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\begin{center}
               {\bf RANDOM SURFACES AND QUANTUM GRAVITY:\\
               \vglue 5pt
               LOOKING FOR THE EMERGENCE OF CONTINUUM \\
               \vglue 5pt
               THEORIES FROM TRIANGULATED MANIFOLDS}\\
\vglue 1.0cm
MARK BOWICK, BERND BR{\"{U}}GMANN, PAUL CODDINGTON,\\
\vglue 5pt
LEPING  HAN, GEOFFREY HARRIS and ENZO
MARINARI\footnote{Talk presented by E. Marinari}$^{(*)}$\\
\baselineskip=25pt
{\it Physics Department, Syracuse University}\\
\baselineskip=14pt
{\it Syracuse, NY 13244 (USA)}\\
\baselineskip=25pt
$^{(*)}$ {\it and Dipartimento di Fisica, Universit\`a di Roma {\em Tor
Vergata}} \\ 
\baselineskip=14pt
{\it V. Ricerca Scientifica, 00133 Roma (Italy)}\\
\vglue .5cm
{\tt (bowick brugman paulc han gharris marinari @ sccs.npac.syr.edu)}
\vglue 0.8cm
{ABSTRACT}
\end{center}
\vglue 0.3cm
{\rightskip=3pc
 \leftskip=3pc
 \tenrm\baselineskip=12pt
 \noindent

We investigate the possible existence of continuum theories
originating from lattice theories defined on triangulated manifolds.
We discuss the $2d$ case, considering theories with extrinsic
curvature coupled to embedding bosonic matter fields, and the $4d$
case, where we find that the effect of a non-trivial measure is likely
to be relevant.
\vglue 0.8cm}

{\bf\noindent 1. Introduction}
\vglue 0.4cm

In this note we will discuss the output of two related fields of
investigation \cite{RS2,RS4}. Both of them are based on the hope
that a critical point may allow a continuum theory to emerge out of a
discrete lattice theory defined on a (diffeormosphism invariant)
triangulated lattice. In the $2d$ case one hopes that the addition to
the {\em free} surface Hamiltonian of an extrinsic curvature dependent
term could favour the emergence of a non-trivial continuum theory. In
$d=4$ the hope is that a non-trivial critical point could emerge
already from the discretized form of the simple Einstein action.  In
this case maybe self-consistency of the theory (for example the
simple demand of the emergence of a $4d$ manifold at large distances)
could help in solving the key point of the determination of the
correct measure factor. In both cases the tools used are from Statistical
Mechanics combined with a very detailed analysis of the features of the
cross-over region.
\vglue .6cm

{\bf \noindent 2. $2d$ Random Surfaces with Extrinsic Curvature}
\vglue 0.4cm

String theory, in a number of guises, has been conjectured to describe the
underlying fundamental physics of a wide variety of physical phenomena and
models. These include the strong interaction at long distances, the
three-dimensional Ising model and unified models incorporating gravity.  In its 
simplest form, the bosonic string, it is a theory of free fluctuating surfaces. 
The functional integral for the Euclideanized bosonic string is just the
partition function for an ensemble of  random fluctuating fluid surfaces. Such
surfaces are also ubiquitous in nature, being found for example in
macro-emulsions and the lipid bilayers that form an important part of cell
membranes \cite{DAVREV}. These systems are fluid because their component
`molecules' are loosely bound. Their constituents are arranged so that the net
surface tension (nearly) vanishes; thus these membranes are subject to large
thermal fluctuations.   In one important respect, however, these
chemical/biological membranes differ fundamentally from the surfaces we discuss
and simulate; they are self-avoiding. The worldsheets of the bosonic string, in
contrast, generically self-intersect.

The bosonic string for surfaces embedded in $26$ dimensional space has been
studied extensively.  Much progress in numerically simulating strings has been
made through the use of Dynamically Triangulated Random Surfaces
(DTRS)\cite{DYTRAS,KAZMIG}. For theories of surfaces embedded in $ D \leq 1$,
analysis of the continuum Liouville theory \cite{KPZ,DADIKA} and of matrix
models has led to consistent predictions for critical exponents and correlation
functions. In the `double scaling limit', in fact, exact nonperturbative
solutions \cite{EXACT} have been found from these matrix models; the functional
integral over surfaces has been essentially summed over all genera.

These analytical techniques have failed, for the most part, in probing the
theory of random surfaces in the more interesting embedding regime $D > 1$.
Indeed the formulas for critical exponents computed in Liouville theory give
complex results when continued to the range $1<D<25$.\footnote{More precisely,
one encounters these instabilities in Liouville theory when the quantity $c -
24\Delta  >1$, where $c$ denotes the central charge of the matter theory which
describes the embedding of the surfaces and $\Delta$ is the conformal weight of
the lowest weight state in this theory \cite{KUTSEI}. Since here we are
considering flat space, $c=D$ and $\Delta=0$.} The matrix models describing
$D>1$ strings have so far been too complicated to solve.

Monte Carlo simulations for $D>1$ \cite{BILDAV,AMNOEX,EALRYM,MINOEX} indicate
that these theories do not appear to describe the fluctuations of two
dimensional {\em smooth} surfaces in the continuum limit.  Extremely spiky,
branched-polymer-like configurations with high (perhaps infinite) Hausdorff
dimension dominate the functional integral\footnote{In the same way singular
configurations dominate the Gaussian theory, which is essentially a theory of
free random walks.}.

Evidence for this pathology was obtained, for instance, when it was
shown that the bare string tension, essentially the amount of work per
unit area needed to perturb a boundary loop on these configurations,
cannot vanish at the critical point \cite{AMNEED}. (This implies that
the renormalized string tension diverges in the continuum limit.) It
has been speculated in ref. \cite{CATES} that the proliferation of
vortex configurations of the internal geometry (the `Liouville mode')
induces the degeneration of these surfaces in the embedding space.  A
related explanation that is often proferred is that a negative
mass-squared particle\footnote{In some contexts, this is referred to
as the `tachyon'.}, which comes on shell in the string theory for $D
>1$, creates instabilities which are made manifest by these singular
configurations.

The tachyon, and apparently these related instabilities, can be eliminated in
particular cases by introducing fermionic coordinates and supersymmetry on
the worldsheet, and implementing an appropriate projection of states.
Presumably, the fermions effectively smooth out the
surfaces.  This would be consistent with what has been observed for
one-dimensional geometries; the random walk of a spin one-half particle has
Hausdorff dimension one and thus appears to be smooth \cite{POLBOO}. Many
authors have proposed an alternative modification of the string action
\cite{CANHAM,HELFRI,POLEXT,KLEINA} via the addition of a term that directly
suppresses extrinsic curvature\footnote{In fact, integrating the fermions out
of the Green-Schwarz superstring yields an action similar to the one we
consider, but with the addition of a complex Wess-Zumino type term
\cite{WIEGMA}.}. 

To write down our action we introduce an explicit parametrization of a
generic surface $\cm$ in $R^{3}$ with coordinates $(\sigma _1, \sigma _2)$
and the embedding $X^{\mu}(\sigma _i)$. $\mu$ runs from $1$ to $3$ (since
we only study the case of a $3d$ embedding space).  The induced metric
(the pullback of the  Euclidean $R^{3}$ metric via the embedding) is given by
%
\be
  h_{ij} = \partial _{\sigma_i} X^{\mu}\partial _{\sigma_j} X_{\mu}\ .
\ee

We will use Greek letters for the embedding space indices; they can be raised
and lowered at will since our background space is flat. Associated with each
point in $\cm$ are tangent vectors  ($t^{\mu}_i {\in} T\cm$) and a normal
vector  $n^{\mu} {\in} T\cm^{\perp}$.  The extrinsic curvature matrix $K_{ij}$
(the second fundamental form) can be defined by
%
\begin{equation}
\protect\label{kijdef}
	 \partial_{i} n^{\mu} = - K_{ij}{t^{\mu}}^j \ .
\end{equation}
The eigenvalues of this matrix are the inverses of the radii of curvature
of $\cm$. One usually describes the geometry of
these surfaces in terms of the mean curvature \cite{CHOBRU,DOCARM}
%
\begin{equation}
            H = \frac{1}{2}h^{ij}K_{ij}\ ,
\end{equation}
and the Gaussian curvature
%
\begin{equation}
           K = \epsilon^{ik}\epsilon^{jl}K_{ij}K_{kl}\ .
\end{equation}

One can show that the Gaussian curvature can be computed solely from
the metric $h_{ij}$, while the mean curvature depends explicitly on
the embedding $X^{\mu}$.

