%Paper: hep-th/9309139
%From: Oscar Diego <emoscar@roca.csic.es>
%Date: Mon, 27 Sep 1993 12:54:06 UTC+0100


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\begin{document}

\title{On the Nature of Nonperturbative Effects \\
            in Stabilized 2D Quantum Gravity}

\author{{\bf Oscar Diego}\thanks{e-mail: imtod67@cc.csic.es} \ and \
        {\bf Jos\'e Gonz\'alez} \thanks{e-mail: imtjg64@cc.csic.es} \\
        {\em Instituto de Estructura de la Materia } \\
        {\em Serrano 123, 28006 Madrid} \\
        {\em Spain } }

\date{\mbox{ }}

\maketitle

\thispagestyle{empty}

\begin{abstract}
We remark that the weak coupling regime of the stochastic
stabilization of 2D quantum gravity has a unique perturbative vacuum,
which does not support instanton configurations. By means of Monte
Carlo simulations we show that the nonperturbative vacuum is also
confined in one potential well. Nonperturbative effects can be
assessed in the loop equation. This can be derived from the Ward
identities of the stabilized model and is shown to be modified by
nonperturbative terms.
\end{abstract}

\begin{flushright}
\vspace{-13.5 cm} {IEM-FT-65/92}
\end{flushright}
\baselineskip=21pt
\vfill
\newpage
\setcounter{page}{1}

{\bf 1.  Introduction}

It is well know that discretized 2-D quantum gravity coupled to conformal
matter with $ c < 1 $ can be represented by a zero-dimensional matrix model
\cite{discrete,kazakov,ds}
and that the
expansion in powers of $ 1/N $ of the matrix model, where $ N $ is the
dimension of the hermitian matrix field, defines the topological expansion of
the original discretized 2-D quantum gravity. The partition function of
zero-dimensional matrix model has the form
\begin{equation}
Z = \int d \Phi\, \exp \Bigl\{ - N Tr W ( \Phi ) \Bigr\}
\label{eq:11}
\end{equation}
where $W$ is the potential and $ \Phi $ is an hermitian matrix field. The
continuum limit of the discretized 2-D quantum gravity is achieved by taking
the
limit of the coupling constants of the potential $W$ to some critical set of
coupling constants for which the partition function
(\ref{eq:11}) has non-analytic
behaviour. For pure gravity the
argument of the integral (\ref{eq:11})
is an increasing function for large values of $\Phi$
and the integral does not exist for finite $N$, but in the
large $N$ limit the integral exists up to some value of the coupling constant.
This is the critical value of the coupling constant, and the non-analytic
behaviour developed makes the partition function above it not well-defined.
The matrix model is defined only in the
large $N$ limit and its expansion in powers of $ 1/N $ represents the
topological expansion of the quantum gravity, but the matrix model does not
define uniquely the non perturbative 2-D quantum gravity \cite{dav}.

Stochastic stabilization \cite{gre,amv} provides, on the other hand, a
way of mapping the model (\ref{eq:11}) into one which is defined
for all values of $N$ and the coupling constant, while
reproducing the perturbative expansion in powers of $ 1/N $.
There has been much hope that this stabilization of the original
matrix model could lead to an unambiguous nonperturbative
definition of quantum gravity. If these expectations have not
been met, it has been in part because of the poor understanding
of nonperturbative effects in the stabilized theory. As we
review afterwards, this is defined in one more dimension than
the original matrix model, setting therefore the problem into the
framework of quantum mechanics. In fact, in the zero-dimensional
model (\ref{eq:11}) the source of nonperturbative effects lies
on the transmission of eigenvalues through the walls of the
confining potential well \cite{daz}, so that one
could expect the same kind
of phenomenon in the stabilized theory. The main point of this
article, however, is that the ground state of the
one-dimensional model does not bear tunnelling between different
wells of the potential. We are able to prove this, for the
simplest matrix model, in perturbation theory as well as in the
nonperturbative regime (using a Monte Carlo approach in this
latter case). This leaves open the proper interpretation of the
nonperturbative effects. Their significance can be assessed
otherwise, since, as also shown in the paper, they modify the
form of the loop equation in the stabilized theory.


{\bf 2.  Perturbative approach }

The stochastic stabilization of the matrix model introduces a positive definite
hamiltonian
\begin{equation}
H = \frac{1}{2} Tr \left\{- \frac{1}{N^2}\frac{\partial^2}{\partial\Phi^2} +
\frac{1}{4} \left( \frac{ \partial W }{ \partial \Phi} \right)^{2} -
\frac{1}{2 N} \frac{\partial^2 W}{\partial\Phi^{2}}  \right\}
\label{eq:12}
\end{equation}
This hamiltonian is well defined for all values of $N$ and the coupling
constants.
The zero mode of the stabilized theory is given by
\begin{equation}
\Psi(\Phi) \sim \exp \left\{ - N \frac{W(\Phi)}{2} \right\}
\end{equation}
its norm being the partition function of the original matrix model.
Hence, the original matrix model is well defined if and only if this
zero energy state is a
normalizable state. When this is the case, corresponding obsevables in both
theories coincide
\begin{equation}
\langle Q \rangle_{stab} = \int Q \mid\Psi\mid^2 d\Phi =
\frac{1}{Z} \int Q \exp\Bigl\{- N W (\Phi)\Bigr\} d\Phi =
\langle Q \rangle_{matrix}
\end{equation}
We can use the stabilized theory to calculate the observables of the
original matrix model, with the ground state energy playing the role of an
order parameter which
gives us information about the range of definition of the integral
(\ref{eq:11})
. The non-analytic
behaviour of the original matrix model becomes now non-analytic behaviour
in the ground state energy of the hamiltonian (\ref{eq:12}).
For pure gravity in the large $N$ limit, below the critical point the zero
energy state is a normalizable state and the ground state
energy is zero, while
above the critical point the zero-dimensional matrix model does not exist and
the ground state energy of the stabilized hamiltonian is greater than zero.
The stabilized theory exists anyway for all values of $N$ and the coupling
constant, even if the original matrix model is ill defined.

