%Paper: hep-th/9303097
%From: Yves Leroyer <LEROYER@FRCPN11.IN2P3.FR>
%Date: Wed, 17 Mar 93 13:26:00 MET

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 \title{{\bf Intermittency in the $q$-state Potts model\vspace{1cm}}}
 \author{Yves LEROYER\vspace{1cm}\\
{\em Laboratoire de Physique Th\'eorique } \\
Unit\'e Associ\'ee au CNRS, U.A. 764 \\
{\em Universit\'e de Bordeaux I} \\
{\em Rue du Solarium }\\
{\em F-33175 Gradignan Cedex}}
\date{ }
\begin{titlepage}
  \maketitle
\thispagestyle{empty}
\begin{center}
{\large {\bf Abstract}}\vspace{0.5cm}
\end{center}
We define a block observable for the $q$-state Potts model which
 exhibits an intermittent behaviour  at the critical point. We express the
intermittency indices of the normalised moments in terms of the magnetic
critical exponent $\beta /\nu$ of the
model. We confirm this relation by a numerical similation of the $q=2$ (Ising)
and $q=3$ two-dimensional Potts model.  \vspace{2.0cm}\\

\noindent LPTB 93-2\\
Mars 1993\\
PACS 05.70.Jk 64.60.Fr	\\
e-mail : LEROYER@FRCPN11.IN2P3.FR
\end{titlepage}
\newpage



\section{Introduction}
Recently, the concept of intermittency has been intoduced in the field of
equilibrium critical phenomena as a tool for studying local fluctuations of a
system undergoing a second order phase transition~\cite{Pesch,Wosiek}. For
example, in the Ising model, as the
the magnetisation fluctuates without limit at the critical point,
one expects that the normalised moments of this observable, measured on
sub-blocks of
the system, scale with the size of the block, with exponents that may depend on
the order of the moment, the so-called intermittency indices.
On the basis of real space renormalisation group arguments,  Satz~\cite{Satz}
established that, for the Ising
model, the intermittency indices are directly connected to the magnetic
critical exponent. However, the normalised moments of the magnetisation used in
Satz's argument
are not suitable for a numerical simulation, as the net magnetisation, which
enters in the denominator of  the moments,  is almost zero
on a (large) finite lattice.
In subsequent numerical simulations, Bambah et. al.\cite{Bambah}
and Gupta et. al.\cite{Gupta},	using a new observable which respects the
$Z_2$ symmetry and remains finite on finite lattices, present clear evidence
for an intermittent behaviour  in the two-dimensional Ising model. However,
they do not observe the
expected relation between the intermittency exponents and the critical ones.
Burda et. al.~\cite{Zalewski}  suggest that this is a consequence of their
choice
for the fluctuating observables, among which the moments have only a
subdominant scaling behaviour in the critical regime.

Therefore, the problem of finding convenient block observables which have
well-defined moments on finite lattices and
which exhibit the expected scaling law,  remains open. In
this paper we address this question in the more general framework of the
$q$-state Potts model.

 In section 2, we define the fluctuating observable
for which we derive the intermittency
indices in terms of the magnetic exponent by generalising the Satz's
argument~\cite{Satz}. In section 3 we give the details of the numerical
simulation and discuss the results. Conclusions are drawn in the last section.


\section{Definition of the observables}
The most convenient definition of the moments for studying critical spin
systems has been thoroughly discussed
in  ref~\cite{Pesch,Gupta}. Following the authors of ref.~\cite{Gupta} we use
the so-called {\em standard block} moments of order $p$ defined as
\begin{equation}
F_p(L;\ell )=\frac{1}{M}\sum_\alpha \frac{<k_\alpha^p>}{<k_\alpha >^p}
\end{equation}
where $L$ is the lattice size, $\ell$ the cell size,  $M=(L/\ell )^d$
the number of cells and $k_\alpha$ the block observable
defined on the $\alpha^{th}$ cell. The symbols $<>$ stands for the
thermodynamical average which will be taken
{\em at the bulk critical temperature} $T_c$ defined by $K_c=J/k_bT_c=\ln
[1+\sqrt{q}]$ for the $q$-state Potts model. We recall that the intermittency
indices $\lambda_p$ are defined according to the behaviour of the moments with
respect to the block size $\ell$~:
$$F_p(L;\ell )\sim \left(\frac{\ell}{L}\right)^{-\lambda_p}\qquad \mbox{for}\;
\frac{\ell}{L}\rightarrow 0$$

