%Paper: hep-th/9506174
%From: wynter@peterpan.ens.fr (WYNTER Tom)
%Date: Tue, 27 Jun 95 00:40:47 +0200
%Date (revised): Tue, 27 Jun 95 16:24:19 +0200

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\Title{\vbox{\baselineskip12pt
\vbox{\hbox{
\hfill\hbox{LPTENS-95/24}}}}}{Almost Flat Planar
Diagrams}
%
\vskip -120pt {\it in memoriam Claude Itzykson} \vskip 80pt
\bigskip
\bigskip
\centerline{Vladimir A. Kazakov}
\smallskip
\centerline{Matthias Staudacher $^\dagger$}
\smallskip
\centerline{{\it and}}
\smallskip
\centerline{Thomas Wynter $^\dagger$}\footnote~{
\hskip -11.5pt $^\dagger$ \hskip 4pt
This work is supported by funds provided by the European Community,
Human Capital and Mobility  Programme.}
\bigskip
\centerline{Laboratoire de Physique Th\'eorique de}
\centerline{l'\'Ecole Normale Sup\'erieure \footnote*{
Unit\'e Propre du
Centre National de la Recherche Scientifique,
associ\'ee \`a l'\'Ecole Normale Sup\'erieure et \`a
l'Universit\'e de Paris-Sud.}}
\bigskip
\noindent
We continue our study of matrix models of dually weighted graphs.
Among the attractive features of these models is the possibility to
interpolate between ensembles of regular and random two-dimensional
lattices, relevant for the study of the crossover from two-dimensional
flat space to two-dimensional quantum gravity.  We further develop the
formalism of large $N$ character expansions. In particular, a general
method for determining the large $N$ limit of a character is
derived. This method, aside from being potentially useful for a far
greater class of problems, allows us to exactly solve the matrix
models of dually weighted graphs, reducing them to a well-posed
Cauchy-Riemann problem.  The power of the method is illustrated by
explicitly solving a new model in which only positive curvature
defects are permitted on the surface, an arbitrary amount of negative
curvature being introduced at a single insertion.
\Date{June 1995}
%
\nref\BIPZ{E.~Br\'ezin, C.~Itzykson, G.~Parisi \& J.-B.~Zuber, Commun.
Math. Phys. 59 (1978), 35.}
\nref\IDiF{P.~Di~Francesco \& C. Itzykson, Ann. Inst. Henri.
Poincar\'e Vol. 59, no. 2 (1993) 117.}
\nref\ITZUB{C.~Itzykson \& J.-B.~Zuber, J.~Math.~Phys.~21(3) (1980) 411.}
\nref\KSW{V.A.~Kazakov, M.~Staudacher \& T.~Wynter, \'Ecole Normale
preprint LPTENS-95/9, accepted for publication in Commun.~Math.~Phys.}
\nref\DAVID{F.~David, Nucl. Phys. B257 (1985) 45.}
\nref\VOL{V.A.~Kazakov, Phys. Lett. B150 (1985) 282.}
\nref\FRO{J.~Fr\" ohlich, in: Lecture Notes in Physics, Vol. 216,
Springer, Berlin, 1985; \hfill\break
J.~Ambj{\o }rn, B.~Durhuus and J.~Fr\" ohlich,
Nucl. Phys. B257[FS14](1985) 433.}
\nref\BYRD{P.F.~Byrd \& M.D.~Friedman, ``Handbook of Elliptic
Integrals for Engineers and Physicists'', Springer, Berlin, 1954.}
\nref\LAWD{D.F.~Lawden, ``Elliptic Functions and Applications'',
Springer, New York, 1989.}

\newsec{Introduction}

Hermitian one matrix models were introduced and for the first time
solved in the large $N$ limit in the seminal paper by
Br\'ezin, Itzykson, Parisi and Zuber \BIPZ. These models generate
ensembles of planar, random graphs whose vertex coordination numbers
are controlled by the matrix potential. By varying the potential,
different classes of diagrams may be obtained, e.g.~random
square or random triangular lattices. However, despite this freedom,
there is a class of physically important lattices that can not
be generated by simply tuning the potential: {\it regular}, flat lattices
with fixed coordination numbers of both vertices and faces.
To attain them it is necessary to study planar
graphs having coordination number dependent weights for both the
vertices and faces. It is straightforward to define modified
hermitian matrix models producing such graphs, but they can no longer
be treated with the methods of \BIPZ. In fact, until very recently
this class of models of dually weighted graphs seemed
intractable. However, an important but little
noticed observation due to Itzykson and Di~Francesco \IDiF\ has made
possible the explicit treatment of dually weighted graphs.
The number of degrees of freedom of these models is crucially
reduced by rewriting the model in the language of group theory.
It should also be noted that this method, based on expanding the
matrix model potential in Weyl characters, was already used
presciently in a special case in another early paper by
Itzykson and Zuber \ITZUB.
In a recent work \KSW\ we demonstrated that this new approach leads
indeed to a problem amenable to mathematical analysis once the
large $N$ limit is taken.

The physical importance of matrix models has been elucidated through
a large body of work over the last ten years.
In \DAVID\ \VOL\ matrix models were first introduced to furnish a description
of two-dimensional quantum gravity and non-critical bosonic
strings and successfully used to calculate the critical properties
of these theories.
This approach is based on the representation of the sum over
world-sheet metrics as a sum over dynamical triangulations as originally
proposed in \DAVID\ \VOL\ \FRO. Studying the crossover from random, dynamical
graphs to regular, static graphs, then, will correspond to suppressing
the curvature fluctuations of the world-sheet metric and result in
a flat two-dimensional metric. Our work, in conjunction with \KSW,
should thus be seen as representing a first attempt towards establishing
a connection between integrable two-dimensional models both coupled to
and decoupled from quantum gravity.

To be precise, let us consider general planar graphs
and introduce a set of couplings
$t^*_1,t^*_2,...t^*_q,...\;$, namely the weights of vertices with
$1,2,...,q,...$
neighbours, and a dual set $t_1,t_2,...t_q,...\,$,
the weights of the dual
vertices (or faces) with appropriate coordination numbers.
The partition function of closed planar graphs $G$ is defined to be
%
\eqn\DWG{
Z(t^*,t)=
\sum_{G} \prod_{v^*_q,v_q \in G} {t_q^*}^{\# v^*_q}\ {t_q}^{ \# v_q}}
%
where $v_q^*,v_q$ are the vertices with $q$ neighbours on the original
and dual graph, respectively, and $\# v_q^*,\# v_q$ are the numbers of
such vertices in the given graph $G$.
Choosing $t_q^*=t_q=\delta_{q,4}$ the only
allowed graphs are regular square lattices (see Fig.~1.a).
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\put{$t_4$} [lB] at  1.812 24.989
\put{(a) flat space} [lB] at  0.358 22.864
\put{(b) positive curvature} [lB] at  3.319 22.693
\put{(c) negative curvature } [lB] at  8.162 22.589
\put{$t_2^*$} [lB] at  4.096 25.066
\put{$t_2$} [lB] at  5.836 25.066
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\endpicture
\vskip 15pt
\centerline{{\bf Fig. 1:} Flat space and curvature defects}
\vskip 20pt
%
However, it is easy to see that a regular square lattice cannot be of
spherical (i.e.~planar) topology.  Positive curvature defects have to
be added in order to be able to close the surface. Considering for the
moment only even couplings, we must therefore ``turn on''
couplings $t_2$ or $t^*_2$, or both (see Fig.~1.b).  Exactly four such
defects are needed to close the square lattice.  Adding more
defects, then requires balancing the total curvature by also adding
negative curvature defects.  The simplest examples for such negative
defects, corresponding to the couplings $t_6$ and $t_6^*$, are shown
in Fig.~1.c.  Allowing for an arbitrary number of positive and
negative curvature defects we expect to generate random graphs which,
at critical values of the couplings, corresponding to very big graphs
dominating the sum in the partition function \DWG , allow us to reach
a continuum limit lying in the universality class of pure
two-dimensional quantum gravity \DAVID\ \VOL. On the other hand, having
``tuned away'' the negative curvature couplings $t_q$, $t_q^*$ with $q >
4$, no such continuum limit is possible. Then, only a small, finite number
of positive curvature defects are allowed; this brings us back to the phase of
essentially flat surfaces. The main physical motivation for studying
the models of dually weighted graphs, then, is to understand the
transition between these two very distinct phases.

In the present paper we continue to develop powerful techniques which
permit us to address this physical problem. Furthermore, we will
present the full and explicit solution of a non-trivial problem: the
case of flat, planar graphs with an arbitrary number of positive
curvature defects and a single negative curvature defect (see
Fig.~2(a)) adjusted to balance the total curvature.  We
call the resulting lattice surfaces ``almost flat planar diagrams''. A
typical surface of this type is shown in Fig. 2(b).

This model illustrates a non-trivial example that can be solved by the
method presented in this paper. This model cannot currently be solved
by standard matrix model techniques
%
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	/
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	/
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	/
\plot 10.952 24.270 	10.873 24.233
	10.816 24.206
	10.746 24.166
	10.695 24.124
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	/
\put{$t_{2q}$} [lB] at  2.656 24.066
\put{(a)} [lB] at  2.557 21.400
\put{(b)} [lB] at  8.867 21.421
\linethickness=0pt
\putrectangle corners at  1.657 25.775 and 11.144 21.323
\endpicture
\vskip 20pt
\centerline{{\bf Fig. 2}
     (a) Negative curvature defect of angle $(2-q)\pi$ and
     (b) a typical surface.}
\vskip 20pt
%

It should be stressed that the methods we develop here are general and
could have applications going beyond the problem under
investigation. Given that the model of dually weighted graphs
seemed entirely inaccessible even a short while ago, we regard the
present approach to be an important step in extending current large
$N$ techniques.

We will quickly recall in the next section some of the results of our
previous paper \KSW\ and precisely define the class of models we are
studying.  Then, in section 3, we demonstrate how to derive the large
$N$ limit of group theoretical characters. The model of almost flat
planar diagrams will be solved and interpreted in section 4. The full
model capturing the transition from flat to random graphs
will be briefly discussed in section 5. We demonstrate how to
reformulate it as a well-posed Cauchy-Riemann problem.  We conclude in
section 6 and present an outlook on how our
approach might be put to further use in the near future.  Technical
details and additional illustrations are included in two
appendices.

\newsec{Review of the character expansion method for matrix models
of dually weighted graphs}

The partition function \DWG\ for dually weighted graphs can be
formulated as the following matrix model:
%
\eqn\DWGmatrix{
Z(t^*,t)=\int\,{\cal D}M\ e^{-{N\over 2} \Tr~M^2\ +\ \Tr~V_B(M A)},}
%
with
%
\eqn\potential{
V_B(M A)=
\sum_{k=1}^{\infty}{1\over k}~\Tr B^k\ (M A)^k .}
%
The matrices $A$ and $B$ are fixed, external matrices encoding the
coupling constants through
%
\eqn\tqAB{
t_q^*= {1\over N}\ \Tr\ B^q
{\rm \hskip 20pt and \hskip 20pt}
t_q=  {1\over N}\ \Tr\ A^q.}
%
The model generalizes, for $A \neq 1$, the standard one matrix model
first solved by Br\'ezin, Itzykson, Parisi and Zuber \BIPZ.
It can no longer be solved by changing to
eigenvalue variables; a reduction to $N$ variables is nevertheless
possible. An expansion of the potential into a sum over invariant
group characters allows all integrations to be performed and \DWGmatrix\
to be reformulated as a statistical mechanics model in ``Young-tableau
weight space''. This reformulation should be called, after its discoverers,
the ``Itzykson-Di~Francesco formula'' \IDiF\ and reads
%
\eqn\IzDiFr{
Z(t,t^*)=c\,\sum_{\{h^e,h^o\}}
{\prod_i(h^e_i-1)!!h^o_i!!\over
\prod_{i,j}(h^e_i-h^o_j)}~\chi_{\{h\}}(A)~\chi_{\{h\}}(B)}
%
Here $c$ is a constant that we can drop, the weights
$\{h^e\}$ are a set of $N/2$ even, increasing, non-negative
integers while the weights $\{h^o\}$ are $N/2$ odd, increasing, positive
integers, and the sum is taken over all such sets. The characters can be
defined through two equivalent formulae. The first is the Weyl formula:
%
\eqn\weylchar{
\chi_{\{h\}}(A)={det_{_{\hskip -2pt (k,l)}}(a_k^{h_l})\over
\Delta(a)},}
%
where the $a_k$ are the eigenvalues of the matrix $A$ and
$\D(a)$ is the Vandermonde determinant. The second definition makes
use of Schur polynomials, $P_n(\theta)$, defined by
%
\eqn\schupol{
e^{\Sigma_{i=1}^{\infty}z^i\theta_i}=\sum_{n=0}^{\infty}z^n
P_n (\theta)\quad {\rm with}\quad \theta_i={1\over i}~\Tr[A^i],}
%
in terms of which the character is
%
\eqn\schuchar{
\chi_{\{h\}}(A)=det_{_{\hskip -2pt (k,l)}}
                 \bigl(P_{h_k+1-l}(\theta)\bigr).}
%

