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Anisotropic Heisenberg ferromagnet, XXZ model, rigidity 
of interfaces, interface excitations, 111 interface, equivalence  
of ensembles
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\begin{document}
{\baselineskip=10pt \thispagestyle{empty} {{\small Preprint UC Davis Math
1999-24}
%{\tt cond-mat/yymmnnnn} and {\tt mp\_arc 99-nnn} 
\hspace{\fill}}

\vspace{20pt}

\begin{center}
{\LARGE \bf Finite-volume excitations of the 111\\
interface in the quantum XXZ model\\[27pt]}
{\large \bf Oscar Bolina, Pierluigi Contucci, Bruno Nachtergaele 
and Shannon Starr\\[10pt]}
{\large  Department of Mathematics\\
University of California, Davis\\
Davis, CA 95616-8633, USA\\[15pt]}
{\normalsize bolina@math.ucdavis.edu, contucci@math.ucdavis.edu, 
bxn@math.ucdavis.edu, sstarr@math.ucdavis.edu}\\[30pt]
%(date)\\[30pt]
\end{center}

\noindent
{\bf Abstract}
We show that the ground states of the three-dimensional XXZ Heisenberg
ferromagnet with a 111 interface have excitations localized in a
subvolume of linear size $R$ with energies bounded by $O(1/R^2)$. As
part of the proof we show the equivalence of ensembles for the 111
interface states in the following sense: In the thermodynamic limit the
states with fixed magnetization yield the same expectation values for
gauge invariant local observables as a suitable grand canonical state
with fluctuating magnetization. Here, gauge invariant means commuting with
the total third component of the spin, which is a conserved quantity of
the Hamiltonian. As a corollary of equivalence of ensembles we also prove
the convergence of the thermodynamic limit of sequences of canonical
states (i.e., with fixed magnetization).
\vspace{8pt}
\newline 
{\small \bf Keywords:} Anisotropic Heisenberg ferromagnet, XXZ model,
rigidity
of interfaces, interface excitations, $111$ interface, equivalence of
ensembles.
\vskip .2 cm
\noindent
{\small \bf PACS numbers:} 05.30.Ch, 05.70.Nb, 05.50.+q 
\newline
{\small \bf MCS numbers:} 82B10, 82B24, 82D40 
\vfill
\hrule width2truein \smallskip {\baselineskip=10pt \noindent Copyright
\copyright\ 1999 by the authors. Reproduction of this article in its entirety, 
by any means, is permitted for non-commercial purposes.\par }}

\newpage

\Section{Introduction and main results}
\label{sec:intro}

A determining factor in the stability of the magnetic state of small 
ferromagnetic particles is the structure of the spectrum of their 
low-lying excitations. Stability against thermal (and quantum) fluctuations
is a major concern when one is interested in increasing the 
density of information stored on magnetic hard disks. Higher density of
information requires smaller magnetic particles to store the bits. The
smaller these particles get, the less stable their magnetic state tends to
be. It is also well-known that ferromagnets spontaneously form domains with 
different orientations of the magnetization. These two facts motivate us
to study the excitation spectrum of finite size ferromagnets with a domain
wall or {\it interface}. From examples, it is known that the presence of an 
interface, in general, has an effect on the low-lying excitation spectrum 
\cite{KN2,KN3}.

We consider the spin 1/2 XXZ Heisenberg model on the three-dimensional
lattice $\Ir^3$. For any finite volume $\Lambda\subset\Ir^3$, the 
Hamiltonian is given by 
\be
H_\Lambda = - \sum_{x,y\in\Lambda\atop \vert x-y\vert=1}
\Delta^{-1} (S_x^{(1)} S_y^{(1)} + S_x^{(2)} S_y^{(2)}) 
+ S_x^{(3)} S_y^{(3)},
\ee
where $\Delta>1$ is the anisotropy. It will be convenient to work with
the usual parametrization $\Delta=(q+q^{-1})/2$, $0<q<1$. Note that
in the limit $\Delta\to\infty$ ($q\to 0$), one recovers the Ising model.
The case $\Delta=1$ ($q=1$) is the XXX Heisenberg model. 

It is well-known that this model has two ferromagnetically ordered
translation invariant ground states. What is less well-known is that
there are also ground states describing an interface between two domains
with opposite magnetization. The 100 interfaces are similar to the
Dobrushin interfaces found in the Ising model. They exist for
sufficiently small temperatures, as was recently proved in \cite{BCF}.
Unlike the Ising model, the XXZ model also possesses ground states with a
rigid 111 interface at zero temperature \cite{KN2}. Its stability at
positive temperatures is still an open problem. 

In this paper we are interested in estimating the low-lying excitation\~s
above the ground state with a 111 interface. It is easy to show that the
excitation spectrum above the translation invariant ground states has a 
non-vanishing gap. In \cite{KN2} it was proved that, in the corresponding
two-dimensional model, the excitations above the 11 interface are gapless.
By an extension of the methods in \cite{LPW}, Matsui \cite{Mat2} showed
that the excitation spectrum has to be gapless in all dimensions $\geq 2$.
Here, we are interested in the nature of the low-lying excitations for the
three-dimensional model, and in particular their dependence on size.
We prove the following bound for the energy of an excitation localized in 
a finite domain $\Lambda_R$ of linear size $R$. \\

{\bf Main Result:}
{\em Excitations localized in $\Lambda_R$ have a gap $\gamma_R$ bounded by
\be
\gamma_R\leq 100\frac{q^{2(1-\delta(q,\nu))}}{(1-q^2)}\frac{1}{R^2},      
\quad \textrm{for}\quad R>70.
\label{main}
\ee
where $\delta(q,\nu)$ is an exponent between $0$ and $1/2$ that depends on
the filling factor $\nu$ of the interface plane (see explanation below),
as well as the parameter $q$.}\\

The meaning of this bound is the following. We consider the model in a finite
volume $\Lambda$, with a fixed magnetization and boundary conditions that
induce an interface.  By perturbing the ground state in a cylindrical
subvolume $\Lambda_R$, with circular cross-section of radius $R$, we then 
construct an orthogonal  state with the same magnetization. The bound 
\eq{main} is an upper bound for the difference in
energy of this state with respect to the ground state  in the limit $\Lambda
\nearrow \Ir^3$. For finite volumes $\Lambda$, the same bound holds as long as
$\Lambda$ is substantially larger than $R$.  When $R$ and the finite volume are
comparable in size, a similar bound holds but with a larger constant factor and
additional error terms (see Section 4).

The dependence on $q$ of the bound \eq{main} has some interesting features,
which we explain next. First, in the limit $q\to 1$, the bound diverges.
This means that our Ansatz for the excitations of the 111 interface does not
work for the isotropic model. This is not surprising as the isotropic model
does not have a rigid 111 interface, although it does possess gapless 
excitations, as is well-known from spinwave theory. In the limit $q\to 0$,
the Ising limit, the bound vanishes. This is to be expected, as the 111
interface contours of the Ising model are highly degenerate. 

In order to explain the role of the exponent $\delta(q,\nu)$ in \eq{main}
we first need to discuss some properties of the interface states 
themselves. For $0<q<1$,
the model has a two-parameter family of pure ground states with an interface
in the 111 direction. One parameter is an angle, playing the same role as
the angles $\phi_x$ in the Ansatz \eq{ansatz} for the excitations. The
second parameter, which is relevant for the present discussion, corresponds
to the mean position of the interface in the lattice. 
If we think of spin up at any site as describing an empty site, and spin down 
as a site occupied by a particle, the third component of the spin becomes
equivalent to the number of particles. In Section 2,
\eq{grand_canonical}, we will introduce the chemical potential $\mu$ 
to control the expected number of particles, alias the third component of 
the total spin. In the limit $q\to 0$, the filling factor $\nu$ of the 
interface has a simple interpretation: $\nu=0$ means that interface separates 
a region entirely filled with particles from a region that is empty. A non-zero $\nu$ means that there is a partially
filled plane in between the filled and the empty region, with filling factor
$\nu$. It turns out that the exponent $\delta(q,\nu)$, can be considered as  
a function of $\mu$ alone. For each value of $\mu\in\Rl$, we get an interface 
state, and $\delta$ is the distance of $\mu$ to the integers, i.e.,
$\delta(\mu)=\min(\vert\mu-\floor{\mu}\vert, \vert 1-\mu+\floor{\mu}\vert)$,
where $\floor{\mu}$ is the integer part of $\mu$. In general, the relation
between $\mu$ and $\nu$ depends nontrivially on $q$. But for all $q$,
$0<q<1$, one has $\delta(q,1/2)=0$ and $\delta(q,0)=1/2$. For further
details on the interdependence of the parameters $q,\delta,\mu$, and $\nu$,
we refer to Section 6.1.

We believe that $O(1/R^2)$ is the true behavior of the low-lying excitations.
There are indications in the physics literature that this should indeed be
the case \cite{HN}. Our rigorous bounds are obtained using the variational 
principle: If $\psi_0$ is a ground state of $H_\Lambda$, and $\psi$ is any 
other state that is linearly independent of $\psi_0$, then
\be
\gamma:=E_1-E_0\leq \frac{ \op{\psi}{H^{(q)}_{\Lambda}}{\psi} }{\|\psi\|^2}
	\cdot \frac{1}{1 
	- \frac{| \ip{\psi_0}{\psi} |^2}{\|\psi_0\|^2\|\psi\|^2}} \; .
\label{vp}
\ee
The first factor in the RHS is the energy of the perturbed state $\psi$.
The second factor is necessary to correct for the non-orthogonality of
$\psi$ and the ground state. In general, one would need to consider the
orthogonal complement of $\psi$ to the entire ground state subspace of
$H_\Lambda$. In the present case however, we know that for each eigenvalue
of the third component of the total spin, $J^{(3)}$, there is exactly one
ground state.  As we will only consider perturbations that commute with
$J^{(3)}$, it is sufficient to take the orthogonal complement of $\psi$ to
$\psi_0$.

Our ansatz for $\psi$ is of the following form
\be
\psi=\prod_{x\in\Lambda_R}e^{i2\phi_x S_x^{(3)}}\psi_0\quad .
\label{ansatz}
\ee
The energy of such a state can be written as follows
\be
\frac{\langle \psi \mid H_{\Lambda} \mid \psi \rangle}{\Vert\psi\Vert^2}
=\sum_{x\in\Lambda_R,y\in \Lambda\atop \vert x-y\vert=1}
P_{x,y}[1 - \cos(\phi_x - \phi_y)].
\label{ene_form}\ee
where the $P_{x,y}$ are probabilities determined by the interface ground
state. $P_{x,y}$ can be interpreted as the probability that the bond
$(x,y)$ belongs to ``the interface contour'', i.e., one of the sites is
occupied by an up spin and one by a down spin. These probabilities decay
exponentially fast as a function of the distance to the expected location
of the interface. In particular, this shows that the interface is rigid
and that the problem of calculating its excitation energies is quasi
two-dimensional. In fact, the next step in our proof makes this explicit. 
We consider excitations of the form \eq{ansatz} with
$$
\phi_x = \mathcal{S}\phi(\frac{x_\perp}{R}), \quad R\geq 1
$$
where $\mathcal{S}$ is a suitable scale factor, $\phi$ is a smooth function 
with compact support in $\Rl^2$, and $x_\perp$ is the component of 
$x\in\Ir^2$, orthogonal to the $111$ direction. It is shown that the energy 
$\gamma_R$ of  such excitations satisfies the bound
$$
\gamma_R\leq \frac{C(q)}{R^2}\frac{\Vert \nabla \phi\Vert_{L^2}^2}{\Vert\phi
\Vert^2_{L^2}}\quad .
$$
In principle, $\phi$ is a map from $\Rl^2$ to the circle, and as such could 
have nontrivial topology. As we will only be considering small perturbations,
this will be of no relevance here. It is, therefore, natural to take for $\phi$
an eigenfunction belonging  to the smallest eigenvalue of $-\Delta$ on a
circular domain with Dirichlet boundary conditions, which minimizes of the
Rayleigh quotient on the  RHS, i.e., the Bessel function $J_0$. This is
different from the so-called superinstanton Ansatz of Patrascioiu and Seiler in
\cite{PS}, where they use the fundamental solution of the Laplace equation,
instead of an eigenfunction.

All our results are for ground states that are eigenstates of the third
component of the total spin, which is a conserved quantity, and for 
thermodynamic limits of such states. We will call this {\em the canonical 
ensemble}. Our derivation, however, relies on an equivalence of ensembles 
result for the interface ground states of the XXZ model.  The state of the
``small'' volume $\Lambda_R$, immersed in the much larger  volume $\Lambda$, is
well approximated by a grand canonical state with  suitable chemical potential
(see Chapter 2 for the precise definitions), which does not have a fixed
magnetization. As expected, this equivalence of  ensembles holds only for
observables that commute with the third component of the total spin which are
analogous to the gauge invariant observables in  particle systems. This
equivalence of ensembles result is non-trivial. Although we only give the proof
in dimensions 3, it is straightforward to generalize the proof to all
dimensions $\geq 3$. Equivalence of ensembles (in the above sense) does not
hold for the one-dimensional model. This can be derived from the results in
\cite{GW}. In two dimensions, our method without modifications, yields the
equivalence of ensembles for volumes that  grow as $\sqrt{L}$ in the $11$
direction and as $L$ in the direction of the  interface. With additional work
one can obtain equivalence of ensembles result  for standard sequences of
increasing volumes.

As another application of equivalence of ensembles we prove the existence 
of the thermodynamic limit of sequences canonical ground states with a given
density, i.e., magnetization per site.

Concerning the gap above diagonal interface states in dimensions other than
three we can make the following comments. First of all, diagonal interface
states exist in all dimensions \cite{ASW}. In one dimension there is a spectral
gap above the ground states \cite{KN1}. In two dimensions an upper bound of
order $1/R$ was proved in \cite{KN2}. The method of this paper can be used to
obtain a bound of order $1/R^2$ also in two dimensions. In all dimensions
greater than three our method can be applied without change to obtain
equivalence of ensembles, the existence of the thermodynamic limit and an upper
bound of order $1/R^2$ for the  excitation energies.

The paper is organized as follows. Chapter 2 introduces the model and the
geometrical setting.  Chapter 3 deals with the equivalence of ensembles result
which is a main ingredient of our proofs. The bound on the excitation energy is
a product of two factors as in  \eq{vp}. A bound on the first factor, called
the {\em energy bound}, is  derived in Section 4. The second factor requires an
estimate for the inner  product of the ground state with the perturbed state,
which is derived in  Section 5. In Section 6 we prove a number of results
for the grand canonical ensemble in one dimension that we use 
in the paper. 

\Section{Interface states of the XXZ model}

Our magnet occupies a volume $\Lambda$ which is a subset of $\Ir^3$.
Let $e_1,e_2,e_3$ denote the standard basis vectors in $\Ir^3$.
(See Figure \ref{fig:cylinder}.)
\begin{figure}[t]
\begin{center}
\resizebox{!}{7truecm}{\includegraphics{mag5.eps}}
\parbox{14truecm}{\caption{\baselineskip=16 pt\small\label{fig:cylinder}
Example of a cylindrical $\Lambda$ embedded in $\Ir^3$.
A small cylindrical subvolume as used in the construction of the perturbed
states is also shown.}
}
\end{center}
\end{figure}
We let $l(x)$ denote the signed distance from the origin: 
$l(x) = x^1 + x^2 + x^3$, where $x = (x^1, x^2, x^3)\in\Ir^3$. 
Then
\be
B(\Lambda)=\{(x_0,x_1) :  | x_0 - x_1 | = 1, l(x_1)=l(x_0)+1\}
\label{set_bonds}\ee
describes the set of oriented bonds in $\Ir^3$. The infinite {\it stick} 
$\Sigma_0^\infty$ is, by definition, the set of vertices of the form
$$
\ldots -e_2-e_3,-e_3,0,e_1,e_1+e_2,e_1+e_2+e_3,e_1+e_2+e_3+e_1,\ldots
$$
For any even integer $L$, the finite stick $\Sigma_0$ of length $L+1$ is 
then given by
$$
\Sigma_0=\{x\in\Sigma_0^\infty\mid -L/2\leq l(x)\leq L/2\}\quad.
$$
We will take for $\Lambda$ is a cylindrical region whose axis points in the 
111 direction, where by {\it cylindrical\/} we mean that $\Lambda$ can be 
obtained from a subset $\Gamma$ of the $l(x)=0$ plane, which we will call the 
base, by adding to all vertices $x\in\Gamma$ the finite stick $\Sigma_0$: 
$$
\Lambda=\{x+y\mid x\in\Gamma, y\in\Sigma_0\}
$$
The equation $l(x)=c$, for any constant c, defines a 
cross-section of $\Lambda$, which contains exactly 
$A=\vert\Gamma\vert$ vertices. Hence, $\vert \Lambda\vert =(L+1)A$.
We refer to these cross-sections as planes.