Let us now define the block observable $k_\alpha$.

We consider a system of Potts spins $s_i=0,1,\ldots,q-1$, interacting
with their nearest neighbours via the hamiltonian
$$-\beta H =K\sum_{<i,j>}\delta_{s_is_j}$$
on a two-dimensional square lattice of linear size $L$.
For a particular configuration, we denote by $Q$ the spin value which occurs
the most frequently and to each site $i$ we assign the variable
$n_i=\delta_{s_iQ}$. We now define
$$k_i= \frac{qn_i-1}{q-1}$$
which is such that
$$\frac{1}{L^d}<\sum_i k_i>=<k_i>=<m_q>$$
is the order parameter for the Potts model~\cite{Wu}. For the Ising model
($q=2$) this definition reduces to
$$k_i=\mbox{sgn}(m_I)\sigma_i$$
where $m_I$ is the Ising magnetisation of the configuration of Ising spins
$\{\sigma_i=\pm 1\}$.\\
We then define the {\em block} observable to be
\begin{equation}
k_\alpha =\frac{1}{\ell^d}\sum_{i\in \alpha} k_i =  \frac{qn_Q^{(\alpha
)}-1}{q-1}
\end{equation}
where  $n_Q^{(\alpha )}$ is the fraction
of spins in the cell $\alpha$ having the value $Q$~\cite{Hajdu}.

Unlike the original definition of ref~\cite{Satz},
 on a {\em finite lattice}, $<k_\alpha >$ is non zero at the
critical point, ($<k_\alpha >=<|m_I|>$ for the Ising model) allowing the
moments of eq.~(2.1) to be measured in a numerical simulation.
Moreover, one can extend to this observable the renormalisation argument which
predicts the intermittent behaviour of the moments.\\
 Let us associate to the blocking procedure the following renormalisation
prescription~: we define a block spin $s_\alpha$ according to the majority rule
in the $\alpha^{th}$ cell, associate the corresponding renormalised site
variable\footnote{For a given
configuration, the value of $Q$ is preserved by the blocking procedure}
 $\tilde{n}^{(\alpha )}_Q=\delta_{s_\alpha Q}$ and then the renormalised
quantity $\tilde{k}_\alpha$
 $$\tilde{k}_\alpha = \frac{q\tilde{n}_Q^{(\alpha )}-1}{q-1}$$
Since $k_\alpha$ is related to the magnetic scaling operator, we expect the
following renormalisation relation
$$k_\alpha = Z(\ell )\tilde{k}_\alpha$$
where $Z(\ell )$ is the renormalisation factor depending only on the block
size $\ell$. Therefore
\begin{eqnarray}
<k_\alpha^p> & = & \left[Z(\ell )\right]^p<\left(\frac{q\tilde{n}_Q^{(\alpha
)}-1}{q-1}\right)^p>\\
 & =  &  \left[Z(\ell )\right]^p \left[ <\tilde{n}_Q^{(\alpha )}>\left(
1+\frac{(-1)^{(p+1)}} {(q-1)^p} \right)+\frac{(-1)^p}{(q-1)^p}\right]
\end{eqnarray}
where we have used $(\tilde{n}_Q^{(\alpha )})^p=\tilde{n}_Q^{(\alpha )}$.
Reintroducing $\tilde{k}_\alpha$ in the left-hand side of eq.(2.4),
 we get the relation \begin{equation}
<k_\alpha^p> = \left[Z(\ell )\right]^p (A_p <\tilde{k}_\alpha >+B_p)
\end{equation}
where the coefficients $A_p$ and $B_p$ are given exactly by
\begin{eqnarray}
A_p & =  & \frac{q-1 }{q} \left( 1+\frac{(-1)^{(p+1)}} {(q-1)^p} \right)\\
B_p & =  & \frac{1 }{q} \left( 1+\frac{(-1)^{(p+1)}} {(q-1)^p} \right)
+\frac{(-1)^p}{(q-1)^p}
\end{eqnarray}

