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%                      P. Ginsparg
%                      last updated 1/91
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%------------------------------begin---------------------------------------
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%-----------------------title-------------------------
% @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
\Title{\vbox{\baselineskip12pt\hbox{SU-4241-505}}}
{{\vbox{\centerline{Quantum Gravity, Random Geometry}
\vskip2pt \centerline{and Critical Phenomena }
}}}
\bigskip
\centerline{ Mark J. Bowick\footnote{}
{$^1$Permanent address: Dipartimento di Fisica and INFN,
Universit\`a di Roma {\it Tor Vergata}, Viale della Ricerca Scientifica,
00173, Roma, Italy.}
{\rm and}  $~$Enzo Marinari$^1$}
\medskip
\centerline{ Physics Department }
\centerline{ Syracuse University}
\centerline{ Syracuse, NY 13244-1130, USA}
\bigskip
\centerline{\bf Abstract}
\medskip
We discuss
the theory of non-critical strings with extrinsic curvature embedded 
in a target space dimension
$d$ greater than one. We emphasize the analogy between
$2d$ gravity coupled to matter and non self-avoiding liquid-like membranes
with bending rigidity. 
We first outline the exact solution for strings in dimensions
$d<1$ via the double scaling limit of matrix models
and then discuss the difficulties of an extension to $d>1$. 
Evidence from recent and ongoing
numerical simulations of dynamically triangulated random surfaces
indicate that there is a non-trivial crossover from a crumpled
to an extended surface as the bending rigidity is increased.
If the cross-over is a true second order phase transition
corresponding to a critical point there is the exciting possibility of
obtaining a well defined continuum string theory for $d>1$.
\noindent
\Date{ March 30, 1992}
\vfill \eject

String Theory is a powerful model, capable of unifying the Yang-Mills
interactions of matter with the universal interaction of gravity.
It softens the short distance (ultraviolet) divergences of Einstein
Hilbert gravity by smearing out points to one-dimensional extended
{\it strings}. These strings sweep out two-dimensional Riemann
surfaces as they evolve in Euclidean time. In the first quantized
description of string theory one may view the string coordinates
describing the embedding of the worldsheet in the target spacetime as
a collection of scalar fields living on the worldsheet.
% Here fig. 1, world sheet with coordinates.
\vbox{\vskip 3.5 truein}
The worldsheet, however, must fluctuate as one is required to
integrate over all admissible metrics to enforce diffeomorphism
(reparametrization) invariance. In this way new intrinsic degrees of
freedom (the conformal modes of the metric) enter the theory. From the
statistical mechanics viewpoint one is thus dealing with an exciting
class of models described by certain order fields living on a
fluctuating substrate. Averaging over metrics corresponds to being in
the universality class of translationally and orientationally
disordered fluctuating surfaces or membranes. These are often called
liquid-like membranes, as opposed to crystalline or hexatic membranes
that are translationally or orientationally ordered
respectively~\nref\book{
{\it Statistical Mechanics of Membranes and Surfaces,}
eds. D.~Nelson, T.~Piran and S.~Weinberg
(World Scientific, Singapore, 1989).}\book.
The remarkable fact is that these statistical
mechanical models defined on
a random mesh are, in a sense, easier to solve than the conventional
models defined on a rigid regular lattice. This is because
diffeomorphism invariance reduces the number of effective degrees of
freedom. It is even possible to admit fluctuations which change the
topology of the surface (growth or collapse of handles).
% Here fig. 2, growth of handles.

\vbox{\vskip 2.5 truein}
%\vskip 3 truein

This is certainly of great interest as a model of gravity but also
provides the basis for an exploration of membranes. Recently particularly
simple models
corresponding to certain types of conformal matter coupled to
$2d$-gravity have been exactly solved including the sum over all
possible topologies. The outline of the solution together with the
relation to $2d$-gravity and extensions to more realistic random
surfaces (flexible liquid-like membranes) is discussed below.

Let us start by considering the extreme case with no matter (order
fields) at all. All that remains is the smile on the Cheshire cat - a
fluctuating $2d$-surface. This is clearly two-dimensional gravity.
Since there are no embedding string coordinates it is also a model of
strings in zero dimensions. The Einstein-Hilbert action with a
cosmological constant term for $2d$ gravity is
%
\eqn\EH{
S[g] = {-1 \over 16 \pi G} \int_\Sigma d^2 \xi \sqrt{g} R
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%
where $g_{\alpha \beta}(\xi_1,\xi_2)$ is the $2d$ metric of the Riemann
surface $\Sigma$ with coordinates $\xi_1$ and $\xi_2$.