As an example, the projection onto the plane $l(x)=0$, of the vertices of 
$\Lambda$ with triangular base 
is shown in Figure \ref{fig:trilat}, with different shades
depending on the value of $l(x)$ modulo 3.
The orientation of the bonds is indicated by arrows, and one may
observe that each site on the interior of $\Lambda$ 
has an equal number of incoming and outgoing bonds.
\begin{figure}
\begin{center}
\resizebox{!}{7truecm}{\includegraphics{trilat1.eps}}
\parbox{14truecm}{\caption{\baselineskip=16 pt\small
\label{fig:trilat}
The projection onto the $111$ plane of a cylindrical volume $\Lambda$ with 
triangular base. The shading of the vertices depends on the value of $l(x)$ 
modulo 3. The orientation of the bonds is indicated by arrows. Observe
that each site has an equal number of incoming and outgoing bonds.}
}
\end{center}
\end{figure}
By construction, $\Lambda$ can be decomposed into one-dimensional sticks 
running parallel to the cylindrical axis, which we will generically call
$\Sigma$. (See Figure \ref{fig:stick}.) One should observe that $\Sigma$ is
comprised entirely of nearest-neighbor pairs so that every site on $\Sigma$ is
connected to every other site by a sequence of bonds. This will allows us
to exploit the well-known properties of the one-dimensional Heisenberg XXZ 
model to describe $\Sigma$.
\begin{figure}
\begin{center}
\resizebox{!}{7truecm}{\includegraphics{mag10.eps}}
\parbox{14truecm}{\caption{\baselineskip=16 pt\small
\label{fig:stick}
The bonds connecting the vertices of a stick $\Sigma$ form a 
one-dimensional subsystem.}
}
\end{center}
\end{figure}
The Hamiltonian for the spin-$\frac{1}{2}$ ferromagnetic $XXZ$ Heisenberg
model is given by
\be\label{ham}
H_\Lambda = \sum_{(x_0,x_1)\in B(\Lambda)} h^{q}_{x_0,x_1},
\ee
where 
\be
h^{q}_{x_0,x_1} = - \Delta^{-1} (S_{x_0}^{(1)} S_{x_1}^{(1)}
+ S_{x_0}^{(2)} S_{x_1}^{(2)}) - S_{x_0}^{(3)} S_{x_1}^{(3)} 
+ \frac{1}{4} + \frac{1}{4} A(\Delta) (S_{x_1}^{(3)} - S_{x_0}^{(3)}).
\ee
and $\Delta \geq 1$ is the ``anisotropic coupling'', 
$A(\Delta) = {1 \over 2} \sqrt{1 - 1/\Delta^2}$, and
$q$, $0<q<1$, is the solution of $\Delta = {1 \over 2}(q + q^{-1})$
The matrices $S_x^{(\alpha)}$ ($\alpha = 1,2,3$) are the Pauli spin
matrices acting on the site $x$,
\be
S^{(1)} = \left[\begin{array}{cc} 0 & 1/2 \\
1/2 & 0 \end{array}\right],\quad
S^{(2)} = \left[\begin{array}{cc} 0 & -i/2 \\
i/2 & 0 \end{array}\right],\quad
S^{(3)} = \left[\begin{array}{cc} 1/2 & 0 \\
0 & -1/2 \end{array}\right].		
\ee
The terms containing $A(\Delta)$ cancel on all sites except at the 
top and bottom plane of the cylinder.
The usefulness of the nearest-neighbor Hamiltonian stems from the 
fact that its action on any bond is given by
\begin{eqnarray*}
h^q \dd = 0, && \quad
h^q \du = {1 \over {q + q^{-1}}} 
\left( q \du - \ud \right) ,\\
h^q \uu = 0, && \quad
h^q \ud = - {1 \over {q + q^{-1}}} 
\left( \du - q^{-1} \ud \right).
\end{eqnarray*}
In other words, $h^q$ is the orthogonal projection on the unit vector
\be
\xi_q = {1 \over \sqrt{1 + q^2}} (q \du - \ud). \label{def:xi}
\ee
There is a $(|\Lambda|+1)$-fold degeneracy in the ground states
with a unique ground state for each value of total third component
of the spin $\sum_{x\in\Lambda} S^{(3)}_x$. The basis vectors of the
Hilbert space $(\Cx^2)^{\otimes \vert \Lambda\vert}$ can be labeled with
particle configurations $\alpha =\{\alpha(x)\}_{x \in \Lambda}$, 
where $\alpha(x)$ is 0 or 1, corresponding to $\ket{\uparrow}$ and
$\ket{\downarrow}$, respectively. We write $\Num$ for the operator
defined by
$$
\Num\ket{\alpha} = (\sum_{x \in \Lambda} \alpha(x))\ket{\alpha},
$$
and let $\A(\Lambda,n)$ denote the collection of all 
configurations with $\Num(\alpha) = n$.

Following \cite{ASW} the ground states are given by
\be
\label{def:gs}
\psi_0(\Lambda,n) = \sum_{\alphavec \in \A(\Lambda,n)} 
\bigotimes_{x \in \Lambda} q^{l(x) \alpha(x)} \ket{\alpha(x)} ,
\ee
Note that the weights of $\alpha$ are invariant under 
any permutation of the sites for which planes are invariant. 
These states describe an interface located, on the average, in the
plane determined by $(L/2+l_x)A=n$ \cite{KN2}.

We denote $\Vert \psi_0(\Lambda,n)\Vert^2$ by $Z(\Lambda,n)$.
This quantity is given by
\be\label{uuyy}
Z(\Lambda,n) =\sum_{\alphavec \in \A(\Lambda,n)} \ 
\prod_{x \in \Lambda} q^{2 l(x) \alpha(x)} 
\ee
We will treat $Z(\Lambda,n)$ as a canonical partition function.
It will be useful to consider, also, its grand canonical analogue:
\be
\ZGC{\Lambda}{\mu} = \sum_{n=0}^L Z(\Lambda,n) q^{-2\mu n}
= \prod_{x \in \Lambda} (1 + q^{2 (l(x)-\mu)}) .
\label{grand_canonical}\ee
Then it is easily seen that $\ZGC{\Lambda}{\mu}$ is the squared-norm
of the grand canonical vector defined by
\be\label{def:gc}
\psi^{GC}(\Lambda,\mu) =\sum_{n=0}^{\vert \Lambda\vert}
q^{-n\mu}\psi_0(\Lambda,n)= \bigotimes_{x \in \Lambda}
(\up + q^{l(x)-\mu} \down).
\ee
Due to the product structure, the thermodynamic limit is simply given by
\be
\avgGC{X}{\Ir^3,\mu}=\bigotimes_{x \in \Ir^3} \frac{\bra{\uparrow} 
  + q^{l(x) - \mu} \bra{\downarrow}}{\sqrt{1 + q^{2(l(x)-\mu)}}}
 \;\; X \;\bigotimes_{x \in \Ir^3} \frac{\ket{\uparrow} 
  + q^{l(x) - \mu} \ket{\downarrow}}{\sqrt{1 + q^{2(l(x)-\mu)}}}
\label{product}\ee
for all local observables $X$.

\Section{Equivalence of Ensembles}
\label{sect:eq:ens}

A key step in our argument is the development of an equivalence of ensembles.
Specifically, we will show that for a gauge-invariant local observable
the canonical expectation is close to the grand canonical expectation
for some suitably chosen chemical potential $\mu$.
Here $\mu$ only depends on the total spin of the canonical ensemble, not on
the form of the observable.
From this, naturally follows a thermodynamic limit for gauge-invariant
observables. We begin with activity bounds that show that the ratio of
two canonical partition functions with different particle numbers is
approximately exponential in the difference of the particle numbers,
i.e.,
$$
Z(\Lambda,n-k)\approx Z(\Lambda,n) q^{-2k\mu}
$$
for $\vert k\vert \ll n$. More precisely, we have the following lemma.

{
\lemma[Activity bounds] 
For every volume ${\Lambda}$, 
$|{\Lambda}|=(L+1)A$, the ratio of canonical partition functions
for different number of particles can be bounded from above and 
below by {\rm activity} bounds as follows. Let $A_0$ be any constant.
%Recall the operator $\Num \ket{\alpha} = \ket{\alpha} \sum_x \alpha(x)$,
%and the 1D stick $\Sigma$.
Suppose $n$, $0\leq n \leq A(L+1)$, and $\mu$ are 
such that
\be\label{activ:hyp1}
n - A\navg \leq \frac{1}{2}A_0 A^{1/2}.
\ee
Then, for every $k$ satisfying
\be\label{activ:hyp2}
\vert k\vert\leq \frac{1}{2}A_0 A^{1/2},
\ee
one has the bounds
\be
\frac{Z(\Lambda,n)}{Z(\Lambda,n-k)} \; \le \;
C(A_0,A)
q^{k[2\frac{n}{A}-2\navg+2\mu a\sigma^2-\frac{k}{A}]/(a\sigma^2)},
\label{rrat}
\ee
and
\be
\frac{Z(\Lambda,n)}{Z(\Lambda,n-k)} \; \ge \;
C(A_0,A)^{-1}
q^{k[2\frac{n}{A}-2\navg+2\mu a\sigma^2-\frac{k}{A}]/(a \sigma^2)},
\label{lrat}
\ee
where $a = 2 |\ln q|$,
$$
\sigma^2:=\sigma^2(\mu,L)=\frac{1}{4}\sum_{l=-L/2}^{L/2} 
\frac{1}{\cosh^2(\frac{a}{2}(l-\mu))},
$$
and
\be
C(A_0,A) = \frac{1 + \frac{A_0}{\sigma^2 A^{1/2}}}
{1 - \frac{A_0}{\sigma^2 A^{1/2}}}.
\ee
Moreover, if $\mu$ is the solution of
$ \frac{n}{A}-\navg=0 $,
then, also using the bounds for $\sigma^2$ given in \eq{lower_sigma}, we
obtain
\be
\label{special:rat1}
C(A_0/2,A)^{-1} q^{-\frac{k^2(1-q^2)}{2 a (1+q^2) A}} \leq
q^{-2k\mu} \frac{Z(\Lambda,n)}{Z(\Lambda,n-k)} \leq 
C(A_0/2,A) q^{-\frac{2 k^2(1-q^2)}{a q^2 A}}.
\ee
Alternatively, if $\mu$ solves $\frac{n-k}{A} - \navg = 0$, then we obtain
\be
\label{special:rat2}
C(A_0/2,A)^{-1} q^{\frac{k^2(1-q^2)}{2 a (1+q^2) A}} \leq
q^{-2k\mu} \frac{Z(\Lambda,n)}{Z(\Lambda,n-k)} \leq 
C(A_0/2,A) q^{\frac{2 k^2(1-q^2)}{a q^2 A}}.
\ee
}
\begin{proof}
This can be obtained as follows. Let consider the grand canonical probability 
\be
p(\mu,{\bf n})=q^{-2\mu |{\bf n}|}\frac{Z({\bf n})}{Z_{GC}(\mu)} \, ;
\label{gcprob}
\ee
with
\be
Z({\bf n})=\sum_{\alpha:\A(\Sigma_1,n_1)\otimes\cdots\otimes
\A(\Sigma_{A-A_0},n_{A-A_0})}q^{w(\alpha)}
\label{cprob}
\ee
where $\Sigma_i$ is the i-th one dimensional stick that we are decomposing
our volume in, and where $Z_{GC}(\mu)$ is the grand-canonical partition
function. Clearly, we have
\be
Z(n) \, = \, \sum_{{\bf n}: |{\bf n}|=n} Z({\bf n}) \; .
\ee
Define
\be
p(\mu,n) \, = \, \sum_{{\bf n}: |{\bf n}|=n} p(\mu,{\bf n}) \; ,
\ee
and we have
\be
\frac{Z(n)}{Z(n-k)}=\frac{p(\mu,n)}{p(\mu,n-k)}q^{2k\mu}
\label{ratio}
\ee
The idea now is to make use of the {\it local central limit theorem}
for the probability distribution of the occupation number in the i-th
stick (see \cite{Fel} Theorem XVI.4.3.).
Let $\xi_i=\sum_{x\in\Sigma_i} \alpha_x$. For any integer $N$, consider,  
the probability
\be
P_\mu(\xi_1=n_1,...,\xi_N = n_N) \, = \, p(\mu,{\bf n}) \, .
\label{defpro}
\ee
Due to the factorization property of $p(\mu,{\bf n})$, the $\xi$'s are
independent identically distributed random variables. 
For centered i.i.d. random variables $X_i$ with variance $\sigma^2$,
the local central limit 
theorem guarantees that the random variable
\be
S_N =\frac{1}{\sigma\sqrt{N}}\sum_{n=1}^N X_n\quad.
\label{lclt}
\ee
is close to a Gaussian in the sense that the quantity
\be
P_N(x):=\Prob(\sum_{n=1}^N X_n=x)
\label{cumu}
\ee
fulfills the bounds
\be
\frac{1}{\sigma\sqrt{2\pi N}}e^{-\frac{x^2}{2\sigma^2 N}}
\left(1-\frac{c}{\sqrt{N}}\right)
\leq P_N(x)\leq\frac{1}{\sigma\sqrt{2\pi
N}}e^{-\frac{x^2}{2\sigma^2 N}}\left(1+\frac{c}{\sqrt{N}}\right)
\label{prob_bounds1}
\ee
where $c$ is the constant
\be
c = \frac{\max(|x|,|x-k|)}{\sigma^2 \sqrt{N}} .
\label{ci}
\ee
By applying \eq{prob_bounds1} to the centered quantity 
$X_n=\xi_n - \langle{\xi_n}\rangle$, we obtain the following bounds on the 
ratio of probabilities:
\be
C(N)^{-1}e^{-k(2x-k)/2\sigma^2 N}\leq
\frac{P_N(x)}{P_N(x-k)}\leq C(N)e^{-k(2x-k)/2\sigma^2 N}
\label{ragio}
\ee
where
\be
C(N)=\frac{1+cN^{-1/2}}{1-cN^{-1/2}}\quad .
\label{cicci}
\ee
In terms of the non-centered variables $\xi_i$ we have
\be
p(\mu,n) \; = \; P_A\left(n-A \navg \right)
\label{ncclt}
\ee
where $\navg$ is the average number of particles of a 1D stick $\Sigma$,
in the grand canonical ensemble with chemical potential $\mu$. 	
From this and the hypotheses \eq{activ:hyp1}, \eq{activ:hyp2}, we obtain
\be
c = \frac{A_0}{\sigma^2}
\quad \textrm{ and } \quad
C(A_0,A) = \frac{1 + \frac{A_0}{\sigma^2 A^{1/2}}}
{1 - \frac{A_0}{\sigma^2 A^{1/2}}}.
\ee
Note that in case $\mu$ is chosen so that $\navg = n/A$
or $\navg = (n-k)/A$ then we can replace $c$ by $c/2$,
with the result that $C(A_0,A)$ may be replaced by
$$
C(A_0/2,A) = \frac{1 + \frac{A_0}{2 \sigma^2 A^{1/2}}}
{1 - \frac{A_0}{2 \sigma^2 A^{1/2}}},
$$
as well.
\newline
Also, from \eq{ncclt} and \eq{ragio}, we have
\be
C(A_0,A)^{-1}e^{-\frac{k(2 n - 2 A \navg - k)}{2 \sigma^2 A}}
\leq \frac{p(\mu,n)}{p(\mu,n-k)} 
\leq C(A_0,A) e^{-\frac{k(2 n - 2 A \navg - k)}{2\sigma^2 A}}.
%\label{ragio}
\ee
Using \eq{ratio} (and observing that $q^{2\mu k} = e^{-a \mu}$), 
we have
\be
\frac{Z(n)}{Z(n-k)} \; \le \; 
  C(A_0,A) e^{-k[2\frac{n}{A}-2\navg+2a\sigma^2 \mu - \frac{k}{A}]/2 \sigma^2}\; ,
\label{rratio2}
\ee
and
\be
\frac{Z(n)}{Z(n-k)} \; \ge \; 
  C(A_0,A) e^{-k[2\frac{n}{A}-2\navg+2a\sigma^2 \mu - \frac{k}{A}]/2 \sigma^2}\; .
\label{lratio2}
\ee
Changing to base $q$ then leads to equations \eq{rrat} and \eq{lrat}
of the theorem.
By the derivation of Section \ref{sect:var_n}, we have the bounds
on the variance for the number of particles in a 1D stick:
\be
\label{var:bounds}
\frac{1}{4}\frac{q^2}{1-q^2}\leq \sigma^2(\mu) \leq \frac{1+q^2}{1-q^2}.
\ee
In conjunction with the remark about replacing $C(A_0,A)$ by 
$C(A_0/2,A)$, this gives equations \eq{special:rat1} and \eq{special:rat2}.
\end{proof}

