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\centerline{\ttlfnt Some Remarks on Pathologies
of Renormalization-Group }
\centerline{\ttlfnt Transformations for the
Ising Model}\vskip 1cm
\author{F. Martinelli\ddag , E.
Olivieri\dag }
\address{\ddag Dipartimento di Matematica
III Universit\`a di Roma, Italy \hfill\break{\rm e-mail:
martin@mercurio.dm.unirm1.it}}
\address{\dag Dipartimento di
Matematica Universit\`a "Tor
Vergata" Roma, Italy \hfill\break{\rm e-mail:
olivieri@mat.utovrm.it}}
\abstract{The results recently obtained by van
Enter, Fernandez
and Sokal [EFS] on non-Gibbsianness of the measure
$\nu\,=\,T_b\,\mu_{\beta ,h}$ arising from the application
of a single decimation transformation $T_b$, with spacing
$b$, to the
Gibbs measure $\mu_{\beta ,h}$ of the Ising model, for
suitably
chosen large inverse temperature $\beta$ and non zero
external field
$h$, are critically analyzed. In particular we show that
if, keeping
fixed the same values of $\beta$, $h$ and $b$, one iterates
a
sufficiently large number of times $n$ the transfomation
$T_b$, one
obtains a new measure $\nu '\,=\,(T_b)^n\,\mu_{\beta
,h}$ which is
Gibbsian and moreover very weakly coupled.} \vskip 1cm\noindent
Work partially supported by grant SC1-CT91-0695
of the Commission of European Communities
\pagina
\vskip 1cm
$\S \; 1$ {\bf Introduction and Results}. \bigskip
This note is motivated by a series of discussions with many
colleagues and, in particular, with Giovanni Gallavotti
and Joel
Lebowitz, about
the relationships between: \medskip\item{ i)} some recent
results by
van Enter,
Fernandez and Sokal (EFS) concerning non-Gibbsianness of
some
measures $T\nu$ obtained by applying a renormalization group
transformation $T$ to a Gibbsian measure $\nu$ (see [EFS])
and
\par
\item{ii)} some recent results obtained by the authors
(see [MO1],[MO2],[MO3]), on the
application of finite size conditions, of the form
originally
introduced in [O]
and [OP], to the study of equilibrium and non equilibrium
properties of
lattice spin systems near to a first order phase transition.
\medskip
Joel Lebowitz suggested us to write a note
to clarify these relationships in some example. \par
We
will consider a simple case: the Ising model in dimensions
$d \geq 3$ at
large inverse temperature $\beta$ and non zero external
magnetic
field $h$ and we will denote by $\mu_{\beta ,h}$ the associated
unique Gibbs state.
\par
In this case EFS prove that the {\it decimation
transformation } $T_b$, on a scale $b$, gives
rise, for suitable
values
of $\beta$ and $h$, depending on $b$, to a non-Gibbsian
distribution.\par
We prove here that, as an immediate consequence of the results
in
[O],[OP] and [MO1], if, for {\it exactly the same} thermodynamical
parameters $\beta$ and $h$ , we apply the decimation
transformation $T_{b'}$, on any sufficiently large
scale $b'$,
we obtain
a measure which not only is Gibbsian but is also weakly coupled
(high
temperature) . In particular one can take $b' =
b^n$ for all
sufficiently
large $n$; namely one can iterate the EFS transformation
to
come
back, in this way, to the set of Gibbs measures. Moreover,
as a corollary,
we obtain that $T_{ b^n}\mu_{\beta ,h}$ converges, at the
level of the interaction, for $n$ tending to
infinity, to the
trivial fixed point corresponding to a free system with
the appropriate magnetization.\par
Let us now give some definitions.\par
The configuration space of the system is
$\Omega = \{-1,1\}^{{\bf Z^d}}$.