The partition function $Z$ then depends on two variables, Newton's
constant $G$ and the cosmological constant $\mu$
%
\eqn\part{
Z[G,\mu] = \int [{\cal D}g]\  e^{-S[g]}\ ,}
%
where the path integral is over all admissible metrics of Riemann
surfaces $\Sigma$. In two dimensions the action \EH\ is simple
since the first term is a topological invariant by the Gauss-Bonnet
theorem
%
\eqn\act{
S={-\chi(\Sigma) \over 4 G} + \mu A(\Sigma)\ ,}
%
where $\chi$ is the Euler characteristic of $\Sigma$ and $A$ is the
area. $\chi$ is related to the number of handles, or genus $h$, by
$\chi = 2 - 2 h$, where for simplicity we are assuming $\Sigma$ to be
closed (without boundaries). The partition function thus reduces to
%
\eqn\parttwo{
Z[G,\mu]=\sum_h \int dA e^{ {\chi \over 4G} } e^{ -\mu A} \Omega_h(A)\ ,}
%
where $\Omega_h(A)$ is the density of states of Riemann surfaces
$\Sigma$ of fixed area $A$ and genus $h$,
%
\eqn\dos{
\Omega_h(A) \equiv \int_{(h;A)} {\cal D} g_{\alpha \beta}.}
%
$\Omega_h(A)$ is very difficult to calculate as $h$ increases and the
sum over genus in \parttwo\ diverges~\nref\grpe{D.~Gross and V.~Periwal,
Phys. Rev. Lett. {\bf 60} (1988) 2105.}\grpe . The above expressions are all,
in fact, ill-defined. To give them meaning we must regularize the path
integrals. One approach is to discretize by replacing $\Sigma$
by a lattice. A particularly concrete and appealing discretization is to
consider all triangulations (or more generally cellular decompositions) of
$\Sigma$. The surface is thus replaced by a discrete set of $n$ points
(vertices) labelled by an index $i$. The connectivity of the lattice is
described by the adjacency matrix
%
\eqn\conn{
  C_{ij} = \bigg\{
  \eqalign{
    1 & \hbox{ if } i \hbox{ and } j \hbox{ are connected by a link}\cr
    0 & \hbox{ otherwise }
  }
}
%
This defines a metric on the lattice by fixing all links to have length
one. Thus all triangles (cells) in the triangulation are equilateral
and of fixed area. The Euler characteristic follows from Euler's
relation $\chi=V-E+F$, for $V$ vertices, $E$ edges (links) and $F$
faces (triangles). Local curvature is defined by means of the deficit
angle
%
\eqn\curv{ R_i = {\pi\over 3} {6-q_i\over q_i}\quad,}
%
where $q_i$ is the coordination number of vertex $i$
%
\eqn\coord{ q_i = \sum_{j} C_{ij}.}
%
To simulate the integral over metrics the adjacency matrix must be allowed
to fluctuate so that the coordination number of a node becomes a dynamical
degree of freedom. The local environment of a node is constantly changing.
This considerably complicates the study of such models from a computational
point of view but also makes them more interesting. These models are called
Dynamically Triangulated Random Surfaces DTRS \nref\da{F.~David,
Nucl. Phys. {\bf B257} (1985) 45.}\nref\kaza{V. Kazakov,
Phys. Lett. {\bf B150} (1985) 28.}\nref\kkm{V. Kazakov, I. Kostov
and A. Migdal, Phys. Lett. {\bf B157} (1985) 295.}\nref\adf{J.~Ambjorn,
B. Durhuus and J. Fr\"ohlich,
Nucl. Phys. {\bf B257} (1985) 433.}\refs{\da{--}\adf}. The basic move to
update $C_{ij}$ is a flip on a fundamental parallelogram of two triangles
sharing a common edge.
% * Fig on Flip Move
\vbox{\vskip 2.0 truein}
The discrete version of the partition function \parttwo\ replaces integrals
over metrics by sums over admissible triangulations and may be written in
the form
%
\eqn\disc{
Z[G,\mu]=\sum_{h=0}^{\infty} e^{ {2-2h} \over {4G} }
\sum_{n=0}^{\infty} e^{-\mu n} Z_{h,n}\ ,}
%
where $Z_{h,n}$ is the number of distinct triangulations with $n$ vertices
and genus $h$. $Z_{h,n}$ is a discrete version of $\Omega_h(A)$, since
$A$ is proportional to the $n$ for fixed area elementary triangles.
The combinatorial problem of computing $Z_{h,n}$ is related to the
quantum field theory problem of calculating the number of distinct
Feynman diagrams of a matrix $\Phi^3$ field theory. To see this one
simply constructs the dual of each triangulation.
% FIG 4 ON DUAL GRAPH HERE
\vbox{\vskip 4 truein}
It may then be shown that the original partition function \disc\ is related
to the solution of the matrix model defined by the integral
%
\eqn\mm{
\zeta(g,N) = \int d^{N^2}\Phi\ {\rm exp} \big\{-N {\rm Tr} ({\Phi^2 \over 2} -
{g \over 3} \Phi^3)\big\} }
%
over $N$x$N$-Hermitian matrices $\Phi$. The exact relation is
%
\eqn\iso{ Z[G,\mu] = \log \zeta(g,N)\quad,}
%
where one must identify
%
\eqn\iden{ N= e^{1 \over 4G}\quad {\rm and}\quad g=e^{-\mu}.}
%
It is necessary to take $\Phi$ to be a matrix to generate topologically
non-trivial triangulations. In fact it is easy to see that $N$ appears
weighted as $N^{\chi}$ for a Feynman diagram of Euler characteristic
$\chi$ in the perturbation expansion of $\zeta$ in a double power series
in $g$ and $N$ \nref\th{G.~`t~Hooft, Nucl. Phys. {\bf B72} (1974) 461.}\th .