We spend a few time showing that, despite the strange critical
exponent for the leading contribution \cite{gon}, the
topological expansion
of the ground state energy can be organized in terms of the
scaling variable of the zero-dimensional matrix model. Dealing
with the $1/N$ expansion we will also show our main conclusion in the
weak coupling regime, i.e. that the eigenvalues of $\Phi $ in
the ground state are all confined in the same well of the
potential in (\ref{eq:12}).
Let us consider the cubic matrix model given by
\begin{equation}
W(\Phi) = Tr \Phi^2 - \frac{2 g}{3} Tr \Phi^3 ,
\end{equation}
where $\Phi$ is the $N$
dimensional hermitian matrix. We consider throughout this section the
large $N$ limit of the model, for which (\ref {eq:11}) is still
defined within a certain range at $g>0$.
It is well-known that the one-dimensional matrix model is
mapped into a gas of free fermions \cite{bipz}, which are the eigenvalues
$\{\lambda_{n}\}$ of the $\Phi$ matrix
variable\footnote{In other matrix models,
the hamiltonian of the stabilized theory
has interaction terms, but one can perform a Hartree-Fock approach
to obtain the exact ground state energy in terms of suitable one-particle
states \cite{gon,mir,die}.}.
The Fokker-Planck hamiltonian of the stabilized model (\ref{eq:12})
becomes the sum of $N$ one-particle hamiltonians \cite{gre,amv,amb,amj,amk}
\begin{equation}
h_{n} = - \frac{1}{2} \frac{1}{N^2} \frac{\partial^2}{\partial\lambda_n^2} +
\frac{1}{2} \{ g^2 \lambda_n^4 - 2g\lambda_n^3 + \lambda_n^2 +
2g\lambda_n - 1 \}
\label{eq:21}
\end{equation}
The one-particle hamiltonian (\ref{eq:21}) has a main well
and a local minimun below the critical coupling constant
$g_{c} = \sqrt{1/(6 \sqrt{3})}$,
and only one well
above it (see figure 1).

The ground state energy of the Fokker-Planck hamiltonian is
\begin{equation}
E=\sum_{n=0}^{N-1} e_{n}  \label{tote}
\end{equation}
where $\{e_{n}\}_{0 \leq n \leq N-1}$ are the first $N$ eigenvalues
of the one particle
hamiltonian (\ref{eq:21}).
In the large $N$ limit we can write down for it an integral
representation, taking into account $1/N$ effects. From
the Euler-Maclaurin sum-formula \cite{for} the total energy is
\begin{equation}
E = \sum_{n=0}^{N-1} e_{n} = \int_{E_{0}^{\ast}}^{E_{F}^{\ast}} dE E \rho(E) -
\frac{1}{2} E_{F}^{\ast} + \frac{1}{2} E_{0}^{\ast} + \frac{1}{12} \frac{1}{
\rho(E_{F}^{\ast})} - \frac {1}{12} \frac{1}{\rho(E_{0}^{\ast})} + \cdots
\label{eq:22}
\end{equation}
where $E_{0}^{\ast}$ is the first eigenvalue $ e_{0} $,
$E_{F}^{\ast}$ is the value of the analytic continuation of $e_{n}$ from
$n \leq N-1$ to $n = N$, and $\rho (E)$ is the density of states.

{}From (\ref{eq:21}) we see that $1/N$ play the role of $\hbar$, and
the large $N$ limit is a semiclassical expansion. The density of states
is given by
\begin{equation}
\rho (E) = \frac{d}{dE} N(E)
\end{equation}
The distribution function of states is given by \cite{par}
\begin{equation}
N(E) = \frac {N}{\pi}\int_{a}^{b} d\lambda \sqrt{2(E-V)}
- \frac{1}{24\pi N}  \int_{a}^{b} d \lambda \frac{d^2V}{d\lambda^2}
\frac{1}{\sqrt{2(E-V)}} + O \left( \frac{1}{N^3} \right)
\label{eq:23}
\end{equation}
$a$ and $b$ being  the classical turning points.
$E_{F}^{\ast}$ and $E_{0}^{\ast}$ obey the
quantization conditions
\begin{equation}
N( E_{0}^{\ast} ) = \frac {1}{2}
\end{equation}
and
\begin{equation}
\tilde{N} ( E_{F}^{\ast} ) = N + \frac{1}{2}
\end{equation}
where $\tilde{N}(E)$ is the distribution function $N(E)$ restricted to the
main well of the stabilized potential.
In general, one has to keep track of higher order terms in the
sum formula (\ref{eq:22}) altogether with the perturbative
expansion (\ref{eq:23}), in order to obtain correctly
the $1/N^2$ expansion of the total energy.