The formal hamiltonian is
$$
H(\sigma) = - {1\over 2} \sum_{\langle i,j\rangle}
\sigma_i\sigma_j
- {1\over 2} h\sum_i \sigma_i \Eq(1.1)
$$
where $\langle i,j\rangle$ stands for a pair of nearest
neighbours in
${\bf Z^d}$ and $h>0$.\par
We use $\Omega_{\Lambda} = \{-1,1\}^{\Lambda}$ to denote
the configuration space in $\Lambda \subset {\bf Z^d}$. \par
Consider a {\it finite} region $ \Lambda $ in ${\bf
Z^d}$ (in this
case we write : $\Lambda \subset \subset {\bf Z^d}$) and
an arbitrary
boundary condition $\tau$ outside $\Lambda$ ($\tau \in
\Omega_{\Lambda^c}$). The energy of a configuration
$\sigma$ in
$\Lambda$ is given by :
$$H_{\Lambda}^\tau(\sigma)\,=\,-{1\over
2}\sum_{x,y\in \Lambda :\; \vert x-y\vert\,=\,1}\sigma
_x\sigma _y
\;-\;{1\over 2}\sum_{x\in
\Lambda}[\;h\;+\;\sum_{y\notin
\Lambda :\; \vert x-y\vert\,=\,1}\tau
_y\;]\sigma _x \Eq(1.2) $$
The finite volume Gibbs
measure in $\Lambda$, with $\tau$ boundary conditions
has the
expression:
$$
\mu_{\Lambda}^{\tau} (\sigma) = \exp(-\beta
H_{\Lambda}^\tau(\sigma)) / \hbox {normalization}
$$
Notice that EFS use a different notation : they call
magnetic field
and denote by $h$ our quantity $\beta h$.\par
The Dobrushin-Lanford-Ruelle (DLR) theory of Gibbs measures
is based on the conditional probabilities $\pi_{\Lambda}$ for
the behaviour of the system in a finite box $\L\subset \subset
{\bf Z^d}$ subject to a specific configuration in the complement
of $\L$. According to [EFS] a probability measure whose
conditional
probabilities for finite subsets $\Lambda \subset \subset
{\bf Z^d}$ :
$(\pi_{\Lambda})_{\Lambda \subset \subset {\bf Z^d}}$
satisfy
$$
\lim _{\Lambda'\uparrow {\bf Z^d}}
\sup _{\omega_1,\omega_2 \in \Omega:
(\omega_1)_{\Lambda'}=(\omega_2)_{\Lambda'}}
|\pi_{\Lambda}f(\omega_1) - \pi_{\Lambda}f(\omega_2)|=0
\Eq(1.3)
$$
(namely the conditional expectations in $\Lambda$ of
any cylindrical
function $f$ corresponding to different boundary conditions
$\omega_1,\omega_2$, coinciding in $\Lambda'
\, \supset \, \Lambda$,
tend to coincide as $\Lambda'$ tends to ${\bf Z^d}$ ), is
called
{\it quasilocal}. \par A quasilocal probability measure
on $\Omega$
satisfying also a so called
{\it nonnullity} condition, i.e. a sort of absence of hard
core
exclusion, is called {\it Gibbsian} (see [EFS] for more
details).\par
In [EFS] it is shown that the above notion of Gibbsianness
of a
measure is equivalent to the usual notion based on absolute
summability
properties of the interaction which gives sense to
DLR equations.
\par
The following Theorem is proved
in [EFS] (see Theorem 4.7.therein).\par
\bigskip
{\bf Theorem 1.}\par
{\it For each $d \geq 3$ and $ b \geq 2$ there is a $\bar
\beta$ and a function $\bar h(\beta )$ with $\bar h( \beta
)\,>\,0$ if $\beta \,>\,\bar \beta$ such that for all $ \beta
> \bar
\beta$ and $h < \bar h$ the following is true: Let $\mu$
be a
Gibbs
measure for the $d$-dimensional Ising model described by the
hamiltonian \equ (1.1) with inverse temperature $\beta$
and magnetic
field $h$.