        We shall be concerned primarily with the Polyakov form of the
string action \cite{POLSTR}, in which an additional  {\em intrinsic
metric} $g_{ij}$ is introduced to describe the surface geometry.  We
discretize our model by triangulating surfaces.  In this construction,
each triangle is equilateral with area $1$ in the intrinsic  metric; the
coordination number at each vertex determines the intrinsic curvature of the
surface.  The coordinates $i$ label the vertices of the triangulation. Then
the discrete analogue of the intrinsic metric is the adjacency matrix
$C_{ij}$ whose elements equal $1$ if $i$ and $j$ label neighbouring nodes of
the triangulation, and vanish otherwise.  Two-dimensional diffeomorphism
invariance reduces to the permutation symmetry of the adjacency matrix at
this discrete level. One of the keys, in fact, to the power of this
construction is the preservation of this symmetry. Each vertex of the
triangulation is embedded in $R^3$ via the mapping $X^{\mu}_i$.  Given the
embedding $X$, we can also associate a unit normal vector
$(n^{\mu})_{\dot{k}} $ with each triangle on the surface (dotted Roman
indices label the triangles).  Note that all of the surface curvature of our
triangulations is concentrated along the links and vertices. The
surface is still flat  in the direction tangent (but not transverse) to each
link, so that the mean curvature has support on the links, while the Gaussian
curvature is non-zero only at the vertices.
The intrinsic curvature $R_i$ at vertex $i$ is
given by the deficit angle determined solely by the triangulation
%
\begin{equation}
     	R_i = \pi\frac{(6 - q_i)}{q_i},
\end{equation}
where $q_i$ denotes the connectivity of the lattice at vertex $i$.
The Gaussian curvature $K$ on the other hand is expressed in terms of
the deficit angle in the embedding space.

We shall study the theory defined by the action
%
\begin{equation}
	 S = S_{Gauss} + \lambda S_E =
  \sum_{i,j,\mu}C_{ij}(X^{\mu}_i - X^{\mu}_j)^2 +
  \lambda\sum_{\dot{k},\dot{l},\mu}C^{\dot{k}\dot{l}}(1 -
  n^{\mu}_{\dot{k}}\cdot n^{\mu}_{\dot{l}})\ .
\end{equation}
Thus, for $\lambda > 0$, we have introduced a ferromagnetic interaction
in the surface normals.
The model defined by this action has been studied
in \cite{CATTER,BAJOWI,BCJW,AMBONE,AMBTWO,CAKORE} and references therein.

From (\ref{kijdef}) and the definition of the induced metric, it follows that
this is a discretization of the continuum action
%
\begin{equation}
  \protect\label{ourcontact}
  S = \int\sqrt{|{\det {g}}|}(g^{ij}\partial_i X^{\mu}
  \partial_j X^{\mu}  + \frac{\lambda}{2}g^{ij}h^{kl}K_{ik}K_{jl})\ .
\end{equation}

Note that the second term in this action is manifestly positive, Weyl
and re\-para\-metri\-z\-a\-ti\-on invariant, and that $\lambda$ is a
dimensionless coupling. So, naively, it is not clear whether it is
relevant or not.  If it were relevant, one would then anticipate that
(since it obeys all of the appropriate symmetries) it should be
effectively generated in any string action, and that it should
engender ordering of the normals. It should then lead to another RG
fixed point at a finite value of $\lambda$, which would characterize a
phase transition between the crumpled phase (observed when
$\lambda=0$) and a `smooth(er)' phase.\footnote{The extrinsic
curvature term is also higher-derivative, indicating that the field
theory described by this action is non-unitary. This fact alone does
not imply that the associated string-scattering amplitudes do not
satisfy unitarity.  Polchinski and Yang \cite{POLYAN} do, however,
contend that in this case the string theory will not be unitary.  Even
if this were so, this model could still be an appropriate description
of the statistical mechanics of fluctuating surfaces, although not one
corresponding to a physical fundamental string theory.}

A renormalization group analysis \cite{HELFRI,KLEINB,FORSTE,POLEXT} indicates,
however, that there should be no phase transition at finite coupling when
such extrinsic curvature dependent operators are added to the action.  The
computations of refs. \cite{HELFRI,KLEINB,FORSTE} use the action
%
\begin{equation}
  \protect\label{acthkf}
     S = \int d^2\sigma\ (\mu_0\sqrt{\det{h}} + \frac{1}{\alpha}
    \sqrt{\det{h}}(h^{ij}K_{ij})^2)\ ,
\end{equation}
in the regime in which the string tension $\mu_0$ is small (unlike the usual
particle physics limit of string theory, which is characterized by large
$\mu_0$). After integrating out fluctuations of the embedding $X^{\mu}$
between momentum scales $\Lambda$ and $\tilde{\Lambda}$, it is found that the
renormalization of the extrinsic curvature coupling is given to one-loop
order by
%
\begin{equation}
  \beta ({\alpha}) \equiv \Lambda \frac{d\alpha}{d\Lambda}
  = - \frac{3}{4\pi}\alpha ^2 \ ,
\end{equation}
so that $\alpha$ is driven to infinity in the infra-red.
This theory thus exhibits asymptotic freedom. Surfaces are smooth
(the normals are correlated) below a persistence length \cite{PELLEI}
%
\begin{equation}
  \protect\label{persist}
  \xi_p \sim \exp(\frac{4\pi}{3\alpha_{bare}})\ ,
\end{equation}
and are disordered above this scale.  Some intuition into this result can be
gained by observing that this theory is similar to the $O(3)$ sigma model, which
is  asymptotically free \cite{POLBOO}. The normals to $M$ are the analogues of
$O(3)$ vectors, though in this case they are constrained to be normal to a
surface governed by the action (\ref{ourcontact}).

Without the extrinsic curvature term, (\ref{acthkf}) is the Nambu-Goto action,
while (\ref{ourcontact}), which we use in our simulations,  is based on the
action quantized by Polyakov.  Classically (when the equations of motion for the
Polyakov action are solved and substituted back into the action), the two
actions are equivalent.  It has also been demonstrated \cite{MORRIS} that the
two quantizations are equivalent in the critical dimension $D=26$.  In lower
dimensions (note that the Nambu-Goto action clearly does not make sense for $D <
2$), it is not so clear that quantizations `based' on the two actions are indeed
the same.  The work of Polchinski and Strominger \cite{POLSTRO} suggests that
there are alternative quantizations. Distler (\cite{DIST3D}) has also questioned
the equivalence of these quantizations in $D=3$.  Indeed, even if the two
quantizations are equivalent, it does not automatically follow that the two
theories are still the same once an extrinsic curvature dependent term has been
added.

In fact, Polyakov in \cite{POLEXT} uses a hybrid form of the action
(\ref{acthkf}) and still obtains the same result for the beta function. He
introduces an intrinsic metric $g_{ij}$, chooses the conformal gauge
$g_{ij}=\rho \delta_{ij}$ and considers
%
\begin{equation}
  \protect\label{polyact}
  S = \frac{1}{2\alpha}\int d^2\sigma( \mu_o \rho + \rho^{-1}
  (\partial^2X^{\mu})(\partial^2X^{\mu}) + \lambda^{ij}(\partial_iX^{\mu}
  \partial_jX^{\mu} - \rho\delta_{ij}))\ .
\end{equation}

Classically, the Lagrange multiplier $\lambda^{ij}$ constrains the intrinsic
metric to equal the induced metric (this equality is not enforced by the
classical equations of motion for the original Polyakov action).  This
constraint should be relaxed quantum mechanically if, as Polyakov \cite{POLBOO}
argues,  the condensate of this Lagrange multiplier assumes a value of the order
of the momentum cutoff.  If this dynamical assumption is correct, then one can
essentially derive the equivalence of this Nambu-Goto-like  and the original
Polyakov quantizations.  In the large $D$ (embedding dimension) limit, saddle
point calculations \cite{DAGUAE} show that $\lambda$ indeed does acquire a large
expectation value, and that for small values of the string tension $\mu_o$, the
coupling $\alpha$ is asymptotically free, as the RG calculations suggest.

There are, however, a couple of caveats and suggestions in the analytic
literature that do allow for the existence of a crumpling transition for
fluid surfaces.   Polyakov remarks that if, in the infrared region,
fluctuations of the internal geometry ($\rho$) are suppressed relative to
fluctuations of the extrinsic metric, then the beta function is proportional
to $\alpha$ and hence the continuum limit of the theory exhibits non-trivial
scaling behaviour; this presumably cannot be the case in the large $D$
limit.  Another RG calculation, performed by Yang \cite{YANG} using the
Polchinski-Strominger action \cite{POLSTRO} with an extrinsic curvature
dependent term, indicates that the two-loop correction (which is proportional
to $\alpha^3$) might be large enough to yield a zero of the beta
function, and thus a non-trivial IR fixed point.  The Polchinski-Strominger
action is  based on the assumption that the Liouville mode $\rho$ effectively
decouples (its mass is much greater than the momentum scale set by the
string tension); it is not clear why this assumption should hold for the
model that we simulate.  Finally, note that these computations are
perturbative (in $1/D$ or $\alpha$). It is possible that
non-perturbative effects could drive a crumpling transition.