{}From (\ref{eq:22}) and (\ref{eq:23}) the total energy is
\begin{equation}
NE = N^{2} \left( E_{0} + E_{2} \frac {1}{N^2} + O \left(
\frac{1}{N^4} \right) \right)
\label{eq:24}
\end{equation}
where
\begin{equation}
E_{0} = E_{F}^{(0)} - \frac{1}{3\pi} \int_{a}^{b} d\lambda
[2(E_{F}^{(0)} - V)]^{\frac{3}{2}}  \label{e0}
\end{equation}
\begin{equation}
E_{2} = \frac{1}{24\pi} \int_{a}^{b} d\lambda \frac{d^2V}{d\lambda^2} \frac{1}
{\sqrt{2(E_{F}^{(0)} - V )}}
- \frac{1}{24} \left[ \frac{1}{\pi} \int_{a}^{b} d\lambda
\frac {1}{\sqrt{2(E_{F}^{(0)} - V )}} \right]^{-1} \label{e2}
\end{equation}
Here $ E_{F}^{(0)} $ is the Fermi energy in the planar limit
and is given by the
condition
\begin{equation}
\frac{1}{\pi} \int_{a}^{b} d\lambda \sqrt{2(E_{F}^{(0)} - V )} = 1
\label{fl}
\end{equation}

We expect the total energy to be zero for $g < g_{c}$ to all
perturbative orders \cite{gre}, which we have actually checked
to second order in $1/N$ expansion. The leading order of the
total energy (\ref{e0}) is zero up to the critical coupling
$g_{c} = \sqrt{1/(6 \sqrt{3})}$, and greater than zero above it.
The critical behaviour of this quantity agrees with the
exponent found in the cuartic model \cite{gon}
\begin{equation}
E_{0}  \sim  (g - g_{c})^{11/4}  \;\;\;\;\;\;\;\;  g > g_{c}
\end{equation}
Regarding the subleading contribution (\ref{e2}) we find the
critical behaviour
\begin{eqnarray}
E_{2}  & = &  0       \;\;\;\;\;\;\;\;\;\;    g < g_{c}  \\
E_{2} & \sim &  (g - g_{c})^{1/4}  \;\;\;\;\;\;\;\;  g > g_{c}
\end{eqnarray}
The appropriate double scaling limit $ g \rightarrow g_{c} $ and
$ N \rightarrow \infty$  is such that $z=N(g_{c}-g)^{\frac{5}{4}}$
remains finite \cite{ds}. We see, in fact, that the topological
expansion of the total energy organizes above the critical point
as a power series in the scaling variable $z$ of the original
matrix model
\begin{equation}
NE = N^{2} ( g - g_{c} )^{\frac{11}{4}} \left( B_{1} + B_{2}
\frac {1}{N^2 ( g - g_{c} )^{\frac{5}{2}}} + \cdots \right)
\end{equation}

The striking result, however, concerns the position of the Fermi
level $E_{F}$. To leading order of the $1/N$ expansion,
$ E_{F}^{(0)} $ is characterized by the condition (\ref{fl}), which places
it precisely at the level of the local minimum of the potential,
all the way up to $g_{c}$ \cite{mar}. The first order correction
$ E_{F}^{(1)} $ can be computed from the quantization condition
\begin{equation}
\frac{N}{\pi} \int_{a}^{b} d\lambda \sqrt{2(e_{n} - V )} +
O\left( \frac{1}{N} \right)  = n + \frac{1}{2}
\label{qu}
\end{equation}
bearing in mind that we have to fill the first $N$ levels from
$e_{0}$ to $e_{N-1}$ as in (\ref{tote}). We find then
\begin{equation}
E_{F}^{(1)} = - \frac{1}{2} \left[ \frac{1}{\pi} \int_{a}^{b} d\lambda
\frac{1}{\sqrt{2(E_{F}^{(0)} - V )}} \right ]^{-1}
\end{equation}
which is a negative quantity. This means that, at least in the
weak coupling regime, the ground state appears to be confined to
the central well of the potential, and that there is no issue
about tunnelling to the region around the local minimum.
One may think that this only points at the
unfeasibility of discussing nonperturbative effects in the very
framework of the $1/N$ expansion. Actually, this approximation
implies taking the limit $N \rightarrow \infty$ from the start,
so that it would be conceivable that a more sophisticated way of
tuning the double scaling could lead to a different physical picture.
In the last section we will return to the question of the
localization of the ground state of the model at finite $N$.


{\bf 3.  Observables and the loop equation}

Below the critical point, the observables of the stabilized theory
and the matrix model have to be the same, and may be calculated
as follows \cite{amb,die}.

We add to the potential $V$ an auxiliary
term $\beta \lambda^{n}$ and, using
the Hellmann-Feynman theorem, write
the following formula for observables of
the stabilized theory
\begin{equation}
\frac{1}{N} \langle tr \Phi^{n} \rangle =
\left ( \frac{\partial E (\beta) }
{\partial \beta } \right)_{\beta = 0}.
\label{eq:31}
\end{equation}

Taking into acount (\ref{eq:24}), (\ref{eq:31}) and results from appendix A,
we perform the calculation of the observables of the theory up
to second order in $\frac{1}{N}$. To first order all observables
are given by
\begin{equation}
\langle tr \Phi^{n} \rangle = N^{2} \int_{a}^{b} d
\lambda \lambda^{n} \rho(\lambda)
\end{equation}
and
\begin{equation}
\rho (\lambda) = \frac{1}{\pi} \sqrt{2(E_{F}^{0} - V )}
\end{equation}
is the fermionic density and is equal to the eigenvalue density of the
original matrix model in the planar limit for $g < g_{c}$ \cite{mir}.
Now we perform the limit $ g \longrightarrow g_{c}$, which represents
the continuum limit of the discretized 2-D quantum gravity,
and calculate numerically the observables. After computing
the most singular part of the observables turns out to be given by
\begin{equation}
\langle tr \Phi^{n} \rangle = N^{2} (g_{c} - g)^{\frac{3}{2}}
\left ( A_{1}^{(n)} + A_{2}^{(n)} \frac{1}{N^{2} ( g_{c} - g )
^{\frac{5}{2}}} + \cdots \right)
\label{eq:32}
\end{equation}
for $g < g_{c}$, and
\begin{equation}
\langle tr \Phi^{n} \rangle = N^{2} (g - g_{c} )^{\frac{3}{2}}
\left ( C_{1}^{(n)} + C_{2}^{(n)} \frac{1}{N^{2} ( g - g_{c} )
^{\frac{5}{2}}} + \cdots \right)
\label{eq:33}
\end{equation}
for $g > g_{c}$.