According to the standard renormalisation analysis, we expect
$Z(\ell )\sim \ell^{-\frac{1}{2}(d-2+\eta )} = \ell^{-\beta /\nu}$. Moreover,
$<k_\alpha >$ and $<\tilde{k}_\alpha>$ are the order parameter of
the system of size $L$ and of the
rescaled system of size $L/\ell$, respectively. At the bulk critical
temperature,
these quantities behave according to the finite size scaling law
\begin{eqnarray}
<k_\alpha > & \sim & L^{-\beta/\nu}\\
<\tilde{k}_\alpha > & \sim & (L/\ell)^{-\beta/\nu}
\end{eqnarray}

Thus, for $\ell\gg 1$ (in lattice units)
 in order to sum up enough degrees of freedom for the renormalisation
arguments to be valid and $\ell \ll L$ to avoid finite size effects, we expect
the moments $F_p$ to behave like
\begin{equation}
F_p(L;\ell )   \sim   \left(\frac{\ell}{L}\right)^{-p\frac{\beta}{\nu}}
\left[A'_p \left(\frac{\ell}{L}\right)^{\frac{\beta}{\nu}} +B'_p\right]
\end{equation}
where the coefficients $A'_p$ and $B'_p$ are proportional to the coefficients
$A_p$ and $B_p$ defined above. \\

For the Ising model $(q=2$), eqs (2.6) and (2.7) give $A_p\equiv 0$ for $p$
even and $B_p\equiv
0$ for $p$ odd, leading to the scaling law, derived previously by
Satz~\cite{Satz} \begin{eqnarray}
F_p(L;\ell ) & \sim & \left(\frac{\ell}{L}\right)^{-p\frac{\beta}{\nu}}
\qquad \mbox{for $p$ even}\\
  & \sim & \left(\frac{\ell}{L}\right)^{-(p-1)\frac{\beta}{\nu}}
\qquad \mbox{for $p$ odd}
\end{eqnarray}

For the three-state Potts model such a cancellation does not occur and the
leading behaviour $(\ell /L )^{-p\beta /\nu}$ is affected by a corrective term
as shown in eq. (2.10). However, similar corrections also appear in the
scaling laws of eqs (2.9), which, when taken into account, modify the exact
exponent $\beta
/\nu$ in eq.(2.10). Therefore, for the three-state Potts model, we expect a
more general form
\begin{equation}
F_p(L;\ell )   \sim   \left(\frac{\ell}{L}\right)^{-p\frac{\beta}{\nu}}
\left[A''_p \left(\frac{\ell}{L}\right)^{w} +B''_p\right]
\end{equation}
where $A''_p$, $B''_p$ and $w$ are unknown parameters.\\

{}From this behaviour we deduce the intermittency indices ~: \\

\noindent For the Ising model\\
\begin{eqnarray}
 \lambda_p & = & \frac{1}{8}\; p   \qquad \hspace{1.2cm} \mbox{for $p$ even}
\nonumber\\
	    & = & \frac{1}{8}(p-1) \qquad \mbox{for $p$ odd}
\end{eqnarray}
\noindent For the three-state Potts model\\
\begin{eqnarray*}
 \lambda_p & = & \frac{2}{15}\; p
\end{eqnarray*}


