cluster $S'$ at some site $i$. Each cluster comes
with its own set $M_S$ or $M_{S'}$ which contains the sites where the ASM has been
modified, the modifications themselves being specified by the matrices $B$ and $B'$.
According to the discussion of the previous section, this probability is equal to a
determinant
\be
\prob\left[S(0),\;S'(i)\right] = \det \left({\Bbb I} + 
\pmatrix{G_{00} & G_{0i} \cr G_{i0} & G_{ii}} 
\pmatrix{B & 0 \cr 0 & B'}\right) = \det\pmatrix{{\Bbb I} + G_{00}\,B & G_{0i}\,B'
\cr G_{i0}\,B & {\Bbb I} + G_{ii}\,B'}\,.
\label{2site}
\ee
The $G$--blocks collectively denotes Green function values evaluated at two sites
belonging to the set $M_{S} \cup M_{S'}$, with in addition $G_{i0} = (G_{0i})^t$.

We do not want to know the exact value of this determinant, but rather the
terms that are dominant in the scaling region, when $i$ is far from the origin. 

Using the standard development of a rank $n$ determinant in terms of the matrix
entries,
\be
\det A = \sum_{\sigma \in S_n} \; \epsilon(\sigma) \, A_{1,\sigma(1)} \, 
A_{2,\sigma(2)} \ldots A_{n,\sigma(n)}\,,
\ee
one may distinguish in (\ref{2site}) several types of terms. 

The permutations $\sigma$ which do not mix the sites of the cluster $S$
with the sites of $S'$, produce terms which do not depend on the distance $|i|$
separating $S$ from $S'$, and thus contribute a term equal to $[\prob (S)][\prob
(S')]$.

The other permutations necessarily involve an even number of entries from the
off--diagonal blocks. As all such entries are combinations of Green functions, they
decay exponentially with the distance. Therefore the 2--point function will be
dominated by those terms in the determinant which are quadratic in the off--diagonal
Green functions. With the help of the formulae in the Appendix A, these Green
functions are all reducible to the single $G(i)=G_{0,i}$, and its derivatives. 

The quadratic terms come from the permutations that send one site of the
first cluster onto one site of the second cluster, and vice--versa (with possibly
two other sites). The contributions of all those permutations can be summed up
to yield a formula written in terms of the minors of the diagonal blocks:
\be
\prob\left[S(0),\;S'(i)\right] 
= \prob(S)\cdot\prob(S') -  {\rm Tr} \left\{\left[{\rm Mi}({\Bbb I} +
G_{00}B)\right]^t \cdot (G_{0i}B') \cdot
\left[{\rm Mi}({\Bbb I} + G_{ii}B')\right]^t \cdot (G_{i0}B) \right\} + \ldots
\label{minors}
\ee
Here ${\rm Mi}(A) = (-1)^{i+j} \, \det (A_{\hat i, \hat j})$ denote, up to
signs, the minors of $A$ of maximal order ($A_{\hat i, \hat j}$ is the matrix $A$
with the $i$--th row and the  $j$--th column removed). Formula (\ref{minors}) is
exact modulo quartic, sextic, ... terms in the off--diagonal Green functions. It has
been used to compute all 2--cluster correlations considered in this article. 

In order to determine the dominant term in the perturbing parameter $t$, 
one still makes an expansion in powers of $\sqrt{t}$ (actually the
expansions of elements of the diagonal blocks $G_{00}$ and $G_{ii}$ involve the two
kinds of terms $t^{k/2}$ and $t^{k/2}\,\log{t}$). To this end, one develops all Green
functions around $t=0$ using the formulae of the appendix, and keeps the first
non--zero term in the trace. Since the mass $m$ or inverse correlation length is
related to $\sqrt{t}$, a first non--zero contribution of the form $t^{(x_1+x_2)/2} \,
F\left(G(i\sqrt{t}),G'(i\sqrt{t}),\ldots) \right)$ determines the scaling limit of
the correlation, hence the corresponding field--theoretic 2--point function, in
terms of two fields of scale dimensions $x_1$ and $x_2$, in the usual way. In this
respect the presence of a logarithmic singularity $\log{t}$ in the final result
would be the signal that the scaling limit is ill--defined. It turns out, in
all the calculations we have performed, that the first non--zero term scales like
$t^2$ (yielding $x_1 + x_2 = 4$). Because the off--diagonal terms start
off like $\sqrt{t}$ ---they are differences of Green functions at neighbouring
sites---, it is enough to expand all Green functions up to order
$t^{3/2}$, as has been done in the appendix\footnote{It would not be the case if the
least economical modifications were chosen (the one that cuts the cluster off the
rest of the lattice). The $B$ matrix would not have all row sums equal to zero, and
consequently the off--diagonal Green functions would have non--zero terms of order 0
in $t$. This would force us to expand everything to order 2 (instead of 3/2) in $t$.
So these modifications appear to be doubly inefficient.}  (the 3--cluster
correlations require the expansions to order $t^2$).

In fact this procedure has anticipated the results on one point. For the purpose
of taking the scaling limit, it is the dominant term in $t$ that we want to
determine, while the above procedure determines the dominant term in $t$ among
the contributions that are quadratic in the Green functions. So one should also
check that no higher than quadratic term in the Green functions brings a $t^2$
contribution. This can easily be done in the following way. Since the off--diagonal
terms start off like $\sqrt{t}$, checking the quartic terms is enough, and one
can stop the expansion of the off--diagonal blocks to the $\sqrt{t}$ order. To that
order, the two blocks $G_{0i}\,B'$ and $G_{i0}\,B$ have all their rows identical.
Indeed, inside a given column, all entries are finite differences of Green functions
evaluated at neighbouring sites, and so differ by second order finite
differences of Green functions, i.e. by terms of order $t$. Thus the
determinant with $G_{0i}\,B'$ and $G_{i0}\,B$ as off--diagonal blocks can be reduced
to a determinant where the two off--diagonal blocks have but their first row non
zero, and equal to linear combinations of Green functions. Such a determinant has no
term that is quartic in the off--diagonal block entries. 

To end this section, we give the expansion analogous to (\ref{minors}) that pertains
to the calculation of 3--cluster correlations. Its proof relies on the same
arguments as above regarding permutations. For three clusters rooted at sites
$i,j,k$, it reads
\bea
\prob\left[S(i),\;S'(j),\;S''(k)\right] &=& -2\,\prob(S) \cdot \prob(S') \cdot
\prob(S'') \nonumber \\
\noalign{\medskip}
&& \hskip -3truecm + \; \prob(S) \cdot \prob[S'(j),S''(k)] +
\prob(S') \cdot \prob[S(i),S''(k)] +
\prob(S'') \cdot \prob[S(i),S'(j)] \nonumber\\
\noalign{\smallskip}
&& \hskip -3truecm + \; {\rm Tr} \left\{
\left[{\rm Mi}({\Bbb I} + G_{ii}B)\right]^t \cdot (G_{ij}B') \cdot
\left[{\rm Mi}({\Bbb I} + G_{jj}B')\right]^t \cdot (G_{jk}B'') \cdot
\left[{\rm Mi}({\Bbb I} + G_{kk}B'')\right]^t \cdot (G_{ki}B)\right\} \nonumber\\
\noalign{\smallskip}
&& \hskip -3truecm + \; {\rm Tr} \left\{
\left[{\rm Mi}({\Bbb I} + G_{ii}B)\right]^t \cdot (G_{ik}B'') \cdot
\left[{\rm Mi}({\Bbb I} + G_{kk}B'')\right]^t \cdot (G_{kj}B') \cdot
\left[{\rm Mi}({\Bbb I} + G_{jj}B')\right]^t \cdot (G_{ji}B)\right\} + \ldots
\label{3site}
\eea
This formula gives all terms of the determinant that are cubic in the off--diagonal
Green functions. They are to be expanded around $t=0$ as discussed above. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Unit height variables}
\label{sec:sec4}

The simplest cluster variable is $S_0$, namely the unit height variable. We give in
this section its multi--site correlation functions, in various configurations, as
computed along the lines exposed above. 

The 1--point function, namely the probability a fixed site has height equal to 1,
poses no problem (and is in any case of little interest for the comparison with a
field theory). Making everything very explicit for once, it is given by
\be
\prob(S_0) \equiv P(1) = \det \left({\Bbb I} +
\pmatrix{G(0,0) & G(1,0) & G(1,0) & G(1,0) \cr G(1,0) & G(0,0) & G(1,1) & G(2,0) \cr
G(1,0) & G(1,1) & G(0,0) & G(1,1) \cr G(1,0) & G(2,0) & G(1,1) & G(0,0)} 
\pmatrix{1-x & 1 & 1 & 1 \cr 1 & -1 & 0 & 0 \cr 1 & 0 & -1 & 0 \cr 1 & 0 & 0 & -1}
\right)\,.
\label{1site}
\ee
Here $G(m,n)$ is $(\Delta^{-1})_{i,0}$ for the site $i=(m,n)$, and we have used the
symmetries of the Green function. The site ordering is O, N, E and S.

This can be easily computed in terms of complete elliptic functions (see the
appendix), although the result is not particularly transparent
\be
P(1) = {1 \over 256} \, \big[(x-4) G(0,0) - 1\big]\,\big[x^2 G(0,0) - 16 G(1,1) -
(x+4)\big] \, \big[(x^2-8) G(0,0) - 8 G(1,1) - (x-4)\big]^2\,.
\ee
It goes to $2[2G(1,1)-2G(0,0)+1]\,[G(1,1)-G(0,0)]^2 = 2(\pi-2)/\pi^3$ in the
limit $x \to 4$. 