In the continuum it has been shown \nref\kpz{V.~Knizhnik, A.~Polyakov
and A.~Zamolodchikov, Mod. Phys. Lett. {\bf A3} (1988) 819.}\nref\davi{
F.~David, Mod. Phys. Lett. {\bf A3} (1988) 207.}\nref\dk{J.~Distler and
H.~Kawai, Nucl. Phys. {\bf B321} (1989) 509.}\refs{\kpz{--}\dk}
that there is a scaling relation
%
\eqn\scal{ \Omega_h(A) \sim e^{\mu_c A} A^{\gamma_h - 3},}
%
where the string susceptibility $\gamma_h$ is
%
\eqn\susc{ \gamma_h = -{1 \over 2} + {5 \over 2}h.}
%
This result can be generalized to include particular kinds of matter
living on the surface. These are the so-called minimal conformal
models \nref\bpz{A.~Belavin, A.~Polyakov
and A.~Zamolodchikov, Nucl. Phys. {\bf B241} (1984) 333.}\bpz\
labelled by two integers $p$ and $q$. A key parameter of these
models is their central charge which measures the response of the
free energy to local curvature $R$ of the substrate and is roughly
a measure of the number of effective degrees of freedom of the model.
For a $(p,q)$ model $c$ is given by
%
\eqn\cc{ c= 1 - {6(p-q)^2 \over pq} .}
%
Note that $c$ is less than one. Since a single scalar field has $c=1$
these models of $(p,q)$ conformal matter coupled to $2d$-gravity correspond
to strings in less than one target-space dimension. The result \susc\ for
the string susceptibility is more generally
%
\eqn\kpz{ \gamma_h = 2 - {(1-h) \over 12} \bigg\{ 25-c +
\sqrt{(1-c)(25-c)}\bigg\}.}
%
It turns out that pure gravity corresponds to $p=2$ and $q=3$. One sees
from \cc\ that $c=0$ as expected. Near the critical cosmological
constant $\mu_c$ (or equivalently critical coupling $g_c$) we see that
%
\eqn\crit{ \int_0^{\infty} dA e^{-\mu A}\Omega_h(A) \sim
(\mu-\mu_c)^{2-\gamma_h}}
%
and the mean surface area
%
\eqn\area{ \langle A \rangle = -{\del \log Z \over \del \mu} }
%
diverges as ${1 \over {\mu - \mu_c}}$. The string susceptibility $\gamma_h$ is
clearly the critical exponent
for the specific heat. Diverging surface area is an indication of criticality.
Near $\mu_c$ one may thus construct  a continuum limit with associated
critical exponents that are universal in the sense that they do not depend
on the fine details of the lattice. The linearity of $\gamma_h$ in the genus
$h$ implies that $Z[G,\mu]$ is actually a function of only one scaling
variable
%
\eqn\scvar{ x = (\mu-\mu_c)\ \exp \bigg\{ {1 \over 4G}
\bigg( 1 - \sqrt{ {1-c \over 25-c}} \ \bigg) \bigg\} .}
%
In the Fall of 1989 it was discovered that the complete partition function
$Z=Z(x)$ may be determined by taking the so-called double-scaling limit
in which $\mu \rightarrow \mu_c$ and $N \rightarrow \infty$ with
$x=(\mu_c-\mu)N^{2m/2m+1}$ held fixed\nref\brka{
E.~Br\'ezin and  V.~Kazakov, Phys. Lett. {\bf B236}
(1990) 144-149.}\nref\dosh{M.~Douglas and S.~Shenker,
Nucl. Phys. {\bf B335} (1990) 635-654.}
\nref\grmi{ D.~Gross and A.~Migdal,
Phys. Rev. Lett. {\bf 64}
(1990) 127-130.}\nref\grmit{ D.~Gross and A.~Migdal, Nucl. Phys. {\bf B340}
(1990) 333-365.}\refs{\brka{--}\grmit}.
To reach the double-scaling limit for a fixed $m$ requires fine tuning
the parameters of a degree $2m$ polynomial potential in the matrix
model. The integer $m$ is called the order of multicriticality.
The critical behavior at the $m^{th}$ multicritical point is
governed by a universal scaling of the density of eigenvalues of the matrix
model at the edge of its support \nref\bobr{M.J.~Bowick and E.~Br\'ezin,
Phys. Lett. {\bf B268} (1991) 21-28.}\bobr.
The order of multicriticality is related to the particular conformal matter
being coupled by $p=2$ and $q=2m-1$. The specific heat
$f(x)=-\del^2{\rm ln}Z/\del\mu^2$ is given in this limit by an ordinary
nonlinear differential equation of Painlev\'e I type.
For $m=2$ (pure gravity), for example, it is
%
\eqn\pone{ f^2(x) + {1 \over 3} f''(x) = x .}
%
The string susceptibility determining the behavior of $f$ around the
critical point $f \sim (\mu_c - \mu)^{-\gamma_h}$, is given by
%
\eqn\strs{ \gamma_h = -{2 \over {p+q-1}} = -{ 1 \over m} . }
%
More general $(p,q)$ models are described by introducing multi-matrix
models.