In the double scaling limit (\ref{eq:32}) defines the puncture
operator of the 2D quantum gravity. As long as the critical exponents
in expresions (\ref{eq:32}) and (\ref{eq:33}) are the same, we can
define an analogous double scaling limit above the critical point given
by $ z = N ( g - g_{c})^{\frac{5}{4}} $. This limit does not
exist in the zero-dimensional matrix model, hence, it defines new effects
beyond the original formulation of matrix model. In reference \cite{amj}
this limit is used to show that the stabilized model does not
satisfy the nonperturbative KdV flow equations.



The loops equations arise from the stabilized model as follows:
We add to the potential $W$ a perturbation
\begin{equation}
W = Tr \Phi^2 - \frac{2g}{3} Tr\Phi^3 - \frac{2 \beta_n}{n} Tr \Phi^n .
\end{equation}
The potential of the stabilized theory becomes now
\begin{eqnarray}
V ( \beta_n) & = & \frac{1}{2} \{ Tr\Phi^{2} + g^{2}Tr\Phi^{4} - 2gTr\Phi^{3} +
2gTr\Phi - N \nonumber  \\
& & \mbox{} + \beta_{n}^{2} Tr\Phi^{2n-2} - 2\beta_{n}Tr\Phi^{n} + 2g\beta_{n}
Tr\Phi^{n+1} \nonumber \\
& & \mbox{} + \frac {\beta_n}{N} \sum_{k=0}^{n-2} Tr\Phi^{k} Tr\Phi^{n-2-k} \}
\end{eqnarray}
now from the Hellmann-Feynmam theorem
\begin{equation}
\left( \frac{\partial E }{\partial \beta_n }
\right)_{\beta_n = 0 } = - \langle
Tr\Phi^n\rangle + g \langle Tr\Phi^{n+3} \rangle + \frac{1}{2N}
\sum_{k=0}^{n-2} \langle Tr \Phi^k Tr \Phi^{n-2-k} \rangle.
\end{equation}


The loop of lenght $L$ is given by
\begin{equation}
W(L) = \frac{1}{N} Tr e ^{L\Phi}
\end{equation}
and is not difficult to prove that
\begin{equation}
\dot {V} \left(\frac{\partial}{\partial L } \right) \langle W (L) \rangle -
\int_{0}^{L} dJ \langle W(J) W( L - J ) \rangle =
- \frac{2}{N} \sum_{n=0}^{\infty} \frac{L^n}{n!}
\left (\frac{\partial E}
{\partial \beta_{n+1}} \right ) _{\beta_{n+1} = 0}
\label{eq:ab1}
\end{equation}
then, if the energy of the ground state is zero for all values of $\beta_{n}$
the Hellmann-Feynmam theorem gives the set of Ward identities
\begin{equation}
\left ( \frac{\partial E}{\partial \beta_{n}} \right )_{\beta_{n} = 0}
= 0  \  \  n = 0,..., \infty
\end{equation}
which are equivalent to the first loop equation
\begin{equation}
\dot {V} \left(\frac{\partial}{\partial L } \right) \langle W (L) \rangle =
\int_{0}^{L} dJ \langle W(J) W( L - J ) \rangle.
\end{equation}
We expect that the other loop equation can be found when we add to the
potential $W$ perturbation like :
\begin{equation}
W = Tr \Phi^2 - \frac{2g}{3} Tr\Phi^3 - \frac{2 \beta_{n_1, n_2,... }}{n_1
n_2...} \prod_{i} Tr \Phi^{n_i}.
\end{equation}

The stabilized hamiltonian (\ref{eq:12})
is the supersymmetric hamiltonian of
reference \cite{mar} restricted to the bosonic sector.
The Ward identities of the supersymetric matrix model becomes
the Swchinger Dyson equation of the zero-dimensional matrix model.


In the case of pure gravity the supersymmetry is broken non perturbatively,
then the Ward identities or the loop equation are modified by non
perturbative corrections.




{\bf 4.  Nonperturbative approach  }


We come back to the question of the localization of the ground state
in the stabilized theory, working near the critical coupling at
finite values of $N$. In order to get
information about the eigenvalue
distribution of the ground state, we apply the Monte Carlo
method in the computation of the path integral for the quantum system.
In reference \cite{amv} a Monte Carlo
method  was also applied to simulate observables of the  matrix
variable $ \Phi $. We consider here an alternative approach  which
optimizes the Monte Carlo calculation reducing the number of
variables to the eigenvalues of $ \Phi $.

The basic object to look for is the probability amplitude
between two states $\Psi_{i} $ and $\Psi_{f} $ at times $0$ and
$T$, respectively. This admits the path integral
representation \cite{path}
\begin{eqnarray}
\langle \Psi_{f}(T) |  \Psi_{i}(0) \rangle
 & = & \int\prod_{i=1}^{N}D\lambda_{i}(t) \Delta(\lambda(0))
\Delta(\lambda(T))  \overline{\Psi}_{f} (\lambda (T) )
   \Psi_{i} (\lambda (0) ) \nonumber \\
        &   & \exp{\left \{-N\int_{0}^{T}dt
\sum_{n=1}^{N}(\frac{1}{2}\dot{\lambda_{n}}(t)+
\frac{1}{2}V_{FP}(\lambda_{n}(t)))\right\}}
\label{eq:41}
\end{eqnarray}
where $\Delta(\lambda)$ is the Van der
Monde determinant
 and $V_{FP}$  the stabilized potential
appearing in (\ref{eq:21}). If both $\Psi_{i}$ and $\Psi_{f}$
have nonvanishing projection over the ground state, this is the
state which dominates at large $T$. The
amplitude behaves, in terms of the ground state energy $E$,
\begin{equation}
\langle \Psi_{f}(T) |  \Psi_{i}(0) \rangle
  \sim  \mbox{\Large $e^{- E T}$}
\end{equation}

