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\vglue.5truecm
{\centerline{\bf THE GLOBAL RENORMALIZATION GROUP TRAJECTORY IN A CRITICAL} }
{\centerline{\bf SUPERSYMMETRIC FIELD THEORY ON THE LATTICE
$\math{Z}^{3}$}}
\vglue1.truecm
{\centerline{{\bf P. K. Mitter}$^{1}$, {\bf B. Scoppola}$^{2}$}}
\vglue.5truecm
{\centerline{\smfnt $^1$Laboratoire de Physique Theorique et Astroparticules,
CNRS-IN2P3-Universit\'e Montpellier 2}}
{\centerline{\smfnt Place E. Bataillon, Case 070,
34095 Montpellier Cedex 05 France}}
{\centerline{\smfnt e-mail: pkmitter@LPTA.univ-montp2.fr}}
\vglue.5truecm
{\centerline{\smfnt $^2$Dipartimento di Matematica,
Universit\'a ``Tor Vergata'' di Roma}}
{\centerline{\smfnt Via della Ricerca Scientifica, 00133 Roma, Italy}}
{\centerline{\smfnt e-mail: scoppola@mat.uniroma2.it}}
\vglue1cm
\\{\bf Abstract}: We consider an Euclidean supersymmetric field theory
in $\math{Z}^{3}$ given by a supersymmetric $\Phi^{4}$ perturbation of an
underlying massless Gaussian measure on scalar bosonic and Grassmann fields
with covariance the Green's function of
a (stable) L\'evy random walk in $\math{Z}^{3}$. The Green's function
depends on the L\'evy-Khintchine parameter $\a={3+\e\over 2}$ with
$0<\a<2$. For $\a ={3\over 2}$ the $\Phi^{4}$ interaction is marginal.
We prove for $\a-{3\over 2}={\e\over 2}>0$ sufficiently small and
initial parameters held in an appropriate domain the existence of a global
renormalization group trajectory uniformly bounded on all renormalization group
scales and therefore on lattices which become arbitrarily fine. At the
same time we establish the
existence of the critical (stable) manifold. The
interactions are uniformly bounded away from zero on all scales and
therefore we
are constructing a non-Gaussian supersymmetric field theory on all scales.
The interest of this theory
comes from the easily established fact that the Green's function
of a (weakly) self-avoiding L\'evy walk in $\math{Z}^{3}$
is a second moment (two point correlation
function) of the supersymmetric measure governing this model. The control
of the renormalization group trajectory is a preparation for the study
of the asymptotics of this Green's function.
\vglue.5truecm
\\{\bf 0. Introduction}
\numsec=0\numfor=0
\vglue.5truecm
It was observed long ago, [PS, Mc ], that the Green's function of weakly
self avoiding simple random
walks (SAW) on a lattice $\math{Z}^d$ can be expressed as a
correlation function in a
supersymmetric field theory. This can be shown rigorously by the same
derivation as
in [BEZ, BI1, BI2] for SAWs on hierarchical lattices.
Consider instead of simple random walks the more
general case of continuous time (stable) L\'evy walks whose scaling
limits are stable L\'evy
distributions, [KG, F]. Such walks can be realized as jump processes with
probability distributions permitting long range jumps, [F].
Their characteristic functions are given by the L\'evy-Khintchine formula
with characteristic exponent $\a$, $0<\a\le 2$, [F], $\a=2$ corresponding to
simple random walks. The Green's function of continuous time weakly
self avoiding L\'evy walks (SALW) can also be realized as a two point
correlation function in a supersymmetric field theory by the same derivation
as in [ BEI, BI1, BI2]. We are
interested in SALWs in $\math{Z}^3$ with the L\'evy parameter
$\a$ in the range
${3\over 2}< \a < 2$ and in particular when $\a$ is very close to ${3\over 2}$
from above. Heuristic reasoning indicates that $\a ={3\over 2}$ corresponds
to mean field behaviour. This paper is concerned with proving the existence of
a uniformly bounded renormalization group (RG) trajectory
for the interactions in the underlying supersymmetric field theory. Uniformity
is with respect to the lattice scale which changes with each step of the
renormalization group map.
This gives the foundation for the study of the Green's function which is
postponed to the sequel.
The supersymmetric field theory in question is a lattice supersymmetric
generalization of the model considered in [BMS]. We describe it informally
here and leave the details for the next section.
Let $\D$ be the standard
Laplacian in $\math{Z}^3$. Then for $x,\> y\in\math{Z}^3$, and $0<\a<2$,
$C(x-y)=(-\D)^{-\a/2}(x-y)$ is the Green's function of a stable L\'evy walk.
Let $\f_1,\>\f_2$ be independent identically distributed Gaussian random
fields in $\math{Z}^3$ with covariance ${1\over 2}C$.
Let $\f=\f_1 +i\f_2$ and $\bar\f$ its complex conjugate. Introduce a pair of
Grassmann fields $\psi, \bar\psi$ of degree $1$ and $-1$ respectively.
Let $\Phi=(\f,\psi)$ and
$\bar\Phi=(\bar\f,\bar\psi)$. The inner product is given by
$(\Phi,\Phi)= \Phi\bar\Phi =\f\bar\f +\psi\bar\psi$. Let $\L\subset\math{Z}^3$
be a finite subset. Define
$$V_{0}(\L,\Phi)= g_{0}\int_{\L}dx (\Phi\bar\Phi)^{2}(x)+ \tilde\m_{0}
\int_{\L}dx \Phi\bar\Phi(x) \Eq(0.61) $$
\\where the coupling constant $g_{0}>0$ and $dx$ is the counting measure in
${\math{Z}^3}$. Then our model in finite volume $\L$ is defined by the
the measure
$$d\m_{\L}(\Phi)= d\m_{C_{\L}}(\Phi) e^{-V_{0}(\L,\Phi)} \Eq(0.62)$$
\\where $C_{\L}$ is the restriction of $C$ to the points of $\L$ and
$d\m_{C_{\L}}(\Phi)$ is the Gaussian measure
$$d\m_{C_{\L}}(\Phi)=\prod_{x\in \L} d\f_1(x)d\f_2(x)d\psi(x)d\bar\psi (x)
\> e^{-(\Phi,C_{\L}^{-1}\bar\Phi)_{L^{2}(\L)}} \Eq(0.63)$$
\\Integration over the Grassmann fields is Berezin integration and
$d\m_{\L}(\Phi)$ is interpreted as a linear functional on the Grassman algebra
(generated by the $\psi,\bar\psi$ with coefficients which are functionals of
the $\f,\bar\f$).
An important
fact is that the potential $V_{0}(\L,\Phi)$ is supersymmetric ( supersymmetry
in this context and some of its consequences are given in the Section~1.1).
As a consequence we have that the measure $d\m_{\L}(\Phi)$ is normalized :
$$\int d\m_{\L}(\Phi)\> 1 =1 \Eq(0.64)$$
We give an informal description of the results of this paper.
We will choose $\a={3+\e \over 2}$ with $0<\e <1$ , in particular we hold
$\e>0$ very small. We will take $\L$ to be
a very large cube. By successive RG transformations we will get a sequence
of measures ( the RG trajectory of measures) living in smaller and smaller
cubes in
finer and finer lattices till we arrive at a fixed small cube in a very
fine lattice. This will take $\log \L$ steps. At every step the measure is
a new gaussian measure times a new supersymmetric density. The Gaussian measure
is characterized by a covariance and the sequence of covariances converge to
a smooth continuum covariance.
The supersymmetric
density incorporates the interactions. The principal information is in the
local interactions incorporated in local potentials of the above type albeit
with new parameters (coupling constants) and on a finer lattice. The other
interactions are contracting ( irrelevant)in an appropriate sense and are
expressed in the form of polymer activities.
The coupling constants and polymer activities
give coordinates of the RG trajectory. The goal of this paper is to study the
RG trajectory of these coordinates in the infinite
volume limit which makes sense for these coordinates. The true infinite volume
limit and the scaling limit will be taken at the level of correlation
functions.
In section~1 we define the model, introduce supersymmetry and develop some of
its consequences. The RG analysis of this paper is based on the finite range
multiscale expansion of covariances of [BGM]. We summarize the basic results
of [BGM] pertinent to this paper in Theoerem~1.1.
This is an alternative to the Kadanoff-
Wilson block spin RG developed extensively by
Gawedzki and Kupiainen [GK 1,2], and
Balaban [Bal 1,2,3]. A crucial simplification arises due
to the finite range of the fluctuation covariances: Cluster expansions are no
longer needed in the control of the fluctuation integration which is an
essential step of RG transformations. As a result all estimates are local in
character. In this section we also define lattice
polymers and polymer activities.
\\In section~2 we introduce norms which will
measure the size of polymer activities. These norms are suggested by those in
the continuum analysis of [BMS]
\footnote{$^{(1)}$}{A. Abdesselam in [A]
corrected an error which occurs in [BDH-eps, BMS] where it
is wrongly asserted that certain normed spaces are complete. This problem
was resolved in [A] by some changes in definitions which fortunately are such
that, as noted in [A], the estimates, theorems and proofs of [BDH-eps, BMS]
remain true without change.
On the lattice however the function space subtleties encountered in [A]
disappear.} but now include the Grassmann variables. The definition that
we have hit upon owes much to a suggestion of David Brydges.
\\In section~3 we define the RG map as we will use it and
in section~4 apply it to our model. In particular we
develop second order perturbation theory. The task is to control
the contributions from the remainder and this is taken up in the next section.
\\Section~5 gives the basic estimates that we will need for the control
of the RG trajectory. These estimates resemble those in [BMS] but now take
account of the presence of Grassmann variables as well as the lattice. The
upshot is Theorem~5.1.
\\In section~6 we prove the existence of a uniformly bounded RG trajectory
at all scales (Theorem~6.2). This establishes the existence of a critical mass
but not its uniqueness.
This is remedied by first iterating the RG map a finite number
$n_0$ of times and then restarting the trajectory. There exists a uniformly
bounded trajectory at all scales $n\ge n_0$. For $n_0$ sufficiently large
the critical mass is a
continuous function of the contracting variables. The stable ( critical)
manifold, appropriately interpreted for a sequence of non-autonomous maps,
is constructed (Theorem~6.4). Finally we observe that the coupling
constant is uniformly bounded away from $0$ at all scales. As a result the
global RG trajectory gives rise to a non-Gaussian field theory. We remark that
in a continuum version of this model with a cutoff modelled on that of [BMS]
one can prove more: the continuum RG trajectory ends at a non-trivial fixed
point. But the notion of a fixed point is devoid of meaning for lattice field
theories because the RG map even in infinite volume does not give an
autonomous action on a fixed Banach space.