Note that the original matrix integral for pure gravity is unbounded from
below at the critical point $g_c = e^{-\mu_c}$ since it corresponds
to a cubic potential. There seems to be no escape from this pathology.
Pure gravity is still not non-perturbatively well-defined by the matrix
model. Models with matter corresponding to $m$ odd are well-defined, however,
and this may be a valuable lesson. It may be necessary to add certain
classes of matter to $2d$-gravity to render the model non-perturbatively
sensible. A non-perturbatively well-defined model may be obtained
by introducing a target space supersymmetry in the one-dimensional
string \nref\mapa{E.~Marinari and G.~Parisi, Phys. Lett. {\bf B240} (1990)
375.}\mapa. The target space has one time $t$ and one anticommuting $(\theta,
\bar \theta)$ dimension. The total central charge vanishes with the
Grassmannian dimension cancelling the $d=1$ contribution.

Suppose now that we wish to describe more realistic string models
corresponding to surfaces embedded in a target space of dimensionality
$d$ greater than one. The surface is given by $x^{\mu}(\xi_1,\xi_2)
(\mu=1,...,d)$. These models obviously have $c > 1$. An immediate problem
is then apparent from eqn.\kpz. According to the continuum results the string
susceptibility is imaginary for $1<c<25$. This suggests that the model has an
inherent instability. A clearer understanding of this instability is gained
by examining the continuum limit of the discrete versions of these models
with the additional matter action given either by the Nambu-Goto action
%
\eqn\ng{ S_{NG} = \int d^2\xi \sqrt{h}\quad ,}
%
where $h$ is the determinant of the induced metric $h_{\alpha \beta} =
\del_{\alpha}x^{\mu}\del_{\beta}x_{\mu}$ and $S_{NG}$ is simply the area
of the surface in the induced metric, or by the Polyakov action
%
\eqn\poly{ S_P = \int d^2\xi \sqrt{g} \nabla x^{\mu}\nabla x_{\mu}\quad . }
%
Analytical and computational investigations clearly establish that the
continuum limit of these models is dominated by surfaces which degenerate
into a branched tree of tubes of diameter of order the lattice spacing. These
are called branched polymer configurations and are more one-dimensional
than two-dimensional.