It was demonstrated in \KSW\ how to take the large $N$ limit of this
expansion. In this limit, the weights ${1 \over N} h_i$ condense to
give a smooth, stationary distribution $dh~\rho(h)$, where $\rho(h)$
is a probability density normalized to one.  For technical reasons we
restrict our attention to models in which the matrices $A$ and $B$ are
such that traces of all odd powers of $A$ and $B$ are zero. This means
that the our random surfaces are made from vertices and faces with
even coordination numbers only. As was discussed in \KSW, this ensures
that the support of the density $\rho(h)$ lies entirely on the real
axis, and thus simplifies the solution of the problem\foot{ We do not
want to suggest that models with odd coordination numbers cannot be
treated with our methods.}.  The matrix $A$ will
satisfy this condition if we introduce an ${N\over 2}\times{N\over 2}$
matrix $\sqrt{a}$ in terms of which $A$ and the character
$\chi_{\{h\}}(A)$ are given by
%
\eqn\Aa{
A=\left[\matrix{\sqrt{a}&0\cr 0&-\sqrt{a}\cr}\right]\quad {\rm and} \quad
\chi_{\{h\}}(A)=
    \chi_{\{{h^e\over 2}\}}(a)\chi_{\{{h^o-1\over 2}\}}(a)
    \,\,\sgn\bigl[\prod_{i,j}(h^e_i-h^o_j)\bigr],}
%

We now focus our attention on three intimately related models which
capture the transition from flat to random graphs.
%
\eqn\models{\eqalign{
& {\rm I. }~V_A(MA)
=\sum_{k=1}^\infty {1\over 2k}~\Tr[A^{2k}]~(MA)^{2k}. \cr
& {\rm II. }~V_{A_4}(MA) ={1\over 4}~(MA)^4. \cr
& {\rm III. }~V_A(MA_4)
=\sum_{k=1}^\infty{1\over 2k}~\Tr[A^{2k}]~(MA_4)^{2k} \cr }}
%
Here $A_4$ is defined to satisfy $\Tr[(A_4)^k]=N \delta_{k,4}$ and $A$
is as defined in \Aa.  The first model is self-dual, i.e.~vertices
and faces having the same coordination number have the same
weights. The second and third models are dual to each other (the
lattice of one corresponds to the dual lattice of the other) and are
in turn related to model I by a simple line map. That is, we place the
diagonal of a square belonging to model III (or alternatively a
four-vertex belonging to II) onto each propagator of model I. Thus the
vertices and face centres of model I become the vertices of model III
(or alternatively the faces of II). We illustrate this in Fig.~3
below\foot{ Note that this line-map is only valid on the sphere. The
${1 \over N}$ corrections of I and III will thus be different.  A
careful analysis shows that the spherical free energy of model I is
precisely twice the free energy of models II and III (since there are
two ways of choosing the diagonal of a square in III, or alternatively
two ways of splitting a four-vertex of model II). Note also that this
non-trivial correspondence is {\it predicted} from our formalism,
since we indeed obtain the same $N=\infty$ equations in all three
cases.}.
%
%
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\vskip 20pt
\centerline{{\bf Fig. 3:} Graphical relationship between models I, II
and III}
\vskip 20pt
%
%
\hskip -19pt {}From this line map one can see that the
expectation values in models I and III are also the same. More
specifically
%
\eqn\expsam{
\langle{1\over N}\Tr[(MA)^{2 k}]\rangle_{I}=
\langle{1\over N}\Tr[(MA_4)^{2 k}]\rangle_{III}}
%
Notice, however, that they are not equivalent to
$\langle{1\over N}\Tr[(MA)^k]\rangle_{II}$ in model II.

We can now return to the discussion of the large $N$ limit and write
the saddlepoint equation for these three models. Looking for the
stationary point in \IzDiFr, one finds from \KSW , in all three cases,
the following equation, valid on an interval $[b,a]$ with $0 \leq b
\leq 1 \leq a$:
%
\eqn\sdpt{
2F(h)+\barint_0^a\ dh'\ {\rho(h') \over h-h'}= -\ln h.}
%
The solution requires, evidently, the knowledge of the large $N$ limit of
the variation of the characters in eq.\Aa:
%
\eqn\defF{
F(h_k)=2{\partial \over \partial h^e_k}~\ln\ {\chi_{\{{h^e\over 2}\}}(a)
\over \Delta(h^e)}.}
%
The determination of $F(h)$ is the subject
of the next section. Let us also recall here the definition
of the resolvent $H(h)$:
%
\eqn\defH{
H(h)=\int_0^a dh'\ {\rho(h') \over h-h'}.}
%
In \KSW\  we demonstrated, via a simple functional inversion, how to
relate the results of the weight formalism to the resolvent
$W(P)=\langle {1\over N} \Tr {1\over P-M}\rangle$ of the matrix model
\DWGmatrix . In the model investigated in this paper, however, it is more
natural to study the correlators $\langle {1\over N} \Tr ((MA)^{2q})
\rangle$ . The results of the following section will provide a simple way
to calculate such moments.

\newsec{Large $N$ limit of the character}
In the saddle point equation \sdpt\ we
introduced the function $F(h)$ defined in eq. \defF\ as the
derivative of the logarithm of a character.
This function $F(h)$ depends upon the moments of the matrix
$A$, i.e.~it contains all the information
on the weights that one assigns to the faces of our discrete
surfaces. In order to proceed with the solution of the saddle point
equation, one would like to take the large $N$ limit of
\defF\ and express $F(h)$ in terms of $H(h)$
(which specifies the Young tableau) and the set of moments $t_{q}$ of
the matrix $A$ (the weights assigned to the faces).

In \KSW\ a contour integral formula
relating $H(h)$, $F(h)$ and the set of moments $t_{q}$ was derived. We
recall  here a single essential step of the derivation, which we will
make use of shortly. We observed that
%
\eqn\trachar{
\Tr[a^q]=\sum_{k=1}^{N/2}{\chi_{\{{\tilde h^e\over 2}\}}(a)\over
                          \chi_{\{{h^e\over 2}\}}(a)}\quad {\rm where}
\quad \tilde h^e_i=h^e_i+2q\delta_{i,k},}
%
and the matrix $a$ is the ${N\over 2}\times {N\over 2}$ matrix
introduced in \Aa .  For notational simplicity we omit an index $k$ on
$\tilde h$.  In the large $N$ limit \trachar\ was then reduced to a simple
contour integral
%
\eqn\tqHF{
t_{2q}={1\over q}\oint\,{dh\over 2\pi i} e^{q(H(h)+F(h))}\quad {\rm where}
                 \quad t_{2q}={2\over N}\Tr[a^q].}
%
Note that the definition of $F(h)$ \defF\ differs from that in the
derivation in \KSW\ since we are now restricting our attention to the
case where only the even moments of the matrix $A$ are non-zero\foot{
Indeed, it might be asked why we do not directly use formula (3.5)
derived in section 3.~of
\KSW. There, the contour integration relation was
derived for the general case where both even and odd moments are
non-zero. However, in the special case where we then set all odd
moments to zero, $e^{F(h)}$ contains a cut overlapping with the
cut of $e^{H(h)}$. In this case defining the
contour encircling the cut of $e^{H(h)}$ is ambiguous.
We have therefore
rederived the result for the reduced case of only even non-zero
moments. The same note of caution applies to formula (3.8) of \KSW.}.

As it stands, formula \tqHF\ is of little direct use. It can however be
dramatically simplified as we sketch out below. We introduce a function $G(h)$
defined as
%
\eqn\defG{
G(h) = e^{H(h) +F(h)},}
%
in terms of which \tqHF\ becomes
%
\eqn\tqG{
t_{2q}={1\over q}\oint\,{dh\over 2\pi i}~G(h)^q.}
%
Changing integration variables from $h$ to $G$ we arrive at
%
\eqn\tqGcont{
t_{2q}=\oint\,{dG\over 2\pi i G}~h(G)~G^q,}
%
where $h(G)$ is the inverse of the equation for $G(h)$ given in
\defG, and the contour in the complex $G$ plane encircles the
origin. We now assume that there are only a finite number of non-zero
couplings $t_q$. We obtain immediately the solution:
%
\eqn\hG{
h-1 = \sum_{q=1}^Q{t_{2q}\over G^q}~+~\psi(G).}
%
Here $\psi(G)$ is an as yet unknown function, analytic in the vicinity
of the origin, with $\psi(0)=0$.  It is trivial to see that this
satisfies \tqGcont. Note that, strictly speaking, we can not solve
equation \tqGcont\ for $q=0$ since \tqG\ is not defined there. The $1$
on the l.h.s. of \hG\ comes from the
normalization of the density $\rho(h)$ (See appendix A).

The unknown function $\psi(G)$ is not fixed by \tqG\ and depends on
the specific model being studied. We now give a very simple physical
interpretation to this function. Let us return to the Schur polynomial
definition of the character \schuchar\ . Differentiating eq. \schupol\
with respect to $\theta_i$ we see that
%
\eqn\difpoly{
{\partial\over\partial\theta_q}P_n(\theta)=P_{n-q}(\theta)\quad
{\rm with}\quad \theta_q={N\over 2q}t_{2q}.}
%
This implies immediately that
%
\eqn\difchar{
{2q\over N}{\partial\over \partial t_{2q}}\ln\Bigl(
\chi_{\{{h^e\over 2}\}}(a)\Bigr)=
\sum_{k=1}^{N/2}{\chi_{\{{\tilde h^e\over 2}\}}(a)\over
                          \chi_{\{{h^e\over 2}\}}(a)}\quad {\rm where}
\quad \tilde h^e_i=h^e_i-2q\delta_{i,k}.}
%
{}From \DWGmatrix, \IzDiFr\ and \Aa ,  we see that the left hand side
of this equation is equivalent to differentiating the
logarithm of the original matrix integral \DWGmatrix\ with respect to
$t_{2q}$. In terms of the dual to this matrix integral (in which the
weights $t_{2q}$ assigned to the faces are now the weights of the
vertices) this is equivalent to differentiating the coupling
constants of the dual potential. So, denoting the dual matrix by
$\tilde M$, the left hand side of eq. \difchar\ is equivalent to
the expectation value $\langle\Tr (\tilde MB)^{2q} \rangle$. Now,
comparing the right hand side
of \difchar\ to equations \trachar\ and \tqHF , we
see that we have the following relation in the large $N$ limit,
%
\eqn\expcont{
\langle {1\over N}\Tr (\tilde MB)^{2q} \rangle=
\oint\,{dG\over 2\pi i G}~h(G)~G^{-q},}
%
$G(h)$ being defined by \defG . It is now simple to follow
identical arguments to those used to simplify \tqG\ to \hG\ to arrive
at
%
\eqn\hGexp{
h-1 = \sum_{q=1}^Q{t_q\over G^q} +
\sum_{q=1}^{\infty}\langle {1\over N}\Tr~(\tilde MB)^{2q} \rangle
{}~G^q.}
%
Given $G(h)$, we have,
after a functional inversion, the correlators of the dual model.

To find $G(h)$ we have to connect eq. \hGexp\ with the saddle point
equation \sdpt .  From
\defH\ we obtain
%
\eqn\Hdef{
H(h)=\ln {h\over h-b}+ \tilde H(h)
\quad{\rm with}\quad \tilde H(h)=\int_b^a\,dh' {\rho(h')\over h-h'},}
%
where the first term on the right is the contribution from the flat
part of the density, i.e.~the empty part of the Young tableau.
The integral from $b$ to $a$ is the contribution from the
``excited'' part of the density, i.e.~the non-empty part of the Young
tableau. Noting, from the definition of $G(h)$ \defG ,
that $\ln G(h)=H(h) + F(h)$, we replace the
integral of \Hdef\ by the contour integral
%
\eqn\Hinva{
\tilde H(h)=\oint\,{dh' \over 2\pi i}~{\ln G(h')\over h-h'}}
%
where the contour encircles the $[b,a]$ part of the cut of $H(h)$. The
discontinuity across this cut is precisely $\pm i\pi\rho(h)$. Note
also that $F(h)$ has -- at least for some range of the couplings --
no cut on the interval $[b,a]$. If we
now change the variables of integration from $h$ to $G$, as previously,
and shrink the contour in the complex $G$ plane catching poles on the
way (see appendix A), we arrive at the following simple relationship
between equation \hG\ and $H(h)$:
%
\eqn\HGprod{
e^{H(h)}={(-1)^{(Q-1)}h\over t_Q}\prod_{q=1}^Q G_q(h).}
%
Some words of explanation are in order to clarify the meaning of this
equation. Inverting eq. \hG\ leads to a multi-sheeted function
$G(h)$. The general picture is illustrated in Fig. 4. One of the
sheets is the physical sheet and has two cuts, one corresponding to
$e^{H(h)}$, the other to $e^{F(h)}$; we label this sheet $G_1(h)$. The
sheets $G_2(h),\,\dots\,,G_Q(h)$ are all the sheets connected to
$G_1(h)$ by the cut of $e^{F(h)}$; there are exactly $Q$ of these
sheets, where $Q$ is the maximum inverse power of $G$ in \hGexp .
%
%
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\vskip 20pt
\centerline{{\bf Fig. 4:} Analytic structure of $G(h)$}
\vskip 20pt

%
\hskip -19pt In appendix B several examples are presented to illustrate
explicitly this general analytic structure.

Equation \HGprod\ together with \hGexp\ contains sufficient
information to find the logarithmic derivative of the character. These
two equations represent a well-defined Cauchy-Riemann problem for
$F(h)$ which can be explicitly solved. We will present the solution
elsewhere.