More interesting is its graph, which shows that $\prob(S_0)$ increases when $x$ goes
away from 4 before falling off algebraically when $x$ keeps growing. The graph of
$\prob(S_0)$ as a function of $x$ is reproduced in Figure 2 as the long--dashed
curve.

\begin{figure}[htb]
\leavevmode
\begin{center}
\mbox{\epsfysize=5.7cm \epsfbox{probas.eps}}
%\mbox{\epsfscale 800 \epsfbox{probas.eps}}
\begin{minipage}{14cm}
\bigskip \bigskip 
\caption{Probabilities of the clusters $S_0$ up to $S_4$ as functions of the
perturbing parameter $x$. In this figure, all probabilities have been normalized to 1
at $x=4$ to prevent some of them to melt into the horizontal axis. The curves from
top to bottom refer to $S_0$ (long dashes) down to $S_4$ (shortest dashes). The two
solid lines correspond to $S_2$ and $S_3$.}
\end{minipage}
\end{center}
\end{figure}

\subsection{Two--point correlation}

The joined probability for having a 1 at the origin say and an other 1 at a site $i$
is equal to the 8--by--8 determinant 
\be
\prob\left[h_0 = h_i = 1\right] = \det  
\pmatrix{{\Bbb I} + G_{00}\,B & G_{0i}\,B \cr G_{i0}\,B & {\Bbb I} + G_{ii}\,B}\,,
\ee
where $B$ is the matrix used in (\ref{1site}). Because the two clusters are
identical, $G_{ii} = G_{00}$.

As mentioned earlier, the expansion of the Green function at arbitrary points tends
to be complicated, so we have restricted ourselves to configurations where the Green
functions close to a principal axis or a diagonal axis only are required. For the
2--site correlation, this only leaves the two possibilities $i=(m,0)$ and $i=(m,m)$.
Using the formula (\ref{minors}), we found in both cases the same answer:
\bea
&\prob& \hskip -0.2truecm \left[h_0 = h_i = 1\right] - [P(1)]^2 = \nonumber \\
&-& \hskip -0.3truecm t^2 \, [P(1)]^2 \,\left\{{1 \over 2} \, K_0''^2(\sqrt{t}|i|) - 
{1 \over 2} \, K_0(\sqrt{t}|i|) K_0''(\sqrt{t}|i|) +
{1 \over 2\pi} \, K_0'^2(\sqrt{t}|i|) +
{1+\pi^2 \over 4\pi^2} \, K_0^2(\sqrt{t}|i|)\right\} + \ldots, 
\label{2s0}
\eea
with $|i|=m$ or $\sqrt{2}m$ depending on whether $i$ is real or on the diagonal. The
function $K_0$ is the modified Bessel function. Note that the $P(1)$ appearing in
the l.h.s. (in the subtraction term) is the off--critical probability, while that in
the r.h.s. can be taken to be the critical one. 

This formula has a number of instructive and comforting features. The spatial
dependence is only through the function $K_0$, that is, the scaling form of the
massive lattice Green function. The other functions, denoted $D_i$ or $P_i$ in 
Appendix A, and representing the lattice corrections to the scaled, continuum Green
function, actually do not enter. Moreover the fact that the answer is the same for
the two positions of $i$ suggests that the probability is invariant under rotations,
in agreement with the rotational invariance of the cluster $S_0$ itself. This is
related to the first point, since the functions $D_{i>0}$ and $P_{i>0}$ represent
anisotropic terms in the lattice Green function.

Another reassuring feature is that the correlation (\ref{2s0}) scales like $t^2$,
to the dominant order, and that all logarithmic terms $\log{t}$ have dropped out
at that order. This requires massive cancellations because logarithmic terms occur in
all entries of the blocks $G_{00}$ and $G_{ii}$, which store Green function values
around the origin (see the appendix). We have also checked that (\ref{2s0}) is exact
up to higher order in $t$: all terms of order lower than $t^2$ vanish identically
(apart from the zero--th order term $P(1)^2$), and there are no term quartic or
higher in the Green function that would contribute a $t^2$ term. Thus (\ref{2s0}) is
exact to order $t^2$. 

That the correlation scales like the fourth power of the mass was expected since the
critical correlation decays like $|i|^{-4}$ \cite{md2}. It is easily recovered from
(\ref{2s0}) by taking the limit $t \to 0$, in which the term in $K_0''^2(|i|\sqrt{t})
\sim {1 \over t^2|i|^4}$ is the only one to survive, reproducing the result
(\ref{2pt}). 

What the above suggests is that the scaled unit height variable goes
over, in the scaling limit, to a massive field $\phi_0$ with scale dimension 2, 
\be
\lim_{a \to 0} \; {1 \over a^2} \left[\delta\left(h_{z/a} - 1\right) -
P(1)\right] = \phi_0(z)\,, \qquad i={z \over a} \to \infty\,, \quad t=a^2M^2 \to 0
\quad {\rm with\ } i\sqrt{t} = Mz,
\ee
and whose 2--point function reads
\be
\la \phi_0(0) \phi_0(z) \ra = - M^4 \, [P(1)]^2 \,\left\{{1 \over
2} \, K_0''^2(M|z|) -  {1 \over 2} \, K_0(M|z|) K_0''(M|z|) 
+ {1 \over 2\pi} \, K_0'^2(M|z|) +
{1+\pi^2 \over 4\pi^2} \, K_0^2(M|z|)\right\}.
\label{2phi0}
\ee

\subsection{Three--point correlation}

We made the same calculations for the 3--site probability, using the formula
(\ref{3site}). The use of the Green functions on the horizontal or the diagonal axis
leave essentially two possibilities: either the three insertion points $i$, $j,$ and
$k$ are aligned, or else they form a isoceles right triangle. In both cases, the
probabilities scale like $t^3$, with all logarithms of $t$ cancelled out. The
explicit results however differ in these two cases. 

When they form a linear arrangement, be it on the horizontal or diagonal axis, the
result for the connected probability (i.e. products of lower correlations are
subtracted) reads
\bea
&\prob& \hskip -0.2truecm \left[h_i = h_j = h_k = 1\right]_{\rm aligned,\ connected}
= \nonumber\\ 
\noalign{\smallskip}
&& {M^6 \over 4} \, [P(1)]^3 \times \Bigg\{\big[K_0^{}(12)-K''_0(12)\big] \, 
\big[K_0^{}(13)-K''_0(13)\big] \, \big[K_0^{}(23)-K''_0(23)\big] 
+ K_0''(12) \, K_0''(13) \, K_0''(23) \nonumber\\
&& \hskip 2.3cm + \; {1 \over \pi} \, 
\Big[K_0''(12)\, K_0'(13) \, K_0'(23) - K_0'(12) \, K_0''(13) \, K_0'(23) +
K_0'(12) \, K_0'(13) \, K_0''(23)\Big] \nonumber\\
&& \hskip 2.3cm - \; {1 \over \pi^2} \, 
\Big[K_0(12)\, K_0'(13) \, K_0'(23) - K_0'(12) \, K_0(13) \, K_0'(23) +
K_0'(12) \, K_0'(13) \, K_0(23)\Big] \nonumber\\
&& \hskip 2.3cm - \; {1 \over \pi^3} \, K_0(12)\, K_0(13) \, K_0(23) \Bigg\}.
\eea
We have written the answer in the scaled form, that is, after the scaling limit in
which the sites $i,j,k$ go over to the macroscopic positions $z_1,z_2$ and $z_3$.
The notation $K_0(ij)$ stands for $K_0(M|z_i-z_j|)$.

For the triangular configuration, we chose the insertion points $i=(0,0)$ and
$k=(2m,0)$ to be real, and put $j=(m,m)$ on the diagonal. The result is
slightly different in this case, and reads, in the same notations, 
\bea
&\prob& \hskip -0.2truecm \left[h_i = h_j = h_k = 1\right]_{\rm triangular,\
connected} = \nonumber\\ 
\noalign{\smallskip}
&& \hskip -5truemm -{M^6 \over 4} \, [P(1)]^3 \times \Bigg\{2\, K_0''(12) \, K_0(13)
\, K_0''(23) - K_0''(12)\, K_0(13) \, K_0(23)  - K_0(12) \, K_0(13) \, K_0''(23)
\nonumber\\ && \hskip 1.8cm + \; {1 \over \pi} \, 
\Big[\sqrt{2} \, [K_0''(12)-K_0(12)] \, K_0'(13) \, K_0'(23) + \sqrt{2}\,
K_0'(12) \, K_0'(13) \, [K_0''(23)-K_0(23)] \nonumber\\
&& \hskip 4.5truecm + K_0'(12) \, [2\,K_0''(13)-K_0(13)] \, K_0'(23) \Big] \nonumber
\\ && \hskip 1.8cm + \; {\sqrt{2} \over \pi^2} \, 
\Big[K_0'(12)\, K_0'(13) \, K_0(23)  - K_0(12) \, K'_0(13) \, K_0'(23) \Big]
+ {2 \over \pi^3} \, K_0(12)\, K_0(13) \, K_0(23) \Bigg\}. 
\eea
Exactly the same result was found, as expected, for the rotated configuration where
$i$ is at the origin, $j=(m,0)$ on the real axis and $k=(m,m)$ on the diagonal. 