\vglue.5truecm
\\{\bf 1.1 Definitions, model, supersymmetry}
\numsec=1\numfor=1
\vskip.5truecm
Let $e_1,e_2,e_3$ be the standard basis of unit vectors specifying the
orientation of $\math{Z}^{3}$. We let $\dpr_{\m}$ denote the forward lattice
derivative in direction $e_{\m}$ and $\dpr_{\m}^{*}$ its $L^{2}(\math{Z}^{3})$
adjoint. The latter is the backward derivative.
Then the lattice Laplacian $\D$ in $\math{Z}^{3}$ is defined by
$$-\D = \dpr_{\m}^{*}\dpr_{\m} \Eq(0.1) $$
\\Let $\hat\D (p)$ be the Fourier transform of the
integral kernel of $\D$ in $\math{Z}^{3}$, namely
$$\hat\D (p) = 2\sum_{\m =1}^{3} (cos(p_{\m})-1)\Eq(0.2)$$.
\\Let $\a={3+\e\over 2} $ be a real number with $0 < \a <2$. Then the
Green's function
of a (stable) L\'evy walk in $\math{Z}^{3}$ is given by
$$C(x-y)= (-\D)^{-\a/2}(x-y)= \int_{[-\pi,\pi]^3} {d^{3}p\over (2\pi)^3}\>
e^{ip(x-y)}\> (-\hat\D (p))^{-\a/2} \Eq(0.3)$$
\\$C$ is positive-definite and therefore qualifies as the covariance of a
Gaussian random field in $\math{Z}^{3}$. We introduce a pair of independent
identically distributed Gaussian random fields $\f_{1},\> \f_{2}$ with mean $0$
and covariance
$$E(\f_{j}(x)\f_{j}(y))={1\over 2} C(x-y)\Eq(0.4) $$
for $j=1,2.$
\\Let $\f(x)=\f_1(x)+i\f_2(x) $ be a complex scalar field and $\bar\f(x)$ its
compex conjugate. On the space of functionals of $\f,\bar\f$ we have the
Gaussian probability measure
$$d\m_{C}(\phi)= d\m_{{1\over 2}C}(\phi_{1})d\m_{{1\over 2}C}(\phi_{2})
\Eq(0.4a) $$
\\Then each of $\f, \bar\f$ has zero covariance and
$$E(\bar\f(x)\f(y))= C(x-y)\Eq(0.5) $$
\\The $\f_j$ are Osterwalder-Schrader (O-S) positive generalised free fields.
To see this notice that for $0<\a<2$ we can write
$$C(x-y)= c_{\a}^{-1} \int_{0}^{\infty} da\> a^{-\a/2}\> G^{a}(x-y) \Eq(0.6)$$
\\ where $c_{\a}= \int_{0}^{\infty} da\> a^{-\a/2}\> (a+1)^{-1}$ and
$$ G^{a}(x-y)= \int_{(-\pi,\pi)^3} {d^{3}p\over (2\pi)^3}\> e^{ip(x-y)}\>
(a -\hat\D (p))^{-1} \Eq(0.7) $$
\\is the resolvent of the simple random walk on $\math{Z}^{3}$. From this
convergent representation for $C$ the O-S positivity follows since
it is well known that $ G^{a}$ is O-S positive.
\vglue0.3cm
\\{\it Grassmann algebra and integration}
\\Let $\L \subset \math{Z}^{3}$ be a bounded subset. ${\cal F}_{\L}$ represents
the algebra of $\math{C}$ valued functionals of the fields
$\{\phi,\bar\phi :\> \L\rightarrow \math{C}\}$.
Let $\psi(x), \bar\psi(x)$
for all $x\in \L$ be the ( anti-commuting) Grassman elements of degree
$1, -1$ respectively. Following standard usage we will refer to them as
(scalar) {\it fermions}. We denote by
${\Omega}_{\L}$ the Grassman algebra generated by the
$\psi(x), \bar\psi(y)$
by multiplication and linear sums for all $x,y\in \L$ with coefficients in
${\cal F}_{\L}$.
The Grassmann algebra is naturally graded
${\Omega}_{\L}= \oplus_{p} {\Omega}_{\L}^{p}$ where the integer $p$ is the
degree and each ${\Omega}_{\L}^{p}$ is a ${\cal F}_{\L}$ module.
${\Omega}_{\L}^{0}$ is an algebra. Because of the
anticommuting property of the generators, and because $\L$ is a finite lattice
an element of ${\Omega}_{\L}^{p}$ is a finite sum of degree $p$ elements
with coefficients in ${\cal F}_{\L}$. An element $F_{\L}$ of
$\Omega_{\L}$
can be represented as
$$F_{\L}(\f,\psi)= \sum_{k\ge 0}\sum_{\bf a}\int_{\L^{n}}dx_{1} ...dx_{n}
F_{\L}^{k,\bf a}(\f;x_{1},...,x_{k})\prod_{j=1}^{k}\psi_{a_{j}}(x_{j})
\Eq(0.81)$$
\\where $a_{j}\in \{-1,1\}$, $\psi_{1}=\psi,\> \psi_{-1}=\bar\psi$,
and ${\bf a}=(a_1,..,a_k)$ is a multi-index,
we take the ordering specified by $a_j=1$ if $j$ is odd and $a_j=-1$
and $dx$ is the counting measure
in $\math{Z}^{3}$ When $\L$ is finite subset the above multiple sum is
obviously finite. If $F_{\L}\in \Omega^{p}_{\L}$ we constrain the $\bf a$
sum to satisfy $a_{1}+...a_{k}=p$.
In the following we will often refer to the coefficients
$F_{\L}^{k,\bf a}$ above as {\it bosonic coefficients}. Here and in the
following we suppress
indicating the dependence of the bosonic coefficients on $\bar\f$.
\\These considerations
are ofcourse valid for a lattice $(\d\math{Z})^{3}$ for any lattice spacing
$\d$ with the corresponding notations $\L_{\d},{\cal F}_{\L_{\d}},
{\Omega}_{\L_{\d}}$, $dx$ being ${\d}^{3}$ times the counting measure in
$(\d\math{Z})^{3}$.
\vglue0.3cm
\\Now we define fermionic expectation (integration) using Berezin
integration which we review briefly and set up our conventions.
\\Berezin integration
is a linear map $\Omega_{\L}\rightarrow \Omega_{\L}$ which satisfies
$$\int d\psi(x) F_{\L}(\psi, \bar\psi,\phi,\bar\phi)=
\pi^{-1/2}{\dpr\over\dpr\psi (x)}F_{\L}
\Eq(0.8)$$
\\where $ F_{\L} \in {\Omega}_{\L} $ and the fermionic derivative
${\dpr\over\dpr\psi (x)}$ is an antiderivation.
Integration with respect to $ d\bar\psi(x)$ is given by the same formula
with $\dpr\over\dpr\bar\psi (x)$ on the right hand side. Multiple integration
is repeated integration using the above rule, keeping in mind that
fermionic derivatives anticommute.
\\Define $C_{\L}(x-y)= C(x-y)\> :\> x,y \in \L$. We consider this as a
$|\L|\times |\L|$ dimensional positive definite symmetric matrix with
$x,y$ labelling the entries. Then we define the
fermionic expectation $E_{f,\L}$
as a linear map ${\Omega}_{\L}\rightarrow {\cal F}_{\L}$ as follows:
Let $F_{\L}\in{\Omega}_{\L}$. We adopt the convention
$F_{\L}(\psi,\phi)\equiv F_{\L}(\psi,\bar\psi,\phi,\bar\phi)$. Then
$$E_{f,\L}(F_{\L})= \int d\m_{C_\L}(\psi) F_{\L}(\psi,\phi) \Eq(0.9)$$
\\where
$$\int d\m_{C_\L}(\psi) F_{\L}(\psi,\phi)) = (det\>\pi C_{\L})^{|\L|}
\int \prod_{x\in \L} (d\psi(x)d\bar\psi (x))\>
e^{-(\psi,C_{\L}^{-1}\bar\psi)_{L^{2}(\L)}}F_{\L}(\psi,\phi) \Eq(0.10)$$
\\We call $d\m_{C_\L}(\psi)$ a fermionic Gaussian measure and use the
terminology measure and expectation interchangeably. It is not difficult to
show that we have a fermionic counterpart of the bosonic gaussian formula,
namely
$$\int d\m_{C_\L}(\psi) F_{\L}(\psi,\phi)= e^{\int_{\L\times\L}dx dy
C_{\L}(x-y) {\dpr\over \dpr\psi(x)}{\dpr\over \dpr\bar\psi(y)}}
F_{\L}(\psi,\phi)\vert_{\psi=\bar\psi=0} \Eq(0.10a) $$
\\where $dx$ is the counting measure.
The fermionic expectation above annihilates the component of
$F_{\L}\notin \Omega_{\L}^{0}$.
\vglue0.3cm
\\Note that the expectation of a product of two $\psi$ or of two $\bar\psi$
vanishes whereas if $x,y \in \L$
$$E_{f,\L}(\bar\psi(x)\psi(y))= C(x-y) \Eq(0.11) $$
\\More generally, if $x_{j}, y_{j} \in \L,\> j=1,2,..n$
$$E_{f,\L}(\> \prod_{j=1}^{n}\bar\psi(x_{j})\psi(y_{j}))=
{\rm det}\> ( C(x_{j}-y_{k}))_{j,k=1}^{n} \Eq(0.11a)$$
\vglue0.3cm
\\We define the field $\Phi(x)$ (called superfield in anticipation)
as the pair
$$\Phi(x)=(\f(x),\psi(x)) \Eq(0.12) $$
\\with the scalar product
$$(\Phi(x),\Phi(y))= \Phi(x)\bar\Phi(y)=\f(x) \bar\f(y) +\psi(x)\bar\psi(y)
\Eq(0.13)$$
\\More generally if $A(x,y)$ is a matrix for $x,y\in\L$ we define
$$(\Phi,A\Phi)_{L^{2}(\L)}= \int_{\L\times\L}dxdy\> \Phi(x)A(x,y)\bar\Phi(y)
=\int_{\L\times\L}dxdy\>(\f(x)A(x,y)\bar\f(y) +\psi(x)A(x,y)\bar\psi(y))
\Eq(0.14)$$
\vglue0.3cm
\\Let $F_{\L}(\Phi)$ belong to $\Omega_{\L}$. $F_{\L}$ also depends on
$\bar\Phi$ but here and in the following this is not explicitly indicated.