\newsec{Almost flat planar diagrams}

We now have all the tools necessary to reduce our model of dually
weighted graphs to a well defined Cauchy-Riemann problem. In this
section we will analyse the case in which only positive curvature
defects are allowed on the surface, arbitrary amounts of negative
curvature being introduced at a single point. This is done by studying
the particular case $t_q=t_2 \delta_{q,2}+ t_4 \delta_{q,4}$, which
generates the flat patches (see Fig.~1(a)) and the positive defects
(see Fig.~1(b)). The correlators \expsam\ then correspond to the
insertion of a single defect of curvature $(2-k)~\pi$ (see also
Fig.~2). They will be extracted using eq.\hGexp, after explicit
calculation of the function $h(G)$.

{}From the analysis of the large $N$ limit of the character in the
previous section, we know that the product in \HGprod\ contains only
two sheets $G_q(h)$ (see Fig.~4). We label the physical sheet $G_1(h)$
and the sheet connected to it by the cut of $e^{F(h)}$, $G_2(h)$.
Taking the logarithm of equation \HGprod, we summarize the information
extracted from the large $N$ limit of the character by
%
\eqn\fpfph{
F_1(h) + F_2(h) + H(h) = -\ln (-{h \over t_4}),}
%
where $\ln G_i(h) = F_i(h) + H(h)$. The two sheets $G_1(h)$ and
$G_2(h)$ are glued together by the square root cut coming from
$F(h)$. The combination $F_1(h) + F_2(h)$, evaluated on the cut of
$F(h)$, is twice the constant part of $F(h)$ on the cut (the
discontinuous part of $F(h)$ is of opposite sign on $F_1(h)$ and
$F_2(h)$ and is therefore canceled). We thus have the two equations
%
\eqn\twofph{\eqalign{
2\cut F(h)+H(h)=&-\ln (-{h \over t_4})\cr
2F(h)+\cut H(h)=&-\ln h,}}
%
the first coming from the large $N$ limit of the character \HGprod\
and the second being the saddlepoint equation \sdpt. These two
equations tell us about the behaviour of the function $2F(h)+H(h)$ on
the cuts of $F(h)$ and $H(h)$ respectively. We have introduced the
notation $\cut F(h)$ to denote the real part on the cut of $F(h)$, and
similarly for $\cut H(h)$. The principal part integral in \sdpt\ is
thus denoted in \twofph\ by $\cut H(h)$.

Our object now is to reconstruct the analytic function $2F(h)+H(h)$
from its behaviour on its cuts. To do this we have to understand the
complete structure of cuts. First we notice from \hG\ that $G(h)$ is
non zero everywhere in the complex $h$ plane except at infinity. The
combination $F(h)+H(h)$ thus has no logarithmic cut point except for
the one which starts from $h=b$. This corresponds to the end of the
flat part of the density $\rho(h)$. We introduce two functions
$\tilde F(h)$ and $\tilde H(h)$ defined by
%
\eqn\tildefh{
F(h)=\tilde F(h)-\ln h\quad{\rm and}\quad H(h) = \tilde H(h) + \ln
{h\over h-b},}
%
in terms of which \twofph\ becomes
%
\eqn\twoftpht{\eqalign{
2\cut \tilde F(h)+\tilde H(h)=&\ln \bigl(-t_4(h-b)\bigr)\cr
2\tilde F(h)+\cut \tilde H(h)=&\ln (h-b).}}
%
These two equations define the behaviour of $2\tilde F(h) + \tilde
H(h)$ on all of its cuts. By standard methods we now generate the
full analytic function $2\tilde F(h) + \tilde H(h)$. We introduce
three cut points, $a$, $b$ and $c$ whose values are fixed by
boundary conditions (the points $a$ and $b$ define the cut of $\tilde
H(h)$ and $c$
defines the starting point of the cut of $\tilde F(h)$ which
goes from $c$ to $-\infty$) and generate the full
analytic function by performing the contour integral
%
\eqn\contint{\eqalign{
2\tilde F(h)+\tilde H(h)=\sqrt{(h-c)(h-b)(h-a)}\biggl[&
\oint_{C_H}\,{ds\over 2\pi i}
          {\ln (s-b) \over (h-s)\sqrt{(s-c)(s-b)(s-a)}}\cr+&
\oint_{C_F}\,{ds\over 2\pi i}
     {\ln \bigl(-t_4(s-b)\bigr) \over
(h-s)\sqrt{(s-c)(s-b)(s-a)}}\biggr].}}
%
The contours $C_H$ and $C_F$ are illustrated in Fig.~5(a).
The slanted zigzag line corresponds to the cut of $\ln(h-b)$.
Expanding the contours, catching poles on the way and using the fact that
logarithmic cuts have a discontinuity of $\pm i\pi$, we arrive at
%
\eqn\cbbaint{\eqalign{
2F(h)+H(h)=\ln {t_4\over h}+\sqrt{(h-c)(h-b)(h-a)}\biggl[&
\int_c^b\,{ds\over (h-s)}{1\over\sqrt{(s-c)(s-b)(s-a)}}\cr+&
\int_b^a\,{ds\over (h-s)}
              {{1\over \pi i}\ln
t_4\over\sqrt{(s-c)(s-b)(s-a)}}\biggr].}}
%
Fig.~5(b) clarifies the sign convention for
$\sqrt{(h-c)(h-b)(h-a)}$ on the real axis above and below the cuts.
Note that, for the cuts of $1/\sqrt{(h-c)(h-b)(h-a)}$  the signs on the
cuts are inverted compared to Fig.~5(b),
i.e. $+i\leftrightarrow -i$. The integrals in \cbbaint\ are
defined to be along the upper side.
%
%
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	 /
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	 /
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	11.561 24.705
	11.583 24.672
	11.605 24.655
	11.627 24.672
	 /
\plot 11.627 24.672 11.648 24.706 /
\put{$c$} [lB] at  6.731 25.015
\put{$b$} [lB] at  8.098 25.030
\put{$a$} [lB] at  8.994 25.030
\put{$C_F$} [lB] at  5.414 24.107
\put{$C_H$} [lB] at  8.424 24.011
\put{$+i$} [lB] at 11.430 24.826
\put{$c$} [lB] at 12.446 24.862
\put{$b$} [lB] at 13.805 24.831
\put{$+$ve} [lB] at 15.456 24.687
\put{$+i$} [lB] at 14.228 24.331
\put{$-i$} [lB] at 11.436 24.352
\put{$a$} [lB] at 14.734 24.831
\put{$-$ve} [lB] at 12.958 24.718
\put{$-i$} [lB] at 14.222 24.816
\put{(a) Contours for \contint\ } [1B] at 6.620 23.067
\put{(b) Sign convention for $\sqrt{(h-c)(h-b)(h-a)}$} [1B] at 14.000 23.067
\linethickness=0pt
\putrectangle corners at  4.949 25.309 and 15.456 22.991
\endpicture
\vskip 20pt
\centerline{{\bf Fig. 5:} Contours and sign conventions for \contint\
and \cbbaint\ }
\vskip 20pt
%
%



To fix the constants $a$, $b$ and $c$, we expand \cbbaint\ for large
$h$ and compare the resulting power series expansion to that obtained
from inverting \hG :
%
\eqn\bcexpn{
2F(h)+H(h)=\ln {t_4\over h}+{1\over\sqrt{h}}{t_2\over\sqrt{t_4}}\quad+\quad
        {\cal O}\bigl({1\over h\sqrt{h}}).}
%
The terms of ${\cal O}\bigl({1\over h\sqrt{h}})$ depend on the as yet
unknown function $\psi(G)$. Expanding \cbbaint\ for large $h$ and
comparing to \bcexpn\ we find the two boundary conditions
%
\eqn\twobc{
t_4=q=e^{-\pi {K'\over K}}\quad{\rm and}\quad
                         {t_2\over \sqrt{t_4}}={\pi\over K}\sqrt{a-c},}
%
with $K$ and $K'$ complete elliptic integrals of the first kind, defined
in terms of their respective moduli $k$ and $k'=\sqrt{1-k^2}$ through
%
\eqn\kdef{
k=\sqrt{{a-b\over a-c}}.}
%
The first condition fixes $k$ and hence the ratio of the distances
separating the cut points, and the second condition fixes $\,\,a-c\,\,$,
i.e.~the scale. The condition needed to fix the position of
the cut points along the real axis is provided by the
condition that the density must be normalized to one.

We now perform the integrals in \cbbaint\ and, after using the first
boundary condition and an identity between elliptic functions\foot{
For this and many other relations between Jacobi's elliptic
functions and theta functions useful for performing the
calculations of this section see e.g.~\BYRD,\LAWD.},
we obtain
%
\eqn\twoFH{
2F(h)+H(h)=-\ln h -{i\pi\over K}
             \sn^{-1}\bigl(\sqrt{{a-h\over a-b}},k\bigr),}
%
where $\sn^{-1}(z,k)$ is the inverse Jacobi elliptic function.
Using the saddle point equation $2F(h)+\cut H(h)=-\ln h$ and the fact
that the resolvent for the Young tableau can be written as $H(h)=\cut
H(h) \mp i\pi\rho(h)$, we can immediately write down the expression for
the density of Young tableau boxes as
%
\eqn\dens{
\rho(h)={1\over K}\sn^{-1}\bigl(\sqrt{{a-h\over a-b}},k\bigr).}
%

The Jacobi elliptic function $\sn(z,k)$ is a generalisation of
$\sin(z)$ with quarter period $K$. In fact, in the limit $k\rightarrow
0$, which corresponds to $t_4\rightarrow 0$, the expression for the
density becomes precisely $(2/\pi) \sin^{-1}(\sqrt{(a-h)/(a-b)})$.

Integrating $\rho(h)$ from $b$ to $a$ and equating the answer to $1-b$
to ensure that the density is normalized to $1$ (the flat portion from $0$
to $b$ gives a contribution $b$), gives the final boundary condition
%
\eqn\abc{
a=1+{t_2^2\over \pi^2 t_4}(K^2-EK),}
%
where $E$ is the complete elliptic integral of the second kind.

{}From the expression for the density we now generate the full Young
tableau resolvent, $H(h)$, in the standard way and obtain
following expression:
%
\eqn\Hsoln{\eqalign{
H(h)=&{h\over h-b}+\int_b^a dh'\ {\rho(h') \over h-h'}\cr
    =&\ln h -{i\pi\over K}\sn^{-1}\sqrt{{a-h\over a-b}}+
2\ln\Biggl({
     \theta_4\Bigl({\pi\over 2K}\sn^{-1}\sqrt{{a-h\over a-b}}\Bigr)
      \over q^{1/4}\bigl[(a-c)(a-b)\bigr]^{1/4}\theta_4(0)}\Biggr).}}
%
Using the above expression for $H(h)$, the eq. \twoFH\ for $2F(h)+H(h)$
and the quasi-periodicity of theta functions, we can
write the expression for $G(h)$,
%
\eqn\Gsoln{
G(h)=-{1\over D}
  \theta_4\Bigl({\pi\over 2K}\sn^{-1}\sqrt{{a-h\over a-b}}+
     {i\pi K'\over K}\Bigr)}
%
and its inverse
%
\eqn\hofG{
h=a-{q^{3/2}\over G^2}\Bigl[{\theta_1\bigl[\theta_4^{-1}(-GD)\bigr]\over
                            \theta_4(0)}\Bigr]^2,}
%
where the constant $D$ is given by
%
\eqn\Ddef{
D={t_2K\sqrt{k}~\theta_4(0)\over\pi q^{5/4}}=
                        {t_2\theta_1'(0)\over 2q^{5/4}}.}
%
To simplify \hofG\ we have used the definition of the Jacobi elliptic
function in terms of theta functions.  In view of eq.\hGexp , we see
that we have now explicitly calculated the generating function for
the correlators for models I and III.

We will now expand eq.\hofG\ and read off the correlators as the
coefficients of the positive powers of $G$.  Notice that
\hofG\ is a multivalued function since the function $\theta_4(z)$ is
periodic as $z$ is varied in the real direction and quasiperiodic in
the imaginary direction. We must thus choose the correct zero of
the $\theta_4(z)$ function about which to expand.
The physical sheet
corresponds to expanding about the zero $z={i\pi K'\over K}$. Using
the definition of the Jacobi elliptic function $\sn(u)$ in terms of
theta functions and shifting the arguments of the theta functions
using their quasi-periodicity, we rewrite \hofG\ as the pair
of equations
%
\eqn\hofGph{
h=a+{q\over G^2}\Bigl(e^{iz}{\theta_4(z)\over \theta_4(0)}\Bigr)^2\quad
{\rm with\,}\,z\,{\rm\,the\,\,solution\,\,of}\quad
{t_2G\over 2q}=ie^{iz}{\theta_1(z)\over \theta_1'(0)}.}
%
Expanding this for small $G$ we find that the first three terms
give (as expected from \hGexp ) $h={q\over G^2}+{t_2\over G}+1+{\cal
O}(G)$. Expanding three orders further, permits us
(using \hGexp ) to read off the first three moments of model III
(which are also the moments of model I):
%
\eqn\moms{\eqalign{
\langle~{1\over N}~\Tr~[(MA_4)^2]~\rangle_{III}=&
       {t_2^3\over 24 q^2}(1+f_3)\cr
\langle~{1\over N}~\Tr~[(MA_4)^4]~\rangle_{III}=&
     {t_2^4\over 192 q^3}\bigl(-8(1+f_3)+3f_2^2-4f_2f_3+f_4\bigr)\cr
\langle~{1\over N}~\Tr~[(MA_4)^6]~\rangle_{III}=&
   {t_2^5\over 1920 q^4}
        \bigl(81+90f_3-30f_2^2+40f_2f_3+10f_3^2-10f_4-f_5\bigr).}}
%
where for convenience we have defined
%
\eqn\fdef{\eqalign{
f_2=&{\theta_4''(0)\over \theta_4(0)}\,\,\,=
{4\over\pi^2}(K^2-EK)\cr
f_4=&{\theta_4''''(0)\over \theta_4(0)}\,=
{16K^2\over\pi^4}\bigl((3-2k^2)K^2-6EK+3E^2\bigr)\cr
f_3=&{\theta_1'''(0)\over \theta_1'(0)}\,\,=
{4\over\pi^2}\bigl((2-k^2)K^2-3EK)\cr
f_5=&{\theta_1'''''(0)\over \theta_1'(0)}=
{16K^2\over\pi^4}\bigl((6-6k^2+k^4)K^2-10(2-k^2)EK+15E^2\bigr).}}
%
and have then expressed these derivatives as combinations of the
complete elliptic integrals $K$, $E$ and their modulus $k$.