The same comments as for the 2--site correlation apply here but for one point. If
indeed the 3--site probability scaling $\sim t^3$ around the critical point is
consistent with the dimension 2 of a unit height variable, one observes that the
probabilities themselves vanish in the critical limit ($M \to 0$). Thus the scaling
limit of three unit height variables in the usual, unperturbed, ASM vanishes:
\be
\lim_{\rm scaling} \, \prob \left[h_{i} = h_{j} = h_{k} = 1\right]_{x=4,\,{\rm
connected}} = 0.
\ee
We have checked this result by using the critical Green functions, and found that the
probability for three sites aligned along the real axis
\bea
&\prob& \hskip -0.2truecm \left[h_i = h_j = h_k = 1\right]_{\rm real,\
connected} = -{P(1)^2 \over \pi^3} \, \Big[{1 \over z_{12}^3z_{23}^3z_{13}^2} -
{1 \over z_{12}^3z_{23}^2z_{13}^3} - {1 \over z_{12}^2z_{23}^3z_{13}^3}\Big]
\nonumber\\ 
&& \hskip -5truemm + {P(1)^3 \over 8} \, \Big[{1 \over z_{12}^4z_{23}^4} + {1 \over
z_{12}^4z_{13}^4}  + {1 \over z_{23}^4z_{13}^4}\Big]
+ {3P(1)^3 \over 4} \, \Big[{1  \over z_{12}^4z_{23}^2z_{13}^2} + {1 \over
z_{12}^2z_{23}^4z_{13}^2} + {1 \over z_{12}^2z_{23}^2z_{13}^4}\Big] + {\rm higher\
order}
\eea
indeed decays like a global power $-8$ of the separation distances. Moreover the
same calculation for the three sites aligned on the diagonal axis produces
different coefficients. Thus the dominant term of the critical lattice 3--point
function is not isotropic, contradicting the expected rotational invariance, and so
should not survive the scaling limit.

\subsection{Four--point correlation}

Finally, we have also determined the 4--site probability for unit height variables,
at the critical point only, as otherwise the number of terms grows quickly. So
in this case we have used throughout the calculations the expansions at $x=4$ of the
Green functions, also given in the appendix. 

We have examined two different arrangements of the insertion points, when they are
all aligned on the real axis, and when they lie at the vertices of a square. 

When they are all aligned on the real axis, the connected 4--site probability takes a
very simple form, at the dominant order, 
\bea
\prob[h_i = h_j = h_k = h_l = 1]_{{\rm real} \atop {\rm connected}} &=& -{P(1)^4
\over 4}\left\{ {1 \over (z_{12}z_{34}z_{13}
z_{24})^2} + {1 \over (z_{13}z_{24}z_{14} z_{23})^2} +
{1 \over (z_{14}z_{23}z_{12} z_{34})^2} \right\} + \ldots \nonumber\\
\label{4real}
\eea
where the dots represent terms of global power smaller than or equal to $-10$ (they
disappear in the scaling limit), and $z_{13}=i-k, \ldots$ (real).

The other case, for which $i=(0,0)$, $j=(m,0)$, $k=(0,m)$, $l=(m,m)$ are the
vertices of a square of side length $m$, is much more rigid as it depends on a
single distance $m$. The result we found for this situation is 
\be
\prob[h_i = h_j = h_k = h_l = 1]_{{\rm square} \atop {\rm connected}} =  - {3 \over
8}\,{[P(1)]^4 \over  m^8} + \ldots
\label{4square}
\ee

Before presenting the results for the other cluster variables of Figure 1, we
examine the above correlations for the unit height random variable from the point of
view of the conformal field theory which is the most natural candidate, namely
the $c=-2$ theory, and its massive extension. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Conformal field theory}
\label{sec:sec5}

The $c=-2$ conformal field theory has been studied first in the context of polymers
\cite{saleur}, and a bit later served as the simplest example of a logarithmic
conformal field theory \cite{gur}. Since then it has been extensively examined by
many authors \cite{kau1,gk1,roh,gfn,gk2,kau2,ivash2}. Ref. \cite{gk2} in
particular presents a clear and rather complete account of the structure the $c=-2$
theory as a rational conformal field theory.  Even if it is considered as the
simplest situation where logarithms can occur, it contains many subtle aspects and
probably possesses many different and inequivalent realizations. The one which is
relevant here is perhaps the most natural one.

The underlying field theory is formulated in terms of a pair of free grassmanian
scalars $\theta^\alpha = (\t,\tb)$ with action
\be
S = {1 \over 2\pi} \int \; \varepsilon_{\alpha\beta} \, \d \t^\alpha \, \db
\t^\beta = {1 \over \pi} \int \;\d\t\, \db\tb\,,
\label{action}
\ee
where $\varepsilon$ is the canonical symplectic form, $\epsilon_{12}=+1$. 

The zero modes of $\t,\tb$, call them $\xi$ and $\bar \xi$, have been much
discussed. Because the action does not depend on them, the
expectation value of anything that does not contain $\t$ and $\tb$ explicitely, but
only their derivatives vanishes identically. In particular the partition function
itself vanishes, so the correlation functions are normalized by $Z' = \int
{\cal D}\t' {\cal D}\tb' \, e^{-S}$, where the primed fields exclude the zero modes
$\xi$ and $\bar \xi$. 

This normalization implies for instance ($\epsilon^{\alpha\beta} =
-\epsilon_{\alpha\beta}$)
\bea
\la && \hspace{-4mm} 1 \ra = 0\,, \qquad \la \bar\xi \xi \ra = 1\,, \\
\noalign{\medskip}
\la && \hspace{-4mm} \t^\alpha(z) \t^\beta(w) \ra = \epsilon^{\alpha\beta}\,, \qquad
\la \t^\alpha(z) \t^\beta(w) \bar\xi \xi \ra = \epsilon^{\alpha\beta} \,
\log{|z-w|}\,, \\ 
\la && \hspace{-4mm} \d\t^\alpha(z) \, \d\t^\beta(w) \ra = 0\,, \qquad 
\la \d\t^\alpha(z) \, \d\t^\beta(w) \bar\xi \xi \ra = {\epsilon^{\alpha\beta} \over
2(z-w)^2}\,, \label{dd}\\ 
\la && \hspace{-4mm} \t^\alpha(z_1) \t^\beta(z_2) \t^\gamma(z_3) \t^\delta(z_4)
\ra =
\epsilon^{\alpha\beta}\epsilon^{\gamma\delta} \, \log{|z_{12}z_{34}|}
- \epsilon^{\alpha\gamma}\epsilon^{\beta\delta} \, \log{|z_{13}z_{24}|}
+ \epsilon^{\alpha\delta} \epsilon^{\beta\gamma} \, \log{|z_{14}z_{23}|}\,.
\eea

As far as derivatives of fields are concerned ---as will be the case in the ASM, at
least at the conformal point---, one can insert the two zero modes in the
correlators, as in (\ref{dd}), to take care of the integral on constant fields. The
functional integral on non--constant fields then yields the usual form for the
correlators, obtained from Wick's theorem and  the kernel of the laplacian.
Equivalently, one can define the functional integral for derivative fields by
keeping the zero modes out, or consider the so--called
$\eta$--$\xi$ system \cite{saleur}. 

The stress--energy tensor components $T = 2:\!\d\t \, \d\tb\!:$ and $\bar T =
2:\!\db\t \, \db\tb\!:$ have OPEs characteristic of a conformal theory with central
charge $c=-2$. The fields $\t$ and $\tb$ are primary fields with conformal dimensions
(0,0), while the bosonic composite field $:\!\t\tb\!:$ has the following OPE with $T$
\be
T(z) :\!\t\tb\!:(w) = {-1 \over 2(z-w)^2} + {\d :\!\t\tb\!:(w) \over z-w} + \ldots
\ee
It shows that the conformal transformations of $:\!\t\tb\!:$ do not close on itself
(and its descendants) but also involve the identity and its descendants, which form
a conformal module on their own. Thus the identity and $:\!\t\tb\!:$ generate a
Virasoro module, that is reducible but not fully reducible. This is a characteristic
feature of logarithmic conformal theories \cite{gur}. The field $:\!\t\tb\!:$ is
called the logarithmic partner of the identity. It is neither a primary field nor a
descendant (see below for a field that is primary and descendant without being
null). 

The fact that there are two fields with zero scaling dimension is the main source of
unusual features (and confusing subtleties !), one of them being the existence of
two degenerate vacua $|0\ra$ and $|\xi \bar\xi \ra$ (there are two more of fermionic
nature, $|\xi\ra$ and $|\bar \xi\ra$). The above prescription about the insertion of
the zero modes can be viewed in the operator formalism as the taking of operator
matrix elements between two distinct in--going and out--going vacua.

In conclusion, the theory specified by the action (\ref{action}) is a logarithmic
conformal theory with central charge $c=-2$. It contains a non--logarithmic
local sector, that retains the central charge value $c=-2$, and in which derivative
fields only are considered. Anticipating the analysis to be given below, our results
suggest that the ASM scaling fields related to height variables precisely lie in
this $c=-2$ non--logarithmic conformal theory.

It should also be noted that either theory, logarithmic or non--logarithmic, contains
additional non--local (twisted) sectors. Although they could play an important role
in the sandpile models, for the description of other lattice variables than heights,
we will not discuss them here, and refer to \cite{gk2} for further details.

