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\centerline{\ttlfnt Instability of Renormalization-Group Pathologies under
decimation}\vskip 0.5cm
\author{F. Martinelli $^{\dag}$ E. Olivieri $^{\ddag}$}
\address{\ninerm \dag Dipartimento di Matematica,
III Universit\`a di Roma,
Italy \hfill\break{\ddag
Dipartimento di Matematica, II Universit\`a di Roma Tor Vergata, Italy }
\hfill\break{
e-mail: martin@mat.uniroma.it \hskip 0.5cm olivieri@mat.utovrm.it}}
\abstract{\ninerm We investigate
the stability and instability of pathologies of renormalization group
transformations for lattice spin
systems under decimation. In particular
we show that, even if the original renormalization group transformation
gives rise to a non
Gibbsian measure, Gibbsiannes may be
restored by applying an extra
decimation trasformation. This fact is illustrated in details for
the block spin transformation
applied to the Ising model.
We also discuss the case of another non-Gibbsian measure with nicely decaying
correlations functions which remains
non-Gibbsian after arbitrary decimation. } \vskip 1cm\noindent {\eightrm Work
partially supported by grant SC1-CT91-0695 of the Commission of European
Communities} \par
\bigskip
\bigskip
{\bf Key words:} Renormalization-group, Decimation, Non-Gibbsianness, Ising Model
\vfill
\eject
\bigskip
\numsec=1\numfor=1
\centerline {\bf Introduction}
\vskip 1cm
In this
note we
discuss some
aspects of the
problem of defining,
on rigorous
grounds, a renormalization group transformation (RGT) for the Gibbs measure
of lattice
spin systems of
statistical
mechanics.
For simplicity
and without a true
loss of generality (see the end of section2), we confine our attention
to the average block
spin transformation
for the 2D Ising
model at low temperature and large positive external field. \par
In the basic
reference [EFS], the
authors discuss
in a very complete and clear
way the possible pathologies that may arise when applying a RGT to a
perfectly well
behaved Gibbs
measure like the
one above. To
be more specific,
suppose that $\mu$ denotes the starting Gibbs measure on a probability space
$\O$ and that
$$\nu \;=\; T_b\mu\Eq(0.1)$$
denotes the transformed
measure, obtained
by applying to $\mu$
a RGT $T_b$ acting on "scale"
$b$ and defined on a new probability space $\O '$. The system
described by the measure
$\mu$ and with configurational
variables with values in
$\O$ is called the "object" system, whereas the system described by $\nu$
with configurational
variables with values in $\O '$ is called the "image" system.\par
As the authors of [EFS]
point out, the main
(rather surprising)
pathology of the above
RGT is that the renormalized measure $\nu$ can very well be {\it
non-Gibbsian},
that is the associated system
of conditional
probabilities is not
compatible with any finite norm potential. That may happen even if the
starting measure $\mu$,
e.g the unique Gibbs
measure of some
finite range interaction,
has all the nice properties describing the one phase region: analitycity,
exponential decay of
correlations, convergent
cluster expansion etc.
\par In [EFS] one can find many
examples of such pathology. Moreover, the same authors
show that the typical
mechanism behind the
{\it non-Gibbsiannes} of the measure
$\nu$ is the appearence of long range order, that is a phase transition,
in the object system
{\it conditioned} to some
particular configurations
of the image system.
Such long range order implies, in particular , that the measure
$\nu$ violates a
necessary
condition for being
Gibbsian, namely
"quasi-locality " of its family of conditional probabilities $\{\p_\L\}_{\L}$ .
Such a condition,
introduced by Kozlov
(see [K] and th. 2.12 in [EFS]),
roughly speaking implies
some sort of uniform continuity of the conditional probabilities
$\p_\L$ with respect to the
conditioning configuration
(see also [BMO] and [FP] for a critical discussion ). \par
An interesting example given in
[EFS] of the above phenomenon
refers to the "decimation trasformation"
on scale "b", $T_b^d$, applied to the Gibbs measure
$\mu_{\beta, h}$ of the Ising
model at low temperature, $\beta>>1$,
and small magnetic field $h$.
Such a transformation associates to the original measure
$\mu_{\beta, h}$ its marginal
(or relativization) on the spin
variables sitting on the sites of the
sublattice $\Z (b)$ of $\Z$ with spacing $b$. In other
words one integrates out all the variables in $\Z\setminus \Z (b)$.\par
In [EFS] it was proved that, for any
given $b$ and for suitable values of $\beta$ and $h$, the measure
$$\nu \;=\; T_b^d\mu_{\beta, h}\Eq(0.2)$$
is {\it non-Gibbsian}.\par
This fact is the consequence of the
degeneracy of the ground state of
the Ising model restricted to $\Z\setminus \Z (b)$ if the conditioning spins
at the sites of $\Z (b)$ are held
fixed
in some suitable, particular configuration, for instance all the spins
equal to -1, and the value
of the magnetic field $h$ is suitably
chosen as a function of the lattice spacing $b$.
Such a degeneracy, using the theory
of Pirogov and Sinai, leads to a
first order phase transition at low
enough tenperature for the same constrained
system.\par
One may say that the above "spurious"
phase transition comes from the
fact that, on a too short length scale $b$ (with respect to the thermodynamic
parameters and mainly to $h$),
the system is reminiscent of the
phase transition taking place at $h=0$.