\section{Numerical tests}
We have performed a simulation of the $q=2$ and $q=3$ Potts models at the bulk
critical temperature, using the Swendsen
Wang dynamics, on two-dimensional lattices of size up to $256\times 256$. For
the largest lattice sizes, the data were taken on four independant runs
of $2\times 10^4$ Monte-Carlo lattice updates, with a spacing between two
consecutive measurements of five MC lattice updates. The error analysis is
realised on this total sample of $1.6\times 10^4$ independent
measurements, leading to quite small error bars (of the order of the width of
the data points). In order to
check the reliability of our simulation, we have tested the scaling law for the
order parameter, $<m_q >$ (eq. 2.8), for lattice sizes
ranging from $L=16$ up to $L=256$. The results are displayed in figure~1, with
a linear fit in log scale, giving $\beta /\nu = 0.124(5)$ for the Ising model
(the exact value is 1/8=0.125) and  $\beta /\nu = 0.131(7)$ for the
three-state Potts model (the exact value is 2/15=0.133).\\

We then determine the intermittency indices $\lambda_p$.
We measure the
moments for $p=2$ to $p=5$ and for $\ell =2,4,8,\ldots, L$. Notice
that the values of the moments for $\ell =L$  approach, as
$L\rightarrow\infty$, the ratio of universal
critical amplitudes. For instance, the ratio \mbox{$[F_2(L,\ell
=L)]^2/F_4(L,\ell =L)$}
 has been thoroughly studied~\cite{Binder} and found to have the value
0.85622 in the Ising
model~\cite{Blote}. Our result is 0.856(4) providing a further check of our
data.

In figure 2 we show the moments for the Ising model on a $256\times 256$
periodic lattice.  The behaviour of $F_p$ depending on
the parity of $p$ depicted by eqs (2.11-12) is quite visible on the data.
According to these equations, we have performed a power law fit of the
moments for $4\leq\ell \leq L/2$, shown as the straight lines in figure 2. We
observe that the odd moments are remarkably well fitted by the power law,
whereas the even moments exhibit deviations from this leading behaviour.
Therefore, we
proceed to a double determination of the exponents~: first, from the exact
power law of eqs.(2.11-12) and second, from a corrected parametrisation of the
form of eq.(2.13).
The results of both fitting methods for the largest lattice size $L=256$ are
shown in the last two lines of table I.
The difference between the values corresponding to the same moment
 give an estimate of the systematic error on the exponents. In addition, we
give the result of the power law fit for the $L=128$ lattice, which shows the
stability of this determination.

\begin{center}
\begin{tabular}{||c|c|c|c|c||}
\hline \hline
 & \multicolumn{4}{c|}{\em p}\\ \hline
$ L $ & 2 & 3 & 4 & 5\\
\hline
128 & 0.111 & 0.124 & 0.117 & 0.123\\
\hline
256 & 0.114 & 0.124 & 0.119 & 0.123\\
    & 0.135 & 0.126 & 0.129 & 0.122\\
\hline \hline
\end{tabular}\vspace{0.5cm}	 \\
  \end{center}
\begin{itemize}
\item[\underline{Table I}] The exponents $\lambda_p/p$ ($p$ even) and
$\lambda_p/(p-1)$ ($p$ odd) for the Ising model on
 lattices of size $L=128,256$ and for the moments of order $p=2,3,4,5$.
The exponents given in the bottom line correspond to a fit of
the $L=256$ moments with a corrected scaling law. The expected
exact value is $\beta /\nu =0.125$.\vspace{0.5cm}
\end{itemize}


The agreement of these results with the prediction of eqs.(2.14)
$\beta /\nu =0.125$, is qualitatively good for the even moments and
excellent for the odd ones.