Then the renormalized measure $T_b\mu$, arising from a decimation
transformation with spacing $b$, is not consistent with any
quasilocal
specification. In particular it is not the Gibbs
measure for
any uniformly
convergent interaction.}\par
(we refer to Definitions
2.1,2.2,2.3,2.4 in
[EFS] for precise definitions concerning interactions)\par
Let us now state our result.\par
\bigskip
{\bf Theorem 2.}\par
{\it For each $d \geq 3$, $h > 0$ there is a
$b_0$ and a $ \beta_0$ such that for all $ \beta >
\beta_0$ and $b'
>b_0$ the following is true: Let $\mu$ be the Gibbs
measure for the $d$-dimensional Ising model described by the
hamiltonian \equ (1.1) with inverse temperature $\beta$
and magnetic
field $h$.
Then the renormalized measure $T_{b'}\mu$ arising from a
decimation
transformation with spacing $b'$ is Gibbsian; moreover the
corresponding interaction is absolutely summable
and the sum
of all but the one body terms tends to zero (in the norm ${\cal
B}^1$ defined in [EFS]) as $b'$ tends to
infinity.}
\bigskip
$\S \; 2$ {\bf Proof of
Theorem2.}\bigskip
We will use definitions and notation of [MO1] to which we
refer for details.\par
Let us first recall the notion of finite volume strong mixing
condition (in its simplest form) that has been introduced in
[MO1].\par
We say that the Gibbs measures $\mu_{\Lambda}^{\tau}$
in $\Lambda$,
with boundary condition $\tau$ outside $\Lambda$,
satisfy the {\it strong mixing
condition} in $\Lambda$, with parameters $ C>0,\, \gamma >0$
and denote it by $SMC(\Lambda,C,\gamma)$, if, for all
$x,y \in \Lambda$:
$$
\sup _{\tau \in \Omega_{\Lambda^{c}}} | \mu_{\Lambda}^{\tau}
(\sigma_x\sigma_y) -
\mu_{\Lambda}^{\tau} (\sigma_x)\mu_{\Lambda}^{\tau}
(\sigma_y)|
\leq C\exp(-\gamma|x-y|) \Eq(1.4)
$$
In [MO1] we have shown that, given $C,\gamma$,
if $SMC(.,C,\gamma)$ is verified for a sufficiently large
cube $Q _L(C,\gamma)$ of side $L$ then there are $C'>0,\,
\gamma' >0$ such that $SMC(\Lambda,C',\gamma')$ is verified
for
all arbitrarily large regions $\Lambda $ which are {\it multiples}
of
the basic cube $Q_L$ ; where, given the odd integer $L$,
a set
$\Lambda$ is said to be a multiple of the basic cube
$Q_L(0)$
(of edge $L$ centered at the origin):
$$Q_L(0)=\{y\in {\bf Z^d};|y_i|\leq\,{L-1\over
2} , \; i=1\dots
, d\}, $$
if it is a union of translated cubes $Q_L(x) \, \equiv\,
Q_L(0) +x ,\, x \in {\bf Z^d},$ with
disjoint interior :
$$\Lambda = \cup_{y\in Y} Q_L(L\ y)$$
for some $Y\subset {\bf Z^d}$\par
This property, namely the propagation to all larger scales
of a
finite volume strong mixing condition is called {\it
effectiveness}.\par\bigskip
{\bf Remark}\par \bigskip
Notice that in [MO1] different notions of strong mixing were
defined in a much more general set-up. The possibility
of using
the particularly simple form given in \equ (1.4) is a consequence
of
the peculiarities of the standard Ising model.