Since $F_{\L}(\Phi)\in \Omega_{\L}$ it has the representation
\equ(0.81). We define the expectation $E_{\L}$ as a linear map
$\Omega_{\L}\rightarrow \bf C$ obtained by combining the bosonic and
fermionic expectations: If $F_{\L}\in\Omega_{\L}$ with $\m_{C}$ integrable
bosonic coefficients then
$$E_{\L}(F_{\L}(\Phi))=\int d\m_{C_{\L}}(\Phi)F_{\L}(\Phi)=
\int d\m_{C_{\L}}(\phi)d\m_{C_{\L}}(\psi)F_{\L}(\Phi) \Eq(0.14.1)$$
\\Thus
$$E_{\L}(F_{\L}(\Phi))= \int\prod_{x\in \L} d\f(x)d\bar\f(x)
\prod_{x\in \L} d\psi(x)\prod_{x\in \L} d\bar\psi (x)
\> e^{-(\Phi,C_{\L}^{-1}\bar\Phi)_{L^{2}(\L)}}F_{\L}(\Phi)
\Eq(0.15)$$
\\Notice that the determinant in the fermionic integration formula \equ(0.10)
has cancelled out with the inverse of the same determinant which appears
in the bosonic integration measure.
\\The expectation defined above is normalized. In other words if
$1_{\L}(\Phi)$ is the indicator function of $\Omega_{\L}$ then
$$E_{\L}(1_{\L}(\Phi))=1 \Eq(0.17) $$
\\We have the natural order relation ${\Omega}_{\L}\subset
{\Omega}_{\L'}$ if $\L\subset\L'$. Moreover if $\L\subset\L'$ and
$F_{\L}\in {\Omega}_{\L}$ then $E_{\L'}(F_{\L})= E_{\L}(F_{\L})$ as is
not difficult to show. We
define ${\Omega}$ as the inductive limit of the ${\Omega}_{\L}$ as
$\L\subset \math{Z}^{3} $ varies over increasing subsets tending to
$\math{Z}^{3}$ respecting
the order relation above. The $\{E_{\L},{\Omega}_{\L}\}$ constitute a
projective family. We denote by $E$ the projective limit: Let
$F\in {\Omega}$ with $\m_{C}$ integrable
bosonic coefficients. We have
$$E(F)=\int d\m_{C}(\Phi)\>F(\Phi)=
\lim_{\L'\uparrow \math{Z}^{3}}E_{\L'}(F) \Eq(0.101) $$
\\and this limit exists since $F\in {\Omega}_{\L}$ for some finite set
$\L$ and therefore $E(F)=E_{\L}(F)$ which exists.
\\{\it Remark} : The above construction is motivated by analogous
considerations in [BEI].
\vglue0.3cm
\\{\it Lattice integration } : In the following and throughout this paper
we will represent lattice sums as integrals where for the $(\d\math{Z})^{3}$
lattice the integration measure is the counting measure in $(\d\math{Z})^{3}$
times a factor $\d^{3}$. Thus if $f$ is a function on $(\d\math{Z})^{3}$ we
define
$$\int_{(\d\math{Z})^{3}}dx\> f(x)=\d^{3}\sum_{x\in (\d\math{Z})^{3}} f(x)
\Eq(0.18a) $$
\vglue0.3cm
\\We now define a Laplacian acting on functionals in $\Omega$
$${\bf\Delta}_{C}=\int_{\math{Z}^{3}\times
\math{Z}^{3}} dx dy\> C(x-y){\partial\over \partial\Phi(x)}\cdot
{\partial\over \partial\bar\Phi(y)}=$$
$$=\int_{\math{Z}^{3}\times
\math{Z}^{3}} dx dy\> C(x-y)[{\partial\over \partial\phi(x)}{\partial\over \partial\bar\phi(y)}+{\partial\over \partial\psi(x)}{\partial\over \partial\bar\psi(y)}]
\Eq(0.18)$$
\\This
integration automatically restricts to $\L \times \L$ when applied to
functionals of $\Phi$ which live in a bounded subset $\L$ and the
fermionic part of the exponential becomes a finite sum by the Grassmann
property. It follows from \equ(0.10a) and its bosonic counterpart that
if $F_{\L}(\Phi)\in \Omega_{\L}$ with integrable bosonic coefficients then
$$E(F_{\L}(\Phi))=
e^{{\bf\Delta}_{C}}F_{\L}(\Phi)\vert_{\f=\bar\f=\psi=\bar\psi=0}
\Eq(1.16a)$$
\\Note that the action of $e^{{\bf\Delta}_{C}}$ is well defined. In fact since
$F_{\L}(\Phi)$ is in $\Omega_{\L}$ and $\L$ is a finite lattice,
it can be expressed as a finite sum of Grassmann elements with
coefficients in ${\cal F}_{\L}$. $e^{{\bf\Delta}_{C}}$ factorises into
bosonic and Grassmann exponentials. The expansion of the Grassman exponential
acting on $F_{\L}(\Phi)$ evaluated at $\psi=\bar\psi=0$ thus terminates and
we are left with a purely bosonic expectation which is well defined.
\\We have in particular
$$E(\Phi(x)\bar\Phi(y))=0 \Eq(1.16b) $$
\\and more generally
$$E(\prod_{j=1}^{n}\Phi(x_{j})\bar\Phi(y_{j}))=0 \Eq(1.16c) $$
\\This can be proved by computation or more simply using supersymmetry
(introduced later). The integrand is supersymmetric and Lemma~1.1 below gives
the result.
\\Wick polynomials $P(\Phi)$ are defined by the formula
$$:P(\Phi):_{C}=e^{-{\bf\Delta}_{C}}P(\Phi) \Eq(0.18b)$$
\\This implies in particular that
$$:\Phi(x)\bar\Phi(y):_{C}=\Phi(x)\bar\Phi(y) \Eq(0.19)$$
\\and
$$:(\Phi\bar\Phi)^{2}:_{C}(x)= (\Phi\bar\Phi)^{2}(x)-
2C(0)(\Phi\cdot\bar\Phi)(x) \Eq(0.20) $$
\\For future reference we note that for $\a=1,2$
$$:(\Phi\bar\Phi)\Phi_{\a}:_{C}(x)=
(\Phi\bar\Phi)\Phi_{\a})(x)-C(0)\Phi_{\a}(x) \Eq(0.19a) $$
\\where $\Phi_{1}=\f,\> \Phi_{2}=\psi$.
\vglue0.1cm
\\{\it Remark} : The considerations from \equ(0.8) to \equ(0.20) remain
valid on a lattice $(\d\math{Z})^{3}$ if we replace in the above
$\L$ by a bounded subset $\L_{\d}\subset (\d\math{Z})^{3}$ and the positive
definite matrix $C$ by an arbitrary positive definite matrix $C_{\d}(x,y)$ with
$x,y\in (\d\math{Z})^{3}$. The functional Laplacian $\D_{C}$ in \equ(0.18)
is replaced by $\D_{C_{\d}}$ with the integration over
$(\d\math{Z})^{3}\times (\d\math{Z})^{3}$.
\vglue0.3cm
\\{\it The model} :
\vglue0.3cm
\\Let $L$ be a dyadic integer,
$L=2^p$ with integer $p\ge 2$. Let
$\L_{N}= [-{L^{N}\over 2},{L^{N}\over 2})^{3}\subset \math{R}^{3}$, with $N$
large be a large half open cube in $ \math{R}^{3}$.
Distances in $ \math{R}^{3}$ and
lattices $(\d\math{Z})^{3}$ will be measured in the norm
$$|x-y|= \max_{1\le j\le 3}|x_{j}-y_{j}| \Eq(0.21a)$$
\\ Define $\L_{N,0}= \L_{N}\cap \math{Z}^{3}$. This is a (large) cube in
$\math{Z}^{3}$ of edge length $L^ {N}$.
The second index $0$ in $\L_{N,0}$ emphasizes that this is a cube in
$\math{Z}^{3}$. The local potential \equ(0.61) will be written in a
$C$-Wick ordered form by using \equ(0.20) and \equ(0.19). This gives
$$V_{0}(\L_{N,0},\Phi)= \int_{\L_{N,0}}dx\> g_{0} :(\Phi\bar\Phi)^{2}:_{C}(x)+
\m_{0}\int_{\L_{N,0}}dx\> :\Phi\bar\Phi:_{C}(x)\Eq(0.21) $$
\\where $\m_{0}= \tilde\m_{0} + 2C(0)g_{0}$.
\\Define
$${\cal Z}_{0}(\L_{N,0},\Phi)= e^{-V(\L_{N,0},\Phi)} \Eq(0.22) $$
\\We define the measure
$$d\m_{N,0} (\Phi)= d\m_{C}(\Phi) {\cal Z}_{0}(\L_{N,0},\Phi) \Eq(0.221) $$
\\Note that the measure is normalized
$$ \int d\m_{N,0} (\Phi) =1 \Eq(0.222) $$
\\This follows from Lemma~1.1 below which exploits supersymmetry introduced
later. However heuristically this is evident if we formally expand the
exponential, integrate term by term and use \equ(1.16c). This measure defines
our model.
\vglue.3truecm
\\{\it Supersymmetry}
\vskip.3truecm
\\The density of the measure $d\m_{N,0} (\Phi)$ as well as its RG evolution
have the important property of being {\it supersymmetric}. This will restrict
considerably the form of the evolved density.
\\A {\it supersymmetry} transformation
${\cal Q} : \Omega_{\L}\rightarrow \Omega_{\L}$ is a derivation on
the bosonic fields and an antiderivation on the grassman fields which
acts on the fields as follows :
$$\eqalign{{\cal Q}\f&=\psi\cr
{\cal Q}\bar\f&=-\bar\psi\cr
{\cal Q}\psi&=\f\cr
{\cal Q}\bar\psi&=\bar\f\cr}\Eq(3.3)$$
\\Let $F_{\L}(\Phi)=F_{\L}(\f,\bar\f,\psi,\bar\psi)$ belong to $\Omega_{\L}$
with bosonic coefficients differentiable in the bosonic fields
$\f(x),\> x\in \L$.
Then the action of $\cal Q$ on $F_{\L}$ is given by a super vector field
denoted by the same symbol ${\cal Q}$
$${\cal Q}F_{\L}= \int_{\L} dx\> \Bigl(\psi(x){\dpr\over \dpr\f(x)} -
\bar\psi (x){\dpr\over
\dpr\bar\f(x)}+\f(x){\dpr\over \dpr\psi(x)} +\bar\f (x){\dpr\over
\dpr\bar\psi(x)}\Bigr)F_{\L} \Eq(3.3.2) $$
\\We say that a functional $F_{\L}$ is supersymmetric if ${\cal Q}F=0$.