It thus appears plausible that the above pathology
could be eliminated and therefore
Gibbsiannes recovered, by choosing a
large enough spacing $b$ for given
fixed values of $\beta$ and $h$; in particular,
it should be sufficient to iterate a
sufficiently large number of times the
same trasformation in order to come back to the space of Gibbsian measures.
This is what we actually proved in [MO]
together with some additional results
like convergence of a cluster expansion for $\nu$ and the convergence of
$(T_b^d)^n \mu_{\beta, h}$ to a trivial fixed point as $n\to \infty$.\par
In [EFS] there is another, more subtle,
example of pathology, referring to
the so called "block averaging transformation" for the 2D Ising model. It
is this example and the associated
pathology the main object of the present note.\par
The trasformation is defined as follows.
Suppose to partition the lattice
${\bf Z^2}$ into 2x2 blocks $Q_i$ and let us denote by $m_i$ one of the
five possible values of the
magnetization (sum of the spins)
inside the block $Q_i$. Then the transformed measure
$$\mu^B(\{m_i\})\;=\;T_2^B\mu_{\beta ,h}$$
is defined simply as the probability
distribution of the variables $m\,\equiv\,\{m_i\}$.\par
Here the violation of quasi-locality and
thus the non Gibbsiannes of $\mu^B(\{m_i\})$
is due to the presence, for large enough $\beta$ and arbitrary
value of $h$, of a first order phase
transition in the multicanonical
model represented by the object
system constrained to have zero magnetization in
each block $Q_i$. Notice that, since the
local magnetizations are fixed,
the value of the magnetic field is irrelevant.
Although the proof of this result
was given only for the case of 2x2 blocks, it seems quite
plausible that it persists for any value of
the side of the blocks $Q_i$, thus excluding the
possibility of restoring Gibbsiannes
by simply enlarging the side of the blocks, in
contrast to what happens for the decimation trasformation.\par
On the other hand, if, for example, the magnetic field is large,
the object system without constraints is very close to a product measure and
the non Gibbsian measure $\mu^B(\{m_i\})$ itself enjoys nice mixing
properties like exponential decay of truncated correlations functions
and, quite
likely, a weaker version of quasi-locality, as
the one introduced in [FP] . Moreover it
is quite obvious that the event of having zero magnetization in
each block is exceptional and thus,
in some sense, the above pathology should be
"unstable" with respect to a little bit of decimation. This kind
of considerations was already suggested,
on a informal level, in [EFS] (see page 1066).\par
In the present note we pursue the above point of view quite
seriously, since, in our opinion, the "stability"
or "instability" of non-Gibbsiannes of a
measure under decimation is a relevant property. Decimation
in fact corresponds to select certain variables, which
are the only "relevant" ones for the kind of
questions one is interested in, and disregard (integrate
out) the "irrelevant" ones; moreover
important thermodynamic quantities like
the free energy (and their analyticity properties)
or the asymptotic behaviour of
the truncated correlations can be computed equally
well with the decimated measure. Thus,
if Gibbsiannes can be restored with the
help of some decimation, then the pathologies decribed above becomes irrelevant
at least as far as certain variables
are concerned . On the contrary,
if the measure $\mu$ under cosideration
is non-Gibbsian and remains of such a type after an
arbitrary decimation, then such a character
becomes, in our opinion, a much more
important feature of the system described by $\mu$, probably related to
some non trivial long range dependence hidden inside the system itself. \par
In this note we illustrate in full details
the above considerations, first for
the block spin Ising model (see section 2) and then for the invariant
measure of a certain stochastic
dynamics on the configuration space
$\{0,1\}^{\bf Z^2}$ (see section 3).
In the first case we prove that, if we decimate
the block spin model on the even blocks
(see section 2 for details) and we take the
external field $h$ large enough, then we end up with a nice, weakly
coupled, Gibbs measure whose potential
is expressed via a convergent cluster
expansion. To perform the calculation we
use the commutativity of the decimation
with the block spin trasformation,
that is we first decimate and then apply the block spin transformation. \par
In the second case we show that,
independently of the side of the blocks of
the decimation, the decimated measure remains non Gibbsian like the starting
measure.\par
After the present paper was completed, we learnt of a recent work by A. v.
Enter, R. Fern\'andez and R. Koteck\'y where, in particular, the authors
establish non-Gibbsianness of the renormalized measure $\n \; = \;
T_b ^{m r} \m _{\b, h}$ obtained by applying the majority rule transformation,
over blocks of side $b$, to the Ising Gibbs measure
$\m _{\b, h}$ for $\b$ and $h$ large
enough.\par
One can easily check that, by the same methods developed in Section 2 of the
present paper , it is possible to restore Gibbsianness by simply
decimating the
measure $\n$ over the even blocks ( see Section 2). In other words a statement
analogous to the one of Theorem 2.1 holds true.\par \bigskip
{\bf Acknowledgements.}\par
We are very grateful to Aernout van Enter for informing us about their work
prior to publication.
\pagina
\numsec=2\numfor=1
{\bf Section 2. The block spin and decimation transformation.}
\vskip 1cm
In this main section we discuss in
details the effect of a decimation over the odd
blocks (see below) on the block spin
Ising model for which {\it non Gibbsiannes} was
proved in [EFS]. For simplicity we restrict
ourselves to the case of large external
magnetic field (but see the remark at the end of the section for more general
situations).\par We show that, after
the decimation, Gibbsiannes is recovered. As
already explained in the
introduction, a key remark is that the two transformations,
the block spin and the decimation,
commute so that we can first decimate the original
Gibbs measure of the Ising
model and then do the block average transformation. The technical tool
is the cluster expansion that
provides naturally all the necessary cancellations.