We have performed, in practice, a discretization of the time $T$,
such that
$ t_{j}  = \epsilon \ j $,
where $ j = 0, \cdots, N_{T} $ and $ t_{N_{T}} = \epsilon N_{T} = T$.
The measure of integration in (\ref{eq:41}) becomes
\begin{eqnarray}
\mbox{\Large $e^{-S}$}
  & = & \exp \left\{- N \sum_{i=1}^{N} \sum_{j=0}^{N_{T}-1}
\left ( \frac{1}{2}
\frac{(\lambda_{i}(t_{j+1}) - \lambda_{i}(t_{j}) )^{2}}{\epsilon} + \epsilon
V_{FP}
(\lambda_{i}(t_{j})) \right )  \right.  \nonumber \\
 &  & +  \sum_{i < j} \log{(
\lambda_{i}(t_{0}) - \lambda_{j}(t_{0}) )}  \nonumber \\
 &  &  \left. +  \sum_{i < j} \log{(
\lambda_{i}(t_{N_{T}}) - \lambda_{j}(t_{N_{T}}) )}  \right\} \label{expo}
\end{eqnarray}
This can be simulated by the Monte Carlo method, as if we were
dealing with a two-dimensional statistical system of size
 $ N \times N_{T} $ . The variables are the
eigenvalues $\lambda_{i} ( t_{j} ) $ where $ i = 1, \ldots, N $,
$ j = 0, \ldots, N_{T} $.
The kinetic term defines a nearest-neighbor interaction along the time
axis and the determinants in (\ref{eq:41}) ---the only vestige
of Fermi statistics in the path integral--- define a long-range interaction
at the
boundaries $t = 0$ and $t = T$. There are no interactions between bulk
variables along the $N$ axis.

The crucial point in the simulation is to reach a large enough
value of the time $T$, in order to measure with confidence
the ground state properties. We have taken steps with $\epsilon
= 0.1$, and found that above $T = 8$ observables measured with
(\ref{expo}) do not show significant contribution from the first
excited states. We have imposed boundary conditions in the form
of constant wave functions at both ends of the time interval
$\Psi_{i}(\lambda) = \Psi_{f}(\lambda)
= const. $ , which have
certainly nonvanishing overlap with the ground state.

We have implemented the Monte Carlo simulation in a VAX 9000 machine
with Metropolis algorithm. We have made a random
choice of the point $t_{i}$ each time, updating then the $N$
variables at that site.
The number of iterations between measures has been $ 2000 $ MC sweeps
and we have left a thermalization period equivalent to
$ 5 \times 10^{4} $ MC sweeps of all the variables in the lattice.
We have simulated systems with $N = 5$ and $N = 10$, starting
with different initial conditions for the set of eigenvalues.
In some of the simulations
part of the eigenvalues were initially confined
in the well around the local minimum and in the others
all of them were in the central well. Irrespective of these
different choices, after the thermalization period we ended up
with a distribution of eigenvalues equal to zero in the well
around the local minimum. This happened even for values of $g$ very
close to $g_{c}$. Figure 2 shows a typical distribution for $N =
5$ and $g - g_{c} = 10^{-4}$ after $ 8 \times 10^{6} $ MC sweeps
of all the lattice. This supports strong evidence that at finite
$N$ the Fermi level is placed below the level of the local
minimum of the potential, all the way up to $g_{c}$.

{\bf 5.  Conclusions  }


We have shown that in the framework of the $1/N$ approximation
there is only one perturbative
vacuum. The Fermi level in the weak coupling regime is below the
local minimum of the
potential and,
hence, instanton configurations which start or end
at the local minimun do not exist.
In the nonperturbative approach we have also made plausible that the
ground state does not bear tunnelling between different wells of
the potential.
The explanation of this picture should be the following.
The stochastic stabilization
may be viewed as the usual stochastic
quantization with asymptotic final and initial states fixed and
given by well defined configurations of the original
model \cite{grep}. Therefore, in the stabilized matrix model
all configurations have to start and end
at the main well of the stabilized potential. The Fermi level
has to be
placed below the local minimun and an instanton which starts or ends at
the local minimun cannot exist.

One possible way of understanding the nonperturbative effects
may be envisaged as follows.
There are two kind of static solutions of the classical
equation in the large $N$ limit: (a)
all the fermions are restricted to the main
well of the potential, and (b) a fermion is placed at the local minimun.
These
two solutions are degenerate at first order in $1/N$.
Hence we expect that the time dependent solution of the
classical equation of motion connects the main well and the local minimun.
The perturbative effects lift the degeneracy, and the solution with a
fermion in the local minimun becomes an excited state. The nonperturbative
effects have to arise from trajectories of one fermion
which start and end at the main well.
These closed trajectories
are given by a sucession of instantons
and anti-instantons \cite{zin}. Hence, the instantonic action of one fermion
is
\begin{equation}
S_{inst} = 2 \int_{a}^{b} \sqrt{2(V-E_{F})}
\label{eq:c1}
\end{equation}
which agrees with the instantonic action calculated in \cite{mir}.