\\{\it Remark}: A super vector field is not a vector field because fermionic
derivatives are antiderivations.
\vglue0.3cm
\\An (infinitesimal) {\it gauge transformation}
${\cal G}: \Omega_{\L}\rightarrow \Omega_{\L} $ is a derivation
whose action is given by
$$\eqalign{{\cal G}\f&=i\f\cr
{\cal G}\bar\f&=-i\bar\f\cr
{\cal G}\psi&=i\psi\cr
{\cal G}\bar\psi&=-i\bar\psi\cr}\Eq(3.3.1)$$
\\This induces on an $\Omega_{\L}$ function $F_{\L}$ the action of a
vector field denoted by the same symbol $\cal G$
$${\cal G}F_{\L}=
i\int_{\L} dx\> \Bigl(\f(x){\dpr\over \dpr\f(x)} -\bar\f(x){\dpr\over
\bar\dpr\f(x)}+\psi(x){\dpr\over \dpr\psi(x)} +\bar\psi(x){\dpr\over
\dpr\bar\psi(x)}\Bigr)F_{\L} \Eq(3.3.12) $$
\\We say that a functional $F_{\L}$ is gauge invariant if ${\cal G}F_{\L}=0$.
\vglue0.3cm
\\From \equ(3.3) we see that ${\cal Q}^2$ engenders an infinitesimal
gauge transformation \equ(3.3.1). Thus acting on
gauge invariant functionals
$${\cal Q}^2 =0\Eq(3.5)$$
\\An important property of the super vector field $\cal Q$ which we will
exploit later is that it commutes with the super Laplacian ${\bf\Delta}_{C}$
defined in \equ(0.18):
$$ [\>{\cal Q},\>{\bf\Delta}_{C}\>]=0$$
\\as is easy to verify.
\\It is easy to verify that any polynomial in $\Phi\bar\Phi$ and their
(lattice) derivatives is supersymmetric. This follows from
$${\cal Q}(\Phi(x)\bar\Phi(y))=
{\cal Q}\prod_{j=1}^{n}(\Phi(x_j)\bar\Phi(y_j))=0\Eq(3.4)$$
\\and lattice derivatives commute with ${\cal Q}$.
\\As a consequence we have ${\cal Q}V(\L,\Phi)=0$ where $V$ is given in
\equ(0.21) and thus the starting interaction potential is supersymmetric.
\vglue0.3cm
\\Let $\G(x,y)$ be any positive definite symmetric matrix. Let
${\bf\Delta}_{\G}$ be a super Laplacian given by \equ(0.18) with $C$ replaced
by $\G$. Let $F_{\L}(\Phi)$ be an $\Omega_{\L}$ functional with $\m_{C}$
integrable bosonic coefficients.
Let $\xi = (\z, \h)$ be another superfield. Define the convolution
$$\m_{\G}*F_{\L}(\Phi)=
\int d\m_{\G}(\xi)F_{\L}(\Phi+ \xi )=e^{{\bf\Delta}_{\G}}F_{\L}(\Phi)
\Eq(0.24)$$
\\Since $\cal Q$ commutes with ${\bf\Delta}_{\G}$, $\cal Q$ {\it also
commutes with convolution with the measure} $\m_{\G}$ :
$$\m_{\G}*{\cal Q}F_{\L}(\Phi)= {\cal Q}\m_{\G}*F_{\L}(\Phi) \Eq(0.2411)$$
\\Therefore if $F_{\L}$ is supersymmetric so is $\m_{\G}*F_{\L}$.
This observation prefigures the supersymmetry invariance of the
renormalization group map which we will introduce later.
\vskip0.2cm
\\It follows by evaluating \equ(0.2411) at $\Phi=0$ that
$$\int d\m_{\G}(\Phi)\> {\cal Q}F_{\L}(\Phi) =0 \Eq(0.241) $$
\\since the left hand side is given by
$(\m_{\G}*{\cal Q}F_{\L}(\Phi))\Bigr|_{\Phi=0}$ and this vanishes by virtue of
\equ(0.2411) since the coefficients
of the super vector field ${\cal Q}$ vanish when the fields vanish. \bull
\vglue0.3cm
\\{\it Lemma~1.1} : Let $F_{\L}(\Phi)$ be a supersymmtric
$\Omega_{\Lambda}$ functional with differentiable bosonic coefficients which
are $\m_{\G}$ integrable. Then
$$\int d\m_{\G}(\Phi)\> F_{\L}(\Phi)=F_{\L}(0) \Eq(0.242)$$
\\{\it Proof} : $\l$ be a real parameter. Define
$$f(\l)=\int d\m_{\G}(\Phi)\> F_{\L}(\l\Phi)\Eq(0.243)$$
\\We will prove
$${d\over d\l}f(\l)=0 \Eq(0.244)$$
\\This implies that $ f(\l)$ is a constant and hence evaluating at $\l=0$
gives \equ(0.242).
\\Taking the $\l$ derivative in \equ(0.243) we get
$${d\over d\l}f(\l)=\int d\m_{\G}(\Phi)\> ({\cal D}F_{\L})(\l\Phi)\Eq(0.245)$$
\\where
$${\cal D}=\int_{\L}dx \Bigl(\phi(x){\dpr\over\dpr\phi(x)} +
\bar\phi(x){\dpr\over\dpr\bar\phi(x)}+\psi(x){\dpr\over\dpr\psi(x)}+
\bar\psi(x){\dpr\over\dpr\bar\psi(x)}\Bigr) \Eq(0.246)$$
\\Note that the four coefficients of ${\cal D}$ can also be written as
$({\cal Q}\psi(x),{\cal Q}\bar\psi(x), {\cal Q}\phi(x),
-{\cal Q}\bar\phi(x))$ which we have taken in the same order as above.
This suggests that we consider the operator
$${\cal L}= \int_{\L}dx \Bigl(\psi(x){\dpr\over\dpr\phi(x)} +
\bar\psi(x){\dpr\over\dpr\bar\phi(x)}+\phi(x){\dpr\over\dpr\psi(x)}-
\bar\phi(x){\dpr\over\dpr\bar\psi(x)}\Bigr) \Eq(0.247)$$
\\and act with ${\cal Q}$ on it. We also consider the action of ${\cal L}$
on ${\cal Q}$. A straight forward computation gives the nice formula
$${\cal D}= {1\over 2} ({\cal Q}{\cal L}+ {\cal L} {\cal Q}) \Eq(0.248) $$
\\We substitute for ${\cal D}$ in \equ(0.245) the right hand side of
\equ(0.248). The contribution of the first term vanishes by \equ(0.241).
The contribution of the second term vanishes because $F_{\L}$ is supersymmetric
by hypothesis. This proves \equ(0.244) and we are done. \bull
\\{\it Remark} : The special case of Lemma~1.1 for a hierarchical lattice
is Lemma~2.1 of [BI]. {\it This Lemma has the important
consequence that no field independant relevant parts ( defined later) will
arise in the renormalization group analysis to follow.}
\vglue0.5cm
\\{\bf 1.2 Lattice renormalization group transformations}
\vglue0.3cm
\\We say
that a function $f(x,y)$ has finite range $L$ if $f(x,y)=0\> :\> |x-y|\ge L$.
Lattice renormalization group transformations will be based on the finite
range multiscale expansion of the covariance $C$ established in [BGM].
\\Let $L$ be a large dyadic integer, i.e.
a large integer power of $2$. Define $\d_{n}= L^{-n}$. We have
a sequence of compatible lattices $(\d_{n}\math{Z})^{3}\subset \math{R}^{3}$,
$(\d_{n}\math{Z})^{3}\subset (\d_{n+1}\math{Z})^{3}$, with $n=0,1,2,...$ and
eventually passing to $\math{R}^{3}$.
$B_{\d_n}= [-{\pi\over \d_n},{\pi\over \d_n}]^3 $ denotes the first Brillouin
zone of the dual of the $\d_n$ lattice. We have the following theorem
which gives
the multiscale expansion of the covariance $C$ on $\math{Z}^{3}$ as a sum of
finite range {\it fluctuation} covariances living on increasingly finer
lattices, together with their properties which we will need later :
\vglue0.3cm
\\{\it Theorem 1.1 (finite range multiscale expansion) :
For $0<\a<2$ , $d_{s} ={(3-\a)\over 2}$ and $ n= 0, 1, 2,...$ there
exist positive definite
functions $\G_{n}(x)$ defined for $x \in (\d_{n}\math{Z})^{3}$ and a smooth
positive definite function $\G_{c,*}$ in $\math{R}^{3}$ such that
for all $k\ge 0$, constants $c_{k,L}$ , $c_{L,m}$
independent of $n$ and $q= {1\over 48}$
$$\leqalignno{ C(x-y) =\sum_{n\ge 0} L^{-2nd_{s}}
\G_{n}\Bigl({{x-y}\over L^{n}}\Bigr) & {\rm \> and \> the\> series\>
converges \> in} \> L^{\infty}(\math{Z}^{3}) & (1) \cr
\G_{n}(x)& =0 \quad {\rm for}\quad |x|\ge {L\over 2} & (2) \cr
\Bigl|\hat\G_{n}(p)\Bigr| & \le c_{k,L} (1+p^{2})^{-2k} \quad {\rm for}\quad
p\in B_{\d_{n}},\quad\forall k\ge 0 & (3) \cr
\hat\G_{c,*}(p) & = {\rm lim}_{n\rightarrow\infty}\hat\G_{n}(p)\quad
{\rm exists\> pointwise\> in }\> p & (4a) \cr
\Bigl|\hat\G_{n}(p)- \hat\G_{c,*}(p)\Bigr| & \le c_{k,L} (1+p^{2})^{-2k}L^{-qn}
\quad \forall n \ge 2,\>\forall k\ge 0 & (4b) \cr
\Vert \dpr_{\d_{n}}^{m}\G_{n}\Vert_{L^{\infty}((\d_{n}\math{Z})^{3})}
&\le c_{L,m}, \forall m\ge 0 & (5a) \cr
\dpr_{c}^{m} \G_{c,*} = {\rm lim}_{n\rightarrow\infty}\dpr_{\d_{n}}^{m}\G_{n}
\quad {\rm exists\> in}\quad
L^{\infty}((\d_{l}\math{Z})^{3})\quad &{\rm for}\>
{\rm any\>fixed}\> l\ge 0 \> {\rm and}\> \forall\> m \ge 0 & (5b)\cr
\hskip3cm\Vert \dpr_{\d_{n}}^{m}\G_{n}- \dpr_{c}^{m} \G_{c,*}
\Vert_{L^{\infty}((\d_{l}\math{Z})^{3})}& \le c_{L,m}L^{-qn},\>\forall n\ge l,
\> \forall m\ge 0 & (5c) \cr}$$
\\where $\dpr_{c}$ is a continuum partial derivative and $\dpr_{\d_{n}}$ is a
forward lattice partial derivative in $(\d_{n}\math{Z})^{3}$. }
\\{\it Proof} : The theorem is for the most part a combination of results
obtained in various theorems in [BGM]. For the reader's convenience we record
them here. The references pertain to [BGM].