     Monte Carlo simulations of the action (\ref{ourcontact}) on dynamically
triangulated random surfaces (DTRS) were first performed by Catterall
\cite{CATTER}, and  shortly thereafter by Baillie, Johnston, and Williams
\cite{BAJOWI,BCJW} and Catterall, Kogut and Renken \cite{CAKORE}.  They
simulated triangulations with the topology of the sphere, and measured the
specific heat
%
\be
  C(\lambda) \equiv \frac{\lambda^2}{N}(<S_E^2> - <S_E>^2)\ ,
\ee
%
on surfaces with up to $N = 144$ nodes (and $N = 288$ nodes in the
last reference).  They found a peak in the specific heat; the peak
size appeared to grow with $N$.  A similar model that can be
vectorized rather straightforwardly was also considered; the set of
planar $\phi^3$ graphs was simulated \cite{CAKORE,CEKR}.  Each vertex
of these $\phi^3$ graph was embedded in $R^3$ and the action
(\ref{ourcontact}) was used; graphs of up to $1000$ nodes were
simulated (these would be dual to $500$ node triangulations).  It was
found that the specific heat peak grew with $N$, albeit slowly, as
%
\begin{equation}
\label{specfit}
            C_{max} = AN^{\omega} + B\ ,
\end{equation}
with $\omega = 0.185(50)$.  Further work in \cite{AMBONE,AMBTWO},
using dynamical triangulations with the topology of the torus and
lattices with up to $N=576$ nodes, indicated that the rate of increase
of the peak height severely diminishes with increasing $N$.  The data
strongly suggests that in fact the specific heat peak height does {\bf
not} diverge as $N\rightarrow\infty$.  These authors also measured the
bare string tension and mass gap, by embedding the torus in a
background toroidal space spanned by a loop, and measuring the
dependence of the free energy on the loop size. They found that these
measurements (when taken for $\lambda$ values near the peak position)
are consistent with the appropriate scaling relations (with vanishing
bare string tension and mass gap) that should characterize a phase
transition to smooth surfaces. This measurement, which may turn out to
be quite important (since it constitutes the best evidence we have, up
to now, that there could really be a phase transition for
$\lambda=\lambda_c$), is still quite an indirect way of measuring
correlation functions.  As we will discuss, these scaling relations
could contradict other observed phenomena (such as the absence of
diverging correlation times and increasing finite size effects at the
pseudo-critical point).

Thus it appears that numerical evidence could allow for the existence of a
crumpling transition (most probably of higher order), while analytical
calculations generally indicate that no such transition should occur.

In  \cite{KROGOM} the peak was measured in a DTRS simulation that
incorporated self-avoidance and the extrinsic curvature term $S_E$, with  a
solid-wall potential substituted for the Gaussian term in the action. The
results for the specific heat turned out to be very similar to those found in
the simulations we have just discussed, for example, in \cite{AMBTWO}.  The
specific heat peak is, in this context, considered to be a lattice
artifact, because the peak height levels off with large $N$ (of order $500$).
These simulations included a crude block-spin measurement that suggests that
the renormalization group flow of $\lambda$ is consistent with the analytical
result of asymptotic freedom.

Simulations using other discretizations for the extrinsic curvature dependent
term have yielded somewhat different results  \cite{CATTER,BAJOWI}.  The
specific heat peak, measured in simulations employing  what is referred to as
the `area discretization', is rather feeble, and levels off for small values of
$N$ (by $N=72$) (the authors interpret this as being indicative of perhaps a
`third' order transition). Actions based on these various discretizations have
been simulated for fixed, triangular meshes.  These systems model tethered or
crystalline membranes, in which the constituent molecules are tightly bound
together.  In the tethered case, the specific heat peak obtained from
simulations of the edge action (\ref{ourcontact}) grows vigorously as a function
of $N$ for very large ($128 \times 128$) lattices \cite{WHEATE}.  This is strong
evidence for the existence  of a second order transition which, in this case, is
in accord with the analytic results -- these calculations are reviewed by Nelson
\cite{NELSON} and David \cite{DAVREV,DAVREZ} and involve mean field and large
$D$ computations which suggest that the $\beta$ function is linear at leading
order, with a zero for finite $\alpha$, i.e. a UV fixed point.   When the
alternate area discretization is used in the tethered case, the specific heat
peak again stops growing.  Recent work has demonstrated that this other
discretization is pathological in the tethered case; the  class of `corrugated'
surfaces, which are singular in one direction and smooth in the other, then
dominates the path integral \cite{WHEATE}.

Thus, given the muddle of somewhat contradictory evidence,  it is unclear
whether or not a crumpling transition exists for fluid surfaces.  We have
pursued this question by taking high statistics measurements of the specific
heat peak, and by measuring other observables describing the geometry of these
surfaces, since observables with different quantum numbers can give quite
different information. For example, in the Ising model the magnetization behaves
quite unlike the internal energy (which is invariant under the standard $Z_2$
transformation).

To analyze and interpret this data, we have applied insights gained from work on
better understood systems, primarily spin models and lattice gauge theories.
Issues of the equivalence of the Nambu-Goto and Polyakov quantizations have also
motivated us to compare the intrinsic and induced geometry of the surfaces that
we simulate.
\vglue 0.6cm

{\bf \noindent 3. Numerical Simulations in $d=2$}
\vglue 0.4cm

To minimize finite size effects, we have considered triangulations with the
topology of the torus.  The action (\ref{ourcontact}) was used, with
the BRST invariant measure utilized also by Baillie, Johnston, and Williams
\cite{BAJOWI}, so that
%
\begin{equation}
  Z = \sum_{G \in T(1)}
  \int\prod_{\mu,i}dX^{\mu}_{i}\prod_{i}q_i^{\frac{d}{2}}
  \exp( -S_{Gauss} - \lambda S_E)\ ,
\end{equation}
where $d=3$, $q_i$ is the connectivity of the $i$th vertex, and $T(1)$ refers
to the set of triangulations of genus $1$. The authors of
\cite{AMBONE,AMBTWO} do not include this connectivity dependent term in
their measure.  The long-distance physics of the simulations is
presumably insensitive to the presence of this term.
Because we have chosen a
different measure, though, our quantitative results cannot be
precisely compared with theirs.

     We measured a variety of quantities that characterize the extrinsic and
intrinsic geometry of these surfaces.  These observables include:

\begin{enumerate}

\item

  The edge curvature $S_E$ and the associated specific heat $C(\lambda)$,
  which is a sensitive indicator of the presence of a phase
  transition.

\item

  The squared radius of gyration $R_G$;
%
  \begin{equation}
    R_G \equiv \frac{1}{N}\sum_{i,\mu}(X_i^{\mu} - X_{{\rm{com}}}^{\mu})^2\ ,
    \protect\label{E_RG}
  \end{equation}
  where the com subscript refers to the center of mass of the surface.
  By measuring the $N$ dependence of the gyration radius, we can extract
  a value for the extrinsic Hausdorff dimension, which is given by
%
  \begin{equation}
    R_G \sim N^{\nu}\sim N^{\frac{2}{d_{\rm{extr}}}}\ \ .
    \protect\label{E_RGD}
  \end{equation}

\item

  The magnitude of the extrinsic Gaussian curvature. We measure
  a discretization of $\int \mid K \mid \sqrt{\mid h \mid}$,
  with
%
  \begin{equation}
    \mid {\cal{K}} \mid = \frac{1}{N}\sum_i\mid 2\pi -
    \sum_{\dot{j}} \phi_i^{\dot{j}}\mid\ .
    \protect\label{E_CK}
  \end{equation}
  Here $\phi_i^{\dot{j}}$ denotes the angle subtended by the $\dot{j}$th
  triangle at the $i$th vertex.  This quantity, therefore, measures the
  magnitude of the deficit angle in the embedding space averaged over
  all vertices. We also record the mean square fluctuation of
  $\mid{\cal{K}}\mid$, denoted by $F[\mid{\cal{K}}\mid]$.

\item

  The corresponding intrinsic quantity, $\mid {\cal{R}} \mid$, given by
%
  \begin{equation}
    \mid {\cal{R}} \mid = \frac{\pi}{3N}\sum_i\mid 6 - q_i \mid\ ,
    \protect\label{E_CR}
  \end{equation}
  and its fluctuations.  When the intrinsic and extrinsic metrics are equal,
  the intrinsic and extrinsic deficit angles are identical, and
  $K = R/2$.