We have repeated the same analysis for
 the three-state Potts model. The moments are shown in figure 3 for the
largest lattice size $L=256$. The fit corresponds to the corrected power law
fit of eq.(2.13)

 $$F_p(L;\ell )=\left(\frac{\ell}{L}\right)^{-px}
\left[ a\left(\frac{\ell}{L}\right)^{w}+b\right]$$
where $x,w,a ,b$ are free parameters.
The results are shown in  table II for the lattice size $L=256$.


 \begin{center}
\begin{tabular}{||c|c|rrrr||}
\hline \hline
L & p  & 2 & 3 & 4 & 5\\
\hline
  & $x$   & 0.141 & 0.124 & 0.131 & 0.125\\
  &	  &$\pm$ 0.015 & $\pm$ 0.009 & $\pm$ 0.009 & $\pm$ 0.011\\
\cline{2-6}
256    & $w$   & 0.359 & 0.270 & 0.307 & 0.235\\
\cline{2-6}
   & $a$ & 0.508 & 0.581 & 0.652 & 0.772\\
\cline{2-6}
   & $b$ & 0.516 & 0.503 & 0.521 & 0.560\\
\hline
128  & $x$   & 0.144 & 0.119 & 0.128 & 0.124\\
  &	  &$\pm$ 0.022 & $\pm$ 0.017 & $\pm$ 0.020 & $\pm$ 0.011\\
\hline \hline
\end{tabular}	  \vspace{0.5cm}   \\
\underline{TABLE II}
\end{center}

\begin{itemize}
\item[\underline{Table II}] The exponents $x=\lambda_p/p$ for the moments of
order $p=2,3,4,5$
for the three-state Potts model on the	$L=256$ lattice. The
expected exact value is $\beta /\nu = 0.133$.
We show the fitted values of the other variables
entering the parametrisation of the  moments
$F_p(z)=z^{-px}(az^{w}+b)$ where $z=\ell /L$. $x$ values from the $L=128$ data
are reported at the bottom line.\vspace{0.5cm}
\end{itemize}

The errors quoted are the  statistical ones corresponding to 95\% confidence
level. They
are quite large due to the freedom allowed by our parametrisation. Actually, we
observe that, if we set
the exponents to the exact value, $x=0.133$, the minimum $\chi^2$ does
not change significantly with respect to its best-fit value.
We give the other variables of our parametrisation  in Table II.
 They are only weakly dependent on the order of the
moment, but the exponents $w$ differ significantly from $x$, indicating, as
expected, a strong contamination of the behaviour of eq.(2.10) by corrections
to the scaling law of eq. (2.9). The exponents obtained from the fit of the
$L=128$ data are reported at the bottom line of table II, showing again the
stability of their determination.

Nevertheless, the agreement with the expected intemittency indices is again
quite good.

\section{Conclusions}
We have defined a block observable for the $q$-state Potts model which exhibits
an intermittent behaviour at the critical point of the model. The moments
of this observable are measurable on finite lattices and have well-defined
scaling behaviour in terms of the block-size. By a numerical simulation, we
confirm the previously observed~\cite{Wosiek,Bambah,Gupta} intermittent
behaviour, and we find
good agreement between the measured exponents and the magnetic exponent $\beta
/\nu$, as predicted by renormalisation group arguments.

Although a critical spin system is presumably not multifractal,
a complete description of its scaling properties requires several exponents.
Besides the thermal and
magnetic critical indices which drive the fluctuations of energy and of
magnetisation, the geometrical aspects of the critical clusters of spins are
characterised by other exponents, not related to the thermodynamics
ones~\cite{Coniglio,VDZande,Knops}. For this reason, the intermittency
phenomenon
in a critical system may assume several aspects depending on the definition of
the block observable.
 This may explain the results of ref~\cite{Gupta},
where the measured intermittency indices coincide with the fractal dimension of
the clusters of spins. However, further theoretical and numerical studies are
needed in order to investigate the geometrical origin of intermittency.\\

\noindent {\Large {\bf Acknowledgements}}\\

\noindent The author thanks R. Peschanski for several valuable discussions and
A. Morel
for his invitation to the Service de Physique Th\'eorique at Saclay (CEA,
France
where this work began. Thanks are also due to E. Pommiers for decisive
contribution in the numerical simulation.