We can now give a simple physical interpretation of these moments. The first
two are directly related to the free energy ${\cal F}(t_2,t_4)$. The latter is
defined as the sum over all possible surfaces with the
topology of a sphere that can be constructed out of flat space and
positive curvature defects. It is impossible to put a flat surface
onto the sphere, so positive curvature defects are needed to close the
surface. Since the defects in this model have a deficit angle of $\pi$
it takes precisely four of them to close the surface into a
sphere. The surfaces are in the form of a cylinder with both ends
flattened. The four $t_2$ defects sit at the corners. Below we
illustrate the free energy for model III:
%
\eqn\fdiag{
{\cal F}(t_2,t_4)=\sum
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	 4.864 24.218
	 4.953 24.241
	 5.043 24.264
	 5.131 24.286
	 5.218 24.307
	 5.301 24.327
	 5.379 24.345
	 5.452 24.361
	 5.577 24.384
	 5.689 24.396
	 5.802 24.401
	 5.908 24.392
	/
\plot  1.854 24.280 	 1.975 24.312
	 2.096 24.351
	 2.216 24.392
	 2.305 24.430
	 2.394 24.467
	 2.457 24.487
	 2.524 24.507
	 2.597 24.528
	 2.674 24.550
	 2.755 24.571
	 2.840 24.593
	 2.928 24.616
	 3.020 24.638
	 3.114 24.661
	 3.210 24.684
	 3.309 24.707
	 3.409 24.731
	 3.511 24.754
	 3.614 24.778
	 3.717 24.801
	 3.821 24.825
	 3.924 24.848
	 4.028 24.872
	 4.130 24.895
	 4.232 24.919
	 4.332 24.942
	 4.431 24.965
	 4.527 24.988
	 4.621 25.011
	 4.713 25.034
	 4.801 25.056
	 4.886 25.078
	 4.967 25.099
	 5.044 25.121
	 5.117 25.142
	 5.184 25.162
	 5.247 25.182
	 5.356 25.223
	 5.463 25.271
	 5.518 25.303
	 5.618 25.368
	/
\linethickness=0pt
\putrectangle corners at  1.776 25.430 and  5.954 24.250
\endpicture
}
\vskip 20pt
%
%
\hskip -20pt
Note that the flattened ends can have an angle of twist between them.
The four $t_2$ defects correspond to vertices ${t_2\over
2}~\Tr~[(MA_4)^2]$, and all other vertices (with four legs) correspond
to the vertex ${t_4\over 4}~\Tr~[(MA_4)^4]$. We see that the
first two moments can be written in terms of the free energy ${\cal
F}(t_2,t_4)$ as
%
\eqn\Fmom{\eqalign{
\langle~{1\over N}~\Tr~[(MA_4)^2]~\rangle_{III}=&
                      2{\partial\over\partial t_2}{\cal F}(t_2,t_4)\cr
\langle~{1\over N}~\Tr~[(MA_4)^4]~\rangle_{III}=&
                      4{\partial\over\partial t_4}{\cal F}(t_2,t_4).}}
%
We thus read off the free energy
%
\eqn\fe{
{\cal F}(t_2,t_4)={t_2^4\over 192 q^2}(1+f_3).}
%
Using \fdef , the identity
${\partial\over\partial q}=
                 {2K^2kk'^2\over\pi^2q}{\partial\over\partial k}$,
along with standard identies for differentiating complete elliptic
integrals with respect to the modulus $k$, it is trivial to verify that
the moment $\langle~{1\over N}~\Tr~[(MA_4)^4]~\rangle_{III}$ given in
\moms\ is indeed four times the derivative of the free energy
with respect to $t_4=q$.