\item

  To study the correlation between intrinsic and extrinsic geometry,
  we also measure the quantity which we refer to as ${\cal{K}}*{\cal{R}}$:
%
  \begin{equation}
    {\cal{K}}*{\cal{R}} \equiv \frac{\int KR}{\sqrt{\int K^2 \int R^2}} =
    \frac{\sum_i(2\pi - \sum_{\dot{j}}\phi_i^{\dot{j}})(6 - q_i)}{\sqrt{
    \sum_i(2\pi - \sum_{\dot{j}}\phi_i^{\dot{j}})^2\sum_i(6 - q_i)^2}}\ .
    \protect\label{E_KR}
  \end{equation}

  This quantity is $1$ when the metrics are equal, $0$ if they are 
  un-correlated, and negative when these curvatures are anti-correlated.

\item

  We measure, finally, the average maximum coordination number of the
  surface vertices,  ${\rm{max}}_i \ q_i$.

\end{enumerate}

In our simulations we have used the standard Metropolis algorithm to update
the embedding fields $X^\mu_i$.  To sweep through the space of
triangulations we performed flips (see reference \cite{EALRYM}) on randomly
chosen links.  Flips were automatically rejected if they yielded a degenerate
triangulation; i.e. one in which a particular vertex has fewer than three
neighbors, or in which a vertex is labeled as its own neighbor, or where more
than one link connects two vertices. (It has been proven in ref.
\cite{EALRYM,MINOEX} that the entire space of graphs of a given topology can
be spanned by only performing these flips.)  After a set of  $3N$ flips was
performed, $3N$ randomly selected embedding coordinates were updated via
random shifts from a flat distribution,
%
\be
  X^{\mu} \rightarrow X^{\mu} + \delta X^{\mu}\ .
\ee
%
The mean magnitude of these shifts
%
\be
  <\delta X^{\mu} \delta X^{\mu}>
\ee
was chosen so that the acceptance rate for updates of the $X^{\mu}$ was
roughly $50$ percent.  Most of the  Monte Carlo simulations were performed on
HP-9000 (720 and 750 series) workstations; we also collected some data by
simulating lattices on each of the 32 nodes of a CM-5.  Our code was in
Fortran, with a Fibonacci random number generator.

\begin{table}
\begin{tabular}{|l||l|l|l|l|l|l|l|l|} \hline
N=144 &.8 &1.25 & 1.35 & 1.40 & 1.45 & 1.50 & 2.0 & $\lambda$\\ \cline{2-8}
 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & $\times 10^6$ sweeps \\ \hline
N=288 & .8 & & 1.375 & 1.40 & 1.425 & 1.475 & 2.0 & $\lambda$\\ \cline{2-8}
 & 14.4 && 21.0 & 15.0 & 16.2 & 13.5 & 14.4 & $\times 10^6$ sweeps \\ \hline
N=576 & .8 & 1.325 & 1.375 & 1.40 & 1.425 & 1.475 & 2.0 & $\lambda$
\\ \cline{2-8}
& 12.0 & 27.0 & 27.0 & 27.0 & 27.0 & 27.0 & 9.6 & $\times 10^6$ sweeps \\ \hline
\end{tabular}
\protect\caption[CT_ONE]{A record of the number of sweeps performed
at each different $\lambda$ value for $3$ different lattice sizes.
\protect\label{T_ONE}}
\end{table}

In Table \ref{T_ONE} we summarize our runs. Note that we have
performed quite
long runs on the larger lattice sizes. We will discuss later
why we believe runs of this length are just sufficient to yield
accurate values of the observables for the largest lattice size ($N=576$).

In all of our figures the different dots will be printed with their associated
statistical error (sometimes too small to be visible). The statistical error
is computed by means of a standard binning procedure. We will explicitly
discuss the cases in which our estimator for the statistical error is not
asymptotic.

The lines in these figures are from a histogram reconstruction (see for example
\cite{ENZOHI,SWEFEA}). We patch different histograms
\cite{SWEFEB,ALBESA,SWESTA} by weighting them with the associated statistical
indeterminacy (which we estimate by a jack-knife binned procedure); this
procedure seems to be very effective and reliable. All of the reconstruction
curve sets ($3$: dotted, dashed and continuous for $3$ surface sizes on each
figure) consist of $3$ curves (which sometimes appear as a single
one). The middle curve is the  histogram reconstruction, and the upper and the
lower ones bound the data within the errors obtained by the
procedure we have just described.

For $N=144$ we have patched the four histograms originating from
$\lambda=1.35$, $1.40$, $1.45$, $1.50$. For $N=288$ we have used
$\lambda=1.375$, $1.40$, $1.425$ and $1.475$. For $N=576$, we chose
$\lambda=1.375$,
$1.40$ and $1.425$.

We have only drawn the reconstructed, patched curves (with their reliable
errors) in the regions where we trust them. For example, close to the
pseudo-critical region we can trust a peak pattern only when we can
reconstruct the peak by starting from both sides of the transition (without
multi-histogram patching). So we have always used single histogram
reconstructions to check these criteria, before constructing the final,
multi-histogram data.

We have measured a large number of local observables. We will see that a mixed
picture emerges from these measurements. For example the observables related to
the dynamical triangulations exhibit a characteristic pattern, to be discussed
in detail below.

\begin{figure}
\vspace{8cm}
\protect\caption[One]{The edge curvature $S_E$ as a function
of $\lambda$.  As in all other pictures, filled circles and a dotted
line correspond to $N=144$, crosses and a dashed line indicate
$N=288$, and empty squares and a solid line represent $N=576$.}
\label{FIG1}
\end{figure}

We start by showing, in Fig. \ref{FIG1}, the edge curvature $S_E$ as
a function of $\lambda$.  The crossover region is around $\lambda
\simeq 1.4$.  For small values of $\lambda$, the surface is crumpled
(see the latter part of this section). In this region, finite size
effects are already negligible for our lattice sizes, and our $3$ data
points are on top of each other. We can see weak finite size effects
by comparing the continuous lines in the transient region. The $N=144$
dotted line is far from the ones of the two larger lattices, which
lie, on the contrary, on top of each other. Finite size effects are
larger in the large $\lambda$ phase. One would expect, close to a
phase transition with a diverging correlation length, an increase of
the finite size effects which we do not observe here. The lattice
should feel the presence of the zero mass excitation, and the finite
size corrections should be larger than everywhere else (in the case of
periodic boundary conditions they would obey a power-law, rather than
decaying exponentially with size). This is surely not firm evidence
against the presence of a phase transition, but it does show that the
putative critical behaviour is atypical.

The errors in the `flat phase' ($\lambda=2.0$) are not under control. Our
estimators do not plateau under repeated iterations of the binning
procedure.  In this regime, correlation times are
large, as we will discuss in next section. This {\em caveat} holds for this
figure and for all the quantities we have measured.

\begin{figure}
\vspace{8cm}
\protect\caption[Two]{
The edge curvature specific heat, $C(\lambda)$.  The edge curvature
$S_E$ as a function of $\lambda$.  As in all other pictures, filled
circles and a dotted line correspond to $N=144$, crosses and a dashed
line indicate $N=288$, and empty squares and a solid line represent
$N=576$.}
\label{FIG2}
\end{figure}

In Fig. \ref{FIG2} we show the related specific heat $C(\lambda)$, in
the same $\lambda$ region. In Fig. \ref{FIG3} we enlarge the
pseudo-critical $\lambda$ region, in order to show the reconstructed
peak of the specific heat. As already noted our reconstruction
procedure is quite reliable here.

\begin{figure}
\vspace{8cm}
\protect\caption[Three]{
As in Fig. \ref{FIG1}, but with the multi-histogram reconstruction in the
transient region.  The edge curvature $S_E$ as a function of
$\lambda$.  As in all other pictures, filled circles and a dotted line
correspond to $N=144$, crosses and a dashed line indicate $N=288$, and
empty squares and a solid line represent $N=576$.}
\label{FIG3}
\end{figure}

\begin{table}
\begin{tabular}{|l|l|l|} \hline
N & $C(\lambda)^{\rm{max}}$ & $\lambda_c$ \\ \hline
144 & 5.37(14) & 1.395(30) \\ \hline
288 & 5.55(7) & 1.410(25) \\ \hline
576 & 5.81(17) & 1.425(30) \\ \hline
\end{tabular}
\protect\caption[CT_TWO]{The maximum of the specific heat and its position,
with their errors, for $3$ different lattice sizes. \protect\label{T_TWO}}
\end{table}

In Table \ref{T_TWO} we give the maximum of the edge curvature specific heat
and its location for the $3$ different lattice sizes.
%
We can extract from these data a specific heat exponent
$\omega = .06\pm .05$, with $\omega$ defined as in equation  (\ref{specfit}),
and the constant $B$ set to zero.  If we estimate an effective exponent from
the two smaller lattice sizes we get $.05\pm.06$, and from the two larger ones
we get $.07\pm.06$; this demonstrates that we do not see, within our
statistical precision, any
sign of a non-pure-power, non-asymptotic behavior. Note that if
the constraint that $B$ vanishes is relaxed, our data is not
accurate enough to yield a meaningful fit to equation (\ref{specfit}).
A very small
(asymptotically finite) correlation length is sufficient to produce
such a small effect on our quite small lattice sizes. These results appear to
be consistent with those of the Copenhagen group \cite{AMBTWO}, and they are
not so far from the ones of the Urbana group  \cite{CAKORE,CEKR}.