Let us start with the details. \par
The Ising hamiltonian in a volume
$\L \in {\bf Z^d}$ with open (empty) or periodic
boundary conditions is given by :
$$ H_{\L} (\s_{\L}) \; = \; - J/2 \;\sum _{x,y \in
\L} \;\s_x \s_y \;\; -
h/2\; \sum _{x \in \L} \; \s_x \Eq(1.1)
$$
where $ \s_{\L} \; \in \; \O_{\L} \; \equiv \; \{ -1, +1 \}^{\L}$
and $h$ is the
external magnetic field. \par
The corresponding finite volume Gibbs measure at inverse temperature $\b$ and
magnetic field $h$ is given by :
$$
\m_{\L} \; = \;{\exp[\; -\b ( H_{\L} (\s_{\L}) ) \;]\over
Z_{\L} } \Eq(1.1a)
$$
where the normalization factor
$$
Z_{\L} \; = \; \sum _{\s_{\L} \in \O_{\L} }
\exp[\; -\b ( H_{\L} (\s_{\L}) ) \;] \Eq(1.1b)
$$
is called {\it partition function}. \par
We will consider the case when
both the inverse temperature $\b$ and the
external magnetic field $h$ are very large.\par
For the sake of simplicity of the exposition we will
assume that the dimension is
$d=2$.\par
Consider the partition of ${\bf Z^2}$ into $2\times 2$ squared
blocks $Q_i$ of side 2
(each containing 4 sites). A block $Q_i$ can be characterized by its leftmost
down site $x(Q_i)$. The $x(Q_i)$ are of the form :
$$
x(Q_i) \equiv x^{(i)} \equiv ( x^{(i)}_1,x^{(i)}_2 )\; ; \;
x^{(i)}_1 = 2 y^{(i)}_1,x^{(i)}_2 = 2 y^{(i)}_2 \;\hbox{with}\;\;
y^{(i)} \; \equiv \; (y^{(i)}_1,y^{(i)}_2) \in {\bf Z^2} \Eq(1.1c)
$$
We write
$$
Q_i = Q(y^{(i)}) \;\; \hbox{if} \;\;y^{(i)} = x(Q_i) /2 \Eq(1.2)
$$
Now we introduce a partition the lattice ${\bf Z^2}$ into
two sublattices ${\bf
Z^2_e}$ and ${\bf Z^2_o}$ ( the subscripts $e,o$, respectively, stand for even
and odd). They are given by :
$$
{\bf Z^2_{e,o}} = \{ y \equiv (y_1,y_2) \in {\bf Z^2} \; : \; y_1+ y_2
= \hbox{even integer, odd integer} \} \Eq(1.3)
$$
Given a $2\times 2$ block $Q_i$ we call it even or odd according to the
sublattice to which
$ y^{(i)} = x(Q_i) /2$ belongs.\par
We decompose the original lattice ${\bf Z^2}$ into the union
$$
{\bf Z^2} = {\cal A} \cup {\cal B} \Eq(1.4)
$$
where ${\cal A} = \cup _i A_i$ is the set of the even
blocks $A_i \;\; \hbox{with}\;\; x(A_i) /2\; \in \;{\bf Z^2_e}$ and
${\cal B} = \cup _i B_i$ is the set of the odd blocks
$B_i \;\; \hbox{with}\;\; x(B_i) /2 \in
{\bf Z^2_o}$. Notice that in our notation
we suppose that the total set
$ {\cal Q} =\cup _i Q_i$ of the $2\times 2$ blocks
as well as the sets ${\cal A} = \cup _i A_i$ , ${\cal B} = \cup _i B_i$
of even and odd, respectively, blocks is given a certain order, for example
the lexicographic one, but this ordering will never be used explicitely.\par
We use the notation $\a_i,\; \b_i$ to denote, respectively, the
spin configurations inside the blocks
$ A_i, B_i $; $\a_i ,\; \b_i$ take $ 2^4 = 16$ possible values.\par
We denote by $e_1,e_2,e_3,e_4$ the unit vectors $ (0,1), (1,0), (-1,0),
(0,-1)$\par
Given a bkock $B_i \equiv Q_{l(i)} \hbox {with} \; x(Q_{l(i)}) /2 \;
=\;y^{(l(i))}$,
we denote
by $ A^1_i, A^2_i, A^3_i, A^4_i$ the four nearest neighbours $A$-blocks
given by $ A^j_i = Q(y^{l((i))}+ e_j); \; j=1,\dots , 4\;$ and
by $\a_i^1,\a_i^2, \a_i^3, \a_i^4$ the