Using the definition of $f_3$ in terms of derivatives of the first
theta function, $\theta_1(z)$, along with the standard definition of
the theta function as an infinite product, allows us to write the free
energy as
%
\eqn\feq{
{\cal F}(t_2,t_4)={-t_2^4\over 8}{\partial\over\partial q^2}
\ln\Bigl[\prod_{n=1}^{\infty}(1-q^{2n})\Bigr]=
\sum \hskip 15pt \raise-20pt\hbox{
%
%
\beginpicture
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\linethickness=1pt
\ellipticalarc axes ratio  1.611:0.961  360 degrees
	from  4.367 24.678 center at  2.756 24.678
%\ellipticalarc axes ratio  0.064:0.064  360 degrees
%	from  3.266 24.007 center at  3.203 24.007
\put{$\bullet$} [1B] at 3.203 23.943
\plot  2.805 24.532  2.796 24.437 /
\plot  2.796 24.437  2.790 24.240 /
\plot  2.790 24.240  2.800 24.043 /
\plot  2.800 24.043  2.822 23.829 /
\plot  2.822 23.829  2.845 23.717 /
\plot  3.488 24.767 	 3.380 24.860
	 3.319 24.907
	 3.220 24.966
	 3.118 25.019
	 3.056 25.043
	 2.976 25.072
	 2.896 25.098
	 2.832 25.116
	 2.733 25.136
	 2.635 25.152
	 2.527 25.169
	 2.419 25.184
	 2.296 25.190
	 2.174 25.193
	 2.059 25.189
	 1.945 25.180
	 1.838 25.155
	 1.734 25.121
	 1.650 25.079
	 1.573 25.023
	 1.529 24.943
	 1.501 24.858
	 1.501 24.772
	 1.516 24.687
	 1.561 24.596
	 1.617 24.513
	 1.695 24.438
	 1.780 24.369
	 1.869 24.315
	 1.962 24.268
	 2.023 24.244
	 2.100 24.217
	 2.178 24.192
	 2.239 24.172
	 2.305 24.154
	 2.387 24.131
	 2.470 24.108
	 2.536 24.092
	 2.597 24.080
	 2.674 24.066
	 2.752 24.053
	 2.813 24.043
	 2.877 24.035
	 2.959 24.026
	 3.040 24.018
	 3.105 24.014
	 3.170 24.011
	 3.252 24.010
	 3.334 24.011
	 3.399 24.014
	 3.473 24.019
	 3.566 24.028
	 3.659 24.039
	 3.732 24.052
	 3.849 24.087
	 3.965 24.128
	 4.046 24.164
	 4.125 24.204
	 4.238 24.291
	/
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	 3.552 24.795
	 3.575 24.867
	 3.560 24.928
	 3.512 24.991
	 3.418 25.089
	 3.322 25.154
	 3.222 25.212
	 3.114 25.254
	 3.006 25.290
	 2.921 25.310
	 2.836 25.328
	 2.758 25.341
	 2.680 25.353
	 2.574 25.362
	 2.470 25.366
	 2.370 25.371
	 2.271 25.372
	 2.191 25.366
	 2.110 25.358
	 1.985 25.349
	 1.891 25.329
	 1.797 25.305
	 1.720 25.280
	 1.645 25.252
	 1.576 25.211
	 1.511 25.165
	 1.420 25.076
	 1.372 24.996
	 1.331 24.903
	 1.308 24.778
	 1.317 24.687
	 1.336 24.598
	 1.371 24.511
	 1.416 24.428
	 1.498 24.339
	 1.587 24.257
	 1.710 24.173
	 1.780 24.129
	 1.837 24.096
	 1.964 24.037
	 2.035 24.007
	 2.093 23.984
	 2.166 23.962
	 2.260 23.937
	 2.354 23.913
	 2.428 23.895
	 2.501 23.880
	 2.593 23.862
	 2.686 23.845
	 2.760 23.834
	 2.833 23.827
	 2.925 23.820
	 3.017 23.816
	 3.090 23.815
	 3.173 23.819
	 3.277 23.827
	 3.381 23.837
	 3.463 23.846
	 3.537 23.859
	 3.596 23.871
	 3.677 23.887
	/
\plot  2.764 24.541 	 2.863 24.530
	 2.917 24.526
	 2.974 24.527
	 3.046 24.531
	 3.119 24.535
	 3.177 24.541
	 3.252 24.549
	 3.347 24.561
	 3.442 24.575
	 3.518 24.589
	 3.627 24.619
	 3.736 24.657
	 3.829 24.710
	 3.920 24.767
	 4.022 24.839
	 4.068 24.903
	 4.102 24.972
	 4.111 25.040
	 4.111 25.076
	 4.108 25.144
	 4.085 25.188
	 4.034 25.265
	 4.030 25.269
	/
\plot  3.283 24.602 	 3.388 24.630
	 3.465 24.651
	 3.560 24.682
	 3.681 24.746
	 3.797 24.835
	 3.876 24.951
	 3.901 25.053
	 3.893 25.129
	 3.869 25.188
	 3.844 25.241
	 3.776 25.313
	 3.721 25.358
	 3.655 25.413
	 3.575 25.461
	 3.512 25.489
	 3.448 25.514
	 3.373 25.543
	 3.296 25.569
	 3.207 25.587
	 3.118 25.603
	 2.993 25.626
	/
\plot  3.355 24.678 	 3.247 24.759
	 3.144 24.818
	 3.037 24.871
	 2.919 24.912
	 2.800 24.947
	 2.686 24.973
	 2.572 24.996
	 2.458 25.009
	 2.343 25.019
	 2.247 25.020
	 2.151 25.019
	 2.036 25.010
	 1.922 24.991
	 1.835 24.953
	 1.757 24.892
	 1.734 24.771
	 1.767 24.691
	 1.814 24.619
	 1.911 24.538
	 2.015 24.469
	 2.073 24.441
	 2.147 24.410
	 2.221 24.382
	 2.280 24.361
	 2.356 24.338
	 2.454 24.313
	 2.552 24.289
	 2.629 24.272
	 2.688 24.261
	 2.764 24.249
	 2.839 24.239
	 2.898 24.232
	 2.973 24.224
	 3.068 24.217
	 3.163 24.211
	 3.238 24.208
	 3.311 24.209
	 3.402 24.212
	 3.493 24.217
	 3.564 24.223
	 3.671 24.240
	 3.776 24.263
	 3.873 24.292
	 3.969 24.325
	 4.062 24.369
	 4.153 24.418
	 4.224 24.477
	 4.290 24.541
	 4.343 24.606
	 4.367 24.701
	/
\plot  1.985 24.731 	 2.053 24.657
	 2.161 24.577
	 2.253 24.543
	 2.347 24.517
	 2.468 24.493
	 2.589 24.473
	 2.681 24.459
	 2.797 24.442
	 2.913 24.427
	 3.006 24.418
	 3.085 24.416
	 3.185 24.417
	 3.285 24.420
	 3.363 24.424
	 3.444 24.430
	 3.546 24.440
	 3.647 24.453
	 3.727 24.469
	 3.785 24.487
	 3.856 24.515
	 3.981 24.570
	 4.046 24.610
	 4.108 24.653
	 4.202 24.727
	 4.250 24.799
	 4.286 24.858
	 4.290 24.911
	 4.290 24.979
	/
\plot  1.649 25.381 	 1.756 25.426
	 1.814 25.447
	 1.940 25.477
	 2.066 25.502
	 2.127 25.510
	 2.204 25.520
	 2.282 25.527
	 2.343 25.531
	 2.428 25.532
	 2.536 25.529
	 2.644 25.524
	 2.728 25.519
	 2.800 25.512
	 2.890 25.502
	 2.980 25.490
	 3.050 25.478
	 3.168 25.449
	 3.283 25.413
	 3.375 25.366
	 3.463 25.313
	 3.551 25.247
	 3.632 25.169
	 3.681 25.059
	 3.696 24.979
	 3.677 24.911
	 3.643 24.826
	 3.596 24.767
	 3.552 24.731
	 3.507 24.701
	 3.427 24.661
	 3.363 24.634
	/
\plot  1.327 25.127 	 1.251 25.004
	 1.223 24.907
	 1.206 24.807
	 1.210 24.700
	 1.223 24.594
	 1.240 24.530
	 1.265 24.452
	 1.294 24.376
	 1.323 24.316
	 1.414 24.196
	 1.520 24.088
	 1.580 24.045
	 1.659 23.996
	 1.740 23.951
	 1.806 23.918
	 1.884 23.887
	 1.985 23.852
	 2.088 23.820
	 2.170 23.798
	 2.266 23.778
	 2.390 23.759
	 2.514 23.742
	 2.612 23.730
	 2.688 23.724
	 2.749 23.721
	 2.832 23.717
	/
\plot  2.572 24.545 	 2.500 24.481
	 2.465 24.409
	 2.441 24.333
	 2.429 24.231
	 2.424 24.128
	 2.427 24.016
	 2.436 23.906
	 2.449 23.826
	 2.479 23.749
	 2.523 23.726
	/
\plot  2.324 24.589 	 2.214 24.549
	 2.166 24.482
	 2.134 24.409
	 2.106 24.317
	 2.087 24.223
	 2.081 24.106
	 2.083 23.990
	 2.099 23.895
	 2.134 23.802
	 2.170 23.781
	/
\plot  2.178 24.630 	 2.070 24.634
	 1.979 24.598
	 1.897 24.545
	 1.833 24.455
	 1.789 24.356
	 1.772 24.247
	 1.770 24.138
	 1.788 24.020
	 1.825 23.906
	 1.854 23.874
	/
\plot  2.057 24.701 	 2.015 24.742
	 1.899 24.759
	 1.823 24.754
	 1.748 24.737
	 1.653 24.673
	 1.573 24.589
	 1.547 24.532
	 1.526 24.460
	 1.512 24.386
	 1.505 24.325
	 1.508 24.264
	 1.519 24.189
	 1.536 24.115
	 1.560 24.056
	 1.649 23.982
	/
\plot  2.034 24.759 	 2.002 24.820
	 1.939 24.871
	 1.869 24.907
	 1.782 24.922
	 1.695 24.926
	 1.616 24.903
	 1.541 24.871
	 1.420 24.771
	 1.369 24.677
	 1.331 24.577
	 1.318 24.480
	 1.317 24.384
	 1.343 24.297
	 1.376 24.215
	 1.412 24.143
	/
\plot  2.087 24.687 	 2.102 24.723
	 2.110 24.782
	 2.098 24.848
	 2.030 24.955
	 1.930 25.027
	 1.812 25.063
	 1.753 25.068
	 1.679 25.071
	 1.606 25.068
	 1.547 25.059
	 1.438 25.007
	 1.340 24.939
	 1.275 24.853
	 1.223 24.759
	 1.202 24.646
	 1.198 24.532
	 1.216 24.449
	 1.238 24.387
	 1.270 24.304
	/
\plot  2.123 24.674 	 2.214 24.701
	 2.303 24.718
	/
\plot  2.214 24.661 	 2.239 24.727
	 2.244 24.792
	 2.239 24.858
	 2.215 24.942
	 2.178 25.023
	 2.097 25.110
	 2.002 25.184
	 1.913 25.218
	 1.820 25.241
	 1.719 25.253
	 1.617 25.254
	 1.541 25.242
	 1.465 25.220
	 1.411 25.189
	 1.317 25.121
	/
\plot  3.061 24.562 	 3.109 24.517
	 3.142 24.454
	 3.169 24.388
	 3.184 24.294
	 3.194 24.200
	 3.202 24.093
	 3.207 23.984
	 3.202 23.893
	 3.194 23.802
	 3.190 23.745
	/
\plot  1.985 24.731 	 1.985 24.782
	 2.030 24.820
	 2.110 24.840
	 2.191 24.848
	 2.292 24.855
	 2.394 24.858
	 2.473 24.854
	 2.553 24.848
	 2.597 24.845
	 2.680 24.839
	/
\plot  2.275 24.695 	 2.339 24.786
	 2.362 24.848
	 2.371 24.960
	 2.365 25.034
	 2.352 25.108
	 2.313 25.180
	 2.267 25.245
	 2.163 25.349
	 2.047 25.390
	 1.926 25.415
	 1.860 25.424
	 1.793 25.425
	 1.739 25.414
	 1.640 25.385
	/
\plot  2.083 25.555 	 2.189 25.544
	 2.246 25.531
	 2.317 25.498
	 2.383 25.453
	 2.446 25.369
	 2.496 25.277
	 2.522 25.174
	 2.536 25.068
	 2.529 24.980
	 2.512 24.892
	 2.470 24.799
	 2.396 24.763
	/
\plot  2.548 25.631 	 2.603 25.582
	 2.648 25.491
	 2.684 25.394
	 2.695 25.316
	 2.701 25.237
	 2.706 25.144
	 2.705 25.051
	 2.696 24.975
	 2.680 24.903
	 2.648 24.839
	 2.629 24.826
	/
\plot  2.877 24.826 	 2.862 24.934
	 2.864 25.025
	 2.868 25.116
	 2.869 25.222
	 2.872 25.328
	 2.878 25.408
	 2.885 25.489
	 2.898 25.590
	 2.925 25.639
	/
\plot  3.086 24.790 	 3.046 24.852
	 3.037 24.979
	 3.043 25.079
	 3.054 25.180
	 3.075 25.283
	 3.105 25.385
	 3.150 25.465
	 3.207 25.538
	 3.296 25.586
	/
\plot  3.247 24.723 	 3.198 24.795
	 3.181 24.920
	 3.187 25.007
	 3.203 25.093
	 3.252 25.199
	 3.315 25.296
	 3.398 25.372
	 3.495 25.434
	 3.590 25.452
	 3.685 25.453
	 3.772 25.425
	/
\plot  3.372 24.657 	 3.334 24.714
	 3.315 24.807
	 3.331 24.913
	 3.363 25.013
	 3.428 25.097
	 3.503 25.169
	 3.597 25.227
	 3.696 25.273
	 3.786 25.301
	 3.880 25.317
	 4.005 25.296
	/
\plot  3.399 24.666 	 3.399 24.746
	 3.412 24.822
	 3.435 24.896
	 3.498 24.968
	 3.571 25.027
	 3.655 25.069
	 3.744 25.099
	 3.860 25.114
	 3.977 25.112
	 4.074 25.081
	 4.166 25.036
	 4.259 24.943
	 4.335 24.826
	 4.367 24.704
	/
\plot  3.497 24.795 	 3.539 24.843
	 3.628 24.892
	 3.699 24.911
	 3.772 24.920
	 3.858 24.909
	 3.941 24.884
	 4.018 24.831
	 4.085 24.767
	 4.140 24.682
	 4.183 24.589
	 4.199 24.477
	 4.202 24.365
	 4.192 24.308
	 4.166 24.204
	/
\plot  3.306 24.610 	 3.418 24.570
	 3.471 24.503
	 3.512 24.428
	 3.552 24.304
	 3.562 24.200
	 3.564 24.096
	 3.562 24.010
	 3.556 23.923
	 3.552 23.846
	/
\plot  3.503 24.737 	 3.587 24.737
	 3.632 24.731
	 3.719 24.695
	 3.829 24.589
	 3.874 24.497
	 3.905 24.401
	 3.917 24.292
	 3.920 24.183
	 3.915 24.104
	 3.901 24.026
	 3.861 23.963
	/
\plot  2.030 24.790 	 2.066 24.691
	 2.151 24.638
	 2.218 24.618
	 2.286 24.602
	 2.359 24.585
	 2.432 24.570
	 2.508 24.558
	 2.584 24.549
	 2.670 24.546
	 2.756 24.545
	 2.870 24.547
	 2.985 24.553
	 3.058 24.565
	 3.131 24.579
	 3.219 24.596
	 3.306 24.617
	 3.363 24.638
	 3.448 24.704
	 3.503 24.799
	/
\plot  2.187 24.638 	 2.288 24.714
	 2.392 24.761
	 2.500 24.799
	 2.576 24.817
	 2.652 24.831
	 2.769 24.835
	 2.857 24.829
	 2.944 24.820
	 3.063 24.795
	 3.135 24.772
	 3.207 24.746
	 3.302 24.687
	 3.342 24.646
	/
\linethickness=0pt
\putrectangle corners at  1.113 25.667 and  4.398 25.000
\endpicture
%1.113 25.667
%4.398 23.690
%
%
}
}
\vskip 20pt
%
\hskip -20pt
In this form we recognize the argument of the logarithm to be the
partition function for the torus. The derivative operator acts to mark
a single point. We have thus found, as illustrated in
 equation \feq , that the free energy can be written
as the free energy for a marked torus. Below we illustrate the
connection between a marked torus and the flattened cylinder
diagrammatically.
%
%
\vskip 20pt
\beginpicture
\setcoordinatesystem units <1.00000cm,1.00000cm>
\linethickness=1pt
\setlinear
\ellipticalarc axes ratio  1.611:0.961  360 degrees
	from  7.383 24.045 center at  5.772 24.045
\put {$\bullet$} [1B] at 6.219 23.300
\plot  5.821 23.899  5.812 23.804 /
\plot  5.812 23.804  5.806 23.607 /
\plot  5.806 23.607  5.817 23.410 /
\plot  5.817 23.410  5.838 23.197 /
\plot  5.838 23.197  5.861 23.084 /
\plot 10.590 23.832 10.560 23.760 /
\plot 10.560 23.760 10.577 23.594 /
\plot 10.577 23.594 10.645 23.351 /
\plot 10.645 23.351 10.702 23.150 /
\plot 10.702 23.150 10.774 23.055 /
\plot 11.242 23.061 10.850 23.724 /
\plot 13.568 23.806 14.389 23.082 /
\plot 17.355 24.909 17.653 23.918 /
\plot  6.505 24.134 	 6.396 24.227
	 6.335 24.274
	 6.236 24.333
	 6.134 24.386
	 6.072 24.411
	 5.992 24.439
	 5.912 24.465
	 5.848 24.483
	 5.749 24.503
	 5.652 24.519
	 5.543 24.537
	 5.436 24.551
	 5.313 24.557
	 5.190 24.560
	 5.075 24.556
	 4.961 24.547
	 4.855 24.522
	 4.750 24.488
	 4.666 24.446
	 4.589 24.390
	 4.545 24.311
	 4.517 24.225
	 4.517 24.139
	 4.532 24.054
	 4.577 23.963
	 4.633 23.880
	 4.712 23.805
	 4.796 23.736
	 4.885 23.682
	 4.978 23.635
	 5.039 23.611
	 5.117 23.584
	 5.194 23.559
	 5.256 23.539
	 5.321 23.521