The critical value of $\lambda$ shifts (very slowly, i.e. in a way
compatible with no increase given our statistical precision) to higher values
for increasing $N$.
%
Also the shape of the specific heat (the width, for example), is basically
unchanged as we go to larger lattices. From Figs. \ref{FIG2} 
and \ref{FIG3} we do not infer any evidence of criticality.

\begin{figure}
\vspace{8cm}
\protect\caption[Foura]{
The gyration radius $R_G$ defined in (\ref{E_RG}), plotted as in Fig.
\ref{FIG1}.  The edge curvature $S_E$ as a function of $\lambda$.  As in all
other pictures, filled circles and a dotted line correspond to
$N=144$, crosses and a dashed line indicate $N=288$, and empty squares
and a solid line represent $N=576$.}
\label{FIG4}
\end{figure}

In Fig. \ref{FIG4} we show the radius of gyration of the surface,
$R_G$, as defined in (\ref{E_RG}). Here obviously the volume scaling
is non-trivial: larger surfaces have larger radius. The histogram
reconstruction ceases to work already for quite low values of
$\lambda$ for the larger lattice.  This effect could be related to the
interesting finite size scaling behavior of this quantity, which we
illustrate in better detail in Fig. \ref{FIG5} and \ref{FIG8}.

\begin{figure}
\vspace{8cm}
\protect\caption[Fourb]{The effective inverse 
Hausdorff dimension $\nu$ as a function of $\lambda$, as defined in
(\ref{E_RGD}). The filled dots and the dashed curve are from a fit to
the $N=288$ and $N=144$ data, while the empty dots and solid curve
represent the fit to $N=576$ and $N=288$.}
\label{FIG5}
\end{figure}

Here we plot
%
\be
  \nu(N) \equiv \frac{\log \frac{R(N)}{R(\frac{N}{2})}} {\log(2)}\ .
\ee
%
This is an effective inverse Hausdorff dimension, which is a function of
$\lambda$. In the large $\lambda$ limit $\nu \to 1$ and $d_{extr}\to 2$, as
expected for flat surfaces. In the low $\lambda$ limit $d_{extr}$ becomes
very large. In the pseudo-critical region $\nu$ is a linear function of
$\lambda$. Curiously enough, the latter curve yields a Hausdorff
dimension of $4$, a value characteristic of branched polymers, near the
location of the specific heat peak.
Due to finite-size effects, this value is not particularly reliable, though
(and a value changing rapidly from $2$ to a large number must to pass through
$4$ somewhere in the crossover region).  In
ref. \cite{AMBTWO} a value compatible with ours is quoted for
the critical theory ($D_H(\lambda_c)>3.4$), Which is compatible with our
findings. We stress however (also here, in complete agreement with remarks
of \cite{AMBTWO}) the fact that the Hausdorff dimension in the pseudo-critical
region depends heavily and in a quite unusual manner on $N$.

In both high and low $\lambda$ regions finite size effects are
quite small (compatible with zero in one standard deviation). On the contrary,
in the pseudo-critical region, finite size effects are strong. This effect
cannot be explained by the shift in $\lambda$ which one gets from the shift of
the peak of the specific heat, which is far too small. This behaviour is very
different from the one which we have discussed for $S_E$, and it seems to
indicate the possibility of some sort
of critical behavior close to $\lambda=1.4$.

\begin{figure}
\vspace{8cm}
\protect\caption[Five]{ 
The extrinsic Gaussian curvature
$\mid{\cal{K}}\mid$ defined in (\ref{E_CK}), plotted as in Fig. \ref{FIG1}.}
\label{FIG6}
\end{figure}


In Fig. \ref{FIG6} we plot the expectation values of the magnitude of
the extrinsic Gaussian curvature $\mid \cal{K} \mid$. If the induced
metric is equal to the intrinsic metric, then $\mid \cal{K}
\mid=\frac{\mid \cal{R} \mid}{2}$.
%
This plot is not substantially
different from that of $S_E$. We note that
finite size effects are somewhat larger in this case than for
the edge action, but they follow the same pattern (exhibiting a
big increase in the flat
phase).

\begin{figure}
\vspace{8cm}
\protect\caption[Six]{ 
The fluctuations of $\mid {\cal{K}} \mid$.}
\label{FIG7}
\end{figure}

The plot of the fluctuations of the extrinsic Gaussian curvature,
$F[{\cal{K}}]$ which we present in Fig. \ref{FIG7}, shows something
very new. A very sharp crossover, with perhaps a peak developing for
large $N$, dominates the pseudo-critical behavior.  Fluctuations do
not seem to depend on $\lambda$ in the crumpled phase, while they drop
dramatically, in a very small $\lambda$ interval, in the flat region.
Here, again, finite size effects are sizeable in the pseudo-critical
region. The position of the crossover does not depend sensitively on
$N$, while the detailed shape at $\lambda_c$ seems to change slightly
with $N$.
%
It is difficult to give a precise interpretation of a plot like this, but,
as we said, the change of regime is very clear here.

\begin{figure}
\vspace{8cm}
\protect\caption[Seven]{
The intrinsic curvature $\mid{\cal{R}}\mid$ defined in (\ref{E_CR}),
plotted as in Fig. \ref{FIG1}.}
\label{FIG8}
\end{figure}


\begin{figure}
\vspace{8cm}
\protect\caption[Eight]{ 
The fluctuations of $\mid {\cal{R}} \mid$.}
\label{FIG9}
\end{figure}


In Fig. \ref{FIG8} we give the intrinsic curvature ${\cal{R}}$, and in
Fig. \ref{FIG9} its fluctuations. Both plots are very similar to the
related, extrinsic curvature $\cal{K}$ plots.  $\mid {\cal{R}} \mid$
drops off rapidly, just as $\mid {\cal{K}} \mid$ also does.  Through
the peak region, though, $\mid {\cal{K}} \mid$ decreases by about a
factor of $5$ while $\mid {\cal{R}} \mid$ diminishes to only about
$.6$ of its value on the left-hand side of the peak.  Since the action
explicitly suppresses mean curvature, and the mean and extrinsic
Gaussian curvature are closely related (for instance, $H^2 >
\frac{K}{2}$), we would expect that for large $\lambda$ extrinsic
fluctuations would be suppressed much more than fluctuations of
intrinsic geometry.

\begin{figure}
\vspace{8cm}
\protect\caption[Nine]{ 
The intrinsic extrinsic curvature
correlation, as defined in (\ref{E_KR}), plotted as in Fig. \ref{FIG1}.}
\label{FIG10}
\end{figure}


In Fig. \ref{FIG10} we plot the intrinsic extrinsic curvature
correlation. The plot of $\cal{K}$*$\cal{R}$ indicates that intrinsic
and extrinsic geometry are strongly correlated for small $\lambda$,
but as one passes through the peak region, they become decorrelated.
This is not particularly surprising, given that the action directly
suppresses only extrinsic fluctuations.

\begin{figure}
\vspace{8cm}
\protect\caption[Tena]{ 
The average maximum connectivity
number of the surface vertices,  ${\rm{max}}_i \ q_i$, plotted as in
Fig. \ref{FIG1}.}
\label{FIG11}
\end{figure}

\begin{figure}
\vspace{8cm}
\protect\caption[Tenb]{ The scaling exponent of ${\rm{max}}_i\ q_i$, 
plotted as in Fig. \ref{FIG1}.}
\label{FIG12}
\end{figure}

In Fig. \ref{FIG11} we plot the expectation value of the maximum correlation
number, which has non-trivial scaling behavior. In Fig. \ref{FIG12} we give
its scaling exponent, defined analogously to the exponent we have
exhibited for the gyration radius.  In the pseudo-critical region
$q_{max}$ scales (for our $3$ lattice sizes) as a power, with an
exponent close to $0.1$; we do not know if this scaling is meaningful.

We have also computed correlation times for different observables. As
we already noticed, correlation times become very large in the large
$\lambda$ region. In agreement with ref. \cite{AMBTWO} 
we do not see any increase of the correlation times close to the
pseudo-critical point.