In this paper we have considered the double scaling limit in the order: first
take
the large $N$ limit and then $ g \rightarrow g_{c} $. This is the only double
scaling limit defined in the zero-dimensional matrix model.
But in the stabilized model it is
possible to perform the double scaling limit from finite values of $N$ and $g -
g_{c}$. We have performed a preliminary numerical calculation.
In order to understand how an alternative double scaling limit
may be  achieved from
finite values of $N$, a more detailed investigation of the
phases in the space of parameters $(N, g)$ should be carried out.

\newpage

{\bf    Appendix A}

In this appendix we show how formulas like
\begin{equation}
\frac{\partial}{\partial\beta} \int_{a}^{b} d\lambda
\frac{1}{\sqrt{2(E_{F}^{(0)} -
V)}}
\end{equation}
which have end points singular integrand, can be calculated.

We define two fixed arbitrary points $\Lambda_1$ and $\Lambda_2$ such that
\begin{eqnarray}
\int_a^b d\lambda \frac{1}{\sqrt
{2(E_{F}^{(0)} - V)}} & = & \int_a^{\Lambda_2} d\lambda
\frac{1}{\sqrt{2(E_{F}^{(0)} - V)}}
+ \int_{\Lambda_2}^{\Lambda_1} d\lambda
\frac{1}{\sqrt{2(E_{F}^{(0)} - V)}} \nonumber  \\
& & \mbox{} + \int_{\Lambda_1}^{b}
d\lambda \frac{1}{\sqrt{2(E_{F}^{(0)} - V)}}
\end{eqnarray}

For $g\neq g_{c}$, $\frac{dV}{d\lambda}$ has
only one
zero in the interval $[a, b]$, then we choose $\Lambda_1$ and $\Lambda_2$
such that the zero of $\frac{dV}{d\lambda}$ is placed between
$\Lambda_1$ and $\Lambda_2$, see $(fig. 1,a)$.

The second integral can be derived directly
\begin{equation}
\frac{\partial}{\partial\beta}\int_{\Lambda_2}^{\Lambda_1} d\lambda
\frac{1}{\sqrt{2(E_{F}^{(0)} - V)}} =
\int_{\Lambda_2}^{\Lambda_1} d\lambda
\frac{\partial}{\partial\beta}\left\{\frac{1}
{\sqrt{2(E_{F}^{(0)} - V)}}\right\}
\end{equation}
and the result is well defined for all values of the coupling constant.

The other integrals are posibly singular and we perform them as follows
\begin{eqnarray}
\int_{\Lambda_1}^{b} d\lambda \frac{1}{\sqrt{2(E_{F}^{(0)} - V)}} & = &
\int_{\Lambda_1}^{b} d\lambda \frac{1}
{\sqrt{2(E_{F}^{(0)} - V)}} \frac{\dot{V}}{\dot{V}} \nonumber\\
 & = & - \int_{\Lambda_1}^{b} d\lambda \frac{1}{\dot{V}}
\frac{d}{d\lambda}(\sqrt{2(E_{F}^{(0)} - V)})
\end{eqnarray}
where the dots are $\lambda$ derivatives, and integrations by parts gives
\begin{equation}
\int_{\Lambda_1}^{b} d\lambda \frac{1}{\sqrt{2(E_{F}^{(0)} - V)}} =
\ Analytic \ terms \ + \int_{\Lambda_1}^{b} d\lambda
\frac{d}{d\lambda} \left (
\frac{1}{\dot{V}} \right ) \sqrt{2(E_{F}^{(0)} - V)}
\end{equation}
the result is well defined for $g\neq g_{c}$ and can be derived directly, but
if $g = g_{c}$ then $\dot{V} (b) = 0$ and the integral is posibly singular.
This is the origin of the critical point $g_c$, which is defined by the
condition $\dot{V} (b) = 0$, see $( fig. 1,c )$.

To all orders in $1/N$, the energy and observables are some
combinations of integrals which have the form
\begin{equation}
\frac{d^{j}}{d\beta^{j}}\frac{d^{k}}{dE^{k}}
\left( \int_{a}^{b} d\lambda \frac{P(\lambda)}
{\sqrt{2(E - V(\beta))}} \right)_{E=E_{F}^{(0)},\beta=0}
\end{equation}
This integrals have the same critical point $g_{c}$. The critical
point $g_{c}$ remains constant to all orders in $1/N$ expansion.




\vfill

\pagebreak


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\pagebreak

\newpage


\centerline{\bf Figure Captions}

\begin{itemize}




\item[Figure 1.] Plot of the stabilized potential a) below the critical point,
b) above the critical point and c) at the critical point

\item[Figure 2.] Plot of the normalized fermionic density:
continuun line.
For $N = 5$, $g - g_{c} =
10^{-4}$, $100$ time intervals . We have performed $4200$ measures.
The planar fermionic density is given by
the dashed line. The vertical dashed
line is placed at the absolute
minimun of the stabilized potential. The local maximun and
the cut of the support of $\rho$ are very near of the local
minimun.