corresponding spin configurations. We use $ \underline \a_i$ to denote the set
of spin configurations $\a_i^1,\a_i^2, \a_i^3, \a_i^4$ in these four
$A$-blocks. \par
By $Z_{B_i}^{ \underline \a^i}$ we denote the partition function in the block
$B_i$ with boundary conditions given by $ \underline \a_i$ .\par
We call $0$ a fixed reference configuration inside a $2 \times 2 $ block; for
instance $0$ can be chosen to be the configuration with all minus
spins inside the
block. We use $\underline 0$ to denote the configuration $\underline \a _i
\;\equiv \a^j_i = 0
\; \forall \; j =1,\dots, 4$ in the set $ A^1_i, A^2_i, A^3_i, A^4_i$ of
nearest neighbours $A$-blocks to a given $B$-block $B_i$.\par
Given an integer $L$ multiple of $4$,
consider the squared box $\L \; \equiv
\L_L \; \equiv \; [-L/2, L/2 +1]^2$.\par
We choose, for simplicity, periodic boundary conditions; namely $\L$ ,
by identifying its opposite sides, becomes a
two-dimensional finite torus. Any other boundary condition could be
considered as well, with only minor changes.\par
Using simply $\a,\;\b$ to denote the global
spin configuration in all the $A$-blocks ,
$B$-blocks, respectively, contained in
$\L$, we can write the following expression for the partition function in $\L$
( with periodic boundary conditions) :
$$
Z_{\L} \;= \; \sum _{\a}
Z_{\L} ( \a) \Eq(1.4a)
$$
where:
$$
Z_{\L} ( \a)\;=\;\exp (
\sum_{A_i\subset \L} H(\a_i) )
\prod_{B_i \subset \L} Z_{B_i}^{\underline \a _i}\Eq(1.4b)
$$
and, for $ \a_i \; = \;\s_{A_i}$,
$H(\a_i) $ is the self-energy inside the block
$A_i$: $$
H(\a_i) \; \equiv \;H_{A_i}(\s_{A_i})\; = \;
- J/2 \;\sum _{x,y \in A_i} \;\s_x \s_y \;\; -
h/2\; \sum _{x \in A_i} \; \s_x \Eq(1.4c)
$$
\equ (1.4a) is simply obtained by a decimation procedure; namely by
first summing over the $\b$-variables keeping fixed the $\a$-variables which
play the role of fixed boundary
conditions . The sum over the $\b$-variables, for
fixed $\a$, is immediately seen to factorize into independent sums over the
single $\b_i$ which are mutually decoupled because of the form (nearest
neighbour) of the Ising interaction.\par
By simple manipulations of the previous expression we get:
$$ Z_{\L}\; = \; \prod_{B_i\subset \L}
({1 \over Z_{B_i}^{\underline 0} })^3 \sum
_{\a} \exp (\sum_i H(\a_{A_i\subset \L} ) )
$$
$$\prod _{B_i\subset \L} ({ Z_{B_i}^{\a_i^1,\a_i^2, \a_i^3, \a_i^4}
Z_{B_i}^{ \underline 0} Z_{B_i}^{ \underline 0} Z_{B_i}^{ \underline 0} \over
Z_{B_i}^{\a_i^1,0, 0, 0} Z_{B_i}^{0,\a_i^2, 0, 0}
Z_{B_i}^{0,0, \a_i^3,0} Z_{B_i}^{0,0,0, \a_i^4}} -1 +1)
Z_{B_i}^{\a_i^1,0, 0, 0} Z_{B_i}^{0,\a_i^2, 0, 0}
Z_{B_i}^{0,0, \a_i^3,0} Z_{B_i}^{0,0,0, \a_i^4} \Eq(1.5)
$$
and we define
$$ \widetilde Z_{\L} \; = \;
\prod _{B_i\subset \L } ({1 \over Z_{B_i}^{\underline 0}
})^3 \prod _{A_i\subset \L} \sum _{\a_i} ( \exp(H(\a_i)
Z_{B_i^3}^{\a_i^1,0, 0, 0} Z_{B_i^4}^{0,\a_i^2,
0, 0} Z_{B_i^2}^{0,0,0, \a_i^3} Z_{B_i^1}^{0,0, \a_i^4, 0} ) \Eq(1.6)
$$
where if $ A_i \equiv Q_{m(i)}$, we set $B^j_i = Q(y^{m((i))}+ e_j)' \; j=1,
\dots , 4 \;$
and we use the short forms $\sum_{\a_i}, \sum_{\a}$,
to denote $\sum_{\a_i\in \O_{A_i}}, \sum_{\a \in \O_{\cal A}}$, respectively.