We will not present precise estimates of correlation times
(exponential or integrated): they are too large to get precise
estimates. We will limit ourselves to a discussion of a few figures,
which give quite a clear idea of what is happening. The comparison
with ref. \cite{AMBTWO} cannot be very direct, since our
action is different, and because their dynamics may be more effective
than ours. Still, the comparison is quite puzzling, since we estimate
and exhibit correlation times which are much (orders of magnitude)
larger than the ones of \cite{AMBTWO}.  Applying usual methods to
estimate $\tau_{int}$ can lead to an underestimate of short
correlation times if more than one time scale is present (that does
surely happen to our data if we integrate our data on a window of a
reasonable size).

\begin{figure}
\vspace{8cm}
\protect\caption[Elea]{
$S_E$ as a function of Monte Carlo time ($80,000$
steps) for $N=144$, $\lambda=1.4$.}
\label{FIG13}
\end{figure}

\begin{figure}
\vspace{8cm}
\protect\caption[Eleb]{ 
$R$ as a function of Monte Carlo time ($300,000$
steps) for $N=144$, $\lambda=1.4$.}
\label{FIG14}
\end{figure}

\begin{figure}
\vspace{8cm}
\protect\caption[Twa]{ 
$S_E$ as a function of Monte Carlo time ($80,000$
steps) for $N=144$, $\lambda=1.5$.}
\label{FIG15}
\end{figure}

\begin{figure}
\vspace{8cm}
\protect\caption[Twb]{Fig. \ref{FIG16}
$R$ as a function of Monte Carlo time ($300,000$
steps) for $N=144$, $\lambda=1.5$.}
\label{FIG16}
\end{figure}

\begin{figure}
\vspace{8cm}
\protect\caption[Thea]{ 
$S_E$ as a function of Monte Carlo time ($300,000$
steps) for $N=576$, $\lambda=1.4$.}
\label{FIG17}
\end{figure}

\begin{figure}
\vspace{8cm}
\protect\caption[Theb]{ 
$R$ as a function of Monte Carlo time ($3,000,000$
steps) for $N=576$, $\lambda=1.4$.}
\label{FIG18}
\end{figure}

\begin{figure}
\vspace{8cm}
\protect\caption[Fourteen]{ 
$S_E$ and $R$ from the same Monte Carlo run, $N=576$,
$\lambda=1.325$, $20,000$ steps.}
\label{FIG19}
\end{figure}

In Fig. \ref{FIG13} we plot $S_E$ for $N=144$, $\lambda=1.4$, and in
Fig. \ref{FIG14} the gyration radius for these values (with a different
time scale). Clearly, the correlation time is at least of order
$40,000$ sweeps in the first case and $100,000$ sweeps in the second
one. In Figs. \ref{FIG15} and \ref{FIG16}, we plot the same quantities for
$\lambda=1.5$.  Here correlation times are larger, of order $50,000$
steps for $S_E$ and larger than $150,000$ steps for $R_G$. In Figs.
\ref{FIG17} and \ref{FIG18} we  draw the same plot on  the largest 
lattice we study ($N=576$) for $\lambda=1.4$. Here we can see dramatic
correlations, with times of at least $100,000$ steps for $S_E$ and of
at least $1,000,000$ steps for $R_G$.

In Fig. \ref{FIG19} we plot, for the same time history and on the same
scale, both $S_E$ and $R_G$. This figure shows a clear
anticorrelation: larger surfaces are flatter, and have smaller
curvature (this is apparent in the region close to the 2000th step).

\vglue 0.6cm

{\bf \noindent 4. A Discussion of the $2d$ Results}
\vglue 0.4cm

We have shown that the transition from crumpled to flat surfaces with
increasing $\lambda$ is quite complex. Our interpretation of what is
going on is, at present, mainly based on these data and on the 
results of ref. \cite{AMBTWO}.

On lattices with up to $576$ vertices we can clearly see a sharp
crossover, but the absence of a diverging specific heat, of diverging
correlation times and of strong finite size effects suggests that we are not
observing a usual second order phase transition. On the other hand,
quantities like the intrinsic curvature, or the mass measurements of
\cite{AMBTWO} show that something non-trivial is happening.

Let us review the crux of our observations again.  This model of
crumpled surfaces appears to exhibit sharp crossover behavior in the
region around $\lambda = 1.4$.  The sharp growth in the gyration
radius and the suppression of curvature fluctuations indicate that the
normals acquire long-range correlations, up to the size of the systems
we examine.  Presumably the zero string tension measurement of
\cite{AMBTWO} also shows that the disordered regime differs from the regime in
which the surfaces are ordered (up to scale of the lattices that are
simulated) by only a small shift in $\lambda$. This evidence might
indicate the presence of a phase transition at this point.  If so, it
is very likely to be of order higher than $2$ (or, rather implausibly,
it could be second order with an extremely low negative specific heat
exponent; our lattices are much too small for us to confidently
extrapolate the value of the specific heat exponent as $N \rightarrow
\infty$).

If the transition were higher order, the peak should exhibit a cusp, but we
would need far more accurate data to detect this.  The existence of this phase
transition would then suggest the existence of a new continuum string theory,
though many other issues would have to be resolved (e.g. unitarity) to
determine if such a theory is physically desirable.

There are other possible interpretations of our data.  We  need to
consider the influence of finite-size effects, since  the surfaces which we
simulate are quite small, even smaller than one might naively assume because
they are not intrinsically smooth.  For instance, random surfaces
characteristic of $D=0$ gravity have a Hausdorff dimension of roughly
$d_{intr} = 2.8$ \cite{KAWNIM,Migtalk}; it has been predicted that surfaces
embedded in $1$ dimension have Hausdorff dimension $2 + \sqrt{2}$
\cite{KAWNIM}.  Thus, for instance, if the surfaces in our simulations had an
intrinsic dimension of $3$, they would have a linear size of fewer than $9$
lattice spacings\footnote{Of course, our lattices are too small,  by one or
two orders of magnitude, to really exhibit a convincing fractal structure.}.

Perhaps the simplest alternative explanation  for the presence of this peak is
suggested by the arguments of Kroll and Gompper \cite{KROGOM}.  They argue
that the peak occurs when the persistence length of the system approaches the
size of the lattice ($\xi_p \sim N^{\frac{1}{d}}$)\footnote{For couplings
above this point, our simulations would simply be measuring finite size
effects.}. For larger $\lambda$, fluctuations on a larger scale become more
important, but when this scale is greater than the lattice size, these
fluctuations are suppressed. Thus one might surmise that the specific heat
will drop for large $\lambda$. (It clearly goes to zero for small $\lambda$
because of the presence of the prefactor $\lambda^2$; the lattice
implements a ultraviolet cutoff that freezes out very short-range
fluctuations.)

The one-loop renormalization group calculation (\ref{persist}) predicts that
the persistence length grows as $\xi_p \sim \exp(C\lambda)$; $C$ is inversely
proportional to the leading coefficient of the  beta function.  We would
expect that the peak position should shift  to the right with increasing $N$
in this scenario as
%
\begin{equation}
  \Delta =
  \frac{\delta\ln N}{d_{intr}C}.
\end{equation}
 
Quite a large value of $C$ is needed to explain the  rapid crossover; roughly
values of $C \sim 10, d_{intr} \sim 3$ are more or less consistent with the
magnitude of the peak shift and crossover width.  The RG calculations using
different forms of the action yield $C = \frac{4\pi}{3}$ (see equation
\ref{persist}), but
this may not apply to the action we simulate.

This reasoning also indicates that the peak should widen as the lattice size
increases; we do not observe this at all. It seems plausible though that
these arguments, based only on the leading term of the high lambda expansion,
are too naive.

      An alternative scenario, which builds on the ideas in the above
paragraph, is suggested by the tantalizing similarities between the results of
our fluid surface simulations and what has been observed for the $d=4$ $SU(2)$
Lattice Gauge Theory \cite{ENZOHI} and for the $d=2$ $O(3)$ model. Let us
discuss the case of the $O(3)$ model.

The $O(3)$ model, which is asymptotically free, exhibits a specific heat peak
near $\beta = 1.4$.  This peak was first measured via Monte Carlo
simulations by Colot \cite{Colot}. It can also be obtained by differentiating
the energy data measured by Shenker and Tobochnik \cite{ShTob,Brout}.  The
origin of this peak is understood \cite{Brout,Brout2}; it is due to the
fluctuations of the sigma particle, a low-mass bound state of the massless
$O(3)$ pions.  The sigma induces short-range order, and contributes to the
specific heat as a degree of freedom only at high temperatures (when the
correlation length in the system becomes smaller than its inverse mass).
The peak thus occurs at the beginning of the crossover regime, when the
correlation length is several lattice spacings.