We will also need the off--critical, massive extension of the above conformal
theory. It corresponds to a perturbation by the logarithmic partner of the
identity
\be
S(M) = {1 \over \pi} \int \;:\! \d\t\, \db\tb\!: + \, {M^2 \over 4} :\! \t \tb \!:\,.
\ee

The zero mode problem no longer arises in the massive theory, so that one
can normalize the correlation functions by the full partition function $Z(M) = \int
{\cal D}\t {\cal D}\tb \, e^{-S(M)}$. One then obtains
\be
\la \t(z) \tb(w) \ra = K_0(M|z-w|)\,, \qquad \la \t(z) \t(w) \ra = \la \tb(z) \tb(w)
\ra = 0,
\ee
and for instance
\be
\la \d\t(z) \, \d\tb(0) \ra = -{M^2 \over 4} \, {\bar z \over z} \,
\big[2K_0''(M|z|) - K_0(M|z|)\big]\,.
\ee
On account of $K_0(x) \sim -\log{x}$ for small arguments, the massless limit of the
previous equation exists and reproduces the expression given in (\ref{dd}) with the
zero modes inserted. This is expected since the effect of the zero mode insertion
is formally to change the normalization factor from $Z'$ to $Z$. On the other
hand, the same does not apply to the correlations of the fields $\t^\alpha$
themselves, as the normalizing functional $Z(M)$ goes to zero as $M \to 0$. 

As mentioned above, the cluster variables we consider in this article are
all related to derivative fields. The previous remark then implies that the
off--critical ASM multi--site probabilities have a smooth massless limit, equal to
the critical probabilities. The scaling form of the off--critical probabilities will
be related to the above massive free theory, while the critical ones will be
computable in terms of the non--logarithmic conformal field theory using the
insertion prescription. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Scaling fields for cluster variables}
\label{sec:sec6}

Let us now reconsider the multi--site probabilities for height 1 computed in
the Section \ref{sec:sec4}. The 2--site probability suggested that the unit height
variable is described by a field with scaling dimension 2, that should be in
addition scalar since a unit height variable is rotationally invariant. If one
assumes that this field is local in $\t, \tb$, the only possibilities are
$:\! \d\t^\alpha \, \db\t^\beta \!:$, $:\! \t \tb \, \d\t^\alpha \, \db\t^\beta
\!:$ and $M^2 :\!\t\tb \!:$. The second set of fields $:\! \t \tb \, \d\t^\alpha \, 
\db\t^\beta \!:$ must be excluded, because as explained above, they would produce
logarithms in correlation functions, contradicting the observation we made in
section \ref{sec:sec2} that, in the massless sandpile model ($x=4$), the multi--site
probabilities are never logarithmic (at least those one can compute from the
Majumdar--Dhar technique, i.e. from finite determinants). 

It is not difficult to see that 
\be
\phi_0 = -P(1) \big[:\! \d\t \db\tb + \db\t \d\tb \!: + \, {M^2 \over 2\pi}
:\!\t\tb\!: \big]
\label{phi0}
\ee
is indeed the right combination: its 2--point function is exactly the
form given in (\ref{2phi0}), which was obtained by taking the scaling limit of the
2--site probability computed on the lattice. 

In order to confirm this identification, the field--theoretic 3--point function of
$\phi_0$ can  be computed and compared with the lattice result. In the same notations
as in Section
\ref{sec:sec4}, one finds for an arbitrary arrangement of the insertion points
\bea
\la && \hspace{-4mm} \phi_0(z_1) \, \phi_0(z_2) \, \phi_0(z_3) \ra = 
-{M^6 \over 16} \times \Bigg\{ 
{1 \over 2} \,\left({z_{13}\bar z_{23} \over \bar z_{13}z_{23}} + {\rm c.c.}\right) 
\, K_0(12) \, [2K_0''(13) - K_0(13)] \, [2K_0''(23) - K_0(23)] + {\rm perm.}
\nonumber\\
&& + \, {\pi-2  \over \pi^2} \, \Big[\Big({z_{13}\bar z_{23} \over
|z_{13}|\,|z_{23}|} + {\rm c.c.}\Big) \, K_0(12) \, K_0'(13) \, K_0'(23) 
+ {\rm perm.} \Big] \label{fff} \\
&& + \, {1 \over \pi} \, \Big[\Big({z^2_{12} \bar z_{13} \bar
z_{23} \over |z_{12}|^2 \, |z_{13}| \, |z_{23}|}\Big) [2K_0''(12) - K_0(12)] \,
K_0'(13) \, K_0'(23) + {\rm perm.} \Big] 
+ \, {\pi^3-4 \over \pi^3} \, K_0(12) \, K_0(13) \, K_0(23) \Bigg\}, \nonumber
\eea
where the permutations that must be added are the two exchanges $z_1 \leftrightarrow
z_3$ and $z_2 \leftrightarrow z_3$. One easily checks that it reproduces the 3--site
probabilities reported in section \ref{sec:sec4} for the two arrangements examined
there. The massless limit of (\ref{fff}) vanishes, as it clearly follows from
(\ref{phi0}) for $M=0$, since the 3--point function will necessarily involve a Wick
contraction of a $\d \t^\alpha$ with some $\db \t^\beta$.

Finally the 4--point function can be compared. For convenience, we give the
field--theoretic result in the massless regime:
\bea
\la \phi_0(z_1) \, \phi_0(z_2) && \hspace{-3mm} \phi_0(z_3) \, \phi_0(z_4)
\ra_{M=0} = {P(1)^4 \over 4|z_{12}|^4 \, |z_{34}|^4} + {P(1)^4 \over 4|z_{13}|^4 \,
|z_{24}|^4} + {P(1)^4 \over 4|z_{14}|^4 \, |z_{23}|^4} \nonumber\\
&& \hspace{1.5cm} - {P(1)^4 \over 8} \, \Big\{ {1 \over (z_{12}\,z_{34}\,\bar z_{13}
\,\bar z_{24})^2} + {1 \over (z_{13}\,z_{24}\,\bar z_{14} \, \bar z_{23})^2} +
{1 \over (z_{14}\,z_{23}\,\bar z_{12} \, \bar z_{34})^2} + {\rm c.c.} \Big\},
\eea
where only the last term within the curly brackets represents the connected
part of the 4--point function. When the four insertions lie on the real
axis, it clearly reproduces the lattice result (\ref{4real}), and when they are the
vertices of a square of side $m$, $z_1=0,\,z_2=m,\,z_3=im,\,z_4=(1+i)m$, it reduces
to
\be
\la \phi_0(z_1) \, \phi_0(z_2) \, \phi_0(z_3) \, \phi_0(z_4) \ra_{M=0,\,{\rm
square},\,{\rm connected}} = - {3 \over 8}{[P(1)]^4 \over m^8} ,
\ee
and again matches the connected 4--site probability (\ref{4square}).

We believe these comparisons provide enough evidence to assert that the unit height
random variable of the sandpile model goes over, in the scaling limit, to the field
$\phi_0$ defined in (\ref{phi0}). In the conformal limit, $\phi_0 \sim :\! \d\t
\db\tb + \db\t \d\tb
\!: = \d\db :\!\t\tb\!:$ is a primary field with conformal dimensions (1,1), but is
also a descendant of $:\!\t\tb\!:$. 

The rest of this section presents analogous results for the other cluster variables
pictured in Figure 1. 

We have repeated, for the other thirteen clusters in Figure 1, the same calculations
we performed for the unit height variable. More precisely, for each of the cluster
variables $S_1$ up to $S_{13}$, we have computed its joint probability with a unit
height, namely $\prob[S_0(0),\,S_k(i)]$, with $i$ on the principal and 
on the diagonal axis. From these two probabilities one can write down an Ansatz for
the field $\phi_k$ to which the cluster $S_k$ gets identified in the
scaling limit. These identifications were subsequently checked to reproduce
all 2--site probabilities $\prob[S_k(0),\,S_\ell(i)]$, for all pairs $k,\ell =
0,1,2,\ldots,13$, both on the principal and the diagonal axes. In addition, at least
one rotated (or mirrored) version of each cluster has been examined, although not
systematically (only the correlation with $S_0$ on both axes). The results we
found for the rotated clusters are in agreement with the rotations of the
fields assigned to the unrotated clusters, so that the field of the rotated cluster
is the rotated field. Finally, mixed 3--cluster probabilities involving unit heights
and $S_1$ clusters have also been computed. They all confirmed the field
identifications.

All calculations have been performed exactly, i.e. not
numerically. The 2--cluster probabilities take a form similar to (\ref{2s0}), where
the coefficients are in general complicated rational expressions of $\pi$. Keeping
these coefficients in an exact form allows the check of the field identifications
to be made in an exact way. For simplicity however, the results presented
below are given numerically. 

The features of the 2--cluster probabilities are the same as for the unit height
variables. We found that all of them scale like $t^2$, with all logarithmic
singularities cancelled out. It implies that all cluster variables go in the
scaling limit to fields with scaling dimension 2:
\be
\lim_{a \to 0} \; {1 \over a^2} \left[\delta\left(S(i)\right) -
P(S)\right] = \phi_S(z)\,, \qquad i={z \over a} \to \infty\,, \quad t\equiv
x-4=a^2M^2 \to 0 \quad {\rm with\ } i\sqrt{t} = Mz.
\ee 
This is somewhat surprising as one could have expected the dimension of the scaling
fields to increase with the size of the clusters. 

All cluster variables we have considered have a scaling limit that corresponds to
a field of the following form:
\be
\phi(z) = -\left\{A:\!\d\t \db\tb + \db\t\d\tb \!: + \,
B_1 :\!\d\t \d\tb + \db\t\db\tb \!: +\,
i B_2 :\!\d\t \d\tb - \db\t\db\tb \!: + \,
C\, P(S)\, {M^2 \over 2\pi} \, :\!\t\tb \!: \right\}.
\label{phi}
\ee
The (real) coefficients $A$, $B_1$, $B_2$ and $C$ are given in the Table for each
cluster. The factor $P(S)$ in front of the term $:\!\t\tb \!:$
is the probability of $S$ evaluated at $x=4$. Note that the field is not invariant
under a rotation of $\pi \over 2$ as soon as $B_1$ or $B_2$ is non--zero, but is
invariant under a rotation of $\pi$ no matter what the coefficients are. So in
particular, the scaling limit of the cluster variables does not yield in general
conformal fields, but sums of pieces with different tensor structures.