In [MO1] this condition was called $SMT(\Lambda,1,C,\gamma)$\par
\bigskip
It was shown in [MO1] that the
following Proposition holds true:\par \bigskip
{\bf Proposition 1.}\par \bigskip
{\it For all $d \geq 2,\, h>0$, there exists $L_0 = L_0(d,h)$and
$\beta_0 = \beta_0( d,h,L)$ such that $SMC(Q_L ,C,\gamma)$
holds
for all $L \geq L_0(d,h)$ provided $\beta > \beta_0 (d,h,L)$
}\par
\bigskip
{\bf Proof.}\par
Let us give here a proof of the above statement less sketchy
than
the one given in Section 5 of [MO1].\par
Consider a cube $ \Lambda = Q_L$ in ${\bf Z^d}$.\par
By F.K.G. inequalities ( see [FKG], [H] ) and by taking the
limit $\beta\,\to\,\infty$ of $\mu_\L^{-\underline 1}$, where
$-\underline 1$ is the configuration identically equal to $-1$,
it follows that, if the ground state configuration of
$H_{\Lambda}^{-\underline 1}(\sigma)$
with minus boundary conditions is identically equal to $+1$ for
all $x\in \L$, then the same holds for the ground state configurations
of $H_{\Lambda}^\tau(\sigma)$ with arbitrary boundary conditions
$\tau$.\par
We want now to prove that if $L > {2d\over h}$:
$$
\min _{\sigma} H_{\Lambda}^{-\underline 1}(\sigma) =
H_{\Lambda}^{-\underline 1}(+\underline 1) ;\ \
H_{\Lambda}^{-\underline 1}(\sigma) >
H_{\Lambda}^{-\underline 1}(+\underline 1)\; \forall \,
\sigma \neq +\underline 1
\Eq(1.6)$$
namely that the configuration with all spins $+1$ in $\Lambda$
is
the unique ground state for $-\underline 1$ boundary conditions.\par
Indeed for every configuration $\sigma \in \Omega_{\Lambda}$
consider the union $C(\sigma)$ of all the closed unit cubes
centered at each site $x \in \Lambda\, :\, \sigma_x =+1$.
Consider, also, the union $D(\sigma)$ of the closed
unit cubes
centered at sites $x \, \in \, {\bf Z^d}\; : \; \sigma_x
=-1$ (we
recall that we
set $\sigma_x =-1 \; \forall \;x \, \in \, {\bf Z^d} \setminus
\Lambda$) and call $D^* = D^*(\sigma)$ the unique infinite
connected
component of $D(\sigma)$. $C(\sigma)$ splits into maximal
connected
components $C_1,\dots,C_k$. Among $C_1,\dots,C_k$
we select the
subset $\bar C_1,\dots,\bar C_j$ of components touching
$D^*$.
We call them {\it outer components} and denote by
$\gamma_1,\dots,\gamma_j$ their exterior
boundaries (i.e. $ \bar
\gamma_i = \bar C_i \cap D^*$). We call $|\gamma_i|$
the measure of
their boundaries $\gamma_i $ and $|\theta (\gamma_i) |$
the measure
(cardinality) of the interior $\theta (\gamma_i)
$ of $\gamma_i$,
namely the set of points that are separated from the boundary
$\partial \Lambda$ by
$\gamma_i$.\par
It is easy to prove the isoperimetric estimate:
$$
\sum _i |\theta (\gamma_i)| \leq (\sum _i
{|\gamma_i|\over 2d} )^{ {d\over d-1}} \Eq (1.6a) $$
(see, for instance, Theorem 1.1 in [T])\par
>From \equ(1.6a) we get, for every $\sigma \in \Omega_\Lambda$:
$$
H_{\Lambda}^{-\underline 1}(\sigma) -
H_{\Lambda}^{-\underline 1}(-\underline 1)
\geq
-h \sum _i |\theta (\gamma_i)| + \sum_i |\gamma_i| \geq
-h \sum _i |\theta (\gamma_i)|
+2d (\sum _i |\theta(\gamma_i)|)^{d-1\over d}.
\Eq (1.7) $$
>From \equ(1.7) we get, for $ L > {2d\over h}$:
$$
H_{\Lambda}^{-\underline 1}(\sigma) -
H_{\Lambda}^{-\underline 1}(-\underline 1)
\geq -h L^d +2d L^{d-1} \equiv
H_{\Lambda}^{-\underline 1}(+\underline 1) -
H_{\Lambda}^{-\underline 1}(-\underline 1) \Eq (1.8) $$
and the first equality in \equ (1.6) is proven ; the
uniqueness of the minimum also follows from \equ (1.7).