\newpage

\begin{thebibliography}{99}
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references therein.
\bibitem{Wosiek} J. Wosiek, {\em Acta Phys. Pol.} {\bf B19} (1988)863
\bibitem{Satz} H. Satz,{\em  Nucl. Phys.} {\bf B326}(1989)613
\bibitem{Bambah} B. Bambah, J. Fingberg, H. Satz {\em Nucl. Phys.}
 {\bf B332}(1990)629
\bibitem{Gupta} S. Gupta, P. Lacock, H. Satz {\em Nucl. Phys.}{\bf
B362}(1991)583
\bibitem{Zalewski} Z. Burda, J. Wosiek, K. Zalewski {\em Phys. Lett.} {\bf
B266}(1991)439
 \bibitem{Wu} F.Y. Wu {\em Rev. Mod. Phys.} {\bf 54}(1982)235
\bibitem{Hajdu} This is an explicit realisation of the formal definition given
by D. Hajdukovi\'{c}, CERN preprint CERN-TH.6440/92
\bibitem{Binder}K. Binder,{\em	Z. Phys.} {\bf B43} (1981) 119
\bibitem{Blote}G. Kamieniarz, HW Bl\"{o}te
   {\em J. Phys.} {\bf A26} (1993)201
\bibitem{Coniglio} A. Coniglio, C. Nappi, F. Peruggi and L. Russo {\em J.
Phys.} {\bf A10} (1977)205
\bibitem{VDZande}C. Vanderzande and A. Stella, {\em J. Phys} {\bf
A22}(1989)L445, C. Vanderzande, {\em J. Phys.} {\bf A25}(1991)L75
\bibitem{Knops}H.W.Bl\"{o}te,
    Y.M.M. Knops and B. Nienhuis {\em Phys. Rev.
Let.} {\bf 68} (1992) 3440

\end{thebibliography}


\newpage

\noindent {\Large {\bf Figure Captions}}\\
\begin{itemize}
\item[\underline{Figure 1}] The order parameter for the $q=2$ (Ising) and
$q=3$ Potts
model, measured at the bulk critical temperature as a function of the lattice
size. The lines result from a power fit, giving the exponent $\beta
/\nu=0.124(5)$ for the Ising model (exact value 0.125) and $\beta /\nu
=0.131(7)$ for the three-state Potts model (exact value 0.133).

\item[\underline{Figure 2}] The moments of order $p=2,3,4,5$ for the Ising
model as a
function of the block size~$\ell$. The straight lines result from a power law
fit.
\item[\underline{Figure 3}] The moments of order $p=2,3,4,5$ for the
three-state Potts model as a
function of the block size $\ell$. The curves result from a corrected power law
fit.
\end{itemize}