	 5.404 23.498
	 5.486 23.476
	 5.552 23.459
	 5.613 23.447
	 5.690 23.433
	 5.768 23.420
	 5.829 23.410
	 5.893 23.402
	 5.975 23.393
	 6.057 23.385
	 6.121 23.381
	 6.186 23.378
	 6.268 23.377
	 6.351 23.378
	 6.416 23.381
	 6.489 23.386
	 6.582 23.395
	 6.675 23.407
	 6.748 23.419
	 6.865 23.454
	 6.981 23.495
	 7.062 23.531
	 7.142 23.571
	 7.254 23.658
	/
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	 9.318 24.196
	 9.318 24.109
	 9.332 24.024
	 9.378 23.933
	 9.436 23.848
	 9.513 23.774
	 9.597 23.705
	 9.687 23.651
	 9.781 23.603
	 9.842 23.579
	 9.920 23.553
	 9.999 23.528
	10.061 23.508
	10.148 23.482
	10.261 23.449
	10.373 23.416
	10.461 23.387
	10.536 23.355
	10.632 23.312
	10.726 23.268
	10.801 23.235
	10.885 23.201
	10.990 23.160
	11.096 23.119
	11.178 23.086
	11.242 23.061
	11.269 23.095
	11.351 23.161
	11.436 23.222
	11.514 23.278
	11.593 23.334
	11.696 23.390
	11.800 23.444
	11.877 23.491
	11.953 23.539
	12.065 23.626
	/
\plot 11.252 24.016 	11.333 24.069
	11.377 24.132
	11.398 24.204
	11.386 24.268
	11.337 24.329
	11.242 24.428
	11.146 24.493
	11.045 24.549
	10.938 24.594
	10.829 24.632
	10.744 24.650
	10.657 24.666
	10.579 24.681
	10.501 24.693
	10.395 24.702
	10.289 24.708
	10.191 24.711
	10.092 24.712
	10.011 24.705
	 9.929 24.697
	 9.804 24.689
	 9.709 24.669
	 9.616 24.644
	 9.538 24.619
	 9.462 24.589
	 9.393 24.550
	 9.326 24.505
	 9.237 24.416
	 9.188 24.335
	 9.148 24.240
	 9.125 24.115
	 9.134 24.024
	 9.152 23.935
	 9.188 23.847
	 9.233 23.764
	 9.314 23.675
	 9.402 23.594
	 9.527 23.509
	 9.599 23.465
	 9.656 23.431
	 9.782 23.372
	 9.854 23.342
	 9.912 23.319
	 9.986 23.297
	10.080 23.272
	10.174 23.248
	10.249 23.230
	10.322 23.215
	10.415 23.197
	10.508 23.180
	10.581 23.167
	10.667 23.152
	10.777 23.132
	10.886 23.113
	10.973 23.099
	11.043 23.091
	11.099 23.086
	11.176 23.078
	/
\plot 10.549 23.827 	10.672 23.732
	10.763 23.664
	10.886 23.588
	10.941 23.575
	10.973 23.616
	11.043 23.685
	11.117 23.751
	11.230 23.835
	11.350 23.912
	11.456 23.953
	11.561 23.992
	11.655 24.048
	11.745 24.107
	11.849 24.177
	11.894 24.240
	11.930 24.312
	11.940 24.380
	11.940 24.416
	11.934 24.481
	11.913 24.528
	11.864 24.604
	11.858 24.608
	/
\plot  9.804 24.069 	 9.872 23.992
	 9.978 23.912
	10.073 23.881
	10.168 23.855
	10.293 23.814
	10.416 23.768
	10.474 23.738
	10.543 23.698
	10.620 23.650
	10.700 23.599
	10.779 23.548
	10.855 23.501
	10.981 23.427
	11.034 23.406
	11.077 23.436
	11.178 23.534
	11.282 23.628
	11.341 23.670
	11.417 23.719
	11.495 23.765
	11.557 23.800
	11.683 23.854
	11.809 23.908
	11.934 23.988
	12.029 24.066
	12.078 24.136
	12.114 24.196
	12.118 24.249
	12.118 24.316
	/
\plot 11.178 24.016 	11.070 24.096
	10.967 24.156
	10.861 24.208
	10.742 24.250
	10.621 24.285
	10.508 24.312
	10.393 24.335
	10.278 24.348
	10.164 24.356
	10.067 24.359
	 9.970 24.356
	 9.855 24.349
	 9.741 24.329
	 9.653 24.291
	 9.574 24.232
	 9.553 24.109
	 9.585 24.029
	 9.633 23.956
	 9.730 23.874
	 9.836 23.804
	 9.893 23.777
	 9.967 23.747
	10.041 23.718
	10.101 23.696
	10.177 23.673
	10.274 23.645
	10.371 23.616
	10.446 23.588
	10.504 23.558
	10.576 23.514
	10.702 23.436
	10.785 23.395
	10.891 23.344
	10.997 23.295
	11.081 23.258
	11.136 23.235
	11.184 23.271
	11.241 23.316
	11.313 23.375
	11.385 23.434
	11.445 23.478
	11.538 23.528
	11.633 23.575
	11.714 23.619
	11.796 23.662
	11.888 23.706
	11.980 23.755
	12.051 23.813
	12.118 23.876
	12.173 23.944
	12.194 24.039
	/
\plot 13.970 23.918 	14.082 23.887
	14.160 23.857
	14.235 23.821
	14.335 23.754
	14.429 23.677
	14.493 23.604
	14.550 23.525
	14.600 23.432
	14.639 23.332
	14.630 23.218
	/
\plot 14.453 24.071 	14.530 24.048
	14.586 24.029
	14.654 23.999
	14.743 23.935
	14.823 23.863
	14.888 23.781
	14.946 23.694
	14.977 23.610
	15.001 23.525
	15.011 23.455
	15.009 23.383
	14.963 23.302
	/
\plot 14.897 24.194 	15.019 24.157
	15.081 24.128
	15.172 24.048
	15.251 23.959
	15.299 23.890
	15.339 23.815
	15.358 23.706
	15.365 23.597
	15.359 23.529
	15.348 23.461
	15.299 23.381
	/
\plot 15.251 24.274 	15.327 24.246
	15.384 24.224
	15.454 24.194
	15.528 24.145
	15.598 24.088
	15.643 24.011
	15.676 23.927
	15.695 23.820
	15.701 23.711
	15.694 23.630
	15.676 23.550
	15.627 23.470
	/
\plot 15.638 24.354 	15.749 24.321
	15.807 24.297
	15.917 24.204
	15.960 24.114
	15.989 24.020
	16.007 23.911
	16.015 23.802
	16.007 23.721
	15.989 23.641
	15.943 23.561
	/
\plot 16.061 24.460 	16.157 24.426
	16.212 24.354
	16.254 24.274
	16.294 24.183
	16.326 24.090
	16.344 23.981
	16.351 23.872
	16.343 23.791
	16.326 23.711
	16.277 23.630
	/
\plot 16.391 24.564 	16.489 24.532
	16.543 24.459
	16.584 24.380
	16.624 24.288
	16.656 24.194
	16.674 24.086
	16.681 23.975
	16.674 23.895
	16.656 23.815
	16.607 23.736
	/
\plot 16.722 24.644 	16.819 24.613
	16.873 24.539
	16.914 24.460
	16.954 24.368
	16.986 24.274
	17.004 24.166
	17.012 24.056
	17.004 23.976
	16.986 23.897
	16.940 23.815
	/
\plot 17.029 24.725 	17.124 24.693
	17.179 24.620
	17.221 24.541
	17.261 24.449
	17.293 24.354
	17.311 24.247
	17.319 24.138
	17.310 24.056
	17.293 23.975
	17.247 23.897
	/
\plot 13.898 23.533 	14.008 23.574
	14.067 23.597
	14.148 23.635
	14.228 23.677
	14.340 23.757
	14.410 23.799
	14.500 23.849
	14.591 23.898
	14.662 23.935
	14.719 23.963
	14.791 23.997
	14.863 24.030
	14.920 24.056
	14.980 24.080
	15.057 24.110
	15.133 24.138
	15.193 24.160
	15.316 24.199
	15.392 24.223
	15.472 24.247
	15.552 24.271
	15.628 24.293
	15.750 24.329
	15.846 24.358
	15.968 24.394
	16.090 24.430
	16.186 24.458
	16.293 24.490
	16.358 24.510
	16.428 24.530
	16.497 24.551
	16.563 24.571
	16.669 24.604
	16.725 24.621
	16.801 24.645
	16.905 24.679
	16.971 24.700
	17.048 24.725
	/
\plot 14.082 23.372 	14.165 23.403
	14.225 23.427
	14.300 23.459
	14.376 23.504
	14.470 23.564
	14.564 23.624
	14.639 23.669
	14.697 23.699
	14.768 23.736
	14.848 23.776
	14.933 23.818
	15.019 23.858
	15.101 23.895
	15.174 23.928
	15.236 23.952
	15.298 23.973
	15.374 23.995
	15.461 24.019
	15.553 24.043
	15.644 24.067
	15.731 24.089
	15.808 24.110
	15.871 24.128
	15.940 24.150
	16.024 24.178
	16.119 24.210
	16.218 24.244
	16.318 24.278
	16.412 24.310
	16.496 24.339
	16.565 24.361
	16.684 24.397
	16.757 24.418
	16.835 24.441
	16.913 24.463
	16.986 24.485
	17.105 24.524
	17.220 24.568
	17.311 24.606
	17.435 24.659
	/
\plot 14.252 23.218 	14.325 23.248
	14.379 23.270
	14.446 23.300
	14.534 23.344
	14.643 23.403
	14.753 23.462
	14.840 23.508
	14.917 23.549
	15.015 23.600
	15.114 23.650
	15.193 23.685
	15.275 23.712
	15.379 23.741
	15.484 23.769
	15.566 23.791
	15.637 23.810
	15.727 23.834
	15.817 23.859
	15.888 23.880
	15.961 23.902
	16.053 23.930
	16.145 23.960
	16.218 23.984
	16.297 24.010
	16.393 24.043
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	16.614 24.120
	16.728 24.161
	16.835 24.199
	16.930 24.235
	17.007 24.265
	17.078 24.295
	17.174 24.338
	17.234 24.366
	17.305 24.399
	17.388 24.438
	17.484 24.483
	/
\plot 13.744 23.677 	13.829 23.703
	13.892 23.722
	13.970 23.749
	14.059 23.793
	14.168 23.852
	14.277 23.913
	14.364 23.959
	14.445 24.000
	14.548 24.051
	14.652 24.100
	14.736 24.136
	14.849 24.173
	14.941 24.200
	15.066 24.234
	/
\plot 14.558 23.169 	14.670 23.229
	14.752 23.272
	14.857 23.324
	14.962 23.369
	15.027 23.396
	15.096 23.424
	15.166 23.451
	15.232 23.477
	15.339 23.516
	15.465 23.558
	15.539 23.581
	15.619 23.605
	15.704 23.630
	15.793 23.657
	15.884 23.683
	15.976 23.710
	16.069 23.737
	16.160 23.763
	16.248 23.789
	16.334 23.815
	16.414 23.839
	16.488 23.862
	16.614 23.904
	16.740 23.950
	16.818 23.980
	16.900 24.012
	16.982 24.045
	17.060 24.076
	17.185 24.128
	17.309 24.183
	17.406 24.228
	17.468 24.256
	17.539 24.289
	/
\plot 17.596 24.105 	17.515 24.069
	17.445 24.039
	17.334 23.992
	17.253 23.959
	17.194 23.935
	17.109 23.905
	17.000 23.869
	16.892 23.834
	16.806 23.806
	16.717 23.778
	16.645 23.756
	16.548 23.726
	/
\plot  5.046 24.158 	 5.082 24.058
	 5.167 24.005
	 5.234 23.985
	 5.302 23.969
	 5.375 23.952
	 5.448 23.937
	 5.524 23.926
	 5.601 23.916
	 5.686 23.913
	 5.772 23.912
	 5.886 23.914
	 6.001 23.920
	 6.074 23.932
	 6.147 23.946
	 6.235 23.964
	 6.322 23.984
	 6.380 24.005
	 6.464 24.071
	 6.519 24.166
	/
\plot  5.203 24.005 	 5.304 24.081
	 5.409 24.128
	 5.516 24.166
	 5.592 24.184
	 5.668 24.198
	 5.785 24.202
	 5.873 24.196
	 5.961 24.187
	 6.079 24.162
	 6.152 24.140
	 6.223 24.113
	 6.318 24.054
	 6.358 24.014
	/
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	10.935 24.957
	11.037 24.943
	11.117 24.928
	11.213 24.903
	11.333 24.867
	11.452 24.828
	11.544 24.793
	11.609 24.761
	11.688 24.718
	11.766 24.672
	11.826 24.632
	11.899 24.569
	11.985 24.484
	12.066 24.395
	12.124 24.316
	12.161 24.222
	12.186 24.124
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	12.109 23.702
	12.068 23.637
	12.021 23.575
	11.907 23.467
	11.786 23.368
	11.684 23.313
	11.580 23.262
	11.516 23.231
	11.435 23.193
	11.354 23.152
	11.292 23.114
	11.246 23.063
	11.178 23.082
	11.086 23.088
	10.994 23.086
	10.880 23.071
	10.765 23.055
	10.666 23.052
	10.541 23.053
	10.416 23.057
	10.317 23.063
	10.236 23.074
	10.134 23.091
	10.032 23.110
	 9.953 23.127
	 9.886 23.144
	 9.801 23.168
	 9.717 23.194
	 9.652 23.218
	 9.540 23.273
	 9.432 23.334
	 9.322 23.409
	 9.216 23.491
	 9.143 23.566
	 9.076 23.647
	 9.026 23.735
	 8.985 23.827
	 8.964 23.934
	 8.956 24.043
	 8.974 24.147
	 9.004 24.249
	 9.049 24.337
	 9.102 24.420
	 9.178 24.504
	 9.260 24.581
	 9.378 24.664
	 9.500 24.740
	 9.621 24.798
	 9.745 24.850
	 9.811 24.871
	 9.894 24.895
	 9.979 24.917
	10.046 24.932
	10.166 24.951
	10.262 24.962
	10.323 24.969
	10.393 24.977
	/
\plot  9.974 23.948 	10.060 24.018
	10.109 24.052
	10.214 24.098
	10.321 24.136
	10.397 24.154
	10.473 24.168
	10.590 24.172
	10.678 24.168
	10.765 24.160
	10.886 24.132
	10.959 24.110
	11.030 24.083
	11.117 24.031
	11.214 23.963
	11.214 23.956
	/
\plot  9.849 24.128 	 9.885 24.031
	 9.970 23.975
	10.037 23.955
	10.105 23.940
	10.179 23.923
	10.253 23.908
	10.329 23.897
	10.406 23.887
	10.474 23.872
	10.541 23.855
	10.601 23.832
	10.675 23.800
	10.749 23.767
	10.808 23.743
	10.837 23.732
	10.869 23.764
	10.936 23.821
	11.005 23.876
	11.129 23.952
	11.189 23.975
	11.273 24.043
	11.328 24.136
	/
\plot 14.398 23.091 	14.518 23.148
	14.582 23.178
	14.683 23.224
	14.785 23.266
	14.912 23.307
	14.987 23.329
	15.068 23.351
	15.154 23.375
	15.244 23.398
	15.337 23.422
	15.431 23.445
	15.524 23.469
	15.617 23.492
	15.707 23.514
	15.794 23.535
	15.876 23.555
	15.952 23.573
	16.020 23.590
	16.080 23.605
	16.203 23.637
	16.275 23.656
	16.353 23.676
	16.435 23.698
	16.521 23.721
	16.609 23.744
	16.699 23.767
	16.788 23.790
	16.877 23.812
	16.963 23.833
	17.046 23.853
	17.125 23.871
	17.198 23.886
	17.323 23.910
	17.435 23.921
	17.547 23.927
	17.653 23.918
	/
\linethickness=2pt
\plot 13.600 23.806 	13.720 23.838
	13.841 23.877
	13.962 23.918
	14.050 23.956
	14.139 23.992
	14.202 24.013
	14.270 24.033
	14.342 24.054
	14.420 24.075
	14.501 24.097
	14.586 24.119
	14.674 24.142
	14.765 24.164
	14.859 24.187
	14.956 24.210
	15.054 24.233
	15.155 24.256
	15.256 24.280
	15.359 24.303
	15.462 24.327
	15.566 24.351
	15.670 24.374
	15.773 24.398
	15.876 24.421
	15.977 24.445
	16.077 24.468
	16.176 24.491
	16.272 24.514
	16.367 24.537
	16.458 24.559
	16.546 24.582
	16.631 24.604
	16.712 24.625
	16.789 24.647
	16.862 24.667
	16.930 24.688
	16.993 24.708
	17.102 24.749
	17.209 24.797
	17.263 24.829
	17.363 24.894
	/
\linethickness=0pt
\putrectangle corners at  4.130 25.034 and 17.700 22.991
\endpicture
\vskip 20pt
\centerline{{\bf Fig. 6} Diagrammatic connection between marked torus
and flattened cylinder.}
\vskip 20pt
%
%
\hskip -19pt
Starting from the mark on the twisted torus,
flatten the torus
across its width (this defines a point on the opposite side), then cut
along the flattening and open out the crimped torus into a cylinder
with flattened ends. The two points at either end of the flattening
on the torus become the four $t_2$ defects of the flattened cylinder.