As an application of this lemma, let us consider the case where
$n$ is replaced by $\rho |\Lambda| - n_0$, $k$ is replaced by 
$\rho |\Lambda_0| - n_0$ and $\Lambda$ is replaced by
$\Lambda_0^c := \Lambda \setminus \Lambda_0$.
This means that in the lemma $A$ is replaced by $A - A_0$,
and $(n-k)/A$ is replaced by 
$\rho (|\Lambda| - |\Lambda_0|)/(A-A_0) = \rho (L+1)$.
Then, direct substitution shows
\begin{eqnarray}
&&\frac{Z(\Lambda_0^c,\rho |\Lambda| - n_0)}{Z(\Lambda_0^c,\rho 
|\Lambda_0^c|)} \nonumber\\
&&\qquad \le C(A_0/2,A-A_0)\, q^{-2 k \mu}
e^{-k[2\rho(L+1) - 2 \navg + \frac{k}{A-A_0}]/2 \sigma^2} ,\\
&&\frac{Z(\Lambda_0^c,\rho |\Lambda| - n_0)}{Z(\Lambda_0^c,\rho 
|\Lambda_0^c|)}\; \nonumber\\
&&\qquad \ge C(A_0/2,A-A_0)^{-1}\, q^{-2 k \mu}
e^{-k[2\rho(L+1) -  2 \navg + \frac{k}{A-A_0}]/2 \sigma^2},
\end{eqnarray}
where we have retained $k$, for the moment.
If, further, we choose $\mu$ so that $\navg = \rho (L+1)$, which is
always possible (see Section \ref{sect:avgexp}), then, 
by equation \eq{special:rat2}, we have
\begin{eqnarray}
q^{2 \mu k} 
\frac{Z(\Lambda_0^c,\rho |\Lambda| - n_0)}
{Z(\Lambda_0^c,\rho |\Lambda_0^c|)}\; \le \; 
  C(A_0/2,A-A_0)\, e^{-\frac{k^2}{2 (A-A_0) \sigma^2}} ,\\
q^{2 \mu k} 
\frac{Z(\Lambda_0^c,\rho |\Lambda| - n_0)}
{Z(\Lambda_0^c,\rho |\Lambda_0^c|)}\; \ge \; 
  C(A_0/2,A-A_0)^{-1}\, e^{-\frac{k^2}{2 (A-A_0) \sigma^2}} .
\end{eqnarray}
Using our bounds for $\sigma^2$, we have
\begin{eqnarray}
\label{spef:rat1}
q^{2 \mu k} \frac{Z(\Lambda_0^c,\rho |\Lambda| - n_0)}
{Z(\Lambda_0^c,\rho |\Lambda_0^c|)}\; \le \; 
  C(A_0/2,A-A_0)\, e^{-\frac{(1-q^2) k^2}{2 (1+q^2) (A-A_0)}},\\
\label{spef:rat2}
q^{2 \mu k} \frac{Z(\Lambda_0^c,\rho |\Lambda| - n_0)}
{Z(\Lambda_0^c,\rho |\Lambda_0^c|)}\; \ge \; 
  C(A_0/2,A-A_0)^{-1}\, e^{-\frac{2(1-q^2)k^2}{2 q^2 (A-A_0)}} .
\end{eqnarray}
By our choice of $\mu$, conditions (\ref{activ:hyp1}) and (\ref{activ:hyp2})
are satisfied as long as the order of $L$ does not exceed the order of
$(A-A_0)^{1/2}$.
This estimate will be of use in the next theorem.

Let $\|X\|_{gs}$ denote the operator-norm of $X$ restricted to the 
subspace of ground states. For observables $X$, localized in $\Lambda$
and commuting with $J^{(3)}$, $\|X\|_{gs}$ is also given by
$$
\|X\|_{gs} = \sup_{0\leq n \leq |\Lambda|}
\vert\avg{X}{\Lambda,n}\vert .
$$ 

{\theorem[Equivalence of Ensembles]\label{thm:eqv}
Consider two cylindrical volumes $\Lambda$ and $\Lambda_0$, $\Lambda_0
\subset \Lambda$, of the type defined in Section 2 (in particular
$\vert\Lambda\vert=A (L+1)$, $\vert\Lambda_0\vert=A_0 (L+1)$), and fix a total
number of particles $n_\Lambda$. Define $\rho=n_\Lambda/\vert \Lambda\vert$.
Suppose $X$ is a local observable in the volume $\Lambda_0$, 
which commutes with $J^{(3)} := \sum_x S^{(3)}_x$. Then we have
\be
|\avg{X}{\Lambda,n}
  -  \avgGC{X}{\Lambda_0,\mu}| \; \le \; 
  \varepsilon \|X\|_{gs} \; ,
\label{equicgrc}
\ee
where 
\be
\label{vare}
\varepsilon
  = \frac{\ln^2(A-A_0) + 2 (1+a^2) A_0^2 + 4}{2(A-A_0)} 
  + \frac{4 A_0}{q^2 (A-A_0)^{1/2} - 2 A_0},
\ee
$a = 2 \vert\ln q\vert$,
and the chemical potential $\mu$ is determined by the equation
\be
 \navg = \rho (L+1).
\label{themu}\ee
In particular, for $\rho = 1/2$ the calculations of Section \ref{sect:navg}
will show that $\mu = 0$.
}

{\corollary[Existence of the Thermodynamic limit]\hfill\break
(i) Suppose we have a sequence of pairs $(\Lambda_k,n_k)$ 
with $\Lambda_k$ cylindrical volumes and $\Lambda_k \nearrow \Ir^3$ in such
a way that the length does not grow faster than the linear size of the 
base. Let $\mu_k$ solve $\avgGC{\Num}{\Lambda_k,\mu_k} = n_k$.
Then the convergence $\mu_k \to \mu$ guarantees the convergence,
of $\avg{.}{\Lambda_k,n_k}$ to $\avgGC{.}{\Ir^3,\mu}$, for all local 
observables $X$ commuting with $J^{(3)}$ :
\be \label{thermo:limit}
\avg{X}{\Lambda_k,n_k} \to \avgGC{X}{\Ir^3,\mu} 
\ee
\newline
(ii) Moreover, for any choice of $\mu$, we may find a sequence of pairs
$(\Lambda_k,n_k)$ such that 
\be \label{thermo:limit2}
\avg{X}{\Lambda_k,n_k} \to \avgGC{X}{\Ir^3,\mu} .
\ee
}

\begin{proof}
(Proof of Corollary)
It follows from the monotonicity of $\navg$ proved in Section \ref{sect:navg},
that the equation
\be
\avgGC{\Num}{\Lambda_k,\mu_k} = n_k
\label{mu_eq}\ee
always has a unique solution for $\mu_k$. Then, (i) follows immediately 
from the inequality \eq{equicgrc}, once we observe that $\epsilon \searrow 0$ 
as $\Lambda \nearrow \Ir^3$ in the sense prescribed in the corollary.
\newline
For (ii), take $\Lambda_k$, with base $A_k$, and $n_k$ such that
$$
n_k=\floor{A_k\navg}\quad .
$$
where $\floor{x}$ denotes the largest integer $\leq x$.
Then, $\mu_k$ solving \eq{mu_eq}, is easily seen to converge to $\mu$,
and \eq{thermo:limit2} follows from (i).
\end{proof}

The interpretation of the condition $\mu_k\to\mu$ in (i) of the Corollary
is that, not only does $n_k/\vert\Lambda_k\vert$ converge to $\rho=1/2$,
but, more precisely
$$
n_k=\rho\vert\Lambda_k\vert+\nu A_k + o(A_k)\quad.
$$
The term proportional to $\vert\Lambda_k\vert$ guarantees that the 
interface is in the center of the volume, the second term fixes its
filling factor.

\begin{proof}
(Proof of Theorem \ref{thm:eqv})
Let $\mu$ be determined by \eq{themu}, and define $\Xi$ as follows:
\be
\Xi=\frac{Z(\Lambda, n_\Lambda) q^{-2\mu \rho |\Lambda_0|}}{Z(\Lambda_0^c,
\rho \vert \Lambda_0^c\vert) Z^{GC}(\Lambda_0, \mu)}
\label{Xi}\ee
where $\Lambda_0^c:=\Lambda\setminus\Lambda_0$.
We will obtain the equivalence of ensembles by combining two facts.
The first is that $\Xi$ is approximately equal to $1$, and the second is
an estimate showing that 
$$
\vert \avg{X}{\Lambda,n_\Lambda}\Xi-\avgGC{X}{\Lambda_0,\mu}\vert 
  \leq \varepsilon\Vert X\Vert_{gs} 
$$
But first, let us recall the definitions of the 
expectation of an observable $X$:
\begin{eqnarray}
  \avg{X}{\Lambda,n} 
  &=& \frac{\op{\psi(\Lambda,n)}{X}{\psi(\Lambda,n)}}
  {\ip{\psi(\Lambda,n)}{\psi(\Lambda,n)}}, \\
  \avgGC{X}{\Lambda,\mu} 
  &=& \frac{\op{\psi^{GC}(\Lambda,\mu)}{X}{\psi^{GC}(\Lambda,\mu)}}
  {\ip{\psi^{GC}(\Lambda,\mu)}{\psi^{GC}(\Lambda,\mu)}}.
\end{eqnarray}
Since $X$ is an observable localized in $\Lambda_0$,
we note that $\avgGC{X}{\Lambda,\mu} = \avgGC{X}{\Lambda_0,\mu}$.
Moreover, we may decompose the grand canonical state into
a superposition of canonical states:
\be
\psi^{GC}(\Lambda_0,\mu) 
  = \sum_{n_0 = 0}^{|\Lambda_0|} q^{-\mu n_0} \psi(\Lambda_0,n_0).
\ee
Since $X$ commutes with $J^{(3)}$, it does not have off-diagonal
matrix elements between these canonical states with all different values
of the total spin. Therefore, 
\be
\avgGC{X}{\Lambda,\mu} 
  = \ZGC{\Lambda}{\mu}^{-1} \sum_{n_0 = 0}^{|\Lambda_0|} 
  q^{-2 \mu n_0} Z(\Lambda_0,n_0) \avg{X}{\Lambda_0,n_0}.
\ee
Note also, that since we have a decomposition
\be
\psi(\Lambda,n) = \sum_{n_0 = 0}^{|\Lambda_0|} 
  \psi(\Lambda \setminus \Lambda_0,n-n_0) \otimes 
  \psi(\Lambda_0,n_0),
\ee
and using the previously described properties, we have
\begin{eqnarray}
\avg{X}{\Lambda,n}
  &=& \sum_{n_0 = 0}^{|\Lambda_0|} 
  \frac{Z(\Lambda\setminus\Lambda_0,n-n_0)
  Z(\Lambda_0,n_0)}{Z(\Lambda,n)} \avg{X}{\Lambda_0,n_0} \\
  &=& Z^{GC}(\Lambda_0,\mu)^{-1} \sum_{n_0=0}^{|\Lambda_0|}
  q^{-2\mu n_0} Z(\Lambda_0,n_0)\avg{X}{\Lambda_0,n_0} \times \nonumber\\
  & & \quad \times \frac{Z(\Lambda_0^c,n-n_0) Z^{GC}(\Lambda_0,\mu)}
	{q^{-2\mu n_0} Z(\Lambda,n)}.
\end{eqnarray}
This differs from the definition of $\avgGC{X}{\Lambda_0,\mu}$ only
by the final factor, which is a ratio of partition functions hence amenable
to our activity bounds.
\newline
In fact, we have
\begin{eqnarray}
\avg{X}{\Lambda,n} \Xi - \avgGC{X}{\Lambda,\mu}
  &=& Z^{GC}(\Lambda_0,\mu)^{-1} \sum_{n_0=0}^{|\Lambda_0|}
    q^{-2\mu n_0} \avg{X}{\Lambda_0,n_0} Z(\Lambda_0,n_0) \times \nonumber\\
  & & \quad \times \left[q^{2\mu(n_0 - \avg{n_0}{})} \
    \frac{Z(\Lambda_0^c,n-n_0)}{Z(\Lambda_0^c,\floor{\rho |\Lambda_0|})} 
    - 1\right] 
\end{eqnarray}
where $\avg{n_0}{}=\avgGC{\Num}{\Lambda_0,\mu}$, which equals 
$\rho |\Lambda_0|$ for our choice of $\mu$. Thus we obtain
$|\avg{X}{\Lambda,n} \Xi - \avgGC{X}{\Lambda,\mu}|
  \leq \|X\|_{gs} \avgGC{|g|}{\Lambda_0,\mu},$
where
\be
  g = q^{2\mu(n_0 - \avg{n_0}{})} \
    \frac{Z(\Lambda_0^c,n-n_0)}{Z(\Lambda_0^c,\floor{\rho |\Lambda_0|})} 
    - 1 .
\ee
Now we use the activity bounds \eq{spef:rat1} and \eq{spef:rat2}, 
but replacing $k$ by its actual value, $\avg{n_0}{} - n_0$.
We arrive at the bounds
\begin{eqnarray}
  g &\leq& g_1 := 
  C(A_0/2,A-A_0) e^{-\frac{(1-q^2) (\avg{n_0}{} - n_0)^2}{2 (1+q^2) (A-A_0)}} - 1,\\
  g &\geq& g_2 := 
  C(A_0/2,A-A_0)^{-1} e^{-\frac{2(1-q^2)(\avg{n_0}{} - n_0)^2}{2 q^2 (A-A_0)}} - 1,
\end{eqnarray}
where
\be
C(A_0/2,A-A_0) = \frac{1 + \frac{A_0}{2 \sigma^2 (A-A_0)^{1/2}}}
{1 - \frac{A_0}{2 \sigma^2 (A-A_0)^{1/2}}}.
\ee
Therefore, $|g| \leq \max(|g_1|,|g_2|) \leq |g_1| + |g_2|$.
\newline
We now use the triangle inequality and the fact that the exponent is
negative to obtain:
\be
|g_1| \leq
  \left|1 - e^{-\frac{(1-q^2)(\avg{n_0}{} - n_0)^2}{2 (1+q^2) (A-A_0)}}\right| 
  + |1 - C(A_0/2,A-A_0)|,
\ee
so that
\be
\label{g1:bound1}
\avg{|g_1|}{\Lambda_0,\mu} \leq  
\avgGC{1 - e^{-\frac{(1-q^2) (\avg{n_0}{} - n_0)^2}{2 (1+q^2) (A-A_0)}}}
{\Lambda_0,\mu} + C(A_0/2,A-A_0) - 1.
\ee
Similarly,
\be
\avg{|g_2|}{\Lambda_0,\mu} \leq  
\avgGC{1 - e^{-\frac{2(1-q^2)(\avg{n_0}{} - n_0)^2}{2 q^2 (A-A_0)}}}
{\Lambda_0,\mu} + 1 - C(A_0/2,A-A_0)^{-1}.
\ee
\newline
We will use the Chebyshev inequality to control the expectation term in 
\eq{g1:bound1}. Specifically, for any $B>0$,
\begin{eqnarray*}
\avgGC{1 - e^{-\frac{(1-q^2) (\avg{n_0}{} - n_0)^2}{2 (1+q^2) (A-A_0)}}}
{\Lambda_0,\mu}
	&\leq& \Prob(2 |n_0 - \avg{n_0}{}| \geq 2 B) 
	+ 1 - e^{-\frac{(1-q^2)B^2}{2(1+q^2)(A-A_0)}} \\
	&\leq& q^{2 B} \avgGC{q^{-2 | n_0 - \avg{n_0}{} |}}{\Lambda_0,\mu}
	+ 1 - e^{-\frac{(1-q^2)B^2}{2(1+q^2)(A-A_0)}}.
\end{eqnarray*}
In Section \ref{sect:avgexp} we show that
$\avgGC{q^{-2 |n_0 - \avg{n_0}{}|}}{\Lambda_0,\mu} \leq 2 (2 q^{-2})^{A_0}$.
One choice for $B$ is $a^{-1} [\ln(A-A_0) + A_0 \ln(2 q^{-2})]$.
This gives the bound
\begin{eqnarray}
\avgGC{ 1 - q^{\frac{(n_0 - \avg{n_0}{})^2}{A-A_0}} }{\Lambda_0,\mu}
  &\leq& \frac{2 + \frac{1-q^2}{a^2 (1+q^2)} 
  \left[2 (1+a^2) A_0^2 + \ln^2(A-A_0)\right]}{A-A_0} \nonumber\\
  &\leq& \frac{2 + (1+a^2) A_0^2 + \frac{1}{2} \ln^2(A-A_0)}{A-A_0} \\
  &=:& C_1(A,A_0,q) \nonumber
\end{eqnarray}
The leading order term in the bound is  
$\frac{\ln^2(A-A_0)}{2(A-A_0)}$ for fixed $q$, strictly between 0 and 1.
Also, let 
\be
C_2(q,A,A_0) 
	= \frac{4 A_0}{q^2 (A-A_0)^{1/2} - 2 A_0},
\ee
which is greater than both $C(A_0/2,A-A_0)-1$ and $1 - C(A_0/2,A-A_0)^{-1}$.
Then 
$|\avg{f}{\Lambda,n} \Xi - \avgGC{f}{\Lambda,\mu}|
\leq (C_1 + C_2) \|X\|_{gs}$.
In particular,
$|\avg{\unity}{\Lambda,n} \Xi - \avgGC{\unity}{\Lambda,\mu}|
\leq (C_1 + C_2) \|\unity\|_{gs}$,
which is to say that
$|\Xi - 1| \leq C_1 + C_2$.
Then, using the triangle inequality, we have
\begin{eqnarray*}
|\avg{X}{\Lambda,n} - \avgGC{X}{\Lambda,\mu}|
  &\leq&  |1 - \Xi| \cdot |\avg{X}{\Lambda,n}|
	+ |\avg{X}{\Lambda,n} \Xi - \avgGC{X}{\Lambda,\mu}| \\
  &\leq&  2 (C_1 + C_2) \|X\|_{gs}.
\end{eqnarray*}
So, defining 
$\varepsilon = 2 C_1(q,\Lambda,\Lambda_0,n) + 2 C_2(q,\Lambda,\Lambda_0)$,
the theorem is proved. 
\end{proof}

Note that the restriction to observables $X$ that commute with the
third component of the total spin $J^{(3)}$ is necessary. E.g., the
expectation of $S^+_x$ obviously vanishes in any canonical state,
while it is easy to see, by direct computation, that it does not
vanish in the grand canonical states. This is entirely analogous to
the restriction to gauge invariant observables in particle systems.