\bigskip
\renewcommand{\arraystretch}{1.8}
%\begin{table}[htp]
\begin{center}
\begin{tabular}[t]{|c|c|c|c|c|c|}
\hline
cluster $S$ & $P(S)$ & $A$ & $B_1$ & $B_2$ & \ \ $C$\ \ \ \cr
\hline \hline
\conf0 & 0.0736362 & 0.0736362 & 0 & 0 & 1 \\
\hline
\conf1 & 0.0103411 & 0.0201433 & $-0.00619014$ & 0 & 2 \\
\hline
\conf2 & 0.00141994 & 0.00449027 & $-0.00208908$ & 0 & 3 \\
\mbox{\conf3} & 0.00134477 & 0.00389417 & 0 & $-0.000534524$ & 3 \\
\hline
\mbox{\conf4} & 0.00019246 & 0.000893234 & $-0.000502885$ & 0 & 4 \\
\mbox{\conf5} & 0.000179829 & 0.000752599 & $-0.000172832$ & $-0.000122784$ & 4\\
\mbox{\conf6} & 0.000173323 & 0.000695941 & $-0.00012949$ & $-0.000153467$ & 4 \\
\mbox{\conf7} & 0.000179829 & 0.000752599 & 0.000172832 & $-0.000122784$ & 4 \\
\mbox{\conf8} & 0.000173106 & 0.000692147 & 0.000135489 & 0 & 4 \\
\mbox{\conf9} & 0.000173106 & 0.000692147 & $-0.000135489$ & 0 & 4 \\
\hline
$S_{10}$ & 0.0000572863 & 0.000255127 & $-0.0000470539$ & $-0.0000259802$ & 5 \\
\hline
$S_{11}$ & 0.00000731457 & 0.000042272 & $-0.0000130978$ & $-0.0000071978$ & 6\\
\hline\hline
$S_{12}=$\conf{12} & 0.00496687 & 0.00969315 & $-0.00129393$ & 0 & 2 \\
\hline
$S_{13}=$\mbox{\conf{13}} & 0.00404859 & 0.010213 & 0 & $-0.00312009$ & 2 \\
\hline
\end{tabular}
\begin{minipage}{14cm}
\bigskip \bigskip 
TABLE I. For each cluster in Figure 1, the Table gives the values of the
parameters $A$, $B$ and $C$ specifying the field that describes the scaling behaviour
of the given cluster (see Eq. (\ref{phi})). Note in particular that the coefficient
$C$ is equal to the size of the cluster.
\end{minipage}
\end{center}
%\end{table}

\bigskip \bigskip
As far as numerical values are concerned, the last column of the Table is
particularly striking: all entries are integers, simply equal to the size of the
cluster. This makes the coefficient of the $:\!\t\tb \!:$ terms particularly simple
and apparently regular. The reason for this is unclear.

The other numbers mentioned in the Table are not in themselves particularly
interesting. As mentionned above, all these numbers are complicated expressions. For
instance the first three numbers on the line corresponding to $S_9$ (the last
cluster of size 4) are in fact equal to
\bea
&& \hspace{-4.5mm} \prob(S_9) = \frac{2621440}{27\,{\pi }^7} - 
\frac{21389312}{81\,{\pi }^6} + \frac{24279040}{81\,{\pi }^5} - 
\frac{14968672}{81\,{\pi }^4} + \frac{1809776}{27\,{\pi }^3} - 
  \frac{258037}{18\,{\pi }^2} + \frac{10061}{6\,\pi } - \frac{663}{8} ,\\
&& \hspace{-4.5mm} A = \left({3\pi - 8 \over \pi^2}\right)
\left(\frac{655360}{27\,{\pi }^5} -
\frac{3389440}{81\,{\pi }^4} + \frac{2259952}{81\,{\pi }^3} - \frac{81566}{9\,{\pi
}^2} + \frac{5765}{4\,{\pi }} - \frac{8647}{96}\right),\\
&& \hspace{-4.5mm} B_1 = \left({3\pi - 8 \over \pi^2}\right) \left(
\frac{305152}{81\,{\pi }^4} - \frac{359056}{81\,{\pi }^3} + 
  \frac{17554}{9\,{\pi }^2} - \frac{13693}{36\,\pi } + \frac{2663}{96} \right).
\eea

A gross feature of the Table is that the (non--zero) numbers are roughly constant
for all clusters of the same size, namely the probabilities and the
coefficients do not change much with the shape of the clusters, but depend
essentially on their size only. Roughly speaking, these numbers (except $C$) get
divided by 10 when the size increases by one. 

The zeroes in the Table or the equality (up to signs) of coefficients can be
understood from the transformations of the clusters and the corresponding fields
under the symmetry group of the lattice. One easily sees that the field $\phi$ in
(\ref{phi}) changes under rotations and reflections according to the following rules:
\be
(A,B_1,B_2,C) \longrightarrow \cases{
(A,-B_1,-B_2,C) & under a ${\pi \over 2}$--rotation, \cr
\noalign{\smallskip}
(A,B_1,-B_2,C) & under a $x$-- or $y$--reflection.}
\label{symm}
\ee
By convention, all clusters are assumed to be anchored to their lower left site. The
rotations are performed about an axis passing through that site.

First of all, the only one to have $B_1 = B_2 = 0$ is the unit height. Indeed it is
the only cluster that preserves its shape under rotations and reflections, and so
one can expect the corresponding field to be a scalar under (continuous) rotations
and reflections. 

There are clusters whose fields have $B_2 = 0$, and they are precisely those clusters
which are invariant under a reflection through the horizontal axis. The same can be
said of the rotated clusters for a reflection through the vertical axis. That $S_9$
has a coefficient $B_2=0$ can be understood along the same lines, although it is not
manisfestly invariant under reflections. An $x$--reflection of $S_9$ followed by a
rotation by $\pi$ and a translation of two lattice sites bring it to itself, except
that a height 2 and a height 1 have been swapped. However, the assignment of
heights within a cluster is irrelevant in the actual computation: the modified ASM
is defined in terms of certains bonds being removed. Since each site whose height
is being constrained looses three out of his four bonds, the actual height assignment
is irrelevant. In effect, the set $M_S$ which includes all the sites affected by the
modifications and the modification matrix $B$ itself can be chosen (have been chosen)
invariant under a $y$--reflection. 

In the same way, one sees that $S_5$ and $S_7$ have equal coefficients, up to signs.
As represented in the Table and in Figure 1, they are related by rotation of $\pi
\over 2$ and an $x$--reflection, with the consequence that their $B_1$ coefficients
are opposite but the $B_2$ are equal. The same can be said of $S_8$ and $S_9$, with
the same remark as above regarding the locations of the height values within the
clusters. 

From these remarks, one easily finds the fields corresponding to different
orientations of a cluster. The cluster $S_6$ for instance comes in eight different
orientations (all anchored to the same site). All of them have the same coefficient
$A \sim 0.000695941$ and $C=4$, whereas pairs of clusters have coefficients
$(B_1,B_2)$, or $(-B_1,B_2)$, or $(B_1,-B_2)$ or $(-B_1,-B_2)$. As a consequence, the
sum over the corresponding eight fields reduces to a projection onto the scalar
part, and involves the $A$ and $C$ terms only.

In a sense, the fact that the fields reflect so well the geometric symmetries of the
clusters is surprising. As discussed at length in Section \ref{sec:sec2}, the
actual calculations are based on adequate modifications of the original ASM on a set
we called $M_S$, which not only contains sites belonging to the cluster itself, but
also sites in its close neighbourhood. Thus each cluster drags with itself an
invisible shadow, made of the sites in the set $M_S \setminus S$. The shadow is a
computational artefact, but is nevertheless crucial. Moreover it usually breaks
or alters the geometric symmetries of the cluster it goes with. The insertion of a
height 1 for instance, somewhere in the lattice, really requires to consider a
4--cluster pictured in Section \ref{sec:sec2}. Here the shadow consists of three
neighbours of the central site, and clearly breaks the rotational invariance. 

We will conclude this section by observing that the height $h$ variables, for $h$
bigger than 4, can be handled in the massive ASM exactly like the unit height
variables, even more simply. The reason is that a height equal to $5,6, \ldots, x$
can never be in a forbidden subconfiguration, so that the set of recurrent
configurations containing a height equal to $h>4$ at some site $i$ is equal to the
set of recurrent configurations on the lattice with $i$ removed. Therefore the
modifications needed to freeze the height of a site to $h>4$ must simply reduce the
threshold at that site to 1, and cut it off from the rest of the lattice. This can
be implemented by the following matrix
\be
B=\pmatrix{1-x & 1 & 1 & 1 & 1 \cr
1 & 0 & 0 & 0 & 0 \cr 1 & 0 & 0 & 0 & 0 \cr 1 & 0 & 0 & 0 & 0 \cr1 & 0 & 0 & 0 & 0}.
\ee

The corresponding probability is simply given by $\prob[h_i=h>4] = G(0,0)$, and is
logarithmically divergent at $x=4$. For that reason, one considers instead the
probability that $h_i$ exceeds 4:
\be
\prob[h_i > 4] = (x-4) \, G(0,0) = {2(x-4) \over \pi x} \, K\left({4 \over
x}\right),
\ee
which goes to 0 when $x \to 4$. (The matrix $B$ corresponding to this has $-4$ as
first diagonal entry, rather than $1-x$.) $K$ is a complete elliptic function (see
Appendix A). 