The origin of these spikes is clear in the Nambu-Goto formulation
since an infinitesimally thin long tube has vanishing area and is
therefore not suppressed by the area action. The large entropy
for such configurations eventually dominates the statistical mechanics
of these surfaces. The Polyakov action has been shown to be in the same
universality class.

A bending rigidity may be added to the action to suppress branched
polymer
configurations~\nref\apol{A.~Polyakov, Nucl. Phys. {\bf B268}
(1986) 406.}\nref\kle{H.~Kleinert, Phys. Lett. {\bf B174} (1986)
335.}\nref\hel{W.~Helfrich, J. Phys. {\bf 46} (1985)
1263.}\refs{\apol{--}\hel}. Consider the extrinsic curvature matrix
(Gauss' second fundamental form) $K_{ij}$ given by
%
\eqn\excu{ K_{ij}^{\mu} = D_iD_jx^{\mu} , }
%
where $D_i$ is the covariant derivative along the surface. This is the
only additional term relevant under rescaling $x \rightarrow \lambda x$
that may be added to the string action and so will eventually be generated
by radiative corrections in any case. In three dimensions the trace of $K$
is the mean curvature $H=1/r_1 + 1/r_2$, where $r_i$ are the principal
radii of curvature of the surface. The extrinsic curvature action is
%
\eqn\excuac{ S_{EC} = \kappa \int d^2\xi \sqrt{g} ({\rm Tr} K)^2  .}
%
Its discrete form may be written as
%
\eqn\disec{ S_{EC} = \kappa \sum_{<ij>}
( 1 - {\bf \hat n}_i\cdot{\bf \hat n}_j)\quad  ,}
%
where $i$ and $j$ represent triangles that share a common edge and
${\bf \hat n}_i$
is the unit normal to triangle $i$. $S_{EC}$ clearly suppresses local
fluctuations in the mean curvature of the surface. But the key question
is whether there is long-range order in the normals to the surface. The
bending rigidity is, in fact, a running coupling --- it depends on the scale
at which it is measured. A perturbative calculation in the inverse
coupling $\kappa^{-1}$ reveals that strings with bending rigidity
are asymptotically free in the same sense as Quantum Chromodynamics.
Fluctuations screen the theory and soften the effective bending rigidity
as the length scale increases. The momentum $p$ dependence of $\kappa$ is
found to be~\apol\
%
\eqn\runk{ \kappa^{-1}(p) = {\kappa^{-1}_0 \over {1 - {d \over 2}
{\kappa^{-1}_0 \over 2 \pi} {\rm log} {\Lambda \over p}}}\quad ,}
%
where $\Lambda$ is the cutoff or inverse lattice spacing and $d$ is the
dimensionality of the target space.
%** BETA FUNCTION FIG HERE**
\vbox{\vskip 2 truein}
At large length scales $\kappa$ tends to zero and there is no suppression of
fluctuations in the alignment of normals to the surface. The two-point function
decays exponentially
%
\eqn\corrln{ <{\bf \hat n}(\xi_1,\xi_2)\cdot{\bf \hat n}(0)> =
e^{ - { |\xi| \over \xi_p} } }
%
with persistence length $\xi_p$. Thus the surface is always disordered
or {\it crumpled} at length scales $r$ exceeding $\xi_p$. This
conclusion is of considerable interest in the study of liquid
membranes as well. A typical example of a liquid membrane found in
nature (which can also be manufactured in the laboratory) is a lipid
bilayer. It consists of two layers of amphiphilic molecules with
polar hydrophilic heads and long hydrophobic hydrocarbon tails. These bilayers
arrange themselves in thin extended sheets. Within the bilayer
individual molecules are quite free to diffuse, so that the in-plane
elastic constants turn out to be very low. Another candidate liquid
membrane is a monolayer of surfactant molecules of an oil-water
interface in a microemulsion. In fact any flexible interface between
three-dimensional phases is a candidate system for a liquid membrane
model. We see here a beautiful interplay between string theory,
quantum gravity and the statistical mechanics of fluctuating liquid
membranes.
% FIGS OF MEMBRANES
\vbox{\vskip 3 truein}