\par
A useful graphical way to describe
the r.h.s. of \equ (1.6) is to
associate to anyone of the four partition
functions appearing in the r.h.s. of the
\equ (1.5),
namely to $Z_{B_i}^{\a_i^1,0, 0, 0}$, $Z_{B_i}^{0,\a_i^2, 0, 0}$,
$Z_{B_i}^{0,0, \a_i^3,0}$, $Z_{B_i}^{0,0,0, \a_i^4}$,
four arrows, emerging from the
block $B_i$ and ending in the blocks
$ A^1_i, A^2_i, A^3_i, A^4_i$ ; namely four arrows parallel to the four unit
vectors $e_1,e_2,e_3,e_4$, respectively. Then in \equ (1.6) appear the terms
(partition functions) corresponding to the four arrows ending into the $A$-block
$A_i$ and emerging from the four nearest neighbour $B$-blocks.\par
Consider now, for every $A$-block $A_i$, the
probability measure on $\a_i$ given by:
$$
\n (\a_i) \; =\;
{ \exp (H(\a_i)) Z_{B_i^3}^{\a_i^1,0, 0, 0} Z_{B_i^4}^{0,\a_i^2,
0, 0} Z_{B_i^2}^{0,0, \a_i^3,0} Z_{B_i^1}^{0,0,0, \a_i^4} )
\over
\sum_{\a_i}
\exp (H(\a_i)) Z_{B_i^3}^{\a_i^1,0, 0, 0} Z_{B_i^4}^{0,\a_i^2,0, 0}
Z_{B_i^2}^{0,0, \a_i^3,0} Z_{B_i^1}^{0,0,0, \a_i^4} ) }
\Eq (1.7)
$$
We set
$$
\hat Z _{A_i}\; =\;
\sum_{\a_i}
\exp (H(\a_i)) Z_{B_i^3}^{\a_i^1,0, 0, 0} Z_{B_i^4}^{0,\a_i^2,
0, 0} Z_{B_i^2}^{0,0, \a_i^3,0} Z_{B_i^1}^{0,0,0, \a_i^4} )
\; \equiv \;
Z^{\underline0}_{V_i}
\Eq(1.8)
$$
where
$$V_i = A_i \cup B_i^1 \cup B_i^2 \cup B_i^3 \cup B_i^4 \Eq(1.9)
$$
>From \equ (1.6), \equ (1.8) we obtain:
$$
\widetilde Z \;= \;
\prod _{B_i \subset \L} ({1 \over Z_{B_i}^{\underline 0} })^3
\prod _{A_i\subset \L} \hat Z _{A_i} \Eq(1.10)
$$
from \equ (1.5) ,\equ (1.7), \equ (1.8), \equ (1.10), we get:
$$
Z_{\L}\; = \;
\widetilde Z _{\L} \sum _{\a} \prod _{A_j \subset \L} \n(\a_j)
\prod _{B_i \subset \L}( \psi _{B_i}
( \a_i^1, \a_i^2, \a_i^3, \a_i^4) + 1) \Eq(1.11)
$$
where
$$\psi _{B_i} ( \a_i^1, \a_i^2, \a_i^3, \a_i^4) \equiv
{ Z_{B_i}^{\a_i^1,\a_i^2, \a_i^3, \a_i^4}
Z_{B_i}^{ \underline 0} Z_{B_i}^{ \underline 0} Z_{B_i}^{ \underline 0} \over
Z_{B_i}^{\a_i^1,0, 0, 0} Z_{B_i}^{0,\a_i^2, 0, 0}
Z_{B_i}^{0,0, \a_i^3,0} Z_{B_i}^{0,0,0, \a_i^4}} -1 \Eq(1.12)
$$
We define the {\it renormalized hamiltonian} as
$$
H^r_{\L}\; = \; \sum _{A_i\subset \L} H(\a_i) + \sum _{B_i \subset \L}
- \;\log Z_{B_i}^{\a_i^1,\a_i^2, \a_i^3, \a_i^4}
\Eq(1.13)
$$
Then after having extracted the one-body part we get:
$$
H^r_{\L}\; = \; \sum _{A_i \subset \L}
H(\a_i) + \sum _{B_i \subset \L} -\;(\log
Z_{B_i^3}^{\a_i^1,0, 0, 0}+ \log Z_{B_i^4}^{0,\a_i^2,
0, 0}+ \log Z_{B_i^2}^{0,0, \a_i^3,0} + \log Z_{B_i^1}^{0,0,0, \a_i^4}) \;
+
$$
$$ +\sum_{B_i \subset \L}- \; \log (
{ Z_{B_i}^{\a_i^1,\a_i^2, \a_i^3, \a_i^4}
Z_{B_i}^{ \underline 0} Z_{B_i}^{ \underline 0} Z_{B_i}^{ \underline 0} \over
Z_{B_i}^{\a_i^1,0, 0, 0} Z_{B_i}^{0,\a_i^2, 0, 0} Z_{B_i}^{0,0, \a_i^3,0}
Z_{B_i}^{0,0,0, \a_i^4} }) + \hbox { const.} \Eq(1.14)
$$
Now we want to use the expression
given in \equ (1.11) to make the second step; namely
the sum over the spin configuration $\a_i \; \equiv \; \s_{A_i}$
with given values $m_i$ of
the magnetizations $ m_{A_i}\; \equiv \; m_{A_i} (\a_i)\;\; \equiv \;
\sum _{x \subset A_i} \s _x$ in the blocks $A_i$.\par
If $N =N(L)$ is the total number of $A$-blocks in $\L$, let
$$
Z_{\L} ( m_1, \dots , m_N)\; = \; \sum_{\a}\prod _{A_i \subset \L}
({\bf 1}_{ m_{A_i} = m_i}(\a_i) ) Z_{\L}(\a) \Eq(1.15)
$$
where
$$
{\bf 1}_{ m_{A_i} = m_i}(\a_i) = 1 \;\;\hbox {if }\;\;
m_{A_i}\; \equiv m_{A_i}(\a_i) = m_i;\;\;\;\;\;
{\bf 1}_{ m_{A_i} = m_i}(\a_i)\; \;=\;0 \;\;\hbox {otherwise}
\Eq(1.15a)
$$and $Z_{\L}(\a)$ has been defined in \equ (1.4b) ).\par
Now we transform our original block spin system,
described by \equ (1.15), into a polymer system; for this purpose we need some