As for the above clusters, one can compute the correlations of this random variable
$\delta(h_i > 4)$ with itself or with the other clusters, and see what
field--theoretic description it has in the scaling limit.  

Again the result is simple. The lattice calculation of its own correlation yields
\be
\prob[h_0>4,\,h_i>4] - \prob[h_0>4]^2 = - {t^2 \over 4\pi^2}\,K_0^2(\sqrt{t}|i|),
\ee
which suggests the scaling limit 
\be
\delta(h_i > 4) - \la \delta(h_i > 4) \ra \quad \stackrel{\rm
scaling}{\longrightarrow} \quad \phi = {M^2 \over 2\pi} \, :\!\t\tb \!:\,.
\ee
Correlations with the other cluster variables confirm this limit. It nicely fits the
expectation that the field should vanish at the critical point. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The height two variable}
\label{sec:sec7}

We have so far focused on the class of weakly allowed cluster variables, whose
correlations can be handled by the technique developped in \cite{md2}, and in turn
computed from a finite determinant. The authors point out in that article that
non weakly allowed cluster variables can in fact be viewed as infinite series of
weakly allowed clusters. It dramatically complicates their treatment, since a
correlation involving a single non weakly allowed cluster requires to compute an
infinite number of correlations of weakly allowed clusters, of finite but
unbounded size. 

In this section, we address the question of the field assignment for the height two
variable, in the light of the results of the previous sections. We will consider the
height two variable, both from the perturbative point of view that we have just
summarized, and from the conformal point of view.

That a height two variable can be treated as an infinite sum of weakly allowed
cluster variables can be seen as follows \cite{md2}. Consider the set of recurrent
configurations $\cal C$ with a height 2, at the origin say. That set can be divided
up into two disjoint subsets according to whether the configurations remain
recurrent when the 2 is replaced by a 1, or become transient upon that replacement.

The number of those that remain recurrent is the same as the number of recurrent
configurations which have a height 1 at the origin, because vice--versa, a recurrent
configuration with a 1 remains recurrent if the 1 is replaced by a 2. So the
contribution to $P(2) \equiv \prob[h_0=2]$ from this first subset is exactly equal to
$P(1)$. 

For those configurations which become transient, it must be that the 2 belongs to a
weakly allowed cluster. This weakly allowed cluster can be of various size and
shape, and a straight enumeration according to their size leads directly to the
clusters of Figure 1 (except the first one and the last two) and their various
orientations. In this way, the second subset is itself divided into a infinite
number of disjoint subsets, according to which weakly allowed cluster $S$ the height
2 at the origin is part of. The subset labelled by $S$ (fixed size, shape and
orientation) contributes $P(2)$ a term equal to $P(S)$. 

Putting all together, one obtains, observing that the number $P(S)$ does not depend
on the orientation of $S$, the formula
\be
P(2) = P(1) + \sum_{{\rm w.a.c.\ }S} P(S) = P(1) + 4P(S_1) + 4P(S_2) + 8P(S_3)
+4P(S_4) + 8P(S_5) + \ldots
\ee
where the summation is over the weakly allowed clusters which are `anchored'
to a height 2. As pointed out in \cite{md2}, the convergence is very
slow. From Table 1, the terms up to $S_9$ furnish the lower bound $P(2) \geq
0.13855$, well below the exact value $P(2) \sim 0.1739$ \cite{priezz}.

The argument recalled above leading to the perturbative formula for $P(2)$ works
similarly for any correlation. The result can be expressed as an identity between
random variables, 
\be
\delta(h_i - 2) = \delta(h_i - 1) + \sum_{{\rm w.a.c.\ }S} \delta(S(i)).
\ee
Modulo the issue of convergence, this identity is valid when inserted in
expectation values.

The results of the previous section suggest that all random variables on the r.h.s.
have the same scaling form, given by the field in (\ref{phi}). Assuming this at all
orders, and taking the scaling limit of the previous identity lead to a scaling
field for the height two of the same form as the scaling field for the height one,
namely
\be
\delta(h_i - 2) \quad \stackrel{\rm scaling}{\longrightarrow} \quad 
\alpha :\!\d\t \db\tb + \db\t\d\tb \!: + \,\beta\,M^2 \, :\!\t\tb \!:\,.
\ee
This follows from the observation we made earlier that the other terms 
$\d\t\d\tb \pm \db\t\db\tb$ change sign under a rotation by $\pi\over 2$. The sum
over the orientations of a cluster make these terms cancel against each other,
leaving a scalar field, as it should. 

The natural conclusion one could draw from this is that, at the critical point, the
heights one and two scale the same way and in fact go over, in the scaling limit, to
the same ---up to normalization--- primary field of conformal dimensions (1,1). This
is the first direct though teneous evidence in favour of such a statement, which has
in fact been made in \cite{ivash}, based on an extrapolation to the bulk of a similar
statement on the corresponding boundary variables, itself relying on the boundary
2--point functions. As plausible and likely as it may be, the extrapolation remains
uncontrolled, as there are well--known examples of lattice observables that go to
different fields, depending on whether they lie on a boundary or in the bulk. Thus
neither argument is convincing, but both point to the same field assignment for the
height two variable (and probably similarly for the heights three and four).

This seems reasonable and likely. It is therefore surprising to observe that it does
not appear to be consistent with a naive interpretation of the operator product
expansions (OPE). To simplify, we consider the critical point, and the corresponding
conformal field theory.

The two lattice variables, a height one and a height two, can be taken far apart, and
subsequently brought closer to each other, until they occupy neighbouring sites,
then forming the cluster variable we called $S_1$. In the field--theoretic picture,
this amounts to taking the two corresponding fields closer and closer to each other,
until they become coincident, at which point they form a new composite field. The
information about what composite fields a pair of fields can form when they come
close to each other and asymptotically coincident, is contained in their operator
product expansion (OPE).

Thus it seems natural to expect that the field assigned to the cluster variable
\mbox{\hskip 2truemm \conf{1}} be in the OPE of the field corresponding to
the height one with the field corresponding to the height two. If one assumes, as
argued above, that the heights one and two scale to the same field, the required OPE
is simply
\be
:\!\d\t \db\tb + \db\t \d\tb \!:(z) \, :\!\d\t \db\tb + \db\t \d\tb \!:(w)
= -{1 \over 2|z-w|^4} + {:\!\db\theta \db\bar\theta\!:(w) \over (z-w)^2} +
{:\!\d\theta \d\bar\theta\!:(w) \over (\bar z-\bar w)^2} 
+ {\rm less\ singular},
\label{ope1}
\ee
where, from dimensional analysis, the less singular terms involve fields of scale
dimension strictly larger than 2. One sees from (\ref{ope1}) that the only fields
with scale dimension 2 which can be formed in the fusion of a height one with a
height two are the non--scalar parts of the field making the cluster \mbox{\hskip
2truemm \conf{1}}. The scalar part of it, $:\!\d\t \db\tb + \db\t \d\tb \!:$, is
missing. (Note that it must be so, since otherwise the unit height variables,
represented by that scalar field, would have a non--zero (connected) 3--point
function.)

One may observe that the only dimension 2 scalar fields whose fusion produces all
field components of \mbox{\hskip 2truemm \conf{1}} are logarithmic fields, like $:\!
\t\tb(\d\t \db\tb + \db\t \d\tb) \!:$. The change for this logarithmic scalar field
has however heavy consequences as correlations involving heights two would
automatically contain logarithmic functions of the separation distances, in addition
to the usual rational functions. 

Note that for exactly the same reasons, one could question the field assignment of
the height one variable itself, despite the fact that the field $\phi_0$ has
successfully passed so many tests. Although one cannot bring two heights one side by
side, one can bring them fairly close to each other, like in the last two clusters of
Figure 1 (or Table 1), in fact close enough so as not to loose the OPE argument. But
then the fields associated to the two clusters $S_{12}$ and $S_{13}$ must be
contained in the fusion of two heights one, i.e. in the fusion (\ref{ope1}), which
we know is not the case. 

Perhaps sandpile models are so special that one would reject the fusion altogether,
on the basis that height variables have hair ! Because a particular height imposes
restrictions on what can stand close to it. For example a height 1 forces all its
neighbours to be higher or equal to 2, and a height 2 does not allow two of its
neighbours to have a height 1. This might explain the unconsistency noticed above,
but at the same time it denies the very possibility of a field assignment. 
We believe that this issue should be clarified. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Conclusion}

The power of conformal field theory could bring a much better
understanding of the sandpile model, if some of its observables could be identified
with conformal fields. This is a non--trivial task even for the height variables,
which are probably the easiest variables to account for in a field--theoretic
setting. In addition, and in order to strengthen the connection with a field theory,
the neighbourhood of the critical point should be investigated. In this article, we
have taken the first steps towards a systematic study of this relationship, at and
off--criticality. 

The off--critical extension of the sandpile that we considered is defined by
allowing dissipation, i.e. loss of sand each time a site topples. The dissipation
rate is controlled by a parameter $t \geq 0$ and corresponds to a relevant
perturbation of the usual abelian undirected sandpile model. 

We have examined multi--site probabilities for the simplest local cluster variables
in the off--critical sandpile model. By explicit calculations, we have shown that
their scaling form can be fully reproduced by a free field theory of massive
grassmanian scalars. In the massless, critical limit, this theory is a logarithmic
conformal field theory with central charge $c=-2$. The local fields assigned to the
various cluster variables however all belong to a non--logarithmic bosonic sector.
The massive regime, with a mass $M \sim \sqrt{t}$ directly related to the perturbing
parameter, corresponds to a thermal perturbation of the conformal theory, i.e. a
mass term specified by a logarithmic field. 