 \end{document}
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%%%%%%%%%  Here begins the postscript file of FIGURE 3 %%%%%%%%%%%%%%%%%%
%!PS-Adobe-2.0
%%Title: /F3       LISTPS   B1 (Portrait A 4)
%%Pages: atend
%%Creator: HIGZ Version 1.15/02
%%CreationDate: 17/03/93   11.30
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 564 959 l 699 987 l 795 1008 l 869 1026 l 930 1040 l 981 1053 l 1025 1065 l
 1065 1075 l 1100 1085 l 1131 1093 l 1160 1101 l 1187 1109 l 1212 1116 l 1235
 1123 l 1256 1129 l 1276 1135 l 1295 1140 l 1313 1146 l 1347 1156 l 1362 1160 l
 1377 1165 l 1405 1173 l 1418 1177 l 1454 1189 l 1476 1195 l 1487 1199 l 1507
 1205 l 1516 1208 l 1526 1211 l 1544 1217 l 1552 1220 l 1561 1223 l 1569 1225 l
 1577 1228 l 1585 1230 l 1593 1233 l 1600 1235 l 1607 1238 l 1615 1240 l 1622
 1242 l 1628 1245 l 1635 1247 l 1642 1249 l 1648 1251 l 1655 1253 l 1685 1263 l
 1690 1265 l 1696 1267 l 1701 1269 l 1707 1271 l 1712 1273 l 1717 1274 l 1722
 1276 l 1728 1278 l 1733 1280 l 1737 1281 l 1747 1285 l 1752 1286 l 1756 1288 l
 1761 1289 l 1766 1291 l 1770 1292 l 1774 1294 l 1779 1295 l 1783 1297 l 1787
 1298 l 1792 1300 l 1796 1301 l 1800 1303 l 1808 1305 l 1812 1307 l 1816 1308 l
 1819 1309 l 1823 1311 l 1831 1313 l 1834 1315 l 1842 1317 l 1845 1318 l 1849
 1320 l 1852 1321 l 1856 1322 l 1862 1324 l 1866 1326 l 1869 1327 l 1872 1328 l
 1876 1329 l 1885 1332 l 1888 1334 l 1894 1336 l 1898 1337 l 1904 1339 l 1906
 1340 l 1924 1346 l 1926 1347 l 1935 1350 l 1937 1351 l 1943 1353 l 1945 1354 l
 1951 1356 l 1953 1357 l 1956 1358 l 1958 1358 l 1961 1359 l 1963 1360 l 1966
 1361 l 1968 1362 l 1971 1363 l 1975 1365 l 1978 1365 l s 333 947 m 333 947 l
 564 1031 l 699 1082 l 795 1120 l 869 1149 l 930 1174 l 981 1195 l 1025 1213 l
 1065 1229 l 1100 1244 l 1131 1257 l 1160 1269 l 1187 1281 l 1212 1291 l 1235
 1301 l 1256 1310 l 1276 1319 l 1295 1327 l 1313 1335 l 1330 1343 l 1347 1350 l
 1362 1357 l 1377 1363 l 1405 1375 l 1418 1381 l 1430 1387 l 1454 1397 l 1487
 1412 l 1497 1417 l 1507 1421 l 1516 1425 l 1526 1430 l 1544 1438 l 1552 1442 l
 1561 1446 l 1569 1449 l 1577 1453 l 1585 1456 l 1593 1460 l 1600 1463 l 1607
 1467 l 1615 1470 l 1622 1473 l 1628 1476 l 1635 1479 l 1642 1483 l 1648 1485 l
 1655 1488 l 1679 1500 l 1685 1502 l 1690 1505 l 1696 1508 l 1701 1510 l 1707
 1513 l 1717 1517 l 1722 1520 l 1728 1522 l 1733 1525 l 1737 1527 l 1747 1531 l
 1752 1534 l 1756 1536 l 1766 1540 l 1774 1544 l 1779 1546 l 1787 1550 l 1792
 1552 l 1816 1564 l 1819 1565 l 1831 1571 l 1834 1572 l 1842 1576 l 1845 1578 l
 1849 1579 l 1852 1581 l 1856 1583 l 1859 1584 l 1862 1586 l 1866 1587 l 1869
 1589 l 1872 1591 l 1876 1592 l 1879 1594 l 1882 1595 l 1885 1597 l 1888 1598 l
 1891 1600 l 1894 1601 l 1898 1602 l 1901 1604 l 1904 1605 l 1906 1607 l 1909
 1608 l 1912 1610 l 1918 1612 l 1921 1614 l 1924 1615 l 1926 1616 l 1929 1618 l