definitions.\par
A ( four-body) {\it bond } $p_i \; \equiv \; p(B_i)$, for a given $B_i$,
is the set $ p_i\; = \;
\{A^1_i, A^2_i, A^3_i, A^4_i\}$ of the four $A$-blocks nearest neighbours to
$B_i$.\par
Its {\it support} $\tilde p_i$ is given by
$ \tilde p_i\; = \; \cup_{j=1} ^4 A_i^j$
\par
A set of bonds $R\; = \; p_i, \dots , p_k$ is called {\it polymer} if
it is {\it connected} in the sense
that for every pair $ p_i, p_j \; \in \; R$ there
exists a chain $ p_{k_1}, \dots , p_{k_l} \; \in \; R; \;$ with $p_{k_1} \;
= p_i, \; \; p_{k_l} \; = \; p_j $
of bonds which are connected
in the sense that $ \tilde p_{k_i}\cap \tilde p_{k_{i+1}}
\neq \emptyset,\; i=1, \dots ,
l$.\par
We call {\it support} of a polymer $R$ and we denote it by $\widetilde R $ the
union of the supports
$ \tilde p_i$ of the bonds $ p_i \; \in \; R$. \par
We say that two polymers $R_1$ and $R_2$ are {\it compatible}
if $\widetilde R_1 \cap \widetilde R_2 \; = \; \emptyset $.\par
We set
$$
\widetilde \z_{R} (\a_{R} )\; =\; \prod _{p(B_i)\in R}
\psi _{B_i} ( \a_i^1, \a_i^2, \a_i^3, \a_i^4) \Eq(1.15b)
$$
Now let :
$$
\bar Z_{\L}(m_1, \dots , m_N) \; =\; \prod _{A_i \subset \L} \bar Z_{A_i} (m_i)
\Eq(1.16)
$$
with
$$ \bar Z_{A_i} (m_i)\; = \; \sum _{\a_i} \n(\a_i)
{\bf 1}_{ m_{A_i} = m_i}(\a_i) \Eq(1.17).
$$
Let
$$
\m_{m_i} (\a_i) \; \equiv \;
1/ \bar Z_{A_i} (m_i) \n (\a_i) {\bf 1}_{ m_{A_i} = m_i}(\a_i).
\Eq(1.18)
$$
>From \equ (1.4a),\equ (1.4b),\equ (1.5),
\equ (1.11),\equ (1.15),\equ (1.16),\equ (1.17) it is easy to get:
$${ Z_{\L} ( m_1, \dots , m_N)\; \over
\widetilde Z_{\L} \bar Z_{\L}(m_1, \dots , m_N) }\; = \;
\prod _{A_i \subset \L} \m_{m_i} (\a_i)
( 1 + \sum _{n \geq 1} \sum _{R_1, \dots ,R_n: \atop
\widetilde R_i \subseteq \L,
\widetilde R_i
\cap \widetilde R_j = \emptyset} \prod_{i=1}^n
\widetilde \z_{R_i} (\a_{R_i} ) )\Eq(1.19)
$$
Now, if we introduce the {\it activity}
$\z (R)\,\equiv\,\z_m(R)$ of a generic polymer
$R$ as : $$
\z (R)\; = \; \prod _{A_i \in \widetilde R} \sum _{ \a_i } \m_{m_i} (\a_i)
\widetilde \z_{R} (\a_{R} ) ,\Eq(1.20)
$$
we can write:
$$
{ Z_{\L} ( m_1, \dots , m_N)\; \over
\widetilde Z_{\L} \bar Z_{\L}(m_1, \dots , m_N) }\; =\;
1 + \sum _{n \geq 1} \sum _{R_1, \dots ,R_n: \atop
\widetilde R_i \subseteq \L,
\widetilde R_i
\cap \widetilde R_j = \emptyset} \prod_{i=1}^n
\z (R_i) \Eq(1.21)
$$
Now we observe that in the region of thermodynamic parameters that we are
considering, namely $h$ and $\b$ large, the activity of our
polymers is indeed very small,
uniformly in the $m_i$'s, in the proper sense so that we can apply
the theory of the cluster expansion and obtain its convergence.
As we will discuss later on, one could simply assume $ h \neq 0$ and $\b$
sufficiently large
provided the $2 \times 2$ blocks are replaced by blocks of large enough side
( depending on $h$and $\b$). \par
In our case of $h$ very large everything is much simpler
since after decimation our system is weakly coupled on scale one.\par
In particular it is easy to get the following statement:\par\bigskip
{\bf Proposition 2.1 .} \par \bigskip
For every $\e >0$ there exists a value $h(\e)$ of the magnetic field
such that if $ h \; > \; h(\e)$ and $\b$ is sufficiently large,
$$ \sup _{ \a_i^1, \a_i^2, \a_i^3, \a_i^4}
|\psi _{B_i} ( \a_i^1, \a_i^2, \a_i^3,\a_i^4) |\; < \; \e \Eq(1.22)
$$
\bigskip
>From the previous proposition one can easily deduce, by standard methods, the
convergence of a cluster expansion of the thermodynamic as well as the
correlation functions .
The elemetary geometric objects of this expansion will be
clusters of incompatible polymers (see, for instance, [GMM] for more details).