Thus the multiscale expansion in part (1) and the
finite range property of part (2) were given in Section 4. The range was
stated as being $6L$ but we can easily arrange matters in section 2 so that
the range is $L/2$ which turns out to be more convenient.
Convergence of (1) in
$L^{\infty}(\math{Z}^{3})$ follows on using $d_{s} >0$, Corollary 5.6 and
the lattice Sobolev embedding inequality with $k$ in the Corollary
sufficiently large. Part (3) follows from (5.10) of
Theorem 5.5 by integration on $a$ with the measure $da\> a^{-\a/2}$ ( see
(4.3) of section 4). Corollary 5.6 and lattice Sobolev embedding gives (5a).
Corollary 6.2 gives parts (4a) and (5b). The
convergence rate estimates of parts (4b) and (5c) which were not given in
[BGM] also follow from the results therein. The proof is given elsewhere,
[M]. \bull
\vglue0.3cm
\\Define for all $n\ge 0$ the positive definite functions $C_{n},\> C_{c,*}$
on $(\d_{n}\math{Z})^{3})$ and $\math{R}^3$ respectively :
$$C_{n}(x)=\sum_{j=0}^{\infty} L^{-2jd_{s}} \>
\G_{n+j}\Bigl({{x}\over L^{j}}\Bigr) \Eq(0.25) $$
$$C_{c,*}(x)=\sum_{j=0}^{\infty} L^{-2jd_{s}} \>
\G_{c,*}\Bigl({{x}\over L^{j}}\Bigr) \Eq(0.26) $$
\vglue0.3cm
\\{\it Corollary 1.2 : The series \equ(0.25) for $C_n$ together with that
for its multiple lattice derivatives in $(\d_{n}\math{Z})^{3}$
converge in $L^{\infty}((\d_{n}\math{Z})^{3})$. For every integer $m\ge 0$ we
have a constant $c_{L,m}$ such that
$$\Vert \dpr_{\d_n}^{m} C_{n}\Vert_{L^{\infty}((\d_n\math{Z})^{3})}\le c_{L,m}
\Eq(0.271) $$
\\The series \equ(0.25) defining $C_{c,*}$
and its multiple continuum derivatives of arbitrary order
converge in $L^{\infty}(\math{R}^3)$
so that $C_{c,*}$ is a smooth continuum function. For all $m\ge 0$ and
$\dpr_{c}$ the continuum partial derivative
$$\sup_{x\in \math{R}^3}\vert \dpr_{c}^{m}C_{c,*}(x)\vert\le c_{L,m}
\Eq(0.271a) $$
\\Moreover
for any fixed $l\ge 0$ and $\forall m \ge 0$, there exists a constant
$c_{L,m}$ such that for all $n\ge l$
$$\Vert \dpr_{\d_l}^{m}C_{n}- \dpr_{c}^{m} C_{c,*}
\Vert_{L^{\infty}((\d_l\math{Z})^{3})} \le c_{L,m}L^{-qn} \Eq(0.27) $$ }
\\{\it Proof} : The first part together with the bound \equ(0.271) follow from
(5a) of Theorem 1.1. To prove the next statement apply Theorem 6.1 of [BGM] to
(4.3) of Section 4, followed by Sobolev embedding . Therefore \equ(0.26) and
its multiple continuum derivatives of arbitrary order converge in
$L^{\infty}(\math{R}^3)$ and thus
$C_{c,*}$ is smooth. Finally the estimate \equ(0.27) follows on using
part (5c) of Theorem 1.1. This also establishes that
$\dpr_{\d_l}^{m}C_{n}\rightarrow \dpr_{c}^{m} C_{c,*}$ in
$L^{\infty}((\d_l\math{Z})^{3})$. \bull
\vglue0.3cm
\\From the definition \equ(0.25) we get for $x,\> y \in (\d_{n}\math{Z})^{3}$
$$C_{n}(x-y)= \G_{n}(x-y) + L^{-2d_{s}} C_{n+1}\Bigl({{x-y}\over L}\Bigr)
\Eq(0.28) $$
\\This relation generates by iteration the multiscale expansion of the
covariance $C=C_{0}$ in part (1) of Theorem 1.1.
\vglue0.2cm
\\We consider the finite sequence of compatible lattices
$\{(\d_{n}\math{Z})^{3}\} $ for $0\le n\le N$. The considerations in Section 1.1
for fields in $\math{Z}^{3}$ remain valid for
every lattice $(\d_{n}\math{Z})^{3}$ provided for the expectations we replace
the covariance $C$ by $C_{n}$. Let the fields $\f,\psi,\bar\psi$ be
defined in $(\d_{N}\math{Z})^{3}$. These fields restrict to the coarser lattices
$(\d_{n}\math{Z})^{3}$ for every $n$ with $0\le n\le N$.
\vglue0.2cm
\\We introduce a parameter $\e$ with $0<\e\le 1$ and define
$$\a={3+\e\over 2} \Eq(1.41a) $$
\\Let $x\in(\d_{n}\math{Z})^{3}$. For every $n\le N-1$ we define the scale
transformation $S_{L}$ by
$$S_{L}\Phi(x)= \Phi_{L^{-1}}(x)= L^{-d_{s}}\Phi({x\over L}) \Eq(1.41) $$
\\where
$$d_{s}={(3-\a)\over 2}= {3-\e\over 4} \Eq(1.42) $$
\\is the {\it dimension} of the field $\Phi$. The fields
$\f,\bar\f,\psi,\bar\psi$ are thus assigned the
same dimension $d_{s}$ and the same transformation law \equ(1.41). Note that
the scale transformed fields now live in $(\d_{n+1}\math{Z})^{3}$.
\\The $C_n$ and $\G_n$ are positive definite and therefore qualify as
covariances of Gaussian measures. For $x,y\in (\d_{n}\math{Z})^{3}$ we define
the scale transformation of the covariance $C_{n+1}$ by
$$S_{L}C_{n+1}(x-y)=L^{-2d_{s}} C_{n+1}({{x-y}\over L})\Eq(1.43)$$.
\\which permits us to write \equ(0.28) as
$$C_{n}(x-y)= \G_{n}(x-y) + S_{L}C_{n+1}(x-y) \Eq(1.44) $$
\\ Let $\L \subset (\d_{n}\math{Z})^{3}$ be a bounded subset. Then \equ(1.44)
implies upon using \equ(1.16a) (with $C$ replaced by $C_{n}$) that
$$\int d\m_{C_{n}}(\Phi) F_{\L}(\Phi) = \int d\m_{S_{L}C_{n+1}}(\Phi)
\int d\m_{\G_{n}}(\xi) F_{\L}(\xi + \Phi) \Eq(1.45)$$
\vglue0.3cm
\\Let $L=2^{p}$ with integer $p\ge 2$ and let
$\L_{m}= [-{L^{m}\over 2},{L^{m}\over 2})^{3}\subset \math{R}^{3}$
be a half open cube in $\math{R}^{3}$ centered at the origin.
We denote by
$$\L_{m,n}= \L_{m}\cap (\d_{n}\math{Z})^{3} \Eq(1.45a)$$
the induced
cube of side length $L^{m}$ in $(\d_{n}\math{Z})^{3}$ centered at the origin.
Let $F_{0}(\L_{N,0}\>,\Phi)$ be
functional of $ \Phi$ and ($\bar\Phi$) belonging to $\Omega^{0}(\L_{N,0})$.
By virtue of \equ(1.45) we have for $n=0$
$$\int d\m_{C_{0}}(\Phi)F_{0}( \L_{N,0}\>, \Phi)=
\int d\m_{C_{1}}(\Phi)F_{1}(\L_{N-1,1}\>,\Phi) \Eq(1.46) $$
\\where
$$F_{1}(\L_{N-1,1}\>,\Phi)=S_{L}\m_{\G_{0}}*F_{0}(\L_{N,0}\>,\Phi)=
\int d\m_{\G_{0}}(\xi)
F_{0}(\L_{N,0}\>,\xi +S_{L}\Phi) \Eq(1.47) $$
\\The final scale transformation takes us to a finer lattice as well as
scaling down the size of the volume.
\\The iteration of \equ(1.47) using \equ(1.46) gives after $n$ steps
$$\int d\m_{C_{0}}F_{0}(\L_{N,0}\>,\Phi)=
\int d\m_{C_{n}}F_{n}(\L_{N-n,n}\>,\Phi)\Eq(1.47a) $$
\\where
$$F_{n}(\L_{N-n,n}\>,\Phi)
= S_{L}\m_{\G_{n-1}}*F_{n-1}(\L_{N-n+1,n-1}\>,\Phi) \Eq(1.48) $$
\\\equ(1.48) defines for $N>0$ fixed and $1\le n\le N-1$ a
sequence of maps
$$T_{N-n,n}\> :\> \Omega^{0}(\L_{N-n+1,n-1})\rightarrow
\Omega^{0}(\L_{N-n,n}) \Eq(1.48a)$$
\\any member of which we call a
{\it renormalization group (RG) transformation}. The map is clearly not
autonomous. The first index refers to the volume which has gotten reduced
because of the rescaling. The second index refers to the lattice spacing
which has gotten finer because of the rescaling. In the following we will
apply the RG transformation iteratively to the (interaction) density
${\cal Z}_{0}(\L_{N,0}\>,\Phi)$ of the measure $d\m_{N,0}(\Phi)$ defined in
\equ(0.221) generating thereby the sequence ${\cal Z}_{n}(\L_{N-n,n}\>,\Phi)$
for $0\le n\le N-1$. After $N-1$ steps we arrive at
${\cal Z}_{N-1}(\L_{1,N-1}\>,\Phi)$ where $\L_{1,N-1}$ is the cube of edge
length $L$ in $(\d_{N-1}\math{Z})^{3}$ centered at the origin. The fundamental
goal in this paper is to control this sequence of transformations when $N$
is indefinitely large in the infinite volume limit (as explained at the end
of section~3).