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

\Section{Bound on the energy}

In this section we will estimate the energy of a class of perturbations
of the ground state $\psi_0$ given in \eq{def:gs}. Let $\Lambda$ and 
$\Lambda_R$ be two cylindrical volumes as described in Section 2, 
$\Lambda_R\subset\Lambda$. E.g., $\Lambda_R$ and $\Lambda$, may have triangular
cross-sections (see Figure \ref{fig:cylinder}). We will generally assume
that the radius $R$ of $\Lambda_R$ is much less than that of $\Lambda$. 
We consider $\psi$ of the form
\be
\label{ps}
\psi(\Lambda,n,\phi) = \sum_{\alphavec \in \A(\Lambda,n)} 
\bigotimes_{x \in \Lambda} e^{i \phi(x) \alpha(x)} 
q^{l(x) \alpha(x)} \ket{\alpha(x)} ,
\ee
where $\supp(\phi) \subset \Lambda_R$.

We will also suppose that 
\be
\phi = \frac{\Sc}{R} \tilde \phi(\tilde{y}_1,\tilde{y}_2)
\ee
where $\tilde\phi$ is a smooth functions of its variables
and $\Sc$ is a parameter, which we will eventually take to zero 
independent of $R$. The coordinates $\tilde{y}^1,tilde{y}^2$, are
defined by
\be
\tilde{y}^1 = \frac{2 x^1 - x^2 - x^3}{\sqrt{6} R}
\quad {\rm and} \quad \tilde{y}^2 = \frac{x^2 - x^3}{\sqrt{2} R},
\ee
and are  to be viewed as rescaled coordinates for $x$ along the plane 
perpendicular to the 111 axis.

There are two points to our assumptions on $\phi$: First, that $\phi$ is
independent of the 111 component of $x$. Second, that $\phi$ is 
associated to a scale-invariant phase $\tilde\phi$ by 
$\phi(x) = R^{-1} \tilde\phi(x/R)$. Ultimately, the constant $\Sc$ will vanish.
The leading term in our estimate of the gap  is independent of $\Sc$ 
as long as $\Sc \ll 1$. 

Let $\Gamma_R$ be the projection of $\Lambda_R$ onto the plane
$l(x)=0$, $A_R=\vert\Gamma_R\vert$, $\Omega_R$ be the convex hull of 
$\Gamma_R$, and $\tilde{\Omega} = \{x \in\Rl^2: R x \in \Omega_R\}$, 
the rescaled region, and let $m(\tilde{\Omega})$ be the area of 
$\tilde{\Omega}$ (for the standard Lebesgue measure on $\Rl^2$). 

We will also use the following notation: 
$\partial_{\tilde y}\tilde\phi$ and $\partial^2_{\tilde y}\tilde\phi$
are the first- and second-derivative tensors of $\tilde{\phi}$, and by the 
$L^\infty$ norm of a tensor we mean the maximum of the $L^\infty$ norms of 
the components.

Then we have the following theorem.

%%%%%%%% theorem : bound on the energy %%%%%%%%
%%%%%%%% theorem : bound on the energy %%%%%%%%
%%%%%%%% theorem : bound on the energy %%%%%%%%

{\theorem[Bound on $\frac{\op{\psi}{H^{(q)}_{\Lambda}}{\psi}
}{\|\psi\|^2}$] \label{energy:bound}
Considering a perturbed state as in (\ref{ps}), the energy is bounded by
\begin{eqnarray}
\label{bound:en}
\frac{\langle \psi \mid H^{(q)}_{\Lambda} \mid \psi
\rangle}{\|\psi\|^2} \leq
2\frac{1+q^2}{1-q^2}\left(\frac{A_R \Sc^2}{R^4} 
\frac{\|\nabla_{\tilde{y}} \tilde\phi\|_{L^2(\tilde\Omega)}^2}{m(\tilde\Omega)}
+ \mathcal{E}_{\textrm{num}}\right)
\end{eqnarray} 
where
\be
\mathcal{E}_{\textrm{num}} = \frac{6 A_R \Sc^2}{R^5} 
\|\partial_{\tilde y}^2 \tilde\phi\|_{L^\infty}
\|\partial_{\tilde y} \tilde\phi\|_{L^\infty}  
\ee
is a correction to the main term which becomes negligible as $R \to \infty$.
}

\begin{proof}
We begin by calculating how a two-site hamiltonian $h_b^q$ 
acts on the perturbed state. 

We consider the decomposition of our lattice into the relevant bond
$b = (x_0,x_1)$ and everything else $\Lambda \setminus b$.
Thus
\be
  h_b^q = \unity_{\Lambda \setminus b} \otimes \ket{\xi_b} \bra{\xi_b},
\ee
where $\xi_b$ is the unit vector from (\ref{def:xi}) on the 
pair $b$, and
\be
\psi(\Lambda,n)=\sum_{n_b=0}^2 \psi(\Lambda \setminus b,n-n_b) \otimes
\psi(b,n_b).
\ee
Here $\psi(b,n_b)$ is as would be defined by (\ref{ps}),
but with $\Lambda$ replaced by $b$ and $n$ replaced by $n_b$.
For example $\psi(b,1) = q^{l(x_0)} e^{i \phi(x_0)} \du 
+ q^{l(x_1)} e^{i \phi(x_1)} \ud$. But $\xi_b$ is orthogonal to 
$\psi(b,0)$ and $\psi(b,1)$, since $\xi_b$ lies in the sector of total spin 1.
And
\be
\ip{\xi_b}{\psi(b,1)} = \frac{1}{\sqrt{1+q^2}} 
q^{l(x_0)+1} e^{i \phi(x_0)} (1 - e^{i [\phi(x_1) - \phi(x_0)]}).
\ee
Now it is straightforward to see
\begin{eqnarray}
&&\op{\psi(\Lambda,n)}{h_b^q}{\psi(\Lambda,n)}\\
&&\quad =\|\psi(\Lambda \setminus b,n-1)\|^2\, |\ip{\xi_b}{\psi(b,1)}|^2 
\nonumber \\
&&\quad =\frac{2}{(q+q^{-1})^2}Z(\Lambda,n)P^{q}(b) (1 - \cos[\phi(x_1) - \phi(x_0)]),
\end{eqnarray}
where we have defined 
\be
P^q(b) = \frac{Z(\Lambda \setminus b,n-1)Z(b,1)}{Z(\Lambda,n)}.
\ee
Then we may write
\be\label{ene}
\frac{\langle \psi \mid H^{(q)}_{\Lambda} \mid \psi \rangle}{Z(\Lambda,n)}
= \frac{2}{(q+q^{-1})^2}~ \sum_{b \in B(\Lambda)} \ P^q(b)
(1 - \cos[\phi(x_1) - \phi(x_0)]).
\ee
Actually, $P^q(b)$ depends on $b$ only through $l(x_0)$.
So from here on, we'll write it as $P^q(l(x_0))$, and observe the following:
\be\label{pb}
\frac{\langle \psi \mid H^{(q)}_{\Lambda} \mid \psi \rangle}{Z(\Lambda,n)}
= \frac{2}{(q+q^{-1})^2} \sum_{l=-L/2}^{L/2-1} P^q(l) 
  \sum_{x \in \Gamma^l_R} \sum_{j=1}^3 
  (1 - \cos[\phi(x+e_j) - \phi(x)]) ,
\ee
where $\Gamma^l_R = \{x \in \Lambda_R : l(x) = l\}$.

Let us estimate the term
$\sum_{x \in \Gamma^l_R} \sum_{j=1}^3 (1 - \cos[\phi(x+e_j) - \phi(x)])$.
We have an inequality
\be
\label{cos:ineq}
 1 - \cos[\phi(x+e_j) - \phi(x)] \leq \frac{1}{2} [\phi(x+e_j) - \phi(x)]^2
\ee
(which is actually an  equality in the limit $R \to \infty$ for our ansatz).
Also,
\be
\label{diff:approx}
\sum_{i=1}^3 [\phi(x+e_j) - \phi(x)]^2 \approx |\nabla_x \phi(x)|^2 
	= \frac{\Sc^2}{R^4} |\nabla_{\tilde y} \tilde \phi|^2
\ee
In fact, using the inequality
\be
|[\tilde\phi(\tilde{y}+v) - \tilde\phi(\tilde y)]^2 - 
[c\cdot\nabla_{\tilde y} \tilde\phi(\tilde{y})]^2|
  \leq \| \partial_{\tilde y}^2 \tilde\phi \|_{L^\infty}
  \| \partial_{\tilde y} \tilde\phi\|_{L^\infty}
  \|v\|_{l^1}^3
\ee
one may conclude that the error in (\ref{diff:approx}) is bounded by 
$\frac{3 \Sc^2}{R^5} \|\partial_{\tilde y}^2 \tilde\phi\|_{L^\infty}
\|\partial_{\tilde y} \tilde\phi\|_{L^\infty}$.

Incorporating this estimate into the inequality of (\ref{cos:ineq}),
we have
\begin{eqnarray}
&& \sum_{x \in \Gamma^l_R} \sum_{j=1}^3 (1 - \cos[\phi(x+e_j) - \phi(x)]) 
  \leq \qquad \qquad \nonumber\\
&& \qquad \qquad 
\frac{1}{2 R^2} \sum_{x \in \Gamma^l_R} 
  |\nabla_{\tilde{y}} \phi(x)|^2
  + \frac{3 \Sc^2 |\Gamma^l_R|}{2 R^5} 
  \|\partial_{\tilde y}^2 \tilde\phi\|_{L^\infty}
  \|\partial_{\tilde y} \tilde\phi\|_{L^\infty} 
  \label{Energ:Ineq1} 
\end{eqnarray}
Finally, as $R \to \infty$, the sum over each $\Gamma^l_R$ becomes increasingly
well-approximated by the integral over $\Omega_R$,
we is proved in Lemma \ref{lem:voronoi} immediately following this proof.
The lemma gives us a bound
\be
\sum_{x \in \Gamma^l_R} 
|\nabla_{\tilde{y}} \phi(x)|^2 
  \leq \frac{\Sc^2 |\Gamma^l_R|}{R^2} \left[\frac{1}{m(\tilde{\Omega})} 
  \int_{\tilde{\Omega}} |\nabla_{\tilde{y}} \tilde\phi|^2\, d^2 y 
  + \frac{\rho}{R} 
  \|\nabla^2_{\tilde{y}} \tilde\phi \nabla_{\tilde{y}} \tilde\phi
	\|_{L^\infty(\tilde{\Omega})}\right] ,
  \label{Energ:Ineq2}
\ee
where $\nabla^2$ is the Laplacian and $\rho = \sqrt{2/3}$ is the
maximum radius for the Voronoi domain.
(Note that by its definition, as the trace of the second-derivative tensor,
the Laplacian enjoys the bounds
\be
\|\nabla^2_{\tilde{y}} \tilde\phi \nabla_{\tilde{y}} \tilde\phi
  \|_{L^\infty(\tilde{\Omega})} 
  \leq 2 \|\partial_{\tilde y}^2 \tilde\phi\|_{L^\infty}
  \|\partial_{\tilde y} \tilde\phi\|_{L^\infty}, 
\label{lasteq}\ee  
which may be combined with error term in (\ref{Energ:Ineq1}).)
Combining \eq{Energ:Ineq2} and \eq{lasteq} gives us the theorem, modulo
the term $\sum_{l=-L/2}^{L/2-1} P^q(l)$, for which we derive the necessary
in Lemma \ref{lem:Pq}.
\end{proof}

\begin{lemma}\label{lem:voronoi}
Suppose $\Gamma$ is a region in a regular lattice.
For each $x \in \Gamma$, let $\Omega_x$ be the Voronoi domain of $x$ with
respect to the whole lattice, and
let $\Omega_\Gamma$ be the union of all the individual domains $\Omega_x$.
If $f$ is a smooth function on $\Omega_\Gamma$, then
\be
\left| \frac{1}{|\Gamma|} \sum_{x \in \Gamma} f(x)
	- \frac{1}{m(\Omega_\Gamma)} \int_{\Omega_\Gamma} f(y)\, dy \right|
	\leq \rho \|\nabla_y f\|_{L^\infty(\Omega_\Gamma)}
\ee
where $\rho$ is the maximum radius of a Voronoi domain.
\end{lemma}

\begin{proof}
For each $x \in \Gamma$,
\begin{eqnarray*}
f(x) - \frac{1}{m(\Omega_x)} \int_{\Omega_x} f(y)\, dy 
  &\leq& -\frac{1}{m(\Omega_x)} \int_{\Omega_x} [f(y) - f(x)]\, dy \\
  &=& -\frac{1}{m(\Omega_x)} \int_{\Omega_x} \int_0^1 
    \frac{d}{dt} f(x + t (y-x))\, dt\, dy \\
  &=& -\frac{1}{m(\Omega_x)} \int_{\Omega_x} \int_0^1 
    \nabla_y f(x + t(y-x)) \cdot (y-x)\, dt\, dy .
\end{eqnarray*}
This clearly leads to the bound
\be
\left| f(x) - \frac{1}{m(\Omega_x)} \int_{\Omega_x} f(y)\, dy \right|
	\leq \rho(\Omega_x) \|\nabla_y f\|_{L^\infty(\Omega_x)}.
\ee
From this, the lemma follows easily.
\end{proof}

Now, we will derive the necessary bound on
$$
\sum_{l=-L/2}^{L/2-1} P^q(l)\quad .
$$
We will rely on bounds for similar quantities in the 
one-dimensional model proved in \cite{BCN}.

\begin{lemma}[Bound on $\sum_{l=-L/2}^{L/2-1} P^q(l)$]\label{lem:Pq}
\be
\sum_{l=-L/2}^{L/2-1} P^q(l) \leq  2\frac{1+q^2}{1 - q^2} .
\ee
\end{lemma}

\begin{proof}
Recall
\be
P^q(l)=\frac{Z(\Lambda \setminus b,n-1) Z(b,1)}
{Z(\Lambda,n)}.
\ee
The ratio of partition functions in the equation above is clear:
It is the probability of finding one particle shared by the 
sites of $b$, and $n-1$ particles shared by the sites of 
$\Lambda \setminus b$, conditioned on finding $n$ total particles on 
$\Lambda$. We consider the operator
$$Y_b = \unity_{\Lambda \setminus b} \otimes 
  \left(\ket{\uparrow \downarrow}_b \bra{\uparrow \downarrow}_b
  + \ket{\downarrow \uparrow}_b \bra{\downarrow \uparrow}_b\right).$$
Then
\be
\frac{Z(\Lambda \setminus b,n-1) Z(b,1)}{Z(\Lambda,n)}
	= \avg{Y_b}{\Lambda,n},
\ee
and
\be
\sum_{l=-L/2}^{L/2-1} P^q(l)
  =\left\langle\sum_{l=-L/2}^{L/2-1} Y_{b(l)}
  \right\rangle_{\Lambda,n}.
\label{PtoY}\ee
where $b(l)= (x_0,x_1)$, where $l(x_0)=l$, and $(x_0,x_1)$ is a bond in the
stick containing the origin, which we denote by $\Sigma_0$.
The restriction of the state in $\Lambda$ with $n$ spins down is of the
form
$$
\langle X\rangle_{\Sigma_0}=\sum_{k=0}^{L+1} c_k \avg{X}{\Sigma_0,k}
$$
where $X$ is any observable commuting with $J^{(3)}=\sum_{x\in\Sigma_0}
S^{(3)}_x$, as is, e.g., $Y_{b(l)}$, and the $c_k$ are non-negative
numbers summing up to one. We will now derive an upper bound for 
$\avg{\sum_{l=-L/2}^{L/2-1} Y_l}{\Sigma_0}$, that is independent of the
coefficients $c_k$.
We start from
\be
\avg{Y_l}{\Sigma_0,k}
\leq \Prob_k( S^{(3)}_l=\uparrow, S^{(3)}_{l+1}=\downarrow)
+\Prob_k( S^{(3)}_l=\downarrow, S^{(3)}_{l+1}=\uparrow)
\label{oneDprob}\ee
where $\Prob_k$ denotes the probability in the ground state with $k$
spins down for a one-dimensional system on $[-L/2,L/2]$, the sites of which
we label by $l$. Each term in the RHS of \eq{oneDprob} can be estimate as
follows.
\be
\Prob_k( S^{(3)}_l=\uparrow, S^{(3)}_{l+1}=\downarrow)
\leq
\min\left(\Prob_k(S^{(3)}_l=\uparrow),\Prob_k(S^{(3)}_{l+1}=\downarrow)\right)
\ee
Theorem 7.1 of \cite{BCN} gives the following bounds
\beann
\Prob_k(S^{(3)}_{l+1}=\downarrow)\leq q^{2(l-(k+1-L/2)}
&\mbox{if}& l\geq k+1-L/2\\
\Prob_k(S^{(3)}_l=\uparrow)\leq q^{2(k+1-L/2-l)}
&\mbox{if}& l < k+1-L/2
\eeann
Combining these inequalities and summing over $l$ yields
\be
\sum_{l=-L/2}^{L/2-1} \avg{Y_l}{\Sigma_0,k}\leq 2\frac{1+q^2}{1-q^2}
\ee
for all $k=0,\ldots,L+1$. Together with \eq{PtoY} this concludes the proof.
\end{proof}