\par
As we already said, from \equ(1.6) we also get that $\forall
\, \tau, \;
+\underline 1$ is the unique minimum for the energy.\par
Now, for every $L > {2d\over h},\, C>0,\, \gamma>0 $ given,
if we choose a sufficiently large $\beta h$ it is easy to
get
Condition $SMC(Q_L, C,\gamma)$ (simply because
$\mu^{\tau}_{\Lambda}(\sigma_x
=-1 \;\hbox {for some} \;x \;\in\;
\Lambda) \to 0 $ as $ \beta \to \infty$ so that the
Gibbs measure in
$\Lambda$ is, for every $\tau$, a small perturbation of a
$\delta$-measure concentrated on the unique ground state
$+\underline 1$).\par
Now, from the effectiveness of $SMC(Q_L,
C,\gamma)$ for $L$ large enough, which has been proven
in [MO1], we
are able to deduce properties of the renormalized interaction
obtained by applying a block decimation transformation
on a scale
$L$. Before stating the result in Proposition 2 below we
need some
more definitions.\par Let $b=2L$ and call $A(x)$ the cubic
block
$Q_L(bx)$ and $\alpha _x \in \Omega_{A(x)}$ the corresponding
spin
configuration. We call $A$ the set of all the
$A(x)$'s and we
identify it with the subset of ${\bf Z^d}$ given by
the union of
the cubes $A(x)$.\par
For $\alpha \, \in \, \Omega _A$ let
$H^{(r)}_A (\alpha) $ be the (formal) renormalized
hamiltonian
obtained by integrating out the spins in ${\bf Z^d}
\setminus A$. To
be more precise consider a big cube $\bar \Lambda \, \equiv
Q_{\bar
L}(0) $ centered at the origin with side $\bar L = (2p+1)L$,
$p$
integer. Choose free (empty) boundary conditions outside
$\bar
\Lambda$ . For every $ \sigma_{\bar \Lambda} \, \in
\, \Omega
_{\bar \Lambda}$ call $$
\alpha_{\bar \Lambda}= \sigma_{A\cap \bar \Lambda}
\; ; \; \eta_{\bar \Lambda} =
\sigma_{\bar \Lambda \setminus A}
$$
Let $H(\sigma_{\bar \Lambda}) \equiv H(\alpha_{\bar \Lambda},
\eta_{\bar \Lambda})$ be given by \equ (1.1) and consider
the
renormalized hamiltonian $H^{(r)}_A (\alpha _{\bar
\Lambda})$ given
by:
$$
\exp(-H^{(r)}_A (\alpha _{\bar \Lambda})) = \sum
_{\eta_{\bar
\Lambda}} \exp (- \beta
H (\alpha _{\bar \Lambda}, \eta_{\bar \Lambda}))
$$ This corresponds to applying to the Gibbs measure $\mu
_{\bar
\Lambda}$ a kind of decimation in $\bar \Lambda \setminus
A$, obtained by
integrating out the spins in $\bar \Lambda \setminus
A$, that is to
construct the {\it relativization} of $\mu _{\bar \Lambda}$
to
$\Omega _{\bar \Lambda \cap A}$.
Call $\mu^{(r)} _{\bar \Lambda}$ the renormalized measure
on
$\Omega _{\bar \Lambda \cap A}$ obtained in this way :
$$
\mu^{(r)} _{\bar \Lambda} =
{ \exp (-H^{(r)}_A (\alpha _{\bar \Lambda}))\over Z_{\Lambda}},
\ \
Z_{\Lambda} = \sum_{\alpha_{\bar \Lambda}}
\exp (-H^{(r)}_A (\alpha _{\bar \Lambda}) ) \, \equiv \,
\sum_{\sigma_{\bar \Lambda}}
\exp (-H (\sigma _{\bar \Lambda}) )
$$
One can repeat the same construction for any boundary condition
$\tau \in \Omega _{\bar \Lambda ^c}$ and get, in this way,
the
renormalized measure $\mu^{(r), \tau} _{\bar \Lambda}$.