In the last few years such systems have been extensively explored via
numerical simulations on a wide range of computers, including parallel
machines \nref\cat{S.~Catterall, Phys. Lett. {\bf B220} (1989)
207.}\nref\bjw{C.~Baillie, D. Johnston and R.~Williams, Nucl. Phys.
{\bf B335} (1990) 469.}\nref\bjcw{
C.~Baillie, S.~Catterall, D. Johnston and R.~Williams, Nucl. Phys.
{\bf B348} (1991) 543.}\nref\ajvip{
J.~Ambjorn, J.~Jurkiewicz, S.~Varsted, A.~Irback and B.~Petersson,
{\it Critical Properties of the Dynamical Random surface with Extrinsic
Curvature}, Niels Bohr preprint NBI-HE-91-14 (1991).}\refs{\cat{--}\ajvip}.
There are some novel but not fully understood results. The
full action which is simulated is given by a quadratic interaction
term plus the extrinsic curvature term
\eqn\azione{S = \sum_{<i,j>}(x^\mu_i-x^\mu_j)^2
    +\kappa\sum_{[i,j]}(1-{\bf \hat n}_i \cdot {\bf \hat n}_j )  }
where the first sum is over nearest neighbors and the second over
adjacent triangles. For $\kappa<\kappa_c\simeq 1.5$ one sees the
expected crumpled surface (see fig. $7$).
The radius of gyration of these surfaces
grows only logarithmically with their area corresponding to infinite
Haussdorf dimension $d_H$ defined by
\eqn\dimensione{R_G^2 \simeq A^{{2 \over d_H}} \quad ,}
where $R_G$ is the radius of gyration.
For $\kappa>\kappa_c$ the surfaces become extended and considerably
smoother with $d_H$ approaching two, which would be the value one
would get for a flat surface (see fig. $8$).
The nature of the cross-over at
$\kappa_c$ is still uncertain. It may be that the system is undergoing
a true thermodynamic phase transition. If it is of second order then
the continuum limit constructed at the critical coupling would be an
interesting string theory corresponding to a real extended $2d$
surface rather than a branched polymer with its largely
one-dimensional character. In this case it must be that the coupling
${1 \over \kappa}$ ceases to vary with scale (there is a fixed point of
the beta function $q{{d\kappa \over dq}}$)
at the critical coupling $\kappa_c$. At this point there is
said to be a {\it crumpling transition}. This is the most exciting possibility
from the string point of view because it would mean that we have
successfully regularized and defined the quantum theory of the string
with more than one embedding dimension without any instability
arising. The challenge would then be to understand the exact nature of
the continuum string theory at the crumpling transition and the origin
of the fixed point.

It may also be that the observed cross-over is not a true phase transition
and that the persistence length is simply reaching the finite size of
the surface that is simulated on the computer. In this case it could
still be that the surface is always crumpled on sufficiently large
distance scales. This is a real possibility for a liquid membrane but
would still leave us without a viable lattice regularization
of a string in $d>1$ dimensions. We are presently performing large-scale
simulations in three and four embedding dimensions to decide which of
the above possibilities is in fact
correct~\nref\syr{C.~Baillie, M.~Bowick,
P.~Coddington, L.~Han, G.~Harris and E.~Marinari, Work in Progress.}\syr.
% FIGS of RANDOM SURFACES at VARIOUS KAPPA

Finally it is of great interest to extend the technique of dynamically
triangulated surfaces to manifolds of higher dimension; in particular
to three and four dimensional manifolds. One can then simulate say
four dimensional Einstein-Hilbert quantum gravity and seek critical
points which provide a non-perturbative definition of a perturbatively
non-renormalizable quantum field theory. This would be a very exciting
development. Preliminary work indeed seems to indicate that there are
indeed phase transitions in $4d$ gravity~\nref\agmi{M.~Agishtein and
A.~Migdal, {\it Simulations of Four-Dimensional Simplicial Quantum
Gravity}, Princeton preprint PUPT-1287 (1991).}\agmi.

%----------------end of text-------------------------


\bigskip\bigskip
\bigskip
\centerline{\bf Acknowledgements}
\bigskip

The research of M.B.
was supported by the Outstanding Junior Investigator Grant DOE
DE-FG02-85ER40231. E.M. has been partially supported by the IBM ACIS
Grant 353-7766. We would like to thank P.~Coddington, L.~Han and
G.~Harris for extensive discussions.

\listrefs

\bye