According to the low temperature expansion, the correlation  length grows as
$\xi \sim \exp(2\pi\beta)/\beta$. Thus one would expect  a fairly rapid
crossover in the $O(3)$ model;  the correlation length should increase by
roughly a factor of $9$ when $\beta$ is shifted by about $.35$\footnote{In
fact,  the presence of the sigma significantly modifies this  low-temperature
expansion result \cite{Brout2}  in this intermediate regime, but does not
qualitatively destroy the rapidity of the crossover.  Indeed, despite heroic
efforts, it has been impossible to extend computationally beyond this regime and
precisely verify the asymptotic low-temperature relation for the correlation
length \cite{Apost,Hasen}.}.  Such a crossover is indeed observed, though it
is somewhat less dramatic than the behavior observed for fluid surfaces; the
specific heat peak we measure is roughly half the width of the peak for the
$O(3)$ model.

Recent simulations of the $O(3)$ model \cite{Kostas} indicate that the
specific heat peak grows significantly when the lattice size $L$ is increased
from $5$ to $15$, and that virtually no growth in peak height is evident as
$L$ is increased further up to $100$.  Also, the peak position shifts to the
right as $L$ grows, and then appears to stabilize for large $L$.  This is more
or less what we observe in our simulations of fluid surfaces, on lattices of
small size.  We point out these similarities largely to emphasize that there
does exist an asymptotically free theory (with low mass excitations) which
exhibits crossover behavior qualitatively similar to that
observed in our simulations.

The analogy is perhaps deeper, though, since the fluid surface action (with
extrinsic curvature) in certain guises looks like a sigma model action.  So,
perhaps it would not be so surprising from this point of view to find a sigma
particle in these theories perhaps associated with ($\hat{n}^2 -1$), in which
$\hat{n}$ denotes the unit normal to our surfaces.

Another additional possibility is  that fluctuations of the intrinsic geometry
(the Liouville mode) are  responsible for short-range order and contribute to
the specific heat peak.

There remains much to be done to clarify whether or not a crumpling transition
occurs for a finite value of the extrinsic curvature coupling $\lambda$.  It
would be interesting (and probably a fair amount of work) to apply Wilson
renormalization group techniques to the actual action (\ref{ourcontact}) which
we simulate, to determine the leading coefficient of the beta function.
Additionally, perhaps  a calculation of $1/D$ corrections to the large $D$
computations already performed could unearth evidence of a sigma-type
excitation in these theories (the effects of the sigma appear as $1/N$
corrections in the $O(N)$ model).

       We also are histogramming our data to examine the behavior of complex
zeroes (in complex $\lambda$ space) of the partition function of our
simulations \cite{winprog}.   It has been shown (in the case of $SU(2)$
lattice gauge theory) that such zeroes, when they are near but do not approach
the real axis in the infinite volume limit, occur in theories which exhibit
specific heat peaks with no associated phase transition \cite{ENZOHI}. High
temperature expansions also indicate that the $O(3)$ model susceptibility
has a complex singularity near the real axis \cite{Butera}-- presumably this
corresponds to a zero of the partition function and is a manifestation of the
sigma.)

Of course, simulations on large lattices, with better statistics, should also
help us evaluate whether a crumpling transition exists.  We are testing
algorithms, such  as simulated tempering \cite{simtemp}, in order to evade the
long  auto-correlation times that have characterized our simulations so far.

      Even if no such transition exists for finite $\lambda$, one
could still attempt to study a continuum theory in the strong coupling
limit, as is done for QCD, for instance.  To do so, we would like
to examine global quantities, such as masses extracted from normal-normal
correlation functions, rather than just the local quantities (energy, e.g.)
that we have measured.  Measuring these correlations requires a definition
of distance on these triangulated lattices; the most successful definition
of the metric is based on the propagation of massive particles (via inversion
of the Laplacian) on these lattices \cite{MigLap}.

We have observed dramatic crossover behavior for particular observables
in our Monte Carlo simulations, but on the other hand, the correlation times
and certain finite-size effects do not behave as one would expect in the
presence of a phase transition.  
The behavior of other lattice models also indicates that it is possible
that we are observing the effects of finite-mass excitations on small
lattices, rather than a phase transition.  We hope that future work
will clarify this murky state of affairs, to determine if there indeed exists
a crumpling transition for fluid surfaces.

\vglue 0.6cm 

{\bf \noindent 5. $4d$ Triangulated Manifolds and Quantum Gravity}
\vglue 0.4cm

Dynamically triangulated random sur\-faces (DTRS) \cite{DYTRAS} play an
important role in the efforts to develop a coeherent description of quantum
gravity. The (Euclidian) space-time is approximated by a $d$-dimensional
simplicial triangulation, where the link length is constant, equal to $1$,
but the connectivity matrix is a dynamical variable.

The most important advances have been obtained in two-dimensional
quantum gravity, where DTRS are simplicial triangulations of a $2d$
manifolds. The analytic success of matrix models, which can be for
example exactly solved in the case of pure $2d$ gravity \cite{EXACT},
has strongly encouraged this approach.  The results obtained in the
triangulated approach and in the continuum lead to consistent
predictions for correlation functions and critical exponents.

Dynamical triangulations are also potentially relevant in four
dimensions. One can hope that a sensible, non-perturbative definition
of the quantum gravity theory can be obtained in some {\em scaling}
limit of the theory of $4d$ hyper-tetrahedra. This approach has much
in common with Regge calculus, where the connectivity is fixed but the
functional integration runs over the link lengths. The underlying
principle is clearly very similar, and one could say that DTRS have
the status of an improved Regge calculus.  We face the usual problem
inherent in discretizing a theory, i.e. the discretization scheme can break
some of the continuous symmetries, which will have to be recovered in
the continuum limit (if there is one). Indeed, Wilson lattice gauge
theories have taught us an important lesson. The fact that gauge
invariance is exactly conserved in the lattice theory, for all values
of the lattice spacing $a$, is in that case crucial: it would have
been very difficult to establish firm numerical results if one would
have had to care about the presence of non gauge-invariant correction,
which would disappear only in the $a \to 0$ limit. In the case of
quantum gravity, diffeomorphism invariance plays such a crucial role,
and DTRS are diffeomorphism invariant by construction, at least on the
space of piecewise flat manifolds. Hence part of the difficulties Regge
calculus has in forgetting about the lattice structure are eliminated
a priori in the DTRS lattice approach.

There are two more important points to stress. The first one is that
in the DTRS approach in $3d$ and $4d$, as opposed to $2d$, we can try
to make sense out of the pure Einstein action, without, for example,
curvature squared terms. Even though the partition function formally
diverges, at fixed volume the local curvature is bounded both from
below and from above. Therefore we can study the theory at fixed (or
better quasi-fixed, see later) volume, and look for the existence of a
stable fixed point in the large volume limit. A second order phase
transition with diverging correlation lengths, in the statistical
mechanics language, would allow us to define a continuum limit which
is universal and is not influenced by the details of the underlying
discrete lattice structure. Precisely this scenario constitutes one of
the best hopes we have to find a consistent quantum theory of gravity.
If Euclidean quantum gravity based on the Einstein action does have
non-perturbative meaning, then we can exhibit it in this way.

The second problem is to determine the measure one should use to
define the quantum theory. This problem, far from being solved in the
continuum, is completely open in the lattice approach (for a good
review see for example \cite{MENOTT}). In quantum Regge calculus the
influence of the measure has been examined in \cite{BGM} but there is
no direct relation to DTRS. The problem of the measure is the main
point we address in this note, and we want to suggest that the DTRS
approach may be powerful enough to solve it.

Recently two groups have pionereed Monte Carlo simulations of DTRS
\cite{AMBJUR,AGIMIA,AGIMIB}.
Numerical simulations (even if on quite a small scale) turned out to
be feasible, and lead to very non-trivial results. One clearly
observes a phase transition structure. Although quantitative
statements are not easy to make, given the limited statistics and the
small lattice size, it is clear that the situation is different from
the $3d$ case, where the phase transition is manifestly of first
order, and there is no continuum theory. In $4d$ \cite{AMBJUR} there
is an open possibility that the transition is second order (although
that cannot be claimed without a much more detailed finite size
study). One does not observe hysteresis cycles, and the crossover is
less sharp then in $3d$. More involved statements about critical
exponents have to be taken at this point, we believe, {\em cum grano
salis}, but there is evidence for the possible presence of a critical
point with a non-trivial continuum theory in the phase diagram for the
Euclidean Einstein action. This observation certainly warrants further
careful investigations.

These first simulations have been run with uniform measure, where all
triangulations have the same weight in the sum which defines the path
integral of the quantum theory. There are no particularly good
reasons for this choice to be the correct one, and in the following we
investigate the changes introduced by defining simplicial quantum
gravity with a non-trivial measure.