\par
\bigskip
{\bf Proposition 2.}\par \bigskip
{\it For $L$ large enough we have:\bigskip
i)
$$
\lim _{\bar \Lambda \uparrow {\bf Z^d}}
\mu^{(r), \tau} _{\bar \Lambda} = \mu^{(r)}
$$
independently of (the sequence of) boundary conditions
$\tau$
\par
\bigskip
ii) $\mu^{(r)}$ is Gibbsian; there exists a corresponding
interaction
$ (\Phi_V)_{V\subset A}$ (see [EFS]) which is absolutely summable
and:
\par \bigskip
iii)}
$$
\sum _{V\ni A(0) : |V|>1} \| \Phi_V\| = o(L)
$$
{\bf Proof.}\par\bigskip
Take any finite cube $\bar \Lambda$ with
side $\bar L = (2p+1)L $ and $\tau $ boundary conditions
outside $\bar \Lambda$ . It is sufficient to notice that
$\bar
\Lambda \setminus A$ is a multiple ol $Q_L$ ; so, by
Proposition 1,
for $L$ sufficiently large the Gibbs measure
$\mu^{ \tau, \alpha} _{\bar \Lambda \setminus A}$ satisfies
$SMC(
\bar \Lambda,C',\gamma') $ for suitable $C', \gamma'$, uniformly
in
$\bar \Lambda$. The same is true for any ( not necessarily
cubic) region $\bar V$ multiple of $Q_L$ ( see [MO1] for
more
details).\par
i), ii) immediately follow from effectiveness. Indeed let
$\mu^{(r)}(\alpha_x|\alpha_y)$
be the conditional probability, with
respect to the measure $\mu ^{(r)}$ on $\Omega_A$, of the
configuration $ \alpha_x$ in $A(x)$ given $ \alpha_y$
in $A(y)$.
Gibbsianness follows from nonnullity and quasilocality
which, in
turn, follows from
$$
\vert \mu^{(r)}(\alpha_x|\alpha_y)\,-\,\mu^{(r)}(\alpha_x)\vert\leq
C" \exp(-\gamma"|x-y|) \Eq (1.9)
$$
for suitable $C", \gamma"$, uniformly in $\bar \Lambda, \tau,
\alpha_x,\alpha_y $.
\par
\equ (1.9) is a direct consequence of the strong mixing
condition
valid uniformly in $\bar \Lambda$ ( effectiveness).\par
To get iii) we need more detailed estimates; it easily follows
from
the arguments developed in [O], [OP], based on the cluster
expansion, from Proposition 1 and Appendix 2 in [MO1].\par\bigskip
Let us now conclude the proof of Theorem 2.
Let us use $\omega_x \in \{ -1,+1\}$ to denote the value
of the
original spin variable $\sigma_{bx}$ at the center $bx$
of the cube
$A(x)$ . We set $ \alpha_x = ( \omega_x,\bar \alpha_x)\,;\,
\bar \alpha_x \,\in \,\{ -1,+1\}^{A(x)\setminus bx}$ is the
restriction of $\alpha_x $ to $A(x)\setminus bx$ .
Let $B= \{ y = bx , x \in {\bf Z^d}\}$ be the sublattice of
${\bf Z^d}$ of spacing $b$. Consider the measure $\nu =
T_b \mu$
obtained by applying the usual decimation transformation in
$ {\bf Z^d}\setminus B$ ( relativization to $\Omega _B$ of
the
original Gibbs measure $\mu$ in $\Omega _{{\bf Z^d}}$).\par
We have:
$$ \nu (\omega_x|\omega_y)\, \equiv \,
\mu (\omega_x|\omega_y) = \sum _{\bar \alpha_x}
\mu^{(r)}(\bar \alpha_x , \omega_x|\omega_y)\,\equiv \,
\sum _{\bar \alpha_x}
\mu(\bar \alpha_x , \omega_x|\omega_y)
\Eq (1.10)
$$
On the other hand
$$
\mu( \alpha_x |\omega_y) =
\sum _{\bar \alpha_y}
[ \mu(\alpha_x |\bar \alpha_y,\omega_y)
-\mu(\alpha_x |\bar \alpha^*_y,\omega^*_y)]
\mu(\bar \alpha_y |\omega_y) +
\mu(\alpha_x |\bar \alpha^*_y,\omega^*_y) \Eq (1.11)
$$
where $\alpha^*_y,\omega^*_y$ denote a reference configuration
(e.g. equal to all $+1$ in $A_y$).\par
From \equ (1.9),\equ
(1.10).\equ (1.11) we get the quasilocality condition
\equ (1.3);
the nonnullity condition is trivially satisfied so that we
get the
desired Gibbs property for $\nu$. Absolute summability
of the
renormalized interaction immediately follows from
the arguments of [O],
[OP], together with the estimate of
the norm of the more than one body interaction, which estimate
vanishes
as L increases to infinity.