 1935 1620 l 1937 1622 l 1943 1624 l 1945 1625 l 1948 1627 l 1951 1628 l 1953
 1629 l 1956 1630 l 1958 1632 l 1961 1633 l 1963 1634 l 1966 1635 l 1968 1636 l
 1971 1638 l 1975 1640 l 1978 1641 l s 333 988 m 333 988 l 564 1118 l 699 1197
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 795 1254 l 869 1299 l 930 1336 l 981 1368 l 1025 1396 l 1065 1421 l 1100 1443
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 1131 1463 l 1160 1482 l 1187 1499 l 1212 1515 l 1235 1529 l 1256 1543 l 1276
 1557 l 1295 1569 l 1313 1581 l 1347 1603 l 1377 1623 l 1405 1641 l 1418 1650 l
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 1735 l 1552 1741 l 1561 1746 l 1577 1758 l 1593 1768 l 1607 1778 l 1615 1783 l
 1622 1788 l 1628 1793 l 1635 1797 l 1642 1802 l 1648 1806 l 1655 1811 l 1685
 1831 l 1690 1835 l 1696 1839 l 1701 1843 l 1707 1847 l 1712 1850 l 1722 1858 l
 1728 1861 l 1733 1865 l 1737 1868 l 1742 1871 l 1747 1875 l 1752 1878 l 1756
 1881 l 1761 1884 l 1766 1888 l 1774 1894 l 1779 1897 l 1787 1903 l 1792 1906 l
 1796 1909 l 1800 1911 l 1816 1923 l 1819 1925 l 1827 1931 l 1831 1933 l 1834
 1936 l 1838 1938 l 1842 1941 l 1845 1943 l 1849 1946 l 1852 1948 l 1856 1951 l
 1859 1953 l 1862 1956 l 1866 1958 l 1869 1960 l 1872 1963 l 1876 1965 l 1882
 1969 l 1885 1972 l 1894 1978 l 1898 1980 l 1901 1982 l 1906 1987 l 1924 1999 l
 1926 2001 l 1935 2007 l 1937 2009 l 1940 2011 l 1943 2012 l 1945 2014 l 1951
 2018 l 1953 2020 l 1956 2022 l 1958 2023 l 1961 2025 l 1963 2027 l 1966 2029 l
 1968 2031 l 1971 2032 l 1975 2036 l 1978 2037 l s 333 1034 m 333 1034 l 564
 1213 l 699 1320 l 795 1396 l 869 1455 l 930 1504 l 981 1546 l 1025 1582 l 1100
 1642 l 1131 1668 l 1160 1692 l 1187 1714 l 1212 1734 l 1235 1753 l 1256 1771 l
 1276 1788 l 1295 1804 l 1313 1819 l 1330 1833 l 1347 1846 l 1377 1872 l 1391
 1883 l 1405 1895 l 1418 1906 l 1454 1936 l 1465 1946 l 1487 1964 l 1497 1972 l
 1507 1981 l 1516 1989 l 1526 1997 l 1535 2004 l 1544 2012 l 1552 2019 l 1561
 2026 l 1585 2047 l 1593 2053 l 1600 2060 l 1607 2066 l 1615 2072 l 1622 2078 l
 1628 2084 l 1635 2089 l 1642 2095 l 1648 2101 l 1655 2106 l 1667 2116 l 1673
 2122 l 1685 2132 l 1690 2136 l 1696 2141 l 1701 2146 l 1707 2151 l 1712 2155 l
 1717 2160 l 1722 2164 l 1728 2168 l 1737 2177 l 1752 2189 l 1756 2193 l 1766
 2201 l 1774 2209 l 1779 2212 l 1787 2220 l 1792 2223 l 1796 2227 l 1800 2230 l
 1804 2234 l 1808 2237 l 1812 2241 l 1816 2244 l 1823 2251 l 1831 2257 l 1834
 2260 l 1842 2266 l 1852 2276 l 1856 2279 l 1859 2281 l 1862 2284 l 1866 2287 l
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 1924 2337 l 1926 2340 l 1929 2342 l 1932 2345 l 1935 2347 l 1940 2352 l 1943
 2354 l 1945 2356 l 1948 2358 l 1953 2363 l 1956 2365 l 1963 2372 l 1966 2374 l
 1968 2376 l 1971 2378 l 1975 2382 l 1978 2384 l s
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