%%%%%%%%\Section{Bound for the denominator}%%%%%%%%
%%%%%%%%\Section{Bound for the denominator}%%%%%%%%
%%%%%%%%\Section{Bound for the denominator}%%%%%%%%
\Section{Bound for the denominator}

Note that $\psi(\Lambda,n)=T(\phi)\psi_0(\Lambda,n)$, where $T(\phi)$ is 
the unitary operator defined by,
\be
T(\phi) = \bigotimes_{x \in \Lambda} (\ket{\uparrow}\bra{\uparrow}
  + e^{i \phi(x)} \ket{\downarrow} \bra{\downarrow}).
\ee
In particular, $\Vert T(\phi)\psi_0(\Lambda,n)\Vert^2=
\Vert\psi(\Lambda,n)\Vert^2=Z(\Lambda,n)$. For convenience, we will
sometimes omit the arguments $\Lambda$ and $n$ from the notation.
In this section we will consider the half-filled system, i.e,
$\rho=n/\vert\Lambda\vert=1/2$. This corresponds to $\mu=0$. 

\begin{theorem}[Bound on $|\frac{<\psi_0|\psi>}{<\psi_0|\psi_0>}|$ ]
\label{den:bounds}
Considering a perturbed state in the volume $\Lambda_0$ defined by
(\ref{ps}) we have that canonical and grand-canonical expectations
of the perturbed state are arbitrarily close for large volumes $\Lambda$
in the sense:
\be
\label{gcbl}
\left\vert\frac{\ip{\psi}{\psi_0}}{\ip{\psi_0}{\psi_0}}
  - \avgGC{T(\phi)}{\Lambda,\mu} \right\vert
  \le \frac{\ln^2(A-A_0) + 2 (1+a^2) A_0^2 + 4}{2(A-A_0)} 
  + \frac{4 A_0}{q^2 A^{1/2} - 2 A_0}.
\ee
Moreover, with the ansatz defined by (\ref{ps}), the grand canonical 
expectation is bounded as
\begin{eqnarray}
\label{deb}
&&\ln \left|\avgGC{T(\phi)}{\Lambda,\mu}\right|^2
  \leq \qquad \qquad \\
&& \qquad \qquad \leq - q^{2 \delta(\mu)} \frac{A_R \Sc^2}{4R^2} 
  \Bigg[\frac{\|\tilde\phi\|_{L^2(\tilde\Omega)}^2}{m(\tilde\Omega)} 
  - \frac{\sqrt{6}}{R} \|\partial_{\tilde{y}} \tilde\phi \|_{L^\infty} 
    \|\tilde\phi\|_{L^\infty}
  - \frac{\Sc^2}{12 R^2} \|\tilde\phi\|_{L^\infty}^4 
\Bigg] \nonumber
\end{eqnarray}
where $\delta(\mu)$ is the distance of $\mu$ from its closest integer
neighbor.
(Recall that we have defined the $L^\infty$-norm of a tensor 
to be the $L^\infty$-norm of its maximum component.)
\end{theorem}

\begin{proof}
The proof of equation (\ref{gcbl}) is a direct consequence of the 
equivalence of ensembles because, since $T(\phi)$ is a unitary operator,
$\|T(\phi)\| = 1$. Let us now consider the proof of equation (\ref{deb}).

We wish to bound the denominator from below;
i.e.\ to demonstrate that $1 - |\avg{T(\phi)}{\Lambda,n}|^2$ is not too small.
This is tantamount to showing that $|\avg{T(\phi)}{\Lambda,n}|^2$
is not too close to 1.
Furthermore, we know this quantity 
lies between 0 and 1.
We estimate the actual canonical average with the grand canonical average,
and take the logarithm in order to exploit the factorization properties
of the grand canonical ensemble.
First, we note
\be\label{rat}
\left|\avgGC{T(\phi)}{\Lambda,\mu}\right|
  = \left| \prod_{x \in \Lambda_0} \frac{1+e^{i \phi(x)} q^{2 (l(x)-\mu)}} 
  {1+q^{2 (l(x)-\mu)}} \right|.
\ee
Recall the definition $a=-2 \ln{q}$. 
This allows us a more convenient form
in place of (\ref{rat})
\begin{eqnarray}
\label{uu}
&&\left| \prod_{x \in \Lambda_0} \frac{1+e^{i \phi(x)} q^{2 (l(x)-\mu)}}
  {1+q^{2 (l(x)-\mu)}} \right|^2 \nonumber\\
  &&\quad = \prod_{x \in \Lambda_0} 
  \frac{e^{2 a (l(x)-\mu)} + 2 \cos \phi(x) e^{a (l(x)-\mu)} + 1}
  {e^{2 a (l(x)-\mu)} + 2 e^{a (l(x)-\mu)} + 1} \\
  &&\quad = \prod_{x \in \Lambda_0} 
  \left(1 - \frac{1}{2}(1-\tanh^2[a (l(x)-\mu)/2])
	(1 - \cos \phi(x))\right).
\nonumber
\end{eqnarray}
We partition the product over planes and estimate the logarithm, thus:
\begin{eqnarray*}
\ln \left|\avgGC{T(\phi)}{\Lambda,\mu}\right|^2
  &=& \ln \left( \prod_{x\in \Lambda_0} 1 - \frac{1}{2}
  (1-\tanh^{2}[a (l(x)-\mu)/2])(1 - \cos \phi(x)) \right ) \\
  &\leq& - \frac{1}{2} \sum_{x \in \Lambda-0} (1 - \tanh^2[a (l(x)-\mu)/2]) 
	(1 - \cos \phi(x)) \\
  &=& - \frac{1}{2} \sum_{l=-L/2}^{L/2} (1 - \tanh^2[a (l-\mu)/2])
	\sum_{x \in \Gamma^l_R} (1 - \cos \phi(x)) .
\end{eqnarray*}
We may approximate $1 - \cos(\phi(x))$
by $\frac{1}{2} \phi(x)^2$, with an error no larger than  
$\frac{1}{24} \|\phi\|_{L^\infty}^4 $
which is the same as $ \frac{\Sc^4}{24 R^4} \|\tilde\phi\|_{L^\infty}^4$.
In this case
\be
\label{sumsum}
\ln \left|\frac{Z_{GC}(\Lambda_0,\mu,\phi)}{Z_{GC}(\Lambda_0,\mu,0)}\right|^2
\leq - \frac{1}{2} \sum_{l=-L/2}^{L/2} (1 - \tanh^2 [a (l-\mu)/2])
  \left[\sum_{x \in \Gamma^l_R} \frac{1}{2} \phi_x^2 
  - \frac{\Sc^4 |\Gamma^l_R|}{24 R^4} \|\tilde\phi\|_\infty^4 \right] .
\ee
We may approximate the sum over $\Gamma^l_R$ with an integral such that 
the error is bounded by 
$\frac{\rho \Sc^2 |\Gamma^l_R|}{R^3} 
\|\nabla_{\tilde{y}} \phi \|_{L^\infty} \|\tilde\phi\|_{L^\infty}$.
We may bound the sum 
$\sum_{l=-L/2}^{L/2} (1 - \tanh^2[a (l-\mu)/2])$
from below by its largest term (since all the terms are positive).
The largest term occurs for that integer $l$ which is closest to $\mu$.
Thus, defining 
$\delta(\mu) = \min(\mu - \lfloor{\mu}\rfloor,\lceil{\mu}\rceil - \mu)$, 
we see
\be
\sum_{l=-L/2}^{L/2} (1 - \tanh^2[a(l-\mu)/2])  
  \geq 1 - \tanh^2[a \delta(\mu)/2] 
  = \frac{4}{(q^{\delta(\mu)} + q^{-\delta(\mu)})^2} 
  \geq q^{2\delta(\mu)},
\ee
Using these bounds, we may continue the estimate of $\eq{sumsum}$.
We arrive at
\begin{eqnarray}
\label{den:exp:bound}
&&\ln \left|\avgGC{T(\phi)}{\Lambda,\mu}\right|^2
  \leq \qquad \qquad \\
&& \qquad \qquad \leq - q^{2 \delta(\mu)}
  \frac{\Sc^2 |\Gamma^l_R| }{4R^2} 
  \Bigg[\frac{\|\tilde\phi\|_{L^2(\tilde\Omega)}^2}{m(\tilde\Omega)} 
  - \frac{\rho}{R} \|\nabla_{\tilde{y}} \tilde\phi \|_{L^\infty} 
    \|\tilde\phi\|_{L^\infty}
  - \frac{\Sc^2}{12 R^2} \|\tilde\phi\|_{L^\infty}^4 
\Bigg] \nonumber
\end{eqnarray}
Since $|\nabla_{\tilde y}\tilde\phi| 
\leq 2 \|\partial_{\tilde y}\tilde\phi\|_{l^\infty}$
and since $\rho = \sqrt{3/2}$, we have equation \eq{deb}.
\end{proof}

%%%%%%%%\subsection{Bound on the Ratio}%%%%%%%%
%%%%%%%%\subsection{Bound on the Ratio}%%%%%%%%
%%%%%%%%\subsection{Bound on the Ratio}%%%%%%%%
\subsection{Bound on the Ratio}
We will now combine the results of the bound on the numerator and the bound 
on the denominator to get a true bound on the spectral gap.
We first allow $\Lambda \nearrow \Ir^3$ in the appropriate fashion so that
$\varepsilon \searrow 0$.
Then we consider the case that $S\to 0$, holding $R$ fixed.
This means that we consider a perturbation to the ground state which is 
very small.
But since the ground state has energy zero, the energy of the 
perturbed state is entirely due to the small perturbation.
In fact it is proportional to the size of the perturbation, and from this we 
obtain a linearized (with respect to amplitude of $\phi$) bound: 
In fact we have, combining \eq{vp}, \eq{bound:en}, and \eq{gcbl}
\be
\label{rb}
\gamma_1 \leq \frac{16 q^{2(1-\delta(\mu))}}{(1-q^2)R^2} \cdot
  \frac{\|\nabla_{\tilde{y}}\tilde\phi\|_{L^2(\tilde\Omega)}^2/m(\tilde\Omega) 
  + \frac{6}{R} \|\partial_{\tilde y}^2\tilde\phi\|_\infty 
  \|\partial_{\tilde y}\tilde\phi\|_\infty}
  {\|\tilde\phi\|_{L^2(\tilde\Omega)}^2/m(\tilde\Omega) - \frac{\sqrt{6}}{R} 
  \|\partial_{\tilde y}\tilde\phi\|_\infty
  \|\tilde\phi\|_\infty}
\ee
Note that this bound is homogeneous with respect to the amplitude of $\phi$,
which is the result of our linearization.
We observe that, whatever the form for $\tilde\phi$, as long as it is smooth
we have the same asymptotic behavior for the bound on the spectral gap.
Namely $\gamma_1 = O(1/R^2)$.
This said, it is certainly worthwhile to find a best bound, which we 
take up presently.

%%%%%%%%\subsection{The Bessel Function Ansatz}%%%%%%%%
%%%%%%%%\subsection{The Bessel Function Ansatz}%%%%%%%%
%%%%%%%%\subsection{The Bessel Function Ansatz}%%%%%%%%

\subsection{The Bessel Function Ansatz}
Let us write the leading-order term in the bound for the spectral gap:
\be
E(\tilde\phi) 
  = \frac{\|\nabla_{\tilde{y}}\tilde\phi\|_2^2}{\|\tilde\phi\|_2^2}.
\ee
In order to minimize the bound on the spectral gap, we will minimize
the functional $E(\phi)$ amongst all functions $\phi$ which possess two
continuous derivatives and which vanish on the boundary of 
the rescaled perturbed region $\tilde\Omega$.
(In order that the ``small'' phase $\phi$ match the external phase 
of $0,\pm2 \pi,\dots$ on $\partial\Omega$, it must be zero there.
Thus $\tilde\phi \equiv 0$ on $\partial\tilde\Omega$.)
Therefore, we consider the first variation
\be
\lim_{\tau \to 0} \frac{1}{\tau}[E(\phi+\tau \phi') - E(\phi)] 
  = \frac{2 \int \nabla\phi \cdot \nabla\phi'}{\int \phi^2}
  - \frac{2 \int \phi \phi' \int |\nabla \phi|^2}{\int\phi^2 \int\phi^2}.
\ee
Setting the first variation to zero for all test functions 
$\phi'$ leads to the eigenvalue problem for Laplace's equation
\be
\label{pde:1}
\left\{ \begin{array}{ll}
  -\nabla^2 \tilde\phi = \lambda \tilde\phi 
  & \textrm{ in } \tilde\Omega, \\
  \tilde\phi = 0 & \textrm{ on } \partial\tilde\Omega,
  \end{array} \right.
\ee
where $\lambda = E(\phi)$.

We choose, for our domain, the unit disk.
We seek the solution to equation (\ref{pde:1}) which minimizes $\lambda$,
but with the restriction that $\phi$ must possess two continuous derivatives.
So the fundamental solution, which is the logarithm,  is disallowed
(and, in fact, has higher energy).
We seek the first eigenstate of the Laplacian above the ground state.
This is a classic problem, found in any elementary PDE text, 
with the Bessel Function for the solution:
$$\tilde\phi(\tilde y) = J_0(z_0 r) ,$$
where $r = |\tilde y|$, $J_0$ is the zeroth Bessel function,
and $z_0 \approx 2.406$ is its first zero.
Now, using this choice for $\phi$ and the bounds \eq{rb}, we obtain
\be
\gamma_1 \leq \frac{16 q^{2(1-\delta(\mu))}}{(1-q^2)R^2} \cdot
  \frac{1.56 + \frac{6}{R} (2.90)(1.40)}
  {0.27 - \frac{\sqrt{6}}{R} (1.40)(1)}.
\ee
Thus,
\be
\gamma_1 \leq \frac{100 q^{2(1-\delta(\mu))}}{(1-q^2)R^2} 
\quad \textrm{for}
\quad R>70.
\ee