\vglue0.3cm
\\1.3. {\bf Polymer gas representation}.
\vglue0.2cm
\\In order to analyze the RG evolution we will write the densities
${\cal Z}_{n}$ in a {\it polymer gas} representation whose form is preserved
under RG transformations.
\vglue0.2cm
\\{\it Polymers} :
\\To this end pave $\math{R}^{3}$ with
{\it half-open cubes $\D$
of edge length 1 and with vertices in $\math{Z}^{3}$ called
unit cubes or $1$-cubes}. $\math{R}^{3}$ is then the union of disjoint
half-open unit cubes. A unit cube $\D\subset\math{R}^{3}$ is of the
form $\D = [m_{1},m_{1}+1)\times [m_{2},m_{2}+1)\times [m_{3},m_{3}+1)$ where
$m_{j}\in \math{Z}$. {\it We say two unit cubes from the paving
are connected if their closures share at least a vertex in
common.} If they are not connected (i.e. their closures are disjoint) we say
that they are {\it strictly disjoint}. A continuum (connected) $1$- polymer
$X$ is a (connected) union of a
finite subset of unit cubes chosen from the paving and is thus half-open.
Henceforth unless otherwise mentioned a polymer is connected by default.
\\We will measure distances
in $\math{R}^{3}$ and in all embedded lattices in the norm
$$|x-y|=\max_{1\le j\le 3}|x_{j}-y_{j}| \Eq(1.220) $$
\\If $\D_1$ and $\D_2$ are two unit cubes from the paving then the distance
between them is
$$d(\D_1,\D_2)=\inf_{x\in \D_1,\>y\in \D_2}|x-y| \Eq(1.220a) $$
\\If $\D_1$ and $\D_2$ are strictly disjoint than $d(\D_1,\D_2)\ge 1$.
\\Let $\d$ be any member of the sequence $\d_{n}$. Recall that
$\d_{n}= L^{-n}$ where $L$ is a dyadic integer. Define the {\it unit block
or $1$-block} in $(\d\math{Z})^{3}$ by
$$\D_{\d}=\D \cap (\d\math{Z})^{3} \Eq(1.22a)$$
\\and the {\it lattice $1$-polymer $X_{\d}$} by
$$X_{\d}=X \cap (\d\math{Z})^{3} \Eq(1.22b)$$
\\where $X$ is a continuum $1$-polymer. Note that as point sets
$X_{\d_{n}}\subset X_{\d_{n+1}}$. We say a lattice polymer $X_{\d_{n}}$
is {\it convex} if the continuum polymer $X$ of which it is a restriction is
a a convex set.
\\We denote by $|X_{\d}|$ the volume of $X_{\d}$ measured
in accordance with \equ(0.18a).
The $1$-blocks are lattice restrictions of
half-open continuum unit cubes defined above. Therefore $|\D_{\d}|=1$ and
$$|X_{\d}| = \sharp \{\D_{\d}\>:\> \D_{\d}\subset X_{\d}\} \Eq(1.22f) $$
\\the total number of $1$-blocks in $X_{\d}$. This is equal to $|X|$ the
total number of $1$-cubes in $X$ by our construction. As a consequence we
have $|X_{\d_n}|=|X_{\d_{n+1}}|$.
\vglue0.2cm
\\{\it We say two $1$-blocks in $X_{\d}$ are connected if the continuum
$1$-cubes of which they are the lattice restrictions are connected
( see above).} If the $1$-blocks are not connected we say that they are
{\it strictly disjoint}. The distance between two strictly disjoint $1$-blocks
is $\ge 1$. The lattice (connected) polymer $X_{\d}$ is a (connected) union of
a finite subset of disjoint $1$-blocks $\D_{\d}$. Let $X_{\d}$ and $Y_{\d}$
be each a connected polymer. {\it We say that $X_{\d},Y_{\d}$ are strictly
disjoint if they are mutually disconnected i.e.
if every $1$-block from $X_{\d}$ is strictly disjoint from every $1$-block
from $Y_{\d}$ }. Then distance$(X_{\d},Y_{\d})\ge 1$.
\\Given a polymer $X_{\d}$ we denote by $\dpr X_{\d}$ the set of its boundary
points. Given an integer $n\ge 1$ we define the $n$-{\it collar} of
$X_{\d}$, denoted $\dpr_{n} X_{\d}$ by
$$\dpr_{n} X_{\d}
=\{y\notin X_{\d}\> : |x-y|\le n\d,\> {\rm some}\> x\in X_{\d}\}
\Eq(1.22c) $$
\\where $|\cdot|$ is the distance function inherited from $\math{R}^{3}$. We
define
$${\tilde X^{(n)}_{\d}}= X_{\d} \cup \dpr_{n} X_{\d} \Eq(1.22d) $$
\\Let $f : (\d\math{Z})^{3}\rightarrow \math{C}$. We define the forward
lattice partial derivative $\dpr_{\d,\m}$ and the backward lattice derivative
$\dpr_{\d,-\m}$ by
$$\dpr_{\d,\m}f(x)=\d^{-1}(f(x+\d e_{\m})-f(x))
\Eq(1.22e)$$
$$\dpr_{\d,-\m}f(x)=\dpr_{\d,\m}^{*}f(x)=\d^{-1}(f(x-\d e_{\m})-f(x))
\Eq(1.22g)$$
\\where $e_{1},e_{2},e_{3}$ is the standard basis of unit vectors which
provides the orientation of $\math{R}^{3}$ and thus of all the embedded lattices
we will encounter. $\dpr_{\d,\m}^{*}$ is the $L^{2}((\d\math{Z})^{3})$ adjoint
of $\dpr_{\d,\m}$.
\vglue0.2cm
\\{\it Polymer activity} :
\\A {\it polymer activity} $K(X_{\d},\Phi)= {\tilde K}(X_{\d},\f, \psi)$, where
it is henceforth understood that it also depends on $\bar\f,\bar\psi$,
is a map
$X_{\d},\Phi \rightarrow \Omega^{0}_{{\tilde X}^{(1)}_{\d}}$ where the fields
$\Phi$ depend only on the points of ${\tilde X}^{(1)}_{\d}$.
\vglue0.2cm
\\{\it Remark} : The reason for attaching a $1$-collar is that
lattice derivatives on fermionic fields at the boundary points of
$X_{\d}$ which occur in the representation for $\tilde K$ (given below) take us
into $\dpr_{1} X_{\d}$.
\vglue0.2cm
\\{\it The polymer activities of this paper are of degree $0$,
gauge invariant and supersymmetric, and invariant under translations and
rotations which leave the lattice invariant. In addition they satisfy
the condition $K(X_{\d},\Phi)= K(X_{\d},-\Phi)$ together with the support
condition : $K(X_{\d},\Phi)=0$ if $X$ is not connected. Furthermore
$K(X_{\d},0)=0.$ }
\vglue0.2cm
\\We write the generic density
${\cal Z}({\L_{\d}})(\Phi)$ in the form
$${\cal Z}(\L_{\d})=\sum_{N=0}^{\infty}{1\over
N!}e^{-V(X_{\d}^{(c)})} \sum_{X_{\d,1},..,X_{\d,N}}
\prod_{j=1}^N K(X_{\d,j})
\Eq(1.22)$$
\\where the connected polymers $X_{\d,j}\subset\L_{\d}$ are strictly disjoint,
$X_{\d}=\cup_1^N X_{\d,j}$, $X_{\d}^{(c)}=\L_{\d}\backslash X_{\d}$
and $V(Y_{\d})=V(Y_{\d},\Phi,C,g,\mu)$ is given
by\equ(0.21)
with parameters $g,\mu$ and integration over $Y_{\d}$ with measure $dx$ defined
as the counting measure in $(\d\math{Z})^{3}$ times $\d^{3}$. The
Wick ordering covariance $C =C_{n}$ (see\equ(0.25)) if $\d=\d_{n}$.
We have
suppressed the field dependence in \equ(1.22). Initially the activities
$K$ vanish but they do arise under RG transformations. The representation
\equ(1.22) remains stable under RG transformations as we will see in Section~3.
\vglue0.3cm
\\Polymer activities $K(X_{\d},\Phi)= {\tilde K}(X_{\d},\f,\psi)$ will be
expanded out as a ( finite) series in the fermionic fields $\psi, \bar\psi$
and their (lattice) derivatives
with coefficients which are functionals of the bosonic fields $\f$. This leads
to the representation
$$K(X_{\d},\Phi)= \tilde K(X_{\d},\f,\psi)=\sum_{p\ge 0}
\sum_{\bf l,a}
\int_{X_{\d}^{2p}} d{\bf x}\ \tilde K^{{\bf l,a}
}_{2p}(X_{\d},\f,{\bf x})\ \dpr_{\d}^{\bf l}
\prod_{j=1}^{2p}\psi_{a_j}
(x_{j})\Eq(2.1)$$
\\with the condition
$$K(X_{\d},0)=0\Eq(2.111)$$
\\where:
\\$2p$ is the number of Grassmann fields.
\\A field $\psi_a$, $a\in\{-1,1\}$, is a two component field
defined by $\psi_1=\psi$, $\psi_{-1}=\bar\psi$. The sum on
${\bf a}=(a_1,...a_{2p})$ with $a_j\in \{-1,1\}$ is constrained by
$\sum_j a_j=0$.
\\${\bf l}=(l_1,...,l_{2p})$, $l_i\in
\{0,1,2,3\}$, $\dpr_{\d}^{\bf l}=\dpr_{\d,l_{1}}.......\dpr_{\d,l_{2p}}$ with
the
convention
$\dpr_{\d,l_{j}}=1$ if $l_{j}=0$ and otherwise
if $l_{j}=\m\not= 0$ then $\dpr_{\d,l_{j}}$ is the forward lattice derivative
in $(\d\math{Z})^{3}$ defined in \equ(1.22e).
\\${\bf x}=(x_{1},...,x_{2p})$, and $d{\bf x}=\prod_{i=1}^{2p}dx_{i}$ where
$dx_{i}$ is the counting measure multiplied by $\d^{3}$ on $(\d\math{Z})^{3}$.
\\The kernel $\tilde K^{\bf l,a }_{2p}$ is gauge invariant as is the Grassmann
monomial of degree $0$.