Many properties for the image block-spin system ( after decimation over the odd
$B$-blocks), described by the probability measure with weights proportional to
$Z_{\L} ( m_1, \dots , m_N)$, can be deduced using this convergent cluster
expansion.\par
We will concentrate now
on one important feature of this measure namely its
{\it Gibbsiannnes } .\par
First of all let us introduce the doubly renormalized hamiltonian, after
decimation on the odd $B$-blocks and block-averaging over the surviving even
$A$-blocks. We will denote it
by $ H^{b,d}_{\L} ( m_1, \dots , m_N)$ where in the
superscript $b,d $ stand for block spin averaging transformations and for
decimation transformation, respectively.\par
We define it as
$$
H^{b,d}_{\L} ( m_1, \dots , m_N) \; = \;
- \;\log ({ Z_{\L} ( m_1, \dots , m_N)\; \over
\widetilde Z_{\L} \bar Z_{\L}(m_1, \dots , m_N) }) \Eq(1.23)
$$
which corresponds to
a particular choice of the zero of the energy for our
doubly renormalized system
(namely obtained by decimation and block spin transformsation).\par
Our doubly renormalized probability measure is :
$$
\m_{\L} ^{b,d} ( m_1, \dots , m_N) ={ Z_{\L} ( m_1, \dots , m_N)
\over
\sum_{m_1, \dots , m_N} Z_{\L} ( m_1, \dots , m_N)}
$$
$$
\equiv
{\exp (- H^{b,d}_{\L} ( m_1, \dots , m_N) )
\over
\sum_{m_1, \dots , m_N} \exp (- H^{b,d}_{\L} ( m_1, \dots , m_N) )}
\Eq(1.23a)$$
We want now to state and prove
our main result about the gibbsianness of our doubly
renormalized measure $ \m ^{b,d}$ in the thermodynamic limit
$\L \to {\bf Z^2}$ .
\par\bigskip
{\bf Theorem 2.1 .}
\par\bigskip
If the external magnetic field $h$ and the inverse temperature $\b$ are
sufficiently
large:
\par\noindent i) There exists the (weak)
thermodynamic limit limit of the finite volume doubly renormalized measure :
$$
\m ^{b,d} ( m_1, \dots , m_N)\;\; = \; \lim _ {\L \to {\bf Z^2}}
\m_{\L} ^{b,d} ( m_1, \dots , m_N)\; \Eq(1.23b)
$$
\noindent
ii) $ \m ^{b,d} ( m_1, \dots , m_N)$
is a Gibbs measure corresponding to a finite
norm, translationally invariant, potential.\par
\bigskip
{\it Proof.}\par \bigskip
The proof of the Theorem
is based on the theory of cluster expansion applied to the
system of polymers described by \equ (1.21)
For this purpose let us
recall a proposition which summarizes the basic results on
cluster expansion that we need to prove Gibbsianness.\par
The proof of the proposition together with more details can be found in [CO].
\par
\bigskip
{\bf Proposition 2.2}
\par \bigskip
Let $\;\Xi_\L\;$ denote the polymer partition function:
$${\Xi}_{\L}=1+\sum_{k\geq 1}\;\sum_{{R_1,\dots ,R_k:\atop{\widetilde
R}_i\subseteq\L,1\leq i\leq k,}\atop{\widetilde R}_i\cap{\widetilde R}_{i'}
=\emptyset ,1\leq i*\, 4$, e.g. $C\,=\,5$. With this
choice one has in fact that the Hamiltonian of a single block
$B_i$ has a unique ground state identically equal to plus one,
independently of the boundary conditions $\a_i^1\dots \a_i^4$. In
particular it follows that $$\lim_{\beta \to \infty}
\psi (\a_i^1\dots \a_i^4)\;=\;0$$
and the convergence of the cluster expansion can again be proved. \par
Of course, in the above argument the fact that the unique ground
state is the special configuration identically equal to plus one is
completely irrelevant. Thus the above method is able to treat
systems at low enough temperature having the property that the
ground state in a large enough volume is unique and independent
of the boundary conditions. \par
One may also want to consider much more general cases in which the
thermodynamic parameters garantee only a strong form of weak
dependence of the finite volume Gibbs measure on the boundary
conditions, that we call {\it Strong Mixing} (see e.g. [MOS] and
references therein). This is the case for example of 2D
ferromagnetic systems just above the critical point [MOS]. In such more general
situations, even if the side of the blocks is large, the decimation over the odd
blocks may be not enough to depress some long range dependence in
the doubly renormalized measure and one is forced to decimated
further. One may stop the extra decimation until each surviving
blocks, namely the blocks of the final block spin transformation, are
separated one from the other by at least one block of the
decimation. We shall omit the details of these computations that are
however quite similar to the ones exposed above. \vskip.5cm
\numsec=3\numfor=1
{\bf Section 3. An example of persistence of non-Gibbsianness under decimation.}
\vskip.5cm \par
\vskip.5cm
In this last section we briefly
discuss another example
of a measure $\mu$
on $\{0,1\}^{{\bf Z}^2}$ with nicely decaying correlations functions, which
is {\it non Gibbsian}
and remains such even after
decimation on blocks of arbitrary side.