\Section{Results from the 1D grand canonical ensemble}

\subsection{The mean number of particles in a stick}
\label{sect:navg}
Recall that $\Sigma$ is a 1D stick running parallel to the 111 axis.
So, it is actually a 1D spin chain.
We wish to estimate the mean number of particles in
$\Sigma$, for the grand canonical ensemble.
This is
\begin{eqnarray}
  \navg := \ZGC{\Sigma}{\mu}^{-1}
	\sum_{n=1}^{L+1} n q^{-2 \mu n} Z(\Sigma,n)\\ 
	= \ZGC{\Sigma}{\mu}^{-1}
	\sum_{n=1}^{L+1} n e^{a \mu n} Z(\Sigma,n) .\nonumber
\end{eqnarray}
where $\Sigma$ is the interval 
$\{-\frac{L}{2},-\frac{L}{2}+1,\dots,\frac{L}{2}\}$.
(Recall $a = - 2 \log q$.)
By a standard calculation, we have
\be
\navg = \frac{1}{a} \frac{\partial}{{\partial \mu}} 
\log \ZGC{\Sigma}{\mu} .
\ee
On the other hand, the grand canonical partition function factorizes, 
as we have seen, so that
\be 
  \navg = \sum_{l=-L/2}^{L/2} 
	{e^{a(\mu - l)} \over {1 + e^{a(\mu - l)}}}
	= \sum_{l=-L/2}^{L/2} \frac{1}{2} 
	\left[1 - \tanh\left(\frac{a}{2} (l - \mu)\right)\right].
\ee
An examination of the graph of the function $x \mapsto 1 - \tanh(x)$
reveals an approximate heaviside function, with support on the negative axis.
We define the function 
\be
\eta(x) = \left\{ 
  \begin{array}{ll}
  1 & x < 0,\\
  1/2 & x = 0,\\
  0 & x > 0.\end{array} \right. 
\ee
Then, as long as $-L/2 \leq \mu \leq L/2$, we remark
\be
\navg = 
  \left\{ \begin{array}{ll}
  \floor{\mu} + \frac{L}{2} & \mu \not\in \Ir,\\
  \mu + \frac{L+1}{2} & \mu \in \Ir\end{array} \right\} 
  +\sum_{l=-L/2}^{L/2} \left({1 \over 2} 
  - {1 \over 2} \tanh\left({a \over 2}(l - \mu)\right) - \eta(l - \mu)\right).
\ee
We make the definition
\be
F_L(\mu) = \navg - \left(\mu + \frac{L+1}{2}\right) 
\ee
For $\mu$ in the range above one may determine (by combining the two tails
in the series and estimating upwards by an integral) that
\be 
|F_\infty(\mu) - F_L(\mu)| 
  \leq \frac{1}{a} \ln\left(\frac{1 + \exp(-\frac{a}{2}(\frac{L}{2} - \mu))}
  {1 + \exp(-\frac{a}{2}(\frac{L}{2} + \mu))}\right)
\ee
Notice that in case $\mu = 0$, there is no error at all in estimating 
$F_L$ by $F_\infty$,
and, furthermore, $F_\infty(0)=0$.
It is clear that $F_\infty(\mu)$ is periodic in $\mu$ with period 1, 
because it is a sum over the entire integer lattice, so it will suffice for 
us to consider $\mu$ in the range $]0,1[$.
A straightforward calculation then yields
\begin{eqnarray*}
F_\infty(\mu) 
  & = & -\mu + \frac{1}{2} - \frac{1}{1 + e^{a \mu}}
  + \sum_{l=1}^\infty \left[ \frac{1}{1 + e^{a(l - \mu)}} -
  \frac{1}{1 + e^{a(l + \mu)}} \right] \\
  & = & -\mu + \frac{1}{2}\tanh(a \mu)
  + \sum_{l=1}^\infty \frac{\sinh(a \mu)}
	{\cosh(a \mu) + \cosh(a l)}  
\end{eqnarray*}
Defining $\{\mu\} = \mu - \floor{\mu}$ we have
\be
F_\infty(\mu) = - \{\mu\} + \frac{1}{2}\tanh(a \{\mu\})
  + \sum_{l=1}^\infty \frac{\sinh(a \{\mu\})}
	{\cosh(a \{\mu\}) + \cosh(a l)}
\label{defF}\ee
for all values of $\mu$.

\begin{lemma}
\label{lem:avgn}
The function $F_\infty$ defined in \eq{defF} has the following properties:
i) $F_\infty$ is periodic with period $1$, i.e, 
$F_\infty(\mu+1)=F_\infty(\mu)$, for all
$\mu\in\Rl$.\newline
ii) $F_\infty$ is odd about $\mu=1/2$, i.e., 
$F_\infty(1-\mu)=-F_\infty(\mu)$, for
all $\mu\in\Rl$.\newline
iii) $-1\leq F_\infty(\mu)\leq 1$, for all $\mu\in\Rl$.\newline
iv) $F_\infty(\mu) = 0$ for $\mu \in \Z$ and $\mu \in \frac{1}{2} + \Z$.
I.e.\ the estimate $\navg = \mu + \frac{L+1}{2}$ is exact for 
half-integer and integer filling.
\end{lemma}

\begin{proof}
The periodicity of $F_\infty$ follows directly from its definition. 
To prove (ii), define $F(\mu)$ for $0<\mu<1$ as
\be
F(\mu)
  = \sum_{k=1}^\infty
	\left[\frac{1}{1+e^{a(l-\mu)}}
	-\frac{1}{1+e^{a(l+\mu)}}\right]
    -\frac{1}{1+e^{a\mu}}
\label{otherF}\ee
Then,
\beann
F(1-\mu)&=&\sum_{l=1}^\infty\left[\frac{1}{1+e^{a(l-1+\mu)}}
-\frac{1}{1+e^{a(l+1-\mu)}}\right]-\frac{1}{1+e^{a(1-\mu)}}\\
&=&\sum_{l=1}^\infty\left[\frac{1}{1+e^{a(l+\mu)}}
-\frac{1}{1+e^{a(l-\mu)}}\right]\\
&&+\frac{1}{1+e^{a\mu}}+\frac{1}{1+e^{a(1-\mu)}}-\frac{1}{1+e^{a(1-\mu)}}\\
&=&-F(\mu)
\eeann
And clearly the remainder term
$$
\cases{
\frac{1}{2}-\{\mu\},& if $\mu \not\in \Ir$\cr
                   0 ,& if $\mu \in \Ir$}
$$
satisfies property (ii).
For the bounds, we first restrict ourselves to $\mu\in[0,1]$. 
For $\mu\geq 0$, we note that \eq{defF} implies
$$
F_\infty(\mu) \geq -\{\mu\}\geq -1.
$$
Then we use property ii) in combination with this bound
to also get the upper bound for $\mu\in[0,1]$.
$$
F_\infty(\mu)=-F_\infty(1-\mu)\leq 1
$$
Due to the peridicity property i), the upper and lower bound
are automatically extended to all real $\mu$.
The special values stated in iv) are straightforward from
\eq{defF} and \eq{otherF}.
\end{proof}
\begin{figure}[t]
\begin{center}
\resizebox{13cm}{!}{\includegraphics{F_sigma.eps}}
\parbox{14truecm}{\caption{\baselineskip=16 pt\small\label{fig:F_sigma}
A plot of the functions $F_\infty(\mu)$ and $\sigma^2(\mu)$, with $q
=e^{-10}$.}
}
\end{center}
\end{figure}
We can define the quantity $\delta(\mu)=\min(\vert\mu-\floor{\mu}\vert,
\vert 1-\mu+\floor{\mu}\vert)$, where $\floor{\mu}$ is the integer part of
$\mu$. In general, the relation between $\mu$ and $\nu$ depends
nontrivially on $q$ and the function $\delta$ can be thought as
$\delta(q,\nu)$. But for all $q$, $0<q<1$, one has $\delta(q,1/2)=0$ and
$\delta(q,0)=1/2$. See Figure \ref{fig:deltanu}.
\begin{figure}[t]
\begin{center}
\resizebox{13cm}{!}{\includegraphics{deltanu.eps}}
\parbox{14truecm}{\caption{\baselineskip=16 pt\small\label{fig:deltanu}
A plot of the function $\delta(\nu,q)$ for four different values of $q$.}
}
\end{center}
\end{figure}

\subsection{The variance of the number of particles in a stick}
\label{sect:var_n}

In the same way as was done above for the mean, we can compute
the variance of the number of particles in a stick in the 
grand canonical ensemble by using the standard formula
\be
\sigma^2(\mu,L) =\avgGC{\Num^2}{\Sigma,\mu} -(\avgGC{\Num}{\Sigma,\mu})^2
= \frac{1}{a^2} \frac{\partial^2}{{\partial \mu^2}} 
\log \ZGC{\Sigma}{\mu} ,
\ee 
which gives
\be
\sigma^2(\mu,L) = \frac{1}{4}\sum_{l=-L/2}^{L/2} \frac{1}{\cosh^2
(\frac{a}{2}
(l-\mu))}
\ee
Define 
\be
\sigma^2(\mu)=\lim_{L\to\infty}\sigma^2(\mu,L)
\ee
Then, the speed of convergence of this limit is bounded as follows:
\be
\vert \sigma^2(\mu) - \sigma^2(\mu,L)\vert
\leq 2\sum_{n=0}^{\infty}e^{-a(n-\mu + L/2)}=\frac{2q^{2(L/2-\mu)}}{1-q^2}
\ee
It is clear that $\sigma^2(\mu)$ is a periodic function of $\mu$ with 
period 1. It is not hard to see that $\sigma^2(\mu,L)$ is $C^\infty$ and 
attains its maximum in all integers and its minimum in the integers $+
1/2$.
It is easy to derive upper and lower bounds for $\sigma^2(\mu,L)$.
An upper bound is given by 
\be
\sigma^2(\mu,L)\leq \sum_{l=-L/2}^{L/2} e^{-\vert a(l-\mu)\vert}
\leq \sum_{l=-L/2}^{L/2} e^{-a\vert l\vert}\leq 1+
\frac{2e^{-a}}{1-e^{-a}}
\label{upper_sigma}\ee
and a lower bound  can be obtained using the crude bound
$2\cosh x\leq 2 e^{\vert x\vert }$:
\be
\sigma^2(\mu,L)\geq \frac{1}{4}\sum_{n=1}^{L} e^{-\vert an\vert}
\geq \frac{1}{4}\frac{e^{-a}-e^{-a (L+1)}}{1-e^{-a}}
\label{lower_sigma}\ee
{From} \eq{upper_sigma} and \eq{lower_sigma} we see that the limit 
$\sigma^2(\mu)$ satisfies the bounds
\be
\frac{1}{4}\frac{q^2}{1-q^2}\leq \sigma^2(\mu) \leq \frac{1+q^2}{1-q^2},
\ee
for all real $\mu$ and where we have again used the relation $e^{-a}=q^2$.

For the afficionados, one can also show that 
\be\
\lim_{q\downarrow 0}\sigma^2(\mu)=\cases{0          & if $\mu\not\in\Ir$\cr
                                         \frac{1}{4}& if $\mu\in\Ir$}					 
\ee
The interpretation is simple. When $\mu\in\Ir$, the interface (kink) in
the one-dimensional system is located at a lattice site, which is occupied
by a particle with probability 1/2. Clearly, the variance of the particle
number is them $1/4$. However, for $\mu\not\in\Ir$, the kink is centered
at a position not belonging to the lattice and the state converges, as
$q\downarrow 0$, to a deterministic configuration with zero variance
for the particle number.

\subsection{Estimating $\avgGC{q^{2|\Num - \avg{\Num}{}|}}{\Sigma,\mu}$}
\label{sect:avgexp}
We begin with the obvious fact
\be
q^{2|\Num - \avg{\Num}{}|} 
  \leq q^{2\Num - 2\avg{\Num}{}} + q^{2\avg{\Num}{} - 2\Num}
\ee
from which it follows that
\be
\avgGC{q^{2|\Num - \avg{\Num}{}|}}{\Sigma,\mu}
  \leq q^{-2\avg{\Num}{}} \avgGC{q^{2\Num}}{\Sigma,\mu}
  + q^{2\avg{\Num}{}} \avgGC{q^{-2\Num}}{\Sigma,\mu}.
\ee
Now, we observe
\be
\label{qexp:avg}
\avg{q^{2\Num}}{\Sigma,\mu} 
  = \frac{\sum_{n=0}^{L+1} q^{2n} q^{-2 \mu n} Z(\Sigma,n)}
{Z^{GC}(\Sigma,\mu)}
  = \frac{Z^{GC}(\Sigma,\mu-1)}{Z^{GC}(\Sigma,\mu)} .
\ee
Since
\be
  Z^{GC}(\Sigma,\mu) = \prod_{l=-L/2}{L/2} (1 + q^{2(l - \mu)})
\ee
equation \eq{qexp:avg} leads us to conclude
\be
\avg{q^{2\Num}}{\Sigma,\mu}
  = \frac{1 + q^{2(L/2+1-\mu)}}{1+q^{-2(L/2+\mu)}} 
  \leq 2 q^{2(L/2+\mu)}.
\ee
Similarly,
\be
\avg{q^{-2\Num}}{\Sigma,\mu}
  = \frac{1 + q^{-2(L/2+1+\mu)}}{1 + q^{2(L/2 - \mu)}}
  \leq 2 q^{-2(L/2+1+\mu)}.  
\ee
Using the results of section \ref{sect:navg}, we then have
\be
\avgGC{q^{2|\Num-\avg{\Num}{}|}}{\Sigma,\mu}
	\leq 4 q^{-1-|F_L(\mu)|}
	\leq 4 q^{-2}.
\ee
If we wish to calculate $\avgGC{q^{2|\Num-\avg{\Num}{}|}}{\Lambda,\mu}$,
where $\Lambda$ is comprised of $A$ sticks, then nothing changes except 
that each estimate is raised to the power $A$.
Thus, $\avgGC{q^{2|\Num-\avg{\Num}{}|}}{\Lambda,\mu} \leq 2^{A+1} q^{-2A}$.

\section*{Acknowledgements}

O.B. was supported by Fapesp under grant 97/14430-2. B.N. was partially
supported by the National Science Foundation under grant \# DMS-9706599.

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\end{document}

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/Mimage
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/Mnodistort true def
/colorimage where
{ pop /Mcolorimage {colorimage} bind def /Mimage {image} bind def}
{
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/Mscale {
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/Plain	/Courier findfont def
/Bold	/Courier-Bold findfont def
/Italic /Courier-Oblique findfont def
/MathPictureStart {
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	Mleft
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/MathSubStart {
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        Mgmatrix setmatrix
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	5 -1 roll
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	Msboxa1
	Mboxrot
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	[
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pop
exch
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mul
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mul
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/Mtempproc exch def
{ Mtempproc pop }
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	Mshadeproc
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/Mimage
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	pop pop pop
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/Mnodistort true def
/colorimage where
{ pop /Mcolorimage {colorimage} bind def /Mimage {image} bind def}
{
/colorimage {Mcolorimage} bind def
} ifelse
/Mleft		0.000000 def
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/Mwidth		548.000000 def
/Mheight	338.562500 def
0 setgray
0 setlinewidth
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[.40476 -0.0125 9 0 ]
[.59524 -0.0125 -9 -9 ]
[.59524 -0.0125 9 0 ]
[.78571 -0.0125 -9 -9 ]
[.78571 -0.0125 9 0 ]
[.97619 -0.0125 -3 -9 ]
[.97619 -0.0125 3 0 ]
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[ 0 0 -0.125 0 ]
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 61.000000 13.812500 moveto
 558.000000 13.812500 lineto
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%%IncludeResource: font Courier
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MathSubEnd
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% End of sub-graphic
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MathPictureEnd
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/shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul
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save
-112.0 236.0 translate
1 -1 scale
.9 .9 scale % to make patterns same scale as in xfig

% This junk string is used by the show operators
/PATsstr 1 string def
/PATawidthshow { 	% cx cy cchar rx ry string
  % Loop over each character in the string
  {  % cx cy cchar rx ry char
    % Show the character
    dup				% cx cy cchar rx ry char char
    PATsstr dup 0 4 -1 roll put	% cx cy cchar rx ry char (char)
    false charpath		% cx cy cchar rx ry char
    /clip load PATdraw
    % Move past the character (charpath modified the
    % current point)
    currentpoint			% cx cy cchar rx ry char x y
    newpath
    moveto			% cx cy cchar rx ry char
    % Reposition by cx,cy if the character in the string is cchar
    3 index eq {			% cx cy cchar rx ry
      4 index 4 index rmoveto
    } if
    % Reposition all characters by rx ry
    2 copy rmoveto		% cx cy cchar rx ry
  } forall
  pop pop pop pop pop		% -
  currentpoint
  newpath
  moveto
} bind def
/PATcg {
  7 dict dup begin
    /lw currentlinewidth def
    /lc currentlinecap def
    /lj currentlinejoin def
    /ml currentmiterlimit def
    /ds [ currentdash ] def
    /cc [ currentrgbcolor ] def
    /cm matrix currentmatrix def
  end
} bind def
% PATdraw - calculates the boundaries of the object and
% fills it with the current pattern
/PATdraw {			% proc
  save exch
    PATpcalc			% proc nw nh px py
    5 -1 roll exec		% nw nh px py
    newpath
    PATfill			% -
  restore
} bind def
% PATfill - performs the tiling for the shape
/PATfill { % nw nh px py PATfill -
  PATDict /CurrentPattern get dup begin
    setfont
    % Set the coordinate system to Pattern Space
    PatternGState PATsg
    % Set the color for uncolored pattezns
    PaintType 2 eq { PATDict /PColor get PATsc } if
    % Create the string for showing
    3 index string		% nw nh px py str
    % Loop for each of the pattern sources
    0 1 Multi 1 sub {		% nw nh px py str source
	% Move to the starting location
	3 index 3 index		% nw nh px py str source px py
	moveto			% nw nh px py str source
	% For multiple sources, set the appropriate color
	Multi 1 ne { dup PC exch get PATsc } if
	% Set the appropriate string for the source
	0 1 7 index 1 sub { 2 index exch 2 index put } for pop
	% Loop over the number of vertical cells
	3 index 		% nw nh px py str nh
	{			% nw nh px py str
	  currentpoint		% nw nh px py str cx cy
	  2 index show		% nw nh px py str cx cy
	  YStep add moveto	% nw nh px py str
	} repeat		% nw nh px py str
    } for
    5 { pop } repeat
  end
} bind def