\vglue0.1cm
\\{\it Remarks} :
\\1.It is ofcourse possible to write the polymer representation
such that the fields in the Grassman monomial is always in a specified order
(thus specifying the $a_j$). The more general form that we have introduced
turns out to be convenient for later purposes.
\\2.We will see that the form \equ(2.1) is preserved
by renormalization group transformations. {\it The RG transformations are
gauge invariant, preserve supersymmetry by virtue of \equ(0.2411),
as well as the vanishing condition \equ(2.111) by virtue of Lemma~1.1.}
\\3.To simplify notation we will often write $K$ instead of
$\tilde K$ if there is no risk of confusion.
\vglue.5truecm
\\{\bf 2. REGULATORS, DERIVATIVES AND NORMS }
\numsec=2\numfor=1
\vskip.5truecm
In this section we will introduce Banach spaces of polymer activities.
These are lattice analogues of the continuum constructions in [BDH-est, BMS,
A] albeit with changes because of the presence of Grassman variable.
The Banach space norms that we will presently introduce measure
differentiability properties of the activities with respect to fields $
\f,\psi$, as well
as the behaviour with respect to large fields $\dpr\f$ and large sets. The
behaviour for large $\f$ itself will be controlled with the help of
lattice Sobolev inequalities and the local potential.
\vskip.5truecm
\\{\it Regulators }
\vglue0.3cm
\\Let $\dpr_{\d,\m}$ and $\dpr_{\d,-\m}$
be respectively the forward and backward lattice derivatives in
$(\d\math{Z})^{3}$ along the unit vector $e_{\m}$ defined in \equ(1.22e)
and \equ(1.22g).
Here as before $\d$ is any member of the sequence $\{\d_{n}\}$
where $\d_{n}=L^{-n}$ and $L=2^{p}$ with integer $p\ge 2$.
Define
$$\dpr^{j}_{\m_1, \m_2,..,\m_j}=
\dpr_{\d,\m_1}\dpr_{\d,\m_2}...\dpr_{\d,\m_j}$$
\\Let $X$ be a connected polymer in $\math{R}^3$ and
$X_{\d}=X\cap (\d\math{Z})^3$. Let
${\tilde X}^{(n)}_{\d}= X_{\d}\cup \dpr_{n} X_{\d}$ as defined earlier
(\equ(1.22c) and \equ(1.22d)). Let
$\f : {\tilde X}^{(5)}_{\d}\rightarrow \math{C}$. We define a norm
$\Vert\cdot\Vert_{X_{\d},1,5}$ :
$$\Vert\f\Vert^2_{X_{\d},1,5}=\sum_{j=1}^{5}\> {1\over 2^{j}}
\sum_{\m_{j}\in S,\forall j}
\int_{X_{\d}}dx\> |\dpr^{j}_{\m_1, \m_2,..,\m_j}\f(x)|^{2} \Eq(2.10reg) $$
\\where $S=\{1,-1,2,-2,3,-3\}$. This is a lattice Sobolev norm of the type
introduced
in Section 5, page 421 of [BGM] but now without the $L^{2}$ piece.
\\We define now the {\it large field regulator}
$$G_{\k}\> :\> X_{\d}\times {\cal F}_{{\tilde X}^{(5)}_{\d}}
\rightarrow \math{R} \Eq(2.11reg) $$
\\where ${\cal F}_{{\tilde X}^{(5)}_{\d}}$ is the
algebra of
$\bf C$ valued functions on $ {\tilde X}^{(5)}_{\d} $ by
$$G_{\k}(X_{\d},\f)=e^{\kappa\Vert\f\Vert^2_{X_{\d},1,5}}
\Eq(2.1reg)$$
\\$G_{\k}$ satisfies the multiplicative property: If $X_{\d}, Y_{\d}$
are disjoint sets then
$$G_{\k}(X_{\d}\cup Y_{\d} ,\f)= G_{\k}(X_{\d},\f) G_{\k}(Y_{\d},\f)
\Eq(2.110) $$
\\$G_{\k}^{-1}$ will be a weight function in polymer activity norms.
The norm
$\Vert\cdot\Vert_{X_{\d},1,5}$ can be used in lattice Sobolev inequalities, in
conjunction with the stability provided by the local potential, to control
$\f$ and its first two lattice derivatives pointwise. The parameter
$\k=\k(L) > 0$ is chosen so that for all $L\ge 2$
the large field regulator satisfies the stability property given in the
following Lemma :
\vglue0.2cm
\\ {\it Lemma~2.1 (stability property)} : There exists a constant
$\k_{0}=\k_{0}(L) >0$
independent of $n$ such that for all $\k$ with $0<\k\le\k_{0}$
$$\int d\mu_{\G_{n}}(\zeta)\> G_{\k}(X_{\d_{n}},\zeta +\f)
\le 2^{|X_{\d}|}G_{2\k} (X_{\d_{n}},\f) \Eq(2.3reg)$$
\\where $|X_{\d_{n}}|$ is the number of unit blocks in $X_{\d_{n}}$.
\\{\it Proof} : \equ(2.3reg)
is proved in exactly the same way as in the proof of the stability property
of the continuum large field regulator in Lemma~3 of [BDH-est]. The proof
uses a flow equation for the measure convolution with interpolated covariance
which remains true for the lattice. Another ingredient is Young's
convolution inequality for functions
which is also true on the lattice. In the cited proof we replace the covariance
$C$ by $\G_{n}$ and continuum derivatives by lattice derivatives. From
the proof of Lemma~3 of [BDH-est] we see that two conditions have to be
satisfied by $\k_{0}$, namely :
1) $\k_{0} \max_{2\le m\le 10}
\Vert\dpr_{\d_{n}}^{m}\G_{n}\Vert_{L^{\infty}((\d_{n}\bf Z)^{3})}$ is
sufficiently small and
2) $\k_{0}\Vert \G_{n}\Vert_{L^{1}((\d_{n}\bf Z)^{3})}$ is sufficiently
small.
Parts (5a) of Theorem~1.1 shows that that 1) and 2) above can be
assured by a $\k_{0}$ independent of $n$. From (5a) we have
$$\k_{0} \max_{2\le m\le 10}
\Vert\dpr_{\d_{n}}^{m}\G_{n}\Vert_{L^{\infty}((\d_{n}\bf Z)^{3})}
\le \k_{0}c_{L}$$
\\and from (5a) and the finite range property
$$\k_{0}\Vert \G_{n}\Vert_{L^{1}((\d_{n}\bf Z)^{3})} \le
\k_{0} L^{3}c'_{L}$$
\\It is sufficient to choose $\k_{0}$ so that the right hand side of both
inequalities are sufficiently small. This is achieved independent of $n$.
\bull
\vglue0.1cm
\\Now hold $L$ sufficiently large and recall that $\a <2$.
Then we get after rescaling
$$\int d\mu_{\Gamma}(\zeta)\> G_{\k}(X_{\d_{n}},\zeta +S_{L}\f)
\le 2^{|X_{\d_{n}}|} G_{\k}(L^{-1}X_{\d_{n+1}},\f) \Eq(2.3.1reg)$$
because from the scaling property of the
fields $\f$ , see \equ(1.41), \equ(1.42)we have
$$\Vert S_{L}\f\Vert^2_{X_{\d_{n}},1,5}\le L^{-(2-\a)}
\Vert\f\Vert^2_{L^{-1}X_{\d_{n+1}} ,1,5} \Eq(2.3.1.sc) $$
\vglue0.3cm
\\Next we introduce a {\it large set regulator}. Let $X_{\d}$ be a connected
$1$-polymer in $(\d\math{Z})^{3}$. This is a connected union of $1$-blocks
defined earlier. We define
$$\AA_{p} (X_{\d})= 2^{p|X_{\d}|}L^{(D+2)|X_{\d}|} \Eq(2.4reg)$$
where for us the dimension of space $D=3$, and $p$ is an integer.
\vglue0.2cm
\\{\it Small sets} :
We call a connected polymer $X_{\d}$ {\it small} if $|X_{\d}|\le 2^D$.
A connected polymer which is not small is called {\it large}.
\vglue0.2cm
\\{\it $L$-polymers and $L$-closure} : Pave
$\math{R}^{3}$ with half-open cubes of edge length $L$ with vertices in
$\math{Z}^{3}$, called $L$-cubes. Each $L$-cube is a union of $1$-cubes.
Take the restriction of these $L$-cubes to
$(\d\math{Z})^{3}$ and call the latter cubes $L$-blocks. Each $L$-block is a
union of $1$-blocks. The paving of $\math{R}^{3}$ by $L$-cubes induces a
paving of $(\d\math{Z})^{3}$ by $L$-blocks. An $L$-polymer is a union of $L$-
blocks.
{\it We define the $L$-closure of the $1$-polymer $X_{\d}$, denoted
$\bar{X_{\d}}^{(L)}$, as the $L$-polymer given by the smallest union of
$L$-blocks containing $X_{\d}$}. The notions of connectedness and strict
disjointness carry over from the case of $1$ blocks and $1$-polymers. Thus
we say two $L$-blocks from the
$L$-paving are connected if the closures of the corresponding continuum
$L$-cubes are connected ( i.e. share at least a vertex in common). If they
are not connected we say that they are strictly disjoint. Strictly disjoint
$L$-blocks are separated by a distance $\ge L$. A connected
$L$-polymer is a connected union of $L$-blocks. If two connected $L$-polymers
are not connected to each other we say they are strictly disjoint. Strictly
disjoint $L$-polymers are separated by a distance $\ge L$.
\vglue0.2cm
\\{\it Lemma~2.2} : Fix any
integer $p\ge 0$ and let $L$ be sufficiently large depending on $p$. Then
for any connected $1$-polymer $X_{\d}$
$$\AA (L^{-1}\bar{X_{\d}}^{(L)}) \le c_{p}\AA_{-p}(X_{\d}) \Eq(2.5reg)$$
For $X_{\d}$ a {\it large} connected $1$-polymer,
$$\AA (L^{-1}\bar{X_{\d}}^{(L)}) \le c_{p}L^{-D-1}\AA_{-p}(X_{\d})\Eq(2.6reg)$$
Here $c_p =O(1)$ is a constant independent of $L$ and $\d$.
\\{\it Remark} : This is the lattice version of Lemma~1 of [BDH-est].
It is purely geometrical and proved in the same way.
\vskip0.3cm
\\{\it Field derivatives and norms}: The polymer activities in question are
degree $0$ gauge invariant supersyymetric functionals
of the complex bosonic fields $\f, \bar\f$ and the fermionic fields
$\psi,\bar\psi$. Lattice field derivatives are partial
derivatives with respect to the fields at different points of the lattice.