\end{itemize}

\end{document}



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5250 1300 M 5250 1200 D S
5288 1300 M 5288 1200 D S
5288 1300 M 5354 1200 D S
5354 1300 M 5354 1200 D S
5393 1300 M 5393 1200 D S
5431 1300 M 5431 1200 D S
5431 1300 M 5469 1200 D S
5507 1300 M 5469 1200 D S
5507 1300 M 5507 1200 D S
5545 1300 M 5545 1229 D 5550 1214 D 5559 1205 D 5574 1200 D
5583 1200 D 5597 1205 D 5607 1214 D 5612 1229 D 5612 1300 D S
5650 1300 M 5650 1200 D S
5650 1300 M 5716 1200 D S
5716 1300 M 5716 1200 D S
3334 1000 M 3334 1100 D S
3334 1200 M 3334 1300 D S
3334 1400 M 3334 1500 D S
3334 1600 M 3334 1700 D S
3334 1800 M 3334 1900 D S
3334 2000 M 3334 2100 D S
3334 2200 M 3334 2300 D S
3334 2400 M 3334 2500 D S
3334 2600 M 3334 2700 D S
3334 2800 M 3334 2900 D S
3334 3000 M 3334 3100 D S
3334 3200 M 3334 3300 D S
3334 3400 M 3334 3500 D S
3334 3600 M 3334 3700 D S
3334 3800 M 3334 3900 D S
3334 4000 M 3334 4100 D S
3334 4200 M 3334 4300 D S
3334 4400 M 3334 4500 D S
3334 4600 M 3334 4700 D S
3334 4800 M 3334 4900 D S
3334 5000 M 3334 5100 D S
3334 5200 M 3334 5300 D S
3334 5400 M 3334 5500 D S
3334 5600 M 3334 5700 D S
3334 5800 M 3334 5900 D S
3334 6000 M 3334 6100 D S
2713 1033 M 2716 1183 D S
2721 1333 M 2723 1371 D 2727 1428 D 2730 1477 D 2730 1483 D S
2743 1632 M 2743 1634 D 2747 1667 D 2750 1698 D 2753 1727 D
2757 1755 D 2760 1781 D S
2781 1930 M 2783 1944 D 2787 1964 D 2790 1984 D 2793 2003 D
2797 2021 D 2800 2039 D 2803 2057 D 2807 2074 D 2807 2077 D S
2840 2224 M 2840 2225 D 2843 2238 D 2847 2251 D 2850 2264 D
2853 2277 D 2857 2289 D 2860 2301 D 2863 2313 D 2867 2325 D
2870 2336 D 2873 2347 D 2877 2358 D 2880 2368 D S
2930 2510 M 2930 2511 D 2933 2519 D 2937 2527 D 2940 2535 D
2943 2543 D 2947 2551 D 2950 2558 D 2953 2566 D 2957 2573 D
2960 2580 D 2963 2587 D 2967 2594 D 2970 2601 D 2973 2608 D
2977 2614 D 2980 2621 D 2983 2627 D 2987 2634 D 2990 2640 D 2993 2646 D S
3077 2770 M 3080 2773 D 3083 2777 D 3087 2781 D 3090 2785 D
3093 2788 D 3097 2792 D 3100 2795 D 3103 2799 D 3107 2802 D
3110 2805 D 3113 2809 D 3117 2812 D 3120 2815 D 3123 2818 D
3127 2821 D 3130 2824 D 3133 2827 D 3137 2829 D 3140 2832 D
3143 2835 D 3147 2837 D 3150 2840 D 3153 2843 D 3157 2845 D
3160 2847 D 3163 2850 D 3167 2852 D 3170 2854 D 3173 2856 D
3177 2858 D 3180 2860 D 3183 2862 D 3187 2864 D 3190 2866 D
3190 2867 D S
3334 2904 M 3337 2904 D 3340 2904 D 3343 2904 D 3347 2904 D
3350 2904 D 3353 2904 D 3357 2904 D 3360 2903 D 3363 2903 D
3367 2903 D 3370 2902 D 3373 2902 D 3377 2901 D 3380 2901 D
3383 2901 D 3387 2900 D 3390 2899 D 3393 2899 D 3397 2898 D
3400 2898 D 3403 2897 D 3407 2896 D 3410 2895 D 3413 2895 D
3417 2894 D 3420 2893 D 3423 2892 D 3427 2891 D 3430 2890 D
3433 2889 D 3437 2888 D 3440 2887 D 3443 2886 D 3447 2885 D
3450 2884 D 3453 2883 D 3457 2882 D 3460 2880 D 3463 2879 D
3467 2878 D 3470 2877 D 3473 2875 D 3477 2874 D 3479 2873 D S
3610 2799 M 3610 2799 D 3613 2797 D 3617 2795 D 3620 2792 D
3623 2790 D 3627 2787 D 3630 2785 D 3633 2782 D 3637 2780 D
3640 2777 D 3643 2775 D 3647 2772 D 3650 2770 D 3653 2767 D
3657 2765 D 3660 2762 D 3663 2759 D 3667 2757 D 3670 2754 D
3673 2751 D 3677 2749 D 3680 2746 D 3683 2743 D 3687 2740 D
3690 2737 D 3693 2735 D 3697 2732 D 3700 2729 D 3703 2726 D
3707 2723 D 3710 2720 D 3713 2717 D 3717 2714 D 3720 2712 D
3723 2709 D 3727 2706 D 3727 2705 D S
3834 2601 M 3837 2598 D 3840 2594 D 3843 2591 D 3847 2587 D
3850 2584 D 3853 2580 D 3857 2577 D 3860 2573 D 3863 2569 D
3867 2566 D 3870 2562 D 3873 2558 D 3877 2555 D 3880 2551 D
3883 2547 D 3887 2544 D 3890 2540 D 3893 2536 D 3897 2533 D
3900 2529 D 3903 2525 D 3907 2521 D 3910 2517 D 3913 2514 D
3917 2510 D 3920 2506 D 3923 2502 D 3927 2498 D 3930 2495 D