One of us has described in \cite{BERND} the structure of the Monte
Carlo simulations, and how the efficiency of the numerical procedure
can be optimized. The programming of dynamical triangulations is
difficult since a dynamical data structure is required, but it is very
relevant in many practical applications.  For the same reason the
implementation of DTRS on parallel computers is a hard problem.  Since
a set of ergodic moves preserving the volume for canonical simulations
is not known, one has to consider a Markov chain which sweeps out the
space of different volume simplicial manifolds.  We have used the
quasi-canonical method introduced in \cite{BAUBER}, which allows us to 
control the systematic distortion arising from the {\em ad hoc}
potential that keeps the system close to a given number of
$4$-simplices \cite{BERND}. All the details about the system we study
and about the numerical procedures are given in \cite{BERND}.

In the continuum the Euclidean Einstein-Hilbert action for a metric
$g_{\mu\nu}$ has the form
\begin{equation}
  \protect\label{E_EH}
  S_E[g] = \int d^4 x \sqrt{g}(\lambda-\frac{R(g)}{G})\ ,
\end{equation}
where $R(g)$ is the Ricci scalar and $\lambda$ and $G$ are the
cosmological and gravitational constants, respectively. We consider a
fixed $S^4$ topology.  On a
triangulation $T$ we discretize according to 
\begin{equation}
  V = \int d^4 x \sqrt{g} \to N_4[T]\ ,
\end{equation}
\begin{equation}
  R = \int d^4 x \sqrt{g}R(g) \to \frac{2 \pi}{\alpha} N_2[T] - 10 N_4[T]\ ,
\end{equation}
where $\alpha\simeq 1.318$, and $N_i[T]$ is the number of $i$-simplices of
the triangulation $T$. The discrete action is then
\begin{equation}
  \protect\label{E_EHL}
  S_E[T] = k_4 N_4[T] - k_2 N_2[T]\ ,
\end{equation}
where $k_4 = \lambda + 10/G$ and $k_2 =  2\pi / \alpha G$.

In the discrete quantum theory there exists a critical line
$k_4=k_4^c(k_2)$ such that if $k_4$ is different from $k_4^c$ for a
given value of $k_2$ then the random walk tends to either zero or
infinite volume. All measurements are made for $k_4=k_4^c(k_2)$.

We have selected not one but a family of measures in order to
investigate the influence of the measure in a rather general setting.
Our choice is guided by diffeomorphism invariance of the measure
\cite{MEASURE} but ignores more sophisticated arguments like BRST
invariance. We have studied, as a function of $n$, a measure
contribution of the form
\begin{equation}
  \prod_x g^{n/2}\ ,
\end{equation}
i.e. in the triangulated theory $S_E[T]$ is replaced by $S[T] = S_E[T]
+ S_M[T]$, where
\begin{equation}
  S_M = - n \sum_a \log \frac{o(a)}{5}\ .
\end{equation}
The sum runs over all $0$-simplices (sites) of the manifold, and
$o(a)$ is the number of $4$-simplices which include the site $a$. We
considered $n$ in the interval from $-5$ to $5$. The case $n=0$
repeats simulations with the trivial, uniform measure, which can be
compared with previous results.

Let us summarize our results. We have confirmed the fact that the phase
transition can be of second order, and that it is plausible that we
will be able to define a sensible theory. We find that the measure
factor plays an important role, and that the critical behavior does
depend on $n$. This is very different from $2d$ quantum gravity (see
for example \cite{RS2}), where modifications of the measure factor
of the same kind we use here do not have any non-trivial effect on the
critical behavior. Varying $n$ does not only change non-universal
quantities, like for example the value of the critical coupling, but
changes the actual (pseudo-)critical behavior.

\begin{figure}
\vspace{8cm}
\protect\caption[20]{Expectation value of $R/V$ for
$n=-5, -1, 0, 1, 5$. The curves are ordered from the smallest $n$ at the
bottom upwards.}
\label{RperV}
\end{figure}

\begin{figure}
\vspace{8cm}
\protect\caption[16]
{Expectation value of $d$ for $n=-5, -1, 0, 1, 5$, $N_4=4000$; curves
ordered as in figure \ref{FIG1}.}
\label{d}
\end{figure}

In figure \ref{RperV} we plot the average curvature $R/V$ for $V\equiv
N_4=4000$ as a function of the coupling $k_2$ for different values of
$n$, $n=-5$ for the lowest curve, then $n=-1$, $0$, $1$ and $n=5$ for
the upper curve. In figure \ref{d} we plot the average distance (in
the internal space) of two 4-simplices. We count the minimum number of
steps from 4-simplex to 4-simplex across 3-simplex faces that connect
a pair of 4-simplices and average over all 4-simplices and random
manifolds.

Both figures show that the measure operator has a pronounced effect.
Increasing the coupling of the measure term leads to a continuous,
monotonous deformation of the curves. Notice that the curves are not
just shifted. In the case of $R/V$, the singularity seems stronger for
$n \simeq 0$, where the jump in $R/V$ is quite sharp. The distance $d$
has a sharper jump for $n=1$, where it seems to jump from one constant
value to another constant. Smaller values of $n$ show a slower
increase in $d$.

For large absolute values of $n$, especially for $n=-5$, the plots
show a weaker singularity. The profile of $R/V$ hints less at a sharp
jump than the former cases, and the distance increases very smoothly
from a critical value of $k_2$, $k_2^c(n)$ on. When $n$ increases to
the value of $5$ the system seems to loose criticality on an absolute
scale.  Its behavior through the crossover is quite smooth.

A critical value of $k_2^c$ can be defined, for example, as the point where
the distance value starts to change. But for the $n=5$ case the transition
point is not very clear. Let us note that such a value of $k_2^c$ changes its
sign as a function of $n$.

\begin{figure}
\vspace{8cm}
\protect\caption[17]
{Expectation value of $R/V$ for
$n=-5, 0, 5$, $N_4=4000,16000$; curves ordered as in figure \ref{RperV}.}
\label{RperV2}
\end{figure}
%____________________________________________________________________
\begin{figure}
\vspace{8cm}
\protect\caption[18]
{Expectation value of $d$ for $n=-5, -1, 0, 1, 5$, $N_4=16000$;
curves ordered as in figure \ref{RperV}.}
\label{d2}
\end{figure}
%____________________________________________________________________

Figure \ref{RperV2} and \ref{d2} 
show results for $N_4=16000$ to indicate what kind of
finite size effects are present. As evident from figure \ref{RperV2}, larger
volumes amplify the effect of large absolute $n$ for $k_2<k_2^c$. 
Figure \ref{d2} for the average distance $d$ displays the same qualitative
behaviour as figure \ref{d} for $k_2<k_2^c$, but for large $k_2$ the average
distance does not remain constant, which might already have been guessed
in figure \ref{d}. The explanation is that the measure term $S_M$ for
positive $n$ introduces a bias towards smaller $o(a)$ which increases
$d$, but for large enough $k_2$ and $k_4$ the contribution of $S_E$
dominates and $d$ approaches its $n=0$ value. This may be the
case for $k_2>k_2^c$, while for $k_2>k_2^c$ the critical value 
$k_4^c(k_2)$ is such that $S_E\approx 0$, and at least for the range of
negative $k_2$ considered here $S_M$ is relevant and $d$ remains
large for $n=5$. The same holds for $R/V$ when $n=5$.
 
Our conclusion has been that the measure term has a strong effect, which
seems difficult to reabsorb in a simple renormalization of the
critical coupling. Always keeping in mind that a precise finite size
study is required before making quantitative statements \cite{INPREP},
we believe there are two basic possibilities. The first possibility is
that there is only one universality class, and that all the theories
we have studied do asymptotically show the same critical behavior. In
this case the rate of approach to the continuum limit is strongly
influenced by $n$. We will select the theory with faster convergence
to the continuum.

The second possibility (which is the most interesting one) is that the
measure factor changes the universality class. Our results, albeit
preliminary, seem to hint in this direction. In this case we could
have a critical value of $n_c$, and transitions belonging to different
universality classes. This is a very appealing scenario, and here the
lattice discrete theory could make its own original contribution. It
could be possible to pick out the correct measure, on the lattice, by
requiring a particular expectation value and scaling behavior of some
physical observable. Such a prescription would be a powerful tool,
turning the discrete version of the theory from a source of
indetermination into a completely determined scheme.

\vglue 0.6cm

{\bf \noindent Acknowledgements}
\vglue 0.4cm
We are much indebted to Geoffrey Fox and NPAC, for making this work
possible.
\vglue 0.6cm

{\bf \noindent References}
\vglue 0.4cm

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\end{thebibliography}

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\end{document}