Higher order moments correspond to inserting a single negative
curvature defect. The lowest order defect of this type is the
insertion of negative curvature of deficit angle $-\pi$ introduced by
the vertex $\Tr[(MA_4)^6]$ (see Fig. 1(c) and eq. \moms).

As a final check of our solution, we expand
$\langle~{1\over N}~\Tr~[(MA_4)^6]~\rangle_{III}$ in powers of $q$
(this can be done directly from the expression for the moment
in terms of theta functions \moms ):
%
\eqn\pert{
\langle~{1\over N}~\Tr~[(MA_4)^6]~\rangle_{III}=
t_2^5~(~9q^2~+~27q^4~+~81q^6~+~\dots ).}
%
It is easy to verify that this correctly counts the number of diagrams.

Further moments can be calculated by expanding \hofGph\ to
higher order. They can always be written as sums of products of
complete elliptic integrals.

We now look for a continuum limit in which the
size of graphs tends to infinity. One can see that the critical
point, at which the size of the graphs diverges, is at $q=1$.
Note, however, that (since the critical $q$ is $1$),
in stark contrast to two-dimensional quantum gravity \DAVID\ \VOL,
the leading behaviour for the growth of diagrams
is {\it not} exponential but merely power-like. To extract the power,
notice that the product
$\prod_{n=1}^{\infty}(1-q^{2n})$ in \feq\ can be written in terms of
the $\eta$ function as
$q^{-1/12}\eta(i\tau)$ where
$q=e^{-\pi\tau}$. Making use of the modular invariance of the
$\eta(\tau)$ function under the the modular transformation
$\tau\rightarrow 1/\tau$, we extract
%
\eqn\freelim{
{\cal F}(t_2,1-\mu)\sim {\pi^2t_2^4\over 192}{1\over\mu^2},}
%
where we have defined a ``continuum cosmological constant'' $\mu$ through
$q=1-\mu$ i.e. $\mu=\pi\tau+{\cal O}(\tau^2)$.
We thus see, changing from fixed cosmological constant to fixed area
by Laplace transform, that the number of graphs grows as a linear power of the
area. Employing the conventions of quantum gravity \DAVID\ \VOL,
this would formally correspond to a ``string susceptibility''
$\gamma_{str}=4$.

We can easily understand this
result by performing the calculation directly in the continuum limit. We
thus integrate over cylinders of all possible lengths $t$ and
circumferences $s$ weighted with a factor $s$ (corresponding to the
modular twist between the two flattened ends) and
a delta function for the area so as to count the number of surfaces of a
given area $A$. We thus perform the following integral:
%
\eqn\contfe{
{\cal F}\sim\int_0^{\infty}~dt~ds~s~\delta(ts-A)=
            A\int_0^{\infty}~{dt\over t^2}.}
%
We see immediately a linear dependence on the area $A$, but also a
divergently large contribution coming from small $t$. The most
important contribution comes from the cylinders which are infinitely
short and thus have the maximum amount of entropy coming from the
modular twist.

It is interesting now to investigate the behaviour of the correlators
in the large area ($q\rightarrow 1$) limit. Quite generally, for matrix
models the correlators correspond to surfaces with a boundary of
length proportional to the power of the correlator, and one seeks a
continuum scaling limit for very long boundaries. The correlators
$\langle~{1\over N}~\Tr~[(MA_4)^{2n}]~\rangle$ in the present model,
however, introduce point-like negative curvature
and we cannot look for a scaling
limit involving long boundaries\foot{In principle, it is possible to
study boundaries of arbitrary length by taking correlators of
$\langle~{1\over N}~\Tr~[(MA_4^2)^{2n}]~\rangle$, which correspond to
a boundary in the form of the end of a cylinder. Technically, however,
we do not at present have the means to calculate such quantities.}.
Nevertheless we can find the limiting behaviour of these negative
curvature insertions in the limit of large area.

Using the modular transformation $\tau\rightarrow 1/\tau$ (with
$q=e^{-\pi\tau}$) for the formula \hofG, we can also extract
a scaling limit for the generating function for the
moments. Specifically, for the theta functions $\theta_1(z)$
and $\theta_4(z)$, we find that the dominant contributions are
%
\eqn\thlim{\eqalign{
\theta_4(z)=&{1\over\sqrt{\tau}}e^{-{\tau\over\pi}y^2}2q'^{1/4}
\bigl(\cosh y + {\cal O}(q'^2)\bigr)\cr
\theta_1(z)=&{1\over\sqrt{\tau}}e^{-{\tau\over\pi}y^2}2q'^{1/4}
\bigl(\sinh y + {\cal O}(q'^2)\bigr),}}
%
where $y=z/\tau$ and $q'=e^{-{\pi\over\tau}}$. Holding $y$ fixed as a
parameter of order 1, we take the limit as
$\tau\rightarrow 0$ (corresponding to $q\rightarrow 1$) and
work to the first two orders in $\tau$. Remembering that
$q=1-\pi\tau+\dots$, the constants $D$ and $a$
to the first two orders in $\tau$ are given by
%
\eqn\Da{
D=t_2{1\over\tau^{3/2}}q'^{1/4}\bigl(1+{5\over 4}\pi\tau\bigr)
\quad {\rm and} \quad
a=1+t_2^2\Bigl[{1\over 4\tau^2}-
  \bigl({1\over 2\pi}-{\pi\over
4}\bigr){1\over\tau}\Bigr],}
%
We can now define a natural rescaled parameter $x={Gt_2\over 2\tau}$
and perform the inversion of the theta function to the first two
orders in $\tau$ to find the generating function for the
correlators. The lowest order term gives the contribution ${q\over G^2}$
in \hGexp\ and also a part that cancels with $a-1$. The next order
gives the contribution ${t_2\over G}$ along with the generating
function which we read off as:
%
\eqn\scalmom{
\sum_{n=1}^{\infty}\langle~{1\over N}~\Tr~[(MA_4)^{2n}]~\rangle G^n=
        {t_2\over\tau}
   \Bigl[{\sin^{-1}x-x\over 2x^2}+{(\sin^{-1}x)^2-x^2\over 2\pi
                                                      x^2}\Bigr]
\quad {\rm with} \quad x={Gt_2\over 2\tau}.}
%
This has a simple square root singularity at the point $x=1$. The
series expansions for $\sin^{-1}x$ and $(\sin^{-1}(x))^2$ then give us
the dominant contribution to the correlators in the large area limit:
%
\eqn\momscal{
\langle~{1\over N}~\Tr~[(MA_4)^{2n}]~\rangle \simeq
                            {C_nt_2^{n+2}\over\mu^{n+1}}
\quad {\rm with} \quad \cases{
C_{2n}={\pi^{2n}(n!)^2\over 2~(2n+1)!~(n+1)}\cr
C_{2n+1}={\pi^{2n+2}(2n+2)!\over 2^{4n+4}((n+1)!)^2(2n+3)},}}
%
where we have again introduced the parameter $\mu$ defined by
$q=1-\mu$ i.e. $\mu=\pi\tau+{\cal O}(\tau^2)$. The number of surfaces
of fixed area $A$ for a correlator $\langle~{1\over
N}~\Tr~[(MA_4)^{2n}]~\rangle$ is thus seen to be of the order of
$A^n$, with entropy coming from modular twists analogous to those for
the free energy. The rather curious structure of the considered
surfaces is thus evident - they consist of cylindrical ``fingers''
growing out from the negative curvature defect (see Fig.~2(b)).  The
square root singularity at $x=1$ means that there is a tree-like
growth of the number of ways to attach the fingers to their base at
the negative defect.  As in the case of the free energy, modular
integrations cause a filamentary structure of very long cylinders to
dominate in the large area limit.


\newsec{The onset of quantum gravity: Adding negative curvature defects}
The introduction of arbitrary numbers of negative defects,
specifically $t_6$, alongside
the components $t_4$, $t_4^*$ and positive
curvature defects, $t_2$,  gives us a model in which we can tune away
the curvature fluctuations of two-dimensional quantum gravity. The large
$N$ limit of the character in section 3 allows us to understand the analytic
structure of the solution and thus reduce the model to
a well defined Cauchy-Riemann problem. The function
$G(h)$ now consists of two sheets below the physical sheet (see Fig. 4).
An extra
sheet which we label $G_3(h)$ is now attached to the sheet $G_2(h)$ of
the previous section by a square root cut. We thus have the following
two equations
%
\eqn\grav{\eqalign{
2F(h) + \cut H(h) =& -\ln h\cr
F_1(h)+F_2(h)+F_3(h)+2H(h)=&-\ln({h\over t_6})}.}
%
The first is the saddle point equation \sdpt. The second comes from
the logarithm of eq. \HGprod, where we define, as before,
$F_i(h)$ by $\ln G_i(h)=H(h)+F_i(h)$.  Along with the boundary
conditions provided by the coefficients of the negative powers of $G$
in \hGexp, the system of equations \grav\ completely determines the solution
to this problem.


\newsec{Conclusions and outlook}

In the present work we have demonstrated that our technique of
character expansions for large $N$ matrix models may be successfully
applied to the study of a novel, up to now inaccessible phase of almost
regular planar diagrams. This required determining -- quite generally --
the large $N$ limit of Weyl characters through the functional equation
\HGprod. Specializing to almost flat graphs, we have then
found the exact generating function \hGexp, \hofG\
of planar square lattices endowed with a single negative curvature insertion
balanced by a number of positive defects.


We feel that our observations
could trigger the investigation of many new phenomena in two-dimensional
physics and the combinatorial theory of planar graphs.
However, most urgent is the understanding of the crossover phenomenon from
the phase of almost flat two-dimensional space to the phase of
two-dimensional quantum gravity. It requires the careful analysis
of the well-posed Cauchy-Riemann problem of the last section.
This investigation is pending. Aside its obvious mathematical interest,
the solution of this problem could help to solve
the hitherto inaccessible problem of $R^2$ quantum gravity in two dimensions.

In addition to the even lattices considered in this paper, our methods allow
the study of the ``melting'' of more general regular, or almost
regular, lattices; e.g.~triangular lattices.

It is well known that there are many intriguing relations between integrable
two-dimensional models on regular lattices and dynamical planar
random lattices. It is tempting to try to unify the two classes of
models, a project one might term $GUT_2$. Our work should be considered
a first attempt into this direction, even though it must be noted
that further methods will have to be developed in order to successfully
treat matter coupled to dually weighted graphs.

Some of the results presented above could be interpreted as insights into
the structure of the group $SU(\infty)$ (see section 3 on
the large $N$ limit of Weyl characters.). Further insights into this
direction might prove very useful for the treatment of higher
dimensional matrix models, e.g.~the principal chiral field, discrete
string theories in physical dimensions and, one hopes, QCD.
\vskip 30pt
\hskip -20pt {\bf Acknowledgements}

We would like to thank E.~Br\'ezin and I. Kostov for
useful discussions.