3933 2491 D 3935 2489 D S
4030 2374 M 4033 2370 D 4037 2366 D 4040 2362 D 4043 2357 D
4047 2353 D 4050 2349 D 4053 2345 D 4057 2341 D 4060 2336 D
4063 2332 D 4067 2328 D 4070 2324 D 4073 2319 D 4077 2315 D
4080 2311 D 4083 2307 D 4087 2302 D 4090 2298 D 4093 2294 D
4097 2289 D 4100 2285 D 4103 2281 D 4107 2276 D 4110 2272 D
4113 2268 D 4117 2263 D 4120 2259 D 4123 2256 D S
4213 2136 M 4213 2135 D 4217 2130 D 4220 2125 D 4223 2121 D
4227 2116 D 4230 2112 D 4233 2107 D 4237 2103 D 4240 2098 D
4243 2094 D 4247 2089 D 4250 2085 D 4253 2080 D 4257 2075 D
4260 2071 D 4263 2066 D 4267 2062 D 4270 2057 D 4273 2053 D
4277 2048 D 4280 2043 D 4283 2039 D 4287 2034 D 4290 2030 D
4293 2025 D 4297 2020 D 4300 2016 D 4301 2014 D S
4389 1893 M 4390 1891 D 4393 1886 D 4397 1881 D 4400 1877 D
4403 1872 D 4407 1868 D 4410 1863 D 4413 1858 D 4417 1854 D
4420 1849 D 4423 1844 D 4427 1840 D 4430 1835 D 4433 1830 D
4437 1826 D 4440 1821 D 4443 1816 D 4447 1812 D 4450 1807 D
4453 1802 D 4457 1798 D 4460 1793 D 4463 1789 D 4467 1784 D
4470 1779 D 4473 1775 D 4476 1771 D S
4564 1649 M 4567 1646 D 4570 1641 D 4573 1637 D 4577 1632 D
4580 1628 D 4583 1623 D 4587 1619 D 4590 1614 D 4593 1609 D
4597 1605 D 4600 1600 D 4603 1596 D 4607 1591 D 4610 1587 D
4613 1582 D 4617 1578 D 4620 1573 D 4623 1569 D 4627 1564 D
4630 1560 D 4633 1555 D 4637 1551 D 4640 1547 D 4643 1542 D
4647 1538 D 4650 1533 D 4653 1529 D 4653 1529 D S
4745 1410 M 4747 1408 D 4750 1403 D 4753 1399 D 4757 1395 D
4760 1391 D 4763 1387 D 4767 1382 D 4770 1378 D 4773 1374 D
4777 1370 D 4780 1366 D 4783 1362 D 4787 1358 D 4790 1354 D
4793 1349 D 4797 1345 D 4800 1341 D 4803 1337 D 4807 1333 D
4810 1329 D 4813 1325 D 4817 1321 D 4820 1317 D 4823 1313 D
4827 1309 D 4830 1305 D 4833 1301 D 4837 1297 D 4840 1294 D S
4940 1182 M 4943 1179 D 4947 1175 D 4950 1172 D 4953 1169 D
4957 1165 D 4960 1162 D 4963 1159 D 4967 1155 D 4970 1152 D
4973 1149 D 4977 1145 D 4980 1142 D 4983 1139 D 4987 1136 D
4990 1132 D 4993 1129 D 4997 1126 D 5000 1123 D 5003 1120 D
5007 1117 D 5010 1114 D 5013 1111 D 5017 1108 D 5020 1105 D
5023 1102 D 5027 1099 D 5030 1096 D 5033 1093 D 5037 1090 D
5040 1087 D 5043 1085 D 5047 1082 D 5049 1080 D S
2633 1000 M 2647 1006 D 2660 1011 D 2673 1006 D 2687 1011 D
2700 1000 D 2713 1000 D 2727 1011 D 2740 1006 D 2753 1006 D
2767 1017 D 2780 1040 D 2793 1074 D 2807 1057 D 2820 1091 D
2833 1103 D 2847 1120 D 2860 1177 D 2873 1251 D 2887 1234 D
2900 1274 D 2913 1354 D 2927 1320 D 2940 1474 D 2953 1497 D
2967 1663 D 2980 1863 D 2993 1840 D 3007 2074 D 3020 2274 D
3033 2377 D 3047 2543 D 3060 2697 D 3073 3011 D 3087 3086 D
3100 3354 D 3113 3600 D 3127 3789 D 3140 4303 D 3153 4394 D
3167 4526 D 3180 4846 D 3193 5189 D 3207 5434 D 3220 5509 D
3233 6011 D 3247 6149 D 3260 5994 D 3273 6194 D 3287 6480 D
3300 6800 D 3313 6623 D 3327 6714 D 3340 6794 D 3353 6749 D
3367 7023 D 3380 7097 D 3393 6623 D 3407 6314 D 3420 6480 D
3433 6423 D 3447 6217 D 3460 6074 D 3473 5709 D 3487 5766 D
3500 5486 D 3513 5246 D 3527 5057 D 3540 4600 D 3553 4503 D
3567 4320 D 3580 4069 D 3593 4194 D 3607 3503 D 3620 3497 D
3633 3120 D 3647 3011 D 3660 2594 D 3673 2549 D 3687 2491 D
3700 2446 D 3713 2274 D 3727 2040 D 3740 2149 D 3753 1909 D
3767 1777 D 3780 1697 D 3793 1594 D 3807 1503 D 3820 1383 D
3833 1417 D 3847 1326 D 3860 1263 D 3873 1257 D 3887 1166 D
3900 1120 D 3913 1183 D 3927 1149 D 3940 1109 D 3953 1051 D
3967 1086 D 3980 1057 D 3993 1074 D 4007 1051 D 4020 1034 D
4033 1040 D 4047 1040 D 4060 1034 D 4073 1006 D 4087 1017 D
4100 1006 D 4113 1011 D 4127 1000 D 4140 1006 D 4153 1006 D
4167 1006 D 4180 1000 D 4193 1000 D 4207 1006 D 4220 1000 D
4233 1000 D 4247 1000 D 4260 1006 D 4273 1000 D S
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