The example comes from a model of random discrete time dynamics
introduced in [MS] and further analyzed in [M].\par
The setting is as follows:
to each point x in the lattice
$ {\bf Z}^2$ we associate an
occupation variable $\sigma (x)$ with values 0 or 1;
given a configuration $\sigma \;
\in\; \{0,1\}^{{\bf Z}^2}$ we then define its clusters as
the maximal connected sets of sites in which the configuration
$\sigma $ is equal to one, where a
set $C\,\subset \, {\bf Z}^2$
is connected if for any pair of sites $x,y\;\in \;C$ there exists a
path $x_1,x_2\dots x_n$ of sites in $C$ such that
$$x_1\,=\,x,\;x_n\,=\,y,\;
\hbox{and } \vert x_i-x_{i+1}\vert\,=\,1\; i=1\dots n-1$$
With this position the
dynamics goes as follows:
given the configuration
$\sigma _t\;\in\; \{0,1\}^{{\bf Z}^2}$ at time t, in order to define the new
configuration $\sigma _{t+1}$ at time t+1,
we first remove each cluster
of $\sigma _{t}$ independently one from the other with probability
1/2 ; as a second step
we create particles in each empty site independently with
probability p. \par
For shortness we will refer to the first part of the updating as
the annihilation of particles and to the second part as the creation
of particles. Note that both processes occur simultaneously
(i.e. the updating is parallel) and that the non trivial
interaction of the model is all contained in the killing process
.\par
The above dynamics is similar to a model considered by Graannan and Swindle
in [GS] although in their model clusters disappear
with a rate proportional to their size. The two main results that we need
from [MS] and [M] are the following:\bigskip
{\bf Theorem 3.1 .}\par
\item{i)} For p sufficiently small there exists a unique invariant
measure $\mu$ on $\{0,1\}^{{\bf Z}^2}$ for the above dynamics
and its truncated correlations decay faster than any inverse power
(see corollary 5.1 in [M])
\item{ii)} If $\Omega_N$ denotes the event that the cube $\L_N$ of
side N centered at the origin
is filled with particles, then there exists a constant $c$ such that:
$$\mu (\Omega_N)\; \geq \; \exp(-cN) $$
(see theorem 5.1 in [MS]). \bigskip
In particular part ii) of the above theorem implies that the measure $\mu$
cannot be the Gibbs measure for any absolutely summable interaction, since
it has {\it wrong} large deviations: an exponential of the surface instead
of an exponential of the volume (see e.g. [EFS] for more details). \bigskip
{\bf Remark} It is important to observe that
the measure $\mu$ is non Gibbsian because of reasons which are very
different from the ones behind the {\it non Gibbsiannes} of the
block spin (without decimation) Ising model discussed in section 2.
There Gibbsiannes is lost due to the fact that, conditioned to the
event of having zero magnetization in each 2x2 block, the original
spin model undergoes a phase transition (see [EFS]). Here, on the contrary,
Gibbsiannes is violated because the measure $\mu$ has zero relative
entropy density with respect to the $\d$-measure concentrated on
the configuration identically equal to plus one (see [EFS]).
Clearly the latter is non Gibbsian because it violates
the nonnullity condition or absence of hard-core interaction (see
[EFS] 4.5.5 and 2.3.3). We actually suspect that $\mu$ itself
violates the nonullity condition but we do not have a formal
proof of this fact.\par Keeping in mind such a difference between the
block spin Ising model and the present one, it is not entirely
surprising that a decimation can suppress the phase transition in
the first one and thus restore the Gibbsiannes, while it is useless
in this new situation where the non Gibbsiannes is due to a much
stronger and rigid phenomenon.\bigskip We want now to apply the
usual decimation described in section 2 to the measure $\mu$ and
show that it leads to a new measure $\mu^{d,b_l}$ which is again non
Gibbsian irrespectively of the side $l$ of the blocks on which the
decimation acts. \par As before let us consider the partition of
${\bf Z^2}$ into $l\times l$ squared
blocks $Q_i$ of integer
side $l$ and let us decompose the lattice ${\bf Z^2}$ into the
two odd and even sublattices ${\bf
Z^2_e}$ and ${\bf Z^2_o}$ namely
$$
{\bf Z^2} = {\cal A} \cup {\cal B} \Eq(3.1)
$$
where ${\cal A} = \cup _i A_i$ is the set of the even
blocks and
${\cal B} = \cup _i B_i$
is the set of the odd blocks (see section 2 for more details).
Let us also call $\mu^{d,b_l}$ the projection or relativization of
the measure $\mu$ to ${\cal A}$. Then we have :
\bigskip {\bf Theorem 3.2 .} \par
For any value of the decimation parameter $l$ the measure
$\mu^{d,b_l}$ is not Gibbsian for any absolutely summable
interaction.\bigskip {\bf Proof} \par
Let us denote with $\Omega_N^A$ the
event that all the even
blocks $A$ contained inside the cube $\L_N$ of side N centered at the origin
are filled with particles. Then we trivially have:
$$\mu^{d,b_l} (\Omega_N^A)\;=\;\mu (\Omega_N^A)\;\geq\;\mu
(\Omega_N)\; \geq \; exp(-cN) $$ Thus also the decimated measure
$\mu^{d,b_l}$ has wrong large deviations and the result
follows.\par \bigskip
{\bf Remark}
\par \bigskip
We want to remark that, by the same argument used before for the decimation in one
of the two L-block sublattices, it is immediate to prove that the non-Gibbsianness of
our measure $\m$ persists under any
{\it extensive} decimation; namely a decimation such that the surviving spins, even
though they can be very sparse, have a well defined volume density.
\vskip 1cm
\centerline{\bf References}\bigskip\noindent
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