We have determined the field assignment for the fourteen cluster variables pictured
in Figure 1, and checked their consistency against the correlation functions. On the
other hand, at the critical point, we have noted a disagreement between these
assignments and the naive fusion rules of the conformal theory.  

We do not claim that all features of the sandpile models will be
comprehensible within a field theory, but some of them definitely are. In this
respect, other issues than the height variables can be raised: boundary phenomena
against boundary conformal field theory, the question of the modular non--invariance
on a torus (with leaking sites), ... Also the relevance and the role of
logarithmic fields and twist fields in the $c=-2$ logarithmic conformal field theory
must be further examined. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\acknowledgments
P.R. heartily thanks Michael Flohr for many stimulating discussions and for
sharing his own insight into the subtleties of logarithmic conformal field theories.
Useful discussions with Deepak Dhar about the draft of this article are also
gratefully acknowledged.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\appendix
\section{Green functions}
\label{sec:appa}

In this appendix, we collect a number of expressions we have used for the
computations of correlations in the sandpile model. 

The central object here is the Green function $G$ of the massive discrete laplacian
on $\Z^2$, that is the solution of the Poisson equation $\Delta G = 1$, with $\Delta$
being the finite difference operator given in (\ref{delta}). The solution is easily
obtained by Fourier transform
\be
G(m,n) = G((m',n'),(m+m',n+n')) = \int\!\!\!\!\int_0^{2\pi} \; {{\rm d}^2k \over
4\pi^2} \; {{\rm e}^{ik_1 m + ik_2 n} \over x - 2\cos{k_1} - 2\cos{k_2}}, \qquad
(m,n),(m',n') \in \Z^2.
\label{green}
\ee

As explained in the text, values of $G$ are needed at points $x$ which are
either close to the origin, or else very far from the origin, and in this
last case, we have restricted ourselves to points close to a principal or a diagonal
axis. We treat these three cases in turn. 

\subsection{The Green function at points close to the origin}

By using the invariance of $G$ under the reflection symmetries of the lattice and
its defining equation $\Delta G = 1$, the Green function can be given everywhere in
terms of its values on a diagonal. By a suitable change of variables and
one integration \cite{spitz}, the diagonal values can be recast into
\be
G(m,m) = {(-1)^m \over \pi x} \; \int_0^\pi \; {\rm d}t \; {\cos{2mt} \over
\sqrt{1-{16 \over x^2}\sin^2{t}}}.
\label{diag}
\ee
This can be resolved in terms of the complete elliptic functions \cite{spol}
\bea
K(p) &=& \int_0^{\pi/2} {\rm d}t \; {1 \over \sqrt{1-p^2 \sin^2{t}}} = 
\left[1 + {q^2 \over 4} + {9q^4 \over 64} + \ldots \right]\,\log{\left({4 \over
q}\right)} - \left[{q^2 \over 4} + {21 q^4 \over 128} + \ldots \right],
\label{k}\\
 E(p) &=& \int_0^{\pi/2} {\rm d}t \; \sqrt{1-p^2 \sin^2{t}} = 
\left[1 - {q^2 \over 4} - {13 q^4 \over 64} - \ldots \right] + 
\left[{q^2 \over 2} + {3q^4 \over 16} + \ldots \right]\,\log{\left({4 \over
q}\right)},
\label{e}
\eea
where $q=\sqrt{1-p^2}$, and where the expansions are given for $p \lesssim 1$ close
to 1. 

In terms of our perturbing parameter $t=x-4$, one finds for instance
\bea
G(0,0) &=& {2 \over \pi x} K({\textstyle {4 \over x}})\,, \\ 
G(1,1) &=& {1 \over 4\pi x} \Big\{\big(x^2 - 8\big) \, K\big({\textstyle {4 \over
x}}\big) - x^2 \, E\big({\textstyle {4 \over x}}\big)\Big\} \,,\\  
G(2,2) &=& {1 \over 24\pi x} \Big\{\big(x^4 - 16 \, x^2 + 48\big) \,
K\big({\textstyle {4 \over x}}\big) - x^2 \, \big(x^2 - 8\big) \, E\big({\textstyle
{4 \over x}}\big)\Big\} \,,\\  
G(3,3) &=& {1 \over 120 \pi x} \Big\{\left(x^6 - 24 \, x^4 + 158 \, x^2 -
240\right) \, K({\textstyle {4 \over x}}) - x^2 \, \left(x^4 - 16 \, x^2 + 46
\right) \, E({\textstyle {4 \over x}}) \Big\}\,,
\eea
which can then be expanded around $x=4$ by using (\ref{k}) and (\ref{e}). They all
have the same logarithmic singularity at $x=4$ as $G(0,0)$, so that the differences
$G(m,n)-G(0,0)$ remain finite when $x \rightarrow 4$. In particular, the critical
limit of the subtracted diagonal Green function is simply \cite{spitz}
\be
\lim_{x \to 4} \; [G(m,m) - G(0,0)] = -{1 \over \pi} \, \sum_{k=1}^{m} \; {1 \over
2k-1}\,.
\label{crit}
\ee

\subsection{The Green function on the far diagonal}

For $m$ large, the use of elliptic functions is impractical to extract the
asymptotic behaviour in $m$. Making the change of variables $z = {\rm e}^{it}$, the
formula (\ref{diag}) becomes a  integral over a contour that can be deformed to
enclose the cut lying in between the two roots $\pm u$ of the denominator, with $u =
{x
\over 4} -
\sqrt{{x^2\over 16}-1}$. This yields
\be
G(m,m) = -{1 \over 2\pi} \; \int_{-u}^u {\rm d}z \; {z^{2m} \over
\sqrt{(z^2-u^2)(z^2-{1 \over u^2})}}\,.
\ee
The asymptotic expansion of this kind of integral was studied in \cite{coywu}, from
which one finds, using their notations
\be
G(m,m) = \sqrt{u \over x(1-u^2)} \, {1 \over \sqrt{2\pi m}} \, u^{2m} \, \Big\{
1 + {\tilde A_{1>} \over 8m} + {3 \over 64}\,{\tilde A_{2>}-{5 \over 6} \over m^2} + 
{15 \over 512}\,{\tilde A_{3>}-{7 \over 6}\tilde A_{1>} \over m^3} + \ldots \Big\},
\label{diag2}
\ee
where the coefficients $\tilde A_{n>}$ are defined from the generating function
\be
\tilde A_>(z) = {1 \over \sqrt{(1+{1-u^2 \over 1+u^2}z)(1+{1+u^2 \over 1-u^2}z)}} = 
\sum_{n=0}^\infty \; \tilde A_{n>} \, z^n\,,
\ee
and are thus themselves infinite (Laurent) series in $u^2$, hence in $\sqrt{t} =
\sqrt{x-4}$. It is not difficult to show that these coefficients start off like 
\be
\tilde A_{n>} = (-1)^n \, {(2n-1)!! \over (2n)!!} \left({2 \over t}\right)^{n/2} +
{\cal O}(t^{-{n \over 2}+1})\,,
\ee
with the consequence that the $m^{-n}$ term in (\ref{diag2}) takes the form
\be
{(2n-1)!! \over 8^n} \; {\tilde A_{n>} + \ldots \over m^n} = 
{(-1)^n \over n!} \, \left({(2n-1)!! \over 2^n}\right)^2 \, {1 \over
(2\sqrt{2t}\, m)^n} \, \big[1 + {\rm series\ in\ }t \big]\,,
\label{a14}
\ee
that is, a first term which has the scaling form times corrections in $t$,
independent of the distance $m$.

By combining the previous expansion with that of the prefactor of (\ref{diag2}),
\be
\sqrt{u \over x(1-u^2)} \, {1 \over \sqrt{2\pi m}} \, u^{2m} = \left({1 \over 8\pi
\sqrt{2t}m}\right)^{1 \over 2} \, {\rm e}^{-m\sqrt{2t} + m\sqrt{2t^3}/48 +
\ldots} \, \big[1 + {\rm series\ in\ }t\big],
\ee
one eventually finds that the Green function can be written as
\be
G(m,m) = \Big\{D_0(m\sqrt{2t}) + t \, D_2(m\sqrt{2t}) + t^2 \, D_4(m\sqrt{2t})
+ \ldots \Big\}\,{\rm e}^{m\sqrt{2t^3}/48 + \ldots}\,,
\label{ser}
\ee
where all functions $D_i$ depend on the single scaling variable $m\sqrt{2t}$ (the
square root of 2 has to be included, since the distance from the origin is $\sqrt{2}
m$). Moreover, from (\ref{a14}), the first function $D_0$ is explicitely given as
\be
D_0(z) = {1 \over \sqrt{8\pi z}} \, {\rm e}^{-z} \,
\sum_{n=0}^\infty \; {(-1)^n \over n!} \, \left({(2n-1)!! \over 2^n}\right)^2 \,  {1
\over (2z)^n} = {1 \over 2\pi} \, K_0(z)\,,
\ee
a modified Bessel function. This is to be expected and confirms that the scaling
limit of the Green function is indeed equal to ${1 \over 2\pi} \, K_0(Mr)$, the
propagator of a massive scalar.