% PATkshow - kshow with the current pattezn
/PATkshow {			% proc string
  exch bind			% string proc
  1 index 0 get			% string proc char
  % Loop over all but the last character in the string
  0 1 4 index length 2 sub {
				% string proc char idx
    % Find the n+1th character in the string
    3 index exch 1 add get	% string proe char char+1
    exch 2 copy			% strinq proc char+1 char char+1 char
    % Now show the nth character
    PATsstr dup 0 4 -1 roll put	% string proc chr+1 chr chr+1 (chr)
    false charpath		% string proc char+1 char char+1
    /clip load PATdraw
    % Move past the character (charpath modified the current point)
    currentpoint newpath moveto
    % Execute the user proc (should consume char and char+1)
    mark 3 1 roll		% string proc char+1 mark char char+1
    4 index exec		% string proc char+1 mark...
    cleartomark			% string proc char+1
  } for
  % Now display the last character
  PATsstr dup 0 4 -1 roll put	% string proc (char+1)
  false charpath		% string proc
  /clip load PATdraw
  neewath
  pop pop			% -
} bind def
% PATmp - the makepattern equivalent
/PATmp {			% patdict patmtx PATmp patinstance
  exch dup length 7 add		% We will add 6 new entries plus 1 FID
  dict copy			% Create a new dictionary
  begin
    % Matrix to install when painting the pattern
    TilingType PATtcalc
    /PatternGState PATcg def
    PatternGState /cm 3 -1 roll put
    % Check for multi pattern sources (Level 1 fast color patterns)
    currentdict /Multi known not { /Multi 1 def } if
    % Font dictionary definitions
    /FontType 3 def
    % Create a dummy encoding vector
    /Encoding 256 array def
    3 string 0 1 255 {
      Encoding exch dup 3 index cvs cvn put } for pop
    /FontMatrix matrix def
    /FontBBox BBox def
    /BuildChar {
	mark 3 1 roll		% mark dict char
	exch begin
	Multi 1 ne {PaintData exch get}{pop} ifelse  % mark [paintdata]
	  PaintType 2 eq Multi 1 ne or
	  { XStep 0 FontBBox aload pop setcachedevice }
	  { XStep 0 setcharwidth } ifelse
	  currentdict		% mark [paintdata] dict
	  /PaintProc load	% mark [paintdata] dict paintproc
	end
	gsave
	  false PATredef exec true PATredef
	grestore
	cleartomark		% -
    } bind def
    currentdict
  end				% newdict
  /foo exch			% /foo newlict
  definefont			% newfont
} bind def
% PATpcalc - calculates the starting point and width/height
% of the tile fill for the shape
/PATpcalc {	% - PATpcalc nw nh px py
  PATDict /CurrentPattern get begin
    gsave
	% Set up the coordinate system to Pattern Space
	% and lock down pattern
	PatternGState /cm get setmatrix
	BBox aload pop pop pop translate
	% Determine the bounding box of the shape
	pathbbox			% llx lly urx ury
    grestore
    % Determine (nw, nh) the # of cells to paint width and height
    PatHeight div ceiling		% llx lly urx qh
    4 1 roll				% qh llx lly urx
    PatWidth div ceiling		% qh llx lly qw
    4 1 roll				% qw qh llx lly
    PatHeight div floor			% qw qh llx ph
    4 1 roll				% ph qw qh llx
    PatWidth div floor			% ph qw qh pw
    4 1 roll				% pw ph qw qh
    2 index sub cvi abs			% pw ph qs qh-ph
    exch 3 index sub cvi abs exch	% pw ph nw=qw-pw nh=qh-ph
    % Determine the starting point of the pattern fill
    %(px, py)
    4 2 roll				% nw nh pw ph
    PatHeight mul			% nw nh pw py
    exch				% nw nh py pw
    PatWidth mul exch			% nw nh px py
  end
} bind def

% Save the original routines so that we can use them later on
/oldfill	/fill load def
/oldeofill	/eofill load def
/oldstroke	/stroke load def
/oldshow	/show load def
/oldashow	/ashow load def
/oldwidthshow	/widthshow load def
/oldawidthshow	/awidthshow load def
/oldkshow	/kshow load def

% These defs are necessary so that subsequent procs don't bind in
% the originals
/fill	   { oldfill } bind def
/eofill	   { oldeofill } bind def
/stroke	   { oldstroke } bind def
/show	   { oldshow } bind def
/ashow	   { oldashow } bind def
/widthshow { oldwidthshow } bind def
/awidthshow { oldawidthshow } bind def
/kshow 	   { oldkshow } bind def
/PATredef {
  MyAppDict begin
    {
    /fill { /clip load PATdraw newpath } bind def
    /eofill { /eoclip load PATdraw newpath } bind def
    /stroke { PATstroke } bind def
    /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def
    /ashow { 0 0 null 6 3 roll PATawidthshow }
    bind def
    /widthshow { 0 0 3 -1 roll PATawidthshow }
    bind def
    /awidthshow { PATawidthshow } bind def
    /kshow { PATkshow } bind def
  } {
    /fill   { oldfill } bind def
    /eofill { oldeofill } bind def
    /stroke { oldstroke } bind def
    /show   { oldshow } bind def
    /ashow  { oldashow } bind def
    /widthshow { oldwidthshow } bind def
    /awidthshow { oldawidthshow } bind def
    /kshow  { oldkshow } bind def
    } ifelse
  end
} bind def
false PATredef
% Conditionally define setcmykcolor if not available
/setcmykcolor where { pop } {
  /setcmykcolor {
    1 sub 4 1 roll
    3 {
	3 index add neg dup 0 lt { pop 0 } if 3 1 roll
    } repeat
    setrgbcolor - pop
  } bind def
} ifelse
/PATsc {		% colorarray
  aload length		% c1 ... cn length
    dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor
  } ifelse } ifelse
} bind def
/PATsg {		% dict
  begin
    lw setlinewidth
    lc setlinecap
    lj setlinejoin
    ml setmiterlimit
    ds aload pop setdash
    cc aload pop setrgbcolor
    cm setmatrix
  end
} bind def

/PATDict 3 dict def
/PATsp {
  true PATredef
  PATDict begin
    /CurrentPattern exch def
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---------------9908191749320
Content-Type: application/postscript; name="trilat1.eps"
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gs  clippath
1286 9860 m 1325 10002 l 1232 9888 l 1330 10077 l 1384 10050 l cp
clip
n 1350 10050 m 1024 9420 l gs col0 s gr gr

% arrowhead
15.000 slw
n 1286 9860 m 1325 10002 l 1232 9888 l 1270 9895 l 1286 9860 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
2411 8060 m 2450 8202 l 2357 8088 l 2455 8277 l 2509 8250 l cp
clip
n 2475 8250 m 2149 7620 l gs col0 s gr gr

% arrowhead
15.000 slw
n 2411 8060 m 2450 8202 l 2357 8088 l 2395 8095 l 2411 8060 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
2936 9035 m 2975 9177 l 2882 9063 l 2980 9252 l 3034 9225 l cp
clip
n 3000 9225 m 2674 8595 l gs col0 s gr gr

% arrowhead
15.000 slw
n 2936 9035 m 2975 9177 l 2882 9063 l 2920 9070 l 2936 9035 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
3461 9935 m 3500 10077 l 3407 9963 l 3505 10152 l 3559 10125 l cp
clip
n 3525 10125 m 3199 9495 l gs col0 s gr gr

% arrowhead
15.000 slw
n 3461 9935 m 3500 10077 l 3407 9963 l 3445 9970 l 3461 9935 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
5036 9035 m 5075 9177 l 4982 9063 l 5080 9252 l 5134 9225 l cp
clip
n 5100 9225 m 4774 8595 l gs col0 s gr gr

% arrowhead
15.000 slw
n 5036 9035 m 5075 9177 l 4982 9063 l 5020 9070 l 5036 9035 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
4436 8060 m 4475 8202 l 4382 8088 l 4480 8277 l 4534 8250 l cp
clip
n 4500 8250 m 4174 7620 l gs col0 s gr gr

% arrowhead
15.000 slw
n 4436 8060 m 4475 8202 l 4382 8088 l 4420 8095 l 4436 8060 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
3986 7235 m 4025 7377 l 3932 7263 l 4030 7452 l 4084 7425 l cp
clip
n 4050 7425 m 3724 6795 l gs col0 s gr gr

% arrowhead
15.000 slw
n 3986 7235 m 4025 7377 l 3932 7263 l 3970 7270 l 3986 7235 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
4511 4460 m 4550 4602 l 4457 4488 l 4555 4677 l 4609 4650 l cp
clip
n 4575 4650 m 4249 4020 l gs col0 s gr gr

% arrowhead
15.000 slw
n 4511 4460 m 4550 4602 l 4457 4488 l 4495 4495 l 4511 4460 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
2353 8707 m 2456 8601 l 2404 8739 l 2518 8559 l 2468 8527 l cp
clip
n 2105 9153 m 2485 8556 l gs col0 s gr gr

% arrowhead
15.000 slw
n 2353 8707 m 2456 8601 l 2404 8739 l 2392 8703 l 2353 8707 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
1828 9607 m 1931 9501 l 1879 9639 l 1993 9459 l 1943 9427 l cp
clip
n 1580 10053 m 1960 9456 l gs col0 s gr gr

% arrowhead
15.000 slw
n 1828 9607 m 1931 9501 l 1879 9639 l 1867 9603 l 1828 9607 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
2878 7807 m 2981 7701 l 2929 7839 l 3043 7659 l 2993 7627 l cp
clip
n 2630 8253 m 3010 7656 l gs col0 s gr gr

% arrowhead
15.000 slw
n 2878 7807 m 2981 7701 l 2929 7839 l 2917 7803 l 2878 7807 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
3403 6907 m 3506 6801 l 3454 6939 l 3568 6759 l 3518 6727 l cp
clip
n 3155 7353 m 3535 6756 l gs col0 s gr gr

% arrowhead
15.000 slw
n 3403 6907 m 3506 6801 l 3454 6939 l 3442 6903 l 3403 6907 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
3928 6007 m 4031 5901 l 3979 6039 l 4093 5859 l 4043 5827 l cp
clip
n 3680 6453 m 4060 5856 l gs col0 s gr gr

% arrowhead
15.000 slw
n 3928 6007 m 4031 5901 l 3979 6039 l 3967 6003 l 3928 6007 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
4453 5107 m 4556 5001 l 4504 5139 l 4618 4959 l 4568 4927 l cp
clip
n 4205 5553 m 4585 4956 l gs col0 s gr gr

% arrowhead
15.000 slw
n 4453 5107 m 4556 5001 l 4504 5139 l 4492 5103 l 4453 5107 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
4992 4216 m 5095 4110 l 5043 4248 l 5157 4068 l 5107 4036 l cp
clip
n 4744 4662 m 5124 4065 l gs col0 s gr gr

% arrowhead
15.000 slw
n 4992 4216 m 5095 4110 l 5043 4248 l 5031 4212 l 4992 4216 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
6028 6007 m 6131 5901 l 6079 6039 l 6193 5859 l 6143 5827 l cp
clip
n 5780 6453 m 6160 5856 l gs col0 s gr gr

% arrowhead
15.000 slw
n 6028 6007 m 6131 5901 l 6079 6039 l 6067 6003 l 6028 6007 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
5503 6907 m 5606 6801 l 5554 6939 l 5668 6759 l 5618 6727 l cp
clip
n 5255 7353 m 5635 6756 l gs col0 s gr gr

% arrowhead
15.000 slw
n 5503 6907 m 5606 6801 l 5554 6939 l 5542 6903 l 5503 6907 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
4978 7807 m 5081 7701 l 5029 7839 l 5143 7659 l 5093 7627 l cp
clip
n 4730 8253 m 5110 7656 l gs col0 s gr gr

% arrowhead
15.000 slw
n 4978 7807 m 5081 7701 l 5029 7839 l 5017 7803 l 4978 7807 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
4453 8707 m 4556 8601 l 4504 8739 l 4618 8559 l 4568 8527 l cp
clip
n 4205 9153 m 4585 8556 l gs col0 s gr gr

% arrowhead
15.000 slw
n 4453 8707 m 4556 8601 l 4504 8739 l 4492 8703 l 4453 8707 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
3928 9607 m 4031 9501 l 3979 9639 l 4093 9459 l 4043 9427 l cp
clip
n 3680 10053 m 4060 9456 l gs col0 s gr gr

% arrowhead
15.000 slw
n 3928 9607 m 4031 9501 l 3979 9639 l 3967 9603 l 3928 9607 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
6028 9607 m 6131 9501 l 6079 9639 l 6193 9459 l 6143 9427 l cp
clip
n 5780 10053 m 6160 9456 l gs col0 s gr gr

% arrowhead
15.000 slw
n 6028 9607 m 6131 9501 l 6079 9639 l 6067 9603 l 6028 9607 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
6553 8707 m 6656 8601 l 6604 8739 l 6718 8559 l 6668 8527 l cp
clip
n 6305 9153 m 6685 8556 l gs col0 s gr gr

% arrowhead
15.000 slw
n 6553 8707 m 6656 8601 l 6604 8739 l 6592 8703 l 6553 8707 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
7078 7807 m 7181 7701 l 7129 7839 l 7243 7659 l 7193 7627 l cp
clip
n 6830 8253 m 7210 7656 l gs col0 s gr gr

% arrowhead
15.000 slw
n 7078 7807 m 7181 7701 l 7129 7839 l 7117 7803 l 7078 7807 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
8128 9607 m 8231 9501 l 8179 9639 l 8293 9459 l 8243 9427 l cp
clip
n 7880 10053 m 8260 9456 l gs col0 s gr gr

% arrowhead
15.000 slw
n 8128 9607 m 8231 9501 l 8179 9639 l 8167 9603 l 8128 9607 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
6046 5342 m 6085 5484 l 5992 5370 l 6090 5559 l 6144 5532 l cp
clip
n 6110 5532 m 5784 4902 l gs col0 s gr gr

% arrowhead
15.000 slw
n 6046 5342 m 6085 5484 l 5992 5370 l 6030 5377 l 6046 5342 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
5521 4442 m 5560 4584 l 5467 4470 l 5565 4659 l 5619 4632 l cp
clip
n 5585 4632 m 5259 4002 l gs col0 s gr gr

% arrowhead
15.000 slw
n 5521 4442 m 5560 4584 l 5467 4470 l 5505 4477 l 5521 4442 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
4996 3542 m 5035 3684 l 4942 3570 l 5040 3759 l 5094 3732 l cp
clip
n 5060 3732 m 4734 3102 l gs col0 s gr gr

% arrowhead
15.000 slw
n 4996 3542 m 5035 3684 l 4942 3570 l 4980 3577 l 4996 3542 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
6571 6317 m 6610 6459 l 6517 6345 l 6615 6534 l 6669 6507 l cp
clip
n 6635 6507 m 6309 5877 l gs col0 s gr gr

% arrowhead
15.000 slw
n 6571 6317 m 6610 6459 l 6517 6345 l 6555 6352 l 6571 6317 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
7065 7158 m 7104 7300 l 7011 7186 l 7109 7375 l 7163 7348 l cp
clip
n 7129 7348 m 6803 6718 l gs col0 s gr gr

% arrowhead
15.000 slw
n 7065 7158 m 7104 7300 l 7011 7186 l 7049 7193 l 7065 7158 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
7590 8058 m 7629 8200 l 7536 8086 l 7634 8275 l 7688 8248 l cp
clip
n 7654 8248 m 7328 7618 l gs col0 s gr gr

% arrowhead
15.000 slw
n 7590 8058 m 7629 8200 l 7536 8086 l 7574 8093 l 7590 8058 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
8115 8958 m 8154 9100 l 8061 8986 l 8159 9175 l 8213 9148 l cp
clip
n 8179 9148 m 7853 8518 l gs col0 s gr gr

% arrowhead
15.000 slw
n 8115 8958 m 8154 9100 l 8061 8986 l 8099 8993 l 8115 8958 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
8715 9858 m 8754 10000 l 8661 9886 l 8759 10075 l 8813 10048 l cp
clip
n 8779 10048 m 8453 9418 l gs col0 s gr gr

% arrowhead
15.000 slw
n 8715 9858 m 8754 10000 l 8661 9886 l 8699 9893 l 8715 9858 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
6540 9858 m 6579 10000 l 6486 9886 l 6584 10075 l 6638 10048 l cp
clip
n 6604 10048 m 6278 9418 l gs col0 s gr gr

% arrowhead
15.000 slw
n 6540 9858 m 6579 10000 l 6486 9886 l 6524 9893 l 6540 9858 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
6015 8958 m 6054 9100 l 5961 8986 l 6059 9175 l 6113 9148 l cp
clip
n 6079 9148 m 5753 8518 l gs col0 s gr gr

% arrowhead
15.000 slw
n 6015 8958 m 6054 9100 l 5961 8986 l 5999 8993 l 6015 8958 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
5490 8058 m 5529 8200 l 5436 8086 l 5534 8275 l 5588 8248 l cp
clip
n 5554 8248 m 5228 7618 l gs col0 s gr gr

% arrowhead
15.000 slw
n 5490 8058 m 5529 8200 l 5436 8086 l 5474 8093 l 5490 8058 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
4965 7158 m 5004 7300 l 4911 7186 l 5009 7375 l 5063 7348 l cp
clip
n 5029 7348 m 4703 6718 l gs col0 s gr gr