\appendix{A}{Derivation of the inversion formula}

We start by proving that the constant coefficient in \hG\ is equal to
1 (the normalization of the density).  To correctly normalize the
density, $\rho(h)$, we have to ensure that
%
\eqn\rhonorm{
1-b=\int_b^a\,dh\rho(h).}
%
Using the fact that $\ln G(h)=H(h)+F(h)$, we replace the integral
by the contour integral
%
\eqn\rhocont{
1-b=\oint_{C_H}\,{dh\over 2\pi i}\ln G(h),}
%
with the contour $C_h$ encircling the $[b,a]$ part of the cut of
$H(h)$, as shown in Fig. 7.  The zig-zag line corresponds to the
logarithmic cut starting at $h=b$. Note that this is not a closed contour
since at $b$ there is a discontinuity across the cut of $\pm i\pi$.
Evaluating $G(h)$ around this contour we see that its argument goes
from $+i\pi$ at $h=b$ (below the cut) all the way around to $-i\pi$ at
$h=b$ (above the cut). We now change integration variables from $h$ to
$G$, with (in light of the comment above) the contour $C_G$ in the complex
$G$ plane encircling the origin (see Fig. 7):
%
\eqn\rhocond{
1-b=-\oint_{C_G}\,{dG\over 2\pi i}h'(G)\ln G(h)=
         -\oint_{C_G}\,{dG\over 2\pi i}
    \bigl[{\partial\over\partial G}(h(G)\ln G)-{h(G)\over G}\bigr],}
%
where $h(G)$ is defined through \hG . The contour starts and finishes
on either side of the cut generated by $\ln G$ illustrated in Fig.7 by
a zig-zag line.  The total derivative term picks up the discontinuity
across the cut giving $-b$. The final term, which picks up the
constant coefficient of $h(G)$, is thus equal to $1$.
%
%
\vskip 15pt
\hskip 10pt
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	 2.142 25.216
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	 6.140 25.184
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	 6.227 25.013
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	/
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\put{$a$} [lB] at  3.040 25.796
\put{0} [lB] at  6.841 25.095
\put{$G(b)$} [lB] at  9.967 25.745
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\putrectangle corners at  0.546 26.285 and 12.226 24.000
\endpicture
\vskip 15pt
\centerline{{\bf Fig. 7:} Definition of contours $C_h$ in the complex
$h$ plane and}
\centerline{ $C_G$ and $C_0$ in the complex $G$ plane.}
\vskip 20pt
%
%

We now complete the derivation of \HGprod .
As discussed in section 3. we start by generating $\tilde H(h)$
(related to the full resolvent by $H(h)= \tilde H(h)+\ln{h\over h-b}$ )
from the contour integral
%
\eqn\Hinva{
\tilde H(h)=\oint_{C_h}\,{dh_1\over 2\pi i}{\ln G(h_1)\over h-h_1}.}
%
Changing integration variables from $h$ to $G$, as above, this can be
written as
%
\eqn\Hinvb{
\tilde H(h)=-\oint_{C_G}\,{dG\over 2\pi i}\ln G {h'(G) \over h-h(G)},}
%
where $h(G)$ is defined through \hG\ and $h'(G)$ is the derivative
with respect to $G$.

We now simplify this contour integral by evaluating it for large
$h$. Knowing the solution in any neighbourhood of $h$ means that, by
analytic continuation, we know it everywhere. For large enough $h$, we
see from \hG\ that the contour in \Hinvb\ will encircle precisely $Q$
zeros of $h(G)$, the zeros corresponding to the inverse powers of
$G$. If we shrink the contour in \Hinvb\ so that the contour hugs
either side of the cut (see Fig. 7, contour $C_0$) we pick up these
$Q$ poles:
%
\eqn\Hinvc{
\tilde H(h)=\sum_{q=1}^Q\ln G_q(h)
-\oint_{C_0}\,{dG\over 2\pi i}\ln G {h'(G) \over h-h(G)}.}
%
The remaining contour integral is relatively easy to evaluate provided
careful attention is paid to the
contribution coming from encircling the origin. The net result is that
the contour $C_0$ contributes $\ln\bigl((-1)^{Q-1}t_{2Q}\bigr)$ from
encircling the origin and $\ln (h-b)$ from the discontinuity across
the end points of the contour. Putting these results together and
making use of the relationship between $H(h)$ and $\tilde H(h)$ we arrive
at \HGprod.


\appendix{B}{Analytic structure of $G(h)$}

As is discussed in Appendix A, the sheets $G_q(h)$ in the product of
\HGprod\ are the physical sheet and all the sheets attached to the
physical sheet by the cut of $e^{H(h)}$.
To clarify this we provide some simple examples.

\subsec{Example 1. $V_B(MA)=0$}

In this simplest case it is immediate from eq.\hGexp\ (since
$B=0$ and thus $\psi(G)=0$) that
%
\eqn\hGzero{
h-1=\sum_{q=1}^Q{t_{2q}\over G^q}.}
%
This is a polynomial equation of degree $Q$. $G(h)$ will thus be a
multivalued analytic function with $Q$ sheets. The different sheets
are connected by square root cuts, represented in Fig. 8
below by the vertical walls.
%
%
\vskip 20pt
\beginpicture
\setcoordinatesystem units <1.00000cm,1.00000cm>
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%\ellipticalarc axes ratio  0.112:0.032  360 degrees
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\plot  5.080 19.571 10.207 18.860 /
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\plot  9.495 19.387 10.791 19.215 /
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\plot  6.788 20.995  6.788 20.570 /
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\plot  6.363 21.025  6.363 20.625 /
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\plot  6.149 20.811  6.149 20.811 /
\plot  6.291 20.868  6.291 20.868 /
\plot  6.433 20.925  6.433 20.925 /
\plot  6.574 20.940  6.574 20.940 /
\plot  6.718 20.925  6.718 20.925 /
\plot  6.860 20.896  6.860 20.896 /
\plot  7.002 20.883  7.002 20.883 /
\plot  5.080 20.426  5.791 20.682 /
\plot 11.015 21.046 10.761 20.983 11.015 20.919 /
\plot 10.761 20.983 12.357 20.983 /
\plot  9.017 21.512 11.335 21.289 /
\plot  6.477 21.717  8.668 21.526 /
\plot  8.802 21.567  8.651 21.353  8.885 21.471 /
\plot  8.651 21.353  9.938 22.464 /
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\plot  9.779 20.170  9.779 19.941 /
\plot 10.065 20.127 10.065 19.928 /
\plot 10.336 20.085 10.336 19.971 /
\plot  6.788 21.311  6.788 21.196 /
\plot  6.505 21.338  6.505 21.224 /
\plot  6.219 21.366  6.219 21.253 /
\plot 10.848 19.628 10.848 19.628 /
\plot  5.935 21.380  5.935 21.296 /
\setsolid
\put{$\bullet$} [1B] at 8.492 21.200
\put{physical sheet} [lB] at 12.700 20.892
\put{$\biggr\lmoustache$} [lB] at  4.255 20.733
\put{$\biggr\rmoustache$} [lB] at  4.255 19.907
\put{$G_q$} [lB] at  3.524 20.288
\put{pole at $h=1$} [lB] at  9.224 22.782
\linethickness=0pt
\putrectangle corners at  3.524 23.112 and 15.043 18.834
\endpicture
\vskip 20pt
\centerline{{\bf Fig. 8:} Analytic structure of $G(h)$ for
$V_B(MA)=0$}
\vskip 20pt
%
%
\hskip -20pt
Note that cube roots and higher order roots are just special cases of
the above structure. For example, a cube root in the diagram above is
generated when the two square root cut points touch.

The $G_q(h)$ that enter the
product in \HGprod\ are precisely all the solutions, i.e.~all the
sheets. It then follows that
%
\eqn\eHzero{
e^{H(h)}={h\over h-1},}
%
which corresponds to a completely flat density $\rho(h)=1$ with
support $[0,1]$.

It is seen from equation \hGzero\ that at $h=1$, $G(h)$ becomes infinite
on one of its
sheets, so there is a pole at $h=1$ on what we call the physical
sheet.  For $\psi(G)$ non zero, the positive powers of $G$ ``soften'' this
pole and stretch it into a cut. The cut corresponds to exciting
boxes in the Young tableau. The next example illustrates this.

\subsec{Example 2. $V_B(MA)=MA$}

Here $B=A_1$.  It follows from eq.\hGexp\ and a simple
diagrammatic inspection that
%
\eqn\hGzero{
h-1=\sum_{q=1}^Q{t_{2q}\over G^q}\quad + \quad G}
%
This increases the degree of the polynomial by one from the
previous example, introducing an extra sheet. The pole that was at
$h=1$ has now opened into a cut (the cut of $e^{H(h)}$) connected
to this extra sheet (see Fig. 9).
%
%
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\plot  5.080 22.136 10.207 21.637 /
\plot 10.207 19.799 11.345 20.426 /
\plot 10.207 20.726 11.345 21.281 /
\plot 10.207 21.637 11.345 22.136 /
\plot  5.080 21.281  6.505 21.736 /
\plot  7.785 22.151  7.785 21.937 /
\plot  7.785 21.780  7.785 21.311 /
\plot  8.926 22.049  8.926 21.838 /
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\plot  7.074 21.380  7.074 21.139 /
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\plot  7.785 22.151  8.926 22.049 /
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\plot  6.574 20.940  6.574 20.940 /
\plot  6.718 20.925  6.718 20.925 /
\plot  6.860 20.896  6.860 20.896 /
\plot  7.002 20.883  7.002 20.883 /
\plot  7.857 21.609  7.857 21.609 /
\plot  7.999 21.579  7.999 21.579 /
\plot  8.143 21.567  8.143 21.567 /
\plot  8.285 21.567  8.285 21.567 /
\plot  8.426 21.552  8.426 21.552 /
\plot  8.568 21.552  8.568 21.552 /
\plot  8.712 21.537  8.712 21.537 /
\plot  8.854 21.510  8.854 21.510 /
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\plot  5.112 22.130  6.477 22.543 /
\plot  6.460 22.540 11.350 22.128 /
\plot 11.153 21.050 10.899 20.987 11.153 20.923 /
\plot 10.899 20.987 12.495 20.987 /
\plot 11.689 22.291 11.462 22.162 11.723 22.169 /
\plot 11.462 22.162 12.541 22.464 /
\plot  9.081 21.495  9.923 21.416 /
\plot 10.668 21.353 11.352 21.289 /
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\plot 10.065 20.127 10.065 19.928 /
\plot 10.336 20.085 10.336 19.971 /
\plot  8.071 22.037  8.071 21.937 /
\plot  8.354 22.022  8.354 21.937 /
\plot  8.640 21.979  8.640 21.865 /
\plot  6.788 21.311  6.788 21.196 /
\plot  6.505 21.338  6.505 21.224 /
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\put{physical sheet} [lB] at 12.810 20.906
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\vskip 20pt
\centerline{{\bf Fig. 9:} Analytic structure of $G(h)$ for
$V_B(MA)=MA$}
\vskip 20pt
%
%
The $G_q(h)$ that go into the product of eq.\HGprod\ are the physical
sheet and all the sheets below. We thus obtain
%
\eqn\eHone{
e^{H(h)}={h\over G^-(h)},}
%
where $G^-(h)$ is the topmost sheet.

\subsec{Example 3. $V_B(MA)=(MA)^2$}
By inspecting the moments of the dual model, we obtain
%
\eqn\hGtwo{
h-1=\sum_{q=1}^Q t_{2q}~\bigl({1\over G^q}+G^q\bigr).}
%
The sheet structure is still polynomial, but now, due to the symmetry
$G \rightarrow G^{-1}$ of
equation \hGtwo, the top sheets are the mirror image inverses of the
bottom sheets.
%
%
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\plot 10.420 23.061 10.420 23.061 /
\plot 10.562 23.048 10.562 23.048 /
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\plot  5.791 20.682  7.074 20.540 /
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\plot  6.291 20.868  6.291 20.868 /
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\plot  6.718 20.925  6.718 20.925 /
\plot  6.860 20.896  6.860 20.896 /
\plot  7.002 20.883  7.002 20.883 /
\plot  5.080 20.426  5.791 20.682 /
\plot  9.259 21.342  9.017 21.241  9.278 21.216 /
\plot  9.017 21.241 12.368 21.749 /
\plot  9.091 21.484  9.832 21.421 /
\plot 10.382 21.368 11.345 21.283 /
\plot 11.184 21.050 10.930 20.987 11.184 20.923 /
\plot 10.930 20.987 12.526 20.987 /
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\plot 10.336 20.085 10.336 19.971 /
\plot  8.071 22.037  8.071 21.937 /
\plot  8.354 22.022  8.354 21.937 /
\plot  8.640 21.979  8.640 21.865 /
\plot  6.788 21.311  6.788 21.196 /
\plot  6.505 21.338  6.505 21.224 /
\plot  6.219 21.366  6.219 21.253 /
\plot  6.788 23.004  6.788 22.947 /
\plot  6.505 23.034  6.505 22.962 /
\plot  6.219 23.048  6.219 22.976 /
\plot  5.935 23.061  5.935 23.004 /
\plot  9.779 23.660  9.779 23.603 /
\plot 10.065 23.630 10.065 23.575 /
\plot 10.348 23.618 10.348 23.588 /
\plot 10.848 19.628 10.848 19.628 /
\plot  5.935 21.380  5.935 21.296 /
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\put{$\biggr\rmoustache$} [lB] at  4.255 19.907
\put{$G_k$} [lB] at  3.524 20.288
\put{$\biggr\rmoustache$} [lB] at  4.255 22.479
\put{$\biggr\lmoustache$} [lB] at  4.255 23.305
\put{$G_q^{-1}$} [lB] at  3.524 22.860
\put{physical sheet} [lB] at 12.764 20.892
\put{cut of $e^{H(h)}$} [lB] at 12.732 21.685
\linethickness=0pt
\putrectangle corners at  3.524 24.155 and 12.764 18.834
\endpicture
\vskip 20pt
\centerline{{\bf Fig. 10:} Analytic structure of $G(h)$ for
$V_B(MA)=(MA)^2$}
\vskip 20pt
%
%
Again, what was a pole at h=1 has opened into a cut connecting
the physical sheet to the mirror image inverses of the bottom sheets.

The above three examples clarify the meaning of
equations \hGexp\ and \HGprod. A simple functional inversion
developed in \KSW\ allows us to relate $H(h)$ to the resolvent,
$\langle \Tr \bigl[{1\over P-M}\bigr]\rangle$, of the matrix model.
To verify the methods of section 3, we have directly
calculated the matrix resolvent of these models using loop equations
and simple diagrammatic arguments.

\listrefs
\end