For calculations in the ASM model, one still needs the Green functions at points
close to the diagonal. The Poisson equation is not sufficient, because it would
require the knowledge of the Green function all the way down to the horizontal axis,
but a simple Ansatz similar to (\ref{ser}) leads to the following expressions, valid
for $0 \leq k \ll m$:
\bea
G(m,m+k) &=& \left\{ D_0(z) + k\,D'_0(z) \sqrt{t \over 2} + \left[D_2(z) + {k^2 \over
4} D_0(z)\right]t \; + \right. \nonumber\\
&& \hskip 2truecm\left. \left[{k \over 96}\,D_0(z) + {k^3 \over 8}\,D'_0(z) - {k^3
\over 12}\,D'''_0(z) +  {k \over 2}\,D_2'(z)\right] \, \sqrt{2t^3} + \ldots \right\}
\, {\rm e}^{zt/48+\ldots}\,,
\eea
where $z=m\sqrt{2t}$ is the scaled distance. At the order where all the calculations
have been performed, the terms shown in the previous expression are all that is
needed.

The critical limit of the above expansions is more conveniently computed from
(\ref{crit}) by using the asymptotic expansion of the $\psi$--function \cite{spol},
or from the integral (\ref{diag}). The result is
\be
\lim_{x \to 4} \; [G(m,m)-G(0,0)] = -{1 \over 2\pi} \,\log{m} - {1 \over \pi}\,
({\textstyle {\gamma \over 2}} + \log{2}) - {1 \over 48\,\pi \,m^2} + {7 \over
1920\,\pi \,m^4} - {31 \over 16128\,\pi \,m^6} + \ldots
\ee
with $\gamma = 0.57721\ldots$ the Euler constant.

\subsection{The Green function on the far principal axis}

The calculations can be repeated on a principal axis. 
The integration of (\ref{green}) over $k_2$ followed by the change of variable
$z={\rm e}^{ik_1}$ gives $G(m,0)$ as a contour integral over the unit circle. It
can again be deformed to encircle the branch cut joining the two roots $v < u$ of the
denominator which lie inside the unit circle, yielding
\be
G(m,0) = {1\over 4\pi} \; \int_0^{2\pi} \; {\rm d}k_1 \; {{\rm e}^{ik_1m} \over
\sqrt{\left({x \over 2}-\cos{k_1}\right)^2-1}} = 
{1 \over i\pi} \; \int_v^u \; {\rm d}z \; {z^m \over 
\sqrt{(z-u)(z-{1 \over u})(z-v)(z-{1 \over v})}} \,,
\label{crit2}
\ee
with $u<1$ and $v<1$ the two roots of $z^2-(x-2)z+1$ and $z^2-(x+2)z+1$
respectively, that is,
\be
u = {1 \over 2}[x-2-\sqrt{x(x-4)}], \qquad v = {1 \over 2}[x+2-\sqrt{x(x+4)}]\,.
\ee

The asymptotic behaviour of this integral for large $m$ can be found again in
\cite{coywu}, with the result
\be
G(m,0) = \sqrt{u \over 4\pi (1-u^2)} \, {u^m \over \sqrt{m}} \, \Big\{
1 + {A_{1>} \over 4m} + {3 \over 16}\,{A_{2>}-{5 \over 6} \over m^2} + 
{15 \over 64}\,{A_{3>}-{7 \over 6}A_{1>} \over m^3} + \ldots \Big\}\,.
\ee
The series within the curly brackets is similar to that of the previous subsection,
with however $m \over 2$ substituted for $m$, and with the coefficients $A_{n>}$ as
defined in \cite{coywu}, namely by (the coefficients $\tilde A_{n>}$ used above
correspond to the present $A_{n>}$ upon the identification $v=-u$)
\be
A_>(z) = {1 \over \sqrt{(1+{1+uv \over 1-uv}z)(1-{1+v/u \over 1-v/u}z)
(1+{1+u^2 \over 1-u^2}z)}} = \sum_{n=0}^\infty \; A_{n>} \, z^n\,,
\ee

The usual expansions around $x=4$ now yields
\be
G(m,0) = \Big\{P_0(m\sqrt{t}) + t \, P_2(m\sqrt{t}) + t^2 \, P_4(m\sqrt{t})
+ \ldots \Big\}\,{\rm e}^{m\sqrt{t^3}/24 + \ldots}\,,
\ee
with $P_0(z) = {1 \over 2\pi} K_0(z)$ as before.

For points close to the horizontal axis, one finds from Poisson equation the
expansions for $k \ll m$
\be
G(m,k) = \left\{ P_0(z) + \left[P_2(z) + {k^2 \over
2} \Big(P_0(z) - P_0''(z)\Big)\right]\,t + \ldots \right\} \, {\rm
e}^{zt/24+\ldots}\,,
\ee
with $z=m\sqrt{t}$ the scaled distance. The ASM calculations also need the values of
$G(m \pm \ell,k)$ for small $\ell$, and those can easily be obtained by expanding the
previous result, yielding a Taylor series in $\sqrt{t}$.

The critical asymptotic expansion of $G(m,0)$ can also be computed from
(\ref{crit2}). One has
\be
[G(m,0)-G(m+1,0)]\Big|_{x=4} = {1\over 4\pi} \; \int_0^{2\pi} \; {\rm d}k_1 \; 
{\rm e}^{ik_1m} F(k_1)\,,
\ee
where $F(x) = {1-{\rm e}^{ix} \over \sqrt{\left(2-\cos{x}\right)^2-1}}$. A repeated
use of integration by parts then leads to 
\be
[G(m,0)-G(m+1,0)]\Big|_{x=4} = -{1 \over 4\pi} \; \sum_{k \geq 1} \; 
{(-1)^k \over (im)^k} \; \Big[{\rm d}^k_x\,F\Big]_0^{2\pi} = {1 \over 2\pi}\,\left\{
{1 \over m} - {1 \over 2m^2} + {1 \over 2m^3} - {1 \over 2m^4} + \dots \right\}\,,
\ee
from which one deduces the subtracted Green function itself as
\be
\lim_{x \to 4} \; [G(m,0)-G(0,0)] = -{1 \over 2\pi} \,\log{m} - {1 \over \pi}\,
({\textstyle {\gamma \over 2}} + {\textstyle {3 \over 4}}\log{2}) + {1 \over 24\,\pi
\,m^2} + {43
\over 480\,\pi \,m^4} + {949 \over 2016\,\pi \,m^6} + \ldots
\ee

\section{About the sink site}
\label{sec:appb}

The evacuation of sand is a crucial ingredient to the self--organized criticality of
the sandpile models. In order for its dynamics to be well--defined ---any unstable
configuration relaxes to a stable one---, each site should be pathwise connected to
a sink, where goes the sand that falls off the pile. The sink is usually omitted in
all discussions, perhaps because in the ordinary ASM, only the boundary sites are
connected to the sink, and that the large volume limit takes them to infinity. In
the massive ASM however, each site is connected to the sink. One might thus worry
about a possible role of it in actual computations. 

We show here that the sink has in fact no effect at all and can be omitted
completely, be it in the usual or massive ASM. The argument is simple and
worth being made explicit.

The discrete dynamics of the ASM recalled in the Introduction uses a toppling matrix
that ignores the sink site s. To include it, one simply defines an extended toppling
matrix $\Delta_{\rm e}$ by adding to $\Delta$ a row and a column, 
\be
\Delta_{\rm e} = \pmatrix{1 & 0 \cr V & \Delta}.
\ee
The diagonal entry $(\Delta_{\rm e})_{\rm s,s}$ is set equal to 1, in order to
freeze the height of the sink site. The rest of the first row is equal to 0, since
the sink has no connection to the sites of the pile. On the other hand, the first
column is not zero: $V_{i,{\rm s}}=-n_i$ if $n_i$ grains of sand fall off the pile
when site $i$ topples. The number $n_i$ is equal to $n_i=-\sum_j \, \Delta_{i,j}$,
so that all row sums of $\Delta_{\rm e}$, except the first one, are zero. In the
usual ASM, $V_{i,{\rm s}}$ is non--zero for the boundary sites only, whereas in the
massive model, all components are non--zero, with $V_{i,{\rm s}}=4-x$ for all bulk
sites. The formula for the number of recurrent configurations remains valid, with
obviously the same result, $\det \Delta_{\rm e} = \det \Delta$. 

The same method for computing probabilities and correlations of weakly allowed
clusters works as before. One uses an extended $B$ matrix specifying the way the ASM
needs be modified
\be
B_{\rm e} = \pmatrix{0 & 0 \cr W & B}.
\ee
The first row is clearly always zero, but the first column can be non--zero,
depending on the modifications. For those called the ``least economical'' ones in
Section \ref{sec:sec2}, in which one cuts the cluster off the rest of the lattice,
each site $i$ of the cluster is left with a sole connection to the sink, so one
sets $W_i = 5-x$.

The probability of a cluster variable $S$ is given by the usual formula, which as
before reduces to a finite determinant
\be
\prob(S) = {\det (\Delta_{\rm e} + B_{\rm e}) \over \det \Delta_{\rm e}} = 
\det ({\Bbb I} + \Delta_{\rm e}^{-1}\,B_{\rm e}) = 
\det ({\Bbb I} + \Delta_{\rm e}^{-1}\,B_{\rm e})\Big|_{M_S \cup \{{\rm s}\}}.
\ee
However, the restriction to $M_S \cup \{{\rm s}\}$ of $\Delta_{\rm e}^{-1}\,B_{\rm
e}$ is particularly simple,
\be
\Delta_{\rm e}^{-1}\,B_{\rm e} = \pmatrix{1 & 0 \cr
-\Delta^{-1}\,V & \Delta^{-1}} \, \pmatrix{0 & 0 \cr W & B} = 
\pmatrix{0 & 0 \cr \Delta^{-1} \, W & \Delta^{-1} \, B},
\ee
and manifestly leads to the usual result, with no sink,
\be
\prob(S) = \det ({\Bbb I} + \Delta^{-1}\,B)\Big|_{M_S}.
\ee



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\end{document}