% arrowhead
15.000 slw
n 4965 7158 m 5004 7300 l 4911 7186 l 4949 7193 l 4965 7158 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
4440 6258 m 4479 6400 l 4386 6286 l 4484 6475 l 4538 6448 l cp
clip
n 4504 6448 m 4178 5818 l gs col0 s gr gr

% arrowhead
15.000 slw
n 4440 6258 m 4479 6400 l 4386 6286 l 4424 6293 l 4440 6258 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
3915 5358 m 3954 5500 l 3861 5386 l 3959 5575 l 4013 5548 l cp
clip
n 3979 5548 m 3653 4918 l gs col0 s gr gr

% arrowhead
15.000 slw
n 3915 5358 m 3954 5500 l 3861 5386 l 3899 5393 l 3915 5358 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
2865 7158 m 2904 7300 l 2811 7186 l 2909 7375 l 2963 7348 l cp
clip
n 2929 7348 m 2603 6718 l gs col0 s gr gr

% arrowhead
15.000 slw
n 2865 7158 m 2904 7300 l 2811 7186 l 2849 7193 l 2865 7158 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
3465 8133 m 3504 8275 l 3411 8161 l 3509 8350 l 3563 8323 l cp
clip
n 3529 8323 m 3203 7693 l gs col0 s gr gr

% arrowhead
15.000 slw
n 3465 8133 m 3504 8275 l 3411 8161 l 3449 8168 l 3465 8133 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
3915 8958 m 3954 9100 l 3861 8986 l 3959 9175 l 4013 9148 l cp
clip
n 3979 9148 m 3653 8518 l gs col0 s gr gr

% arrowhead
15.000 slw
n 3915 8958 m 3954 9100 l 3861 8986 l 3899 8993 l 3915 8958 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
4440 9858 m 4479 10000 l 4386 9886 l 4484 10075 l 4538 10048 l cp
clip
n 4504 10048 m 4178 9418 l gs col0 s gr gr

% arrowhead
15.000 slw
n 4440 9858 m 4479 10000 l 4386 9886 l 4424 9893 l 4440 9858 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
2340 9858 m 2379 10000 l 2286 9886 l 2384 10075 l 2438 10048 l cp
clip
n 2404 10048 m 2078 9418 l gs col0 s gr gr

% arrowhead
15.000 slw
n 2340 9858 m 2379 10000 l 2286 9886 l 2324 9893 l 2340 9858 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
1890 9033 m 1929 9175 l 1836 9061 l 1934 9250 l 1988 9223 l cp
clip
n 1954 9223 m 1628 8593 l gs col0 s gr gr

% arrowhead
15.000 slw
n 1890 9033 m 1929 9175 l 1836 9061 l 1874 9068 l 1890 9033 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
4473 3930 m 4329 3900 l 4473 3870 l 4260 3870 l 4260 3930 l cp
clip
n 4275 3900 m 4950 3900 l gs col0 s gr gr

% arrowhead
15.000 slw
n 4473 3930 m 4329 3900 l 4473 3870 l 4449 3900 l 4473 3930 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
5073 4830 m 4929 4800 l 5073 4770 l 4860 4770 l 4860 4830 l cp
clip
n 4875 4800 m 5550 4800 l gs col0 s gr gr

% arrowhead
15.000 slw
n 5073 4830 m 4929 4800 l 5073 4770 l 5049 4800 l 5073 4830 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
3948 4830 m 3804 4800 l 3948 4770 l 3735 4770 l 3735 4830 l cp
clip
n 3750 4800 m 4425 4800 l gs col0 s gr gr

% arrowhead
15.000 slw
n 3948 4830 m 3804 4800 l 3948 4770 l 3924 4800 l 3948 4830 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
4473 5730 m 4329 5700 l 4473 5670 l 4260 5670 l 4260 5730 l cp
clip
n 4275 5700 m 4950 5700 l gs col0 s gr gr

% arrowhead
15.000 slw
n 4473 5730 m 4329 5700 l 4473 5670 l 4449 5700 l 4473 5730 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
5598 5730 m 5454 5700 l 5598 5670 l 5385 5670 l 5385 5730 l cp
clip
n 5400 5700 m 6075 5700 l gs col0 s gr gr

% arrowhead
15.000 slw
n 5598 5730 m 5454 5700 l 5598 5670 l 5574 5700 l 5598 5730 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
3423 5730 m 3279 5700 l 3423 5670 l 3210 5670 l 3210 5730 l cp
clip
n 3225 5700 m 3900 5700 l gs col0 s gr gr

% arrowhead
15.000 slw
n 3423 5730 m 3279 5700 l 3423 5670 l 3399 5700 l 3423 5730 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
3948 6630 m 3804 6600 l 3948 6570 l 3735 6570 l 3735 6630 l cp
clip
n 3750 6600 m 4425 6600 l gs col0 s gr gr

% arrowhead
15.000 slw
n 3948 6630 m 3804 6600 l 3948 6570 l 3924 6600 l 3948 6630 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
2898 6630 m 2754 6600 l 2898 6570 l 2685 6570 l 2685 6630 l cp
clip
n 2700 6600 m 3375 6600 l gs col0 s gr gr

% arrowhead
15.000 slw
n 2898 6630 m 2754 6600 l 2898 6570 l 2874 6600 l 2898 6630 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
4998 6630 m 4854 6600 l 4998 6570 l 4785 6570 l 4785 6630 l cp
clip
n 4800 6600 m 5475 6600 l gs col0 s gr gr

% arrowhead
15.000 slw
n 4998 6630 m 4854 6600 l 4998 6570 l 4974 6600 l 4998 6630 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
6048 6630 m 5904 6600 l 6048 6570 l 5835 6570 l 5835 6630 l cp
clip
n 5850 6600 m 6525 6600 l gs col0 s gr gr

% arrowhead
15.000 slw
n 6048 6630 m 5904 6600 l 6048 6570 l 6024 6600 l 6048 6630 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
6573 7530 m 6429 7500 l 6573 7470 l 6360 7470 l 6360 7530 l cp
clip
n 6375 7500 m 7050 7500 l gs col0 s gr gr

% arrowhead
15.000 slw
n 6573 7530 m 6429 7500 l 6573 7470 l 6549 7500 l 6573 7530 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
5523 7530 m 5379 7500 l 5523 7470 l 5310 7470 l 5310 7530 l cp
clip
n 5325 7500 m 6000 7500 l gs col0 s gr gr

% arrowhead
15.000 slw
n 5523 7530 m 5379 7500 l 5523 7470 l 5499 7500 l 5523 7530 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
4473 7530 m 4329 7500 l 4473 7470 l 4260 7470 l 4260 7530 l cp
clip
n 4275 7500 m 4950 7500 l gs col0 s gr gr

% arrowhead
15.000 slw
n 4473 7530 m 4329 7500 l 4473 7470 l 4449 7500 l 4473 7530 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
3423 7530 m 3279 7500 l 3423 7470 l 3210 7470 l 3210 7530 l cp
clip
n 3225 7500 m 3900 7500 l gs col0 s gr gr

% arrowhead
15.000 slw
n 3423 7530 m 3279 7500 l 3423 7470 l 3399 7500 l 3423 7530 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
2373 7530 m 2229 7500 l 2373 7470 l 2160 7470 l 2160 7530 l cp
clip
n 2175 7500 m 2850 7500 l gs col0 s gr gr

% arrowhead
15.000 slw
n 2373 7530 m 2229 7500 l 2373 7470 l 2349 7500 l 2373 7530 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
1848 8430 m 1704 8400 l 1848 8370 l 1635 8370 l 1635 8430 l cp
clip
n 1650 8400 m 2325 8400 l gs col0 s gr gr

% arrowhead
15.000 slw
n 1848 8430 m 1704 8400 l 1848 8370 l 1824 8400 l 1848 8430 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
2973 8430 m 2829 8400 l 2973 8370 l 2760 8370 l 2760 8430 l cp
clip
n 2775 8400 m 3450 8400 l gs col0 s gr gr

% arrowhead
15.000 slw
n 2973 8430 m 2829 8400 l 2973 8370 l 2949 8400 l 2973 8430 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
4023 8430 m 3879 8400 l 4023 8370 l 3810 8370 l 3810 8430 l cp
clip
n 3825 8400 m 4500 8400 l gs col0 s gr gr

% arrowhead
15.000 slw
n 4023 8430 m 3879 8400 l 4023 8370 l 3999 8400 l 4023 8430 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
5073 8430 m 4929 8400 l 5073 8370 l 4860 8370 l 4860 8430 l cp
clip
n 4875 8400 m 5550 8400 l gs col0 s gr gr

% arrowhead
15.000 slw
n 5073 8430 m 4929 8400 l 5073 8370 l 5049 8400 l 5073 8430 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
6048 8430 m 5904 8400 l 6048 8370 l 5835 8370 l 5835 8430 l cp
clip
n 5850 8400 m 6525 8400 l gs col0 s gr gr

% arrowhead
15.000 slw
n 6048 8430 m 5904 8400 l 6048 8370 l 6024 8400 l 6048 8430 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
7173 8430 m 7029 8400 l 7173 8370 l 6960 8370 l 6960 8430 l cp
clip
n 6975 8400 m 7650 8400 l gs col0 s gr gr

% arrowhead
15.000 slw
n 7173 8430 m 7029 8400 l 7173 8370 l 7149 8400 l 7173 8430 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
7623 9330 m 7479 9300 l 7623 9270 l 7410 9270 l 7410 9330 l cp
clip
n 7425 9300 m 8100 9300 l gs col0 s gr gr

% arrowhead
15.000 slw
n 7623 9330 m 7479 9300 l 7623 9270 l 7599 9300 l 7623 9330 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
6573 9330 m 6429 9300 l 6573 9270 l 6360 9270 l 6360 9330 l cp
clip
n 6375 9300 m 7050 9300 l gs col0 s gr gr

% arrowhead
15.000 slw
n 6573 9330 m 6429 9300 l 6573 9270 l 6549 9300 l 6573 9330 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
5598 9330 m 5454 9300 l 5598 9270 l 5385 9270 l 5385 9330 l cp
clip
n 5400 9300 m 6075 9300 l gs col0 s gr gr

% arrowhead
15.000 slw
n 5598 9330 m 5454 9300 l 5598 9270 l 5574 9300 l 5598 9330 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
4473 9330 m 4329 9300 l 4473 9270 l 4260 9270 l 4260 9330 l cp
clip
n 4275 9300 m 4950 9300 l gs col0 s gr gr

% arrowhead
15.000 slw
n 4473 9330 m 4329 9300 l 4473 9270 l 4449 9300 l 4473 9330 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
3498 9330 m 3354 9300 l 3498 9270 l 3285 9270 l 3285 9330 l cp
clip
n 3300 9300 m 3975 9300 l gs col0 s gr gr

% arrowhead
15.000 slw
n 3498 9330 m 3354 9300 l 3498 9270 l 3474 9300 l 3498 9330 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
2373 9330 m 2229 9300 l 2373 9270 l 2160 9270 l 2160 9330 l cp
clip
n 2175 9300 m 2850 9300 l gs col0 s gr gr

% arrowhead
15.000 slw
n 2373 9330 m 2229 9300 l 2373 9270 l 2349 9300 l 2373 9330 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
1398 9330 m 1254 9300 l 1398 9270 l 1185 9270 l 1185 9330 l cp
clip
n 1200 9300 m 1875 9300 l gs col0 s gr gr

% arrowhead
15.000 slw
n 1398 9330 m 1254 9300 l 1398 9270 l 1374 9300 l 1398 9330 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
798 10230 m 654 10200 l 798 10170 l 585 10170 l 585 10230 l cp
clip
n 600 10200 m 1275 10200 l gs col0 s gr gr

% arrowhead
15.000 slw
n 798 10230 m 654 10200 l 798 10170 l 774 10200 l 798 10230 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
1923 10230 m 1779 10200 l 1923 10170 l 1710 10170 l 1710 10230 l cp
clip
n 1725 10200 m 2400 10200 l gs col0 s gr gr

% arrowhead
15.000 slw
n 1923 10230 m 1779 10200 l 1923 10170 l 1899 10200 l 1923 10230 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
2973 10230 m 2829 10200 l 2973 10170 l 2760 10170 l 2760 10230 l cp
clip
n 2775 10200 m 3450 10200 l gs col0 s gr gr

% arrowhead
15.000 slw
n 2973 10230 m 2829 10200 l 2973 10170 l 2949 10200 l 2973 10230 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
3948 10230 m 3804 10200 l 3948 10170 l 3735 10170 l 3735 10230 l cp
clip
n 3750 10200 m 4425 10200 l gs col0 s gr gr

% arrowhead
15.000 slw
n 3948 10230 m 3804 10200 l 3948 10170 l 3924 10200 l 3948 10230 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
5073 10230 m 4929 10200 l 5073 10170 l 4860 10170 l 4860 10230 l cp
clip
n 4875 10200 m 5550 10200 l gs col0 s gr gr

% arrowhead
15.000 slw
n 5073 10230 m 4929 10200 l 5073 10170 l 5049 10200 l 5073 10230 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
6048 10230 m 5904 10200 l 6048 10170 l 5835 10170 l 5835 10230 l cp
clip
n 5850 10200 m 6525 10200 l gs col0 s gr gr

% arrowhead
15.000 slw
n 6048 10230 m 5904 10200 l 6048 10170 l 6024 10200 l 6048 10230 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
7098 10230 m 6954 10200 l 7098 10170 l 6885 10170 l 6885 10230 l cp
clip
n 6900 10200 m 7575 10200 l gs col0 s gr gr

% arrowhead
15.000 slw
n 7098 10230 m 6954 10200 l 7098 10170 l 7074 10200 l 7098 10230 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
7.500 slw
gs  clippath
8148 10230 m 8004 10200 l 8148 10170 l 7935 10170 l 7935 10230 l cp
clip
n 7950 10200 m 8625 10200 l gs col0 s gr gr

% arrowhead
15.000 slw
n 8148 10230 m 8004 10200 l 8148 10170 l 8124 10200 l 8148 10230 l  cp gs 0.00 setgray ef gr  col0 s
7.500 slw
% Ellipse
n 5175 3900 75 75 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr

% Ellipse
n 3622 4822 75 75 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr

% Ellipse
n 3600 6600 75 75 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr

% Ellipse
n 5175 7500 75 75 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr

% Ellipse
n 6750 8400 75 75 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr

% Ellipse
n 5175 9300 75 75 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr

% Ellipse
n 3600 8400 75 75 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr

% Ellipse
n 2025 7500 75 75 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr

% Ellipse
n 450 10200 75 75 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr

% Ellipse
n 3600 10200 75 75 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr

% Ellipse
n 6750 10200 75 75 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr

% Ellipse
n 5175 5700 75 75 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr

% Ellipse
n 2025 9300 75 75 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr

% Ellipse
n 6750 6600 75 75 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr

% Ellipse
n 8325 9300 75 75 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr

% Ellipse
n 4125 3900 75 75 0 360 DrawEllipse gs col7 0.50 shd ef gr gs col0 s gr

% Ellipse
n 5700 4800 75 75 0 360 DrawEllipse gs col7 0.50 shd ef gr gs col0 s gr

% Ellipse
n 4072 5722 75 75 0 360 DrawEllipse gs col7 0.50 shd ef gr gs col0 s gr

% Ellipse
n 2550 6600 75 75 0 360 DrawEllipse gs col7 0.50 shd ef gr gs col0 s gr

% Ellipse
n 5700 6600 75 75 0 360 DrawEllipse gs col7 0.50 shd ef gr gs col0 s gr

% Ellipse
n 4125 7500 75 75 0 360 DrawEllipse gs col7 0.50 shd ef gr gs col0 s gr

% Ellipse
n 2572 8422 75 75 0 360 DrawEllipse gs col7 0.50 shd ef gr gs col0 s gr

% Ellipse
n 7200 7500 75 75 0 360 DrawEllipse gs col7 0.50 shd ef gr gs col0 s gr

% Ellipse
n 5700 8400 75 75 0 360 DrawEllipse gs col7 0.50 shd ef gr gs col0 s gr

% Ellipse
n 7275 9300 75 75 0 360 DrawEllipse gs col7 0.50 shd ef gr gs col0 s gr

% Ellipse
n 4125 9300 75 75 0 360 DrawEllipse gs col7 0.50 shd ef gr gs col0 s gr

% Ellipse
n 2550 10200 75 75 0 360 DrawEllipse gs col7 0.50 shd ef gr gs col0 s gr

% Ellipse
n 975 9300 75 75 0 360 DrawEllipse gs col7 0.50 shd ef gr gs col0 s gr

% Ellipse
n 8850 10200 75 75 0 360 DrawEllipse gs col7 0.50 shd ef gr gs col0 s gr

% Ellipse
n 5700 10200 75 75 0 360 DrawEllipse gs col7 0.50 shd ef gr gs col0 s gr

$F2psEnd
rs

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