This concludes the proof of Theorem2.\par
\bigskip
$\S \; 3$ {\bf
Conclusions.}\bigskip As it was noticed
in [EFS] the non existence
of the renormalized interaction is a consequence of the presence
of a
first order phase transition for the original model in ${\bf
Z^d}\setminus B$ for particular values of $(\omega_x)_{x\in
B}$ and
suitable $h$ and $\beta$ ; for example $\omega_x = -1
\, \forall x $
and uniform positive $h$ , exponentially in $\beta$ near
to the
value $h^*(b)$ which is needed to compensate, in ${\bf
Z^d}\setminus B$, the effect of the $-1$'s in $B$
and to give rise
to a degeneracy in the ground state in ${\bf Z^d}\setminus
B$ (see also [I]).\par
It seems clear, from the above
analysis, that this pathology comes from the fact that,
on a too
short spatial scale $b$ (with respect to the thermodynamic
parameters and mainly the magnetic field $h$), the system is
reminiscent of the existence of a phase transition
for $h = 0$ .\par
One needs to analyze the system on a large enough scale to
put in
evidence the uniqueness of the phase and the absence
of long range
order. This scale, on which bulk effects become dominant
with
respect to surface effects, corresponds to the formation
of a
critical droplet of the stable phase; in other words
it is necessary
to go to distances of this order to be sure that the boundary
conditions have been screened out.
The fact that on shorter distances the system is sensible
to the
boundary conditions and ordered is somehow related to the
phenomenon of metastability taking place near to a
first order
phase transition.\par
The general philosophy suggested by the outcome of our Theorem
2 is
that, when applying a renormalization group transformation,
the
system behaves as if it was weakly coupled, provided
the scale of the transformation is chosen, depending on the
thermodynamic parameters, in such a way that our strong
mixing condition becomes effective; however it is important
to stress
that the relevant length scale near to a low temperature
coexistence
line is not the correlation length of the unique pure phase
but,
rather, the length of the critical droplet of the stable
phase
inside the metastable one.\par Finally we want to underline
the fact
that Theorem 2 is based on a finite size condition related
to a
particularly simple shape: a cube. \par As we discussed
in [MO1] an
{\it effective} condition \`a la Dobrushin and Shlosman,
implying
their Complete Analyticity ( see, for instance, [DS]),
could not be
verified in the region of thermodynamic parameters that
we are
considering here. Indeed Dobrushin-Shlosman's finite
size condition
involves the consideration of {\it arbitrary shapes};
it is clear
that to exploit the presence of a positive magnetic field
as a
mechanism of screening we need, say, a plurirectangle with
sufficiently large minimal edge. For not sufficiently
"fat" and
regular regions (as, for instance, pathological regions
with many
holes in the bulk) it is conceivable that not only a finite
size
condition coming from the screening effect of $h$ does
not hold
but, also, that , for special values of $h$ and $\beta$, the
Dobrushin-Shlosman complete analyticity can
even fail. This is
actually what EFS prove, in some cases, as a direct
consequence of
their methods to show non-Gibbsianness of some renormalized
measures.\par At the same time the equivalent of
complete analyticity, not stated for {\it all} regions but,
rather, for arbitrarily large {\it but sufficiently regular}
domains
directly follows from [O],[OP] and the above described
finite size
condition on a suitable cube.
\vskip 1cm
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CARR}{25}{Roma}{1992} \refj{MO2}{ F.Martinelli,
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\refj{O}{E.Olivieri}{On a Cluster
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Convergence.}{Journ. Stat. Phys.}{50}{1179-1200}{1988}
\refj{OP}{
E.Olivieri, P.Picco}{Cluster Expansion for D-Dimensional
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