The fermionic derivative is an antiderivation. We see
from the definition of the polymer activities in \equ(2.1)
that the fermionic field derivatives are taken at the origin of the
fermionic field space. However in order to measure the size of the lattice
field derivatives it turns out to be useful to generalize the notion of
field derivatives as directional derivatives (directional in field space).
For the bosonic coefficient this is the lattice transcription
of that given in [BDH].
For the fermionic part there is no clear sense of direction and the definition
we give below suggested to us by David Brydges turns out to be particularly
useful.
\vglue0.2cm
\\Let $X_{\d}\subset (\d\math{Z})^{3}$ be a connected polymer.
Let $f_{j}$ for $j=1,....,m$ be $\bf C$ valued functions on
${\tilde X}^{(2)}_{\d} = X_{\d}\cup \dpr_{2}X_{\d}$.
Let $g_{2p}$ be a $\bf C$ valued function on
$({\tilde X}^{(2)}_{\d})^{2p}$. A polymer activity $K(X_{\d},\Phi)$ has the
representation
\equ(2.1). We consider it as a function of $\f,\bar\f,\psi,\bar\psi$ denoted
as ${K}(X_{\d},\f,\psi)$ where we have suppressed
the dependence on $\bar\f,\bar\psi$. We define:
$$D^{2p,m}K(X_{\d},\f,0;f^{\times m},g_{2p})=
\sum_{{\bf l,a}}\int_{X_{\d}^{2p}} d{\bf x}D^m
K^{{\bf l,a}
}_{2p}(X_{\d},\f,{\bf x};f^{\times m}) \dpr_{\d}^{\bf l}g_{2p}(x_{1},
...x_{2p}) \Eq(2.2)$$
\\where $f^{\times m}=(f_1,..,f_m)$ and
$$D^m K^{{\bf l,a}}_{2p}(X_{\d},\f,{\bf x};f^{\times m})
=\dpr_{s_{1}}....\dpr_{s_{m}}K^{\bf l,a}_{2p}
(X_{\d},\f,{\bf x};\f+s_{1}f_{1},....,\f+s_{m}f_{m})\vert_{s_1=..=s_m =0}
\Eq(2.211)$$
\\and the $s_j$ are real parameters.
\\We endow the linear space of $\bf C$ valued functions $f$ as above
with the norm
$$\Vert f\Vert_{C^{2}(X_{\d})}= \sup_{1\le \m,\n \le 3}
(\Vert \dpr_{\d,\m}f\Vert_{L^{\infty}(X_{\d})},
\Vert \dpr_{\d,\m}\dpr_{\d,\n} f\Vert_{L^{\infty}(X_{\d})}) \Eq(2.111a)$$
\\and call the resulting normed space $C^{2}(X_{\d})$. $\dpr_{\d,\m}$ is the
lattice derivative defined in \equ(1.22e).
\\Let $\dpr_{\d,\m_{j}}$ acting on $g_{2p}(x_{1},...,x_{2p})$ denote the
lattice derivative with respect to $x_j$ in the direction $e_{\m_{j}}$.
We endow the linear space of $\bf C$ valued functions $g_{2n}$ with the norm
$$\Vert g_{2p}\Vert_{C^{2}(X_{\d}^{2p})}= \sup_{1\le \m_{j},\m_{k},\le 3
\atop {1\le j,k\le 2p}}
(\Vert \dpr_{\d,\m_{j}} g_{2p}\Vert_{L^{\infty}(X_{\d}^{2p})},
\Vert \dpr_{\d,\m_{j}}\dpr_{\d,\m_{k}} g_{2p}\Vert_{L^{\infty}(X_{\d}^{2p})})
\Eq(2.112)$$
\\and call the resulting normed space $C^{2}(X_{\d}^{2p})$. The above norms
always exist for lattice functions since $X_{\d}$ is a finite set.
\\ \equ(2.2) then defines a $\bf C$ valued multilinear functional on
$ C^{2}(X_{\d})^{m} \times C^{2}(X_{\d}^{2p})$ whose norm is defined to be
$$\Vert D^{2p,m}K(X_{\d},\f,0)\Vert =
\sup_{\Vert f_j\Vert_{C^{2}(X_{\d})}\le 1
\atop{ \Vert g_{2p}\Vert_{C^{2}(X_{\d}^{2p})}
\le 1\atop \> \forall 1\le j\le m}}\left\vert
D^{2p,m}K(X_{\d},\f,0;f^{\times m},g_{2p}) \right\vert\Eq(2.3)$$
\\The space of $\bf C$ valued multilinear functionals defined in \equ(2.2)
which are bounded in the norm \equ(2.3) is complete and thus a Banach space.
The completeness follows on using the completeness of the number field
$\bf C$ by a standard argument.
\vskip0.2cm
\\Let ${\bf h}=(h_F,h_B)$ where $h_F,h_B >0$ are strictly positive real
numbers. We define the following set of norms. The $\bf h$ norm is defined by
$$\Vert K(X_{\d},\f,0)\Vert_{\bf h}=
\sum_{p=0}^\io\sum_{m=0}^{m_0}h_F^{2p}{h_B^m\over m!}
\Vert D^{2p,m} K(X_{\d},\f,0)\Vert \Eq(2.4) $$
\\In addition we define a {\it kernel} norm with ${\bf h}_*=(h_F,h_{B*})$
$$\vert K(X_{\d})\vert_{\bf h_*} =
\sum_{p=0}^\io\sum_{m=0}^{m_0}{h}_F^{2p}{h_{B*}^m\over m!}
\Vert D^{2p,m} K(X_{\d},0,0)\Vert \Eq(2.5)$$
\\${\bf h},{\bf h}_*$ will be chosen later in Section~5. We now define
the ${\bf h}, G_{\k}$ norm by
$$\Vert K(X_{\d})\Vert_{{\bf h},G_{\k}}=
\sup_{\f\in {\cal F}_{\tilde X_{\d}^{(5)}}}
\Vert K(X_{\d},\f,0)\Vert_{\bf h}G^{-1}_{\k}(X_{\d},\f)\Eq(2.6)$$
\\Let $\AA(X_{\d})$ be the large set regulator defined earlier. We then have
our final set of norms
$$\Vert K\Vert_{{\bf h},G_{\k},\AA,\d}=
\sup_{\Delta_{\d}}\sum_{X_{\d}\supset\Delta_{\d}}
\Vert (K(X_{\d})\Vert_{{\bf h},G_{\k}}\AA (X_{\d}) \Eq(2.611) $$
\\where $\Delta_{\d}=\D\cap (\d\math{Z})^{3}$ and $\D$ is a unit cube in
$\math{R}^{3}$ as defined earlier, and
$$\vert K\vert_{{\bf h}_{*},\AA,\d}=
\sup_{\Delta_{\d}}\sum_{X_{\d}\supset\Delta_{\d}}\vert K(X_{\d})
\vert_{{\bf h}_*}\AA (X_{\d}) \Eq(2.612) $$
\\The index $\d$ in our final norms \equ(2.611) and \equ(2.612) indicate that
the large set
norm is being taken over polymers in $(\d\math{Z})^{3}$. Under each of above
norms we have Banach spaces.
Moreover it is easy to verify that the {\it multiplicative (Banach algebra)
property} holds
for the polymer activities $\tilde K(X_{\d})$ under the {\bf h}-norm \equ(2.4),
the kernel norm \equ(2.5), and, for activities supported on disjoint
polymers , under the $\bf h, G_{\k} $ norm.
The multiplicative property
plays a very important role in the estimates in the rest of the paper. We
therefore state it as {\it Proposition 2.3} below and supply a proof.
\vglue0.2cm
\\{\it Proposition 2.3} : Let $X_{\d,1}$, $X_{\d,2}$ denote two connected
polymers, not necessarily disjoint, and $K_1(X_{\d,1},\f,\psi)$,
\\$K_2(X_{\d,2},\f,\psi)$ the
corresponding polymer activities. Define a new polymer activity
$${\bf K}(X_{\d,1}\cup X_{\d,2},\f,\psi)=
K_1(X_{\d,1},\f,\psi)K_2(X_{\d,2},\f,\psi)$$
\\Then
$$\Vert{\bf K}(X_{\d,1}\cup X_{\d,2},\f, 0)\Vert_{\bf h}\le
\Vert K_1(X_{\d,1},\f, 0)\Vert_{\bf h}
\Vert K_2(X_{\d,2},\f, 0)\Vert_{\bf h}$$
\\If $X_{\d,1}$ and $X_{\d,2}$ are disjoint
$$\Vert{\bf K}(X_{\d,1}\cup X_{\d,2})\Vert_{\bf h, G_{\k}}\le
\Vert K_1(X_{\d,1})\Vert_{\bf h, G_{\k}}
\Vert K_2(X_{\d,2})\Vert_{\bf h, G_{\k}}$$
\\{\it Proof}
\\Let ${\tilde X}^{(2)}_{\d,j}=X_{\d,j}\cup \dpr_{2}X_{\d,j}$.
Let $f_j,\ j=1,...,m$ be functions on
${\tilde X}^{(2)}_{\d,1}\cup {\tilde X}^{(2)}_{\d,2}$
and $g_p$ a function on
$({\tilde X}^{(2)}_{\d,1}\cup {\tilde X}^{(2)}_{\d,2})^p$. By definition
$$\Vert D^{ p,m}{\bf K}(X_{\d,1}\cup X_{\d,2},\f,0)\Vert=\sup_{\Vert f_j\Vert_{C^2(X_{\d,1}\cup X_{\d,2})}\le 1
\atop \Vert g_p\Vert_{C^2((X_{\d,1}\cup X_{\d,2})^p)}\le 1}\vert D^{ p,m}
{\bf K}(X_{\d,1}\cup X_{\d,2},\f,0;f^{\times m},
g_p)\vert$$
We denote now ${\bf x}_{j,p_j}=(x_{j,1},...,x_{j,p_j})$, $f^{\times m}=(f_1,...,f_m)$,
$f^{\times |I|}=\{f_i\}_{i\in I}$ where $I\subset \{1,2,.....,p\}$.
Using the representation \equ(2.1) for the polymer activities and the the definition \equ(2.2)
for the derivatives we get
$$D^{ p,m}{\bf K}(X_{\d,1}\cup X_{\d,2},\f,0;f^{\times m},g_p)=\sum_{p_1+p_2=p}
\sum_{m_1+m_2=m}\sum_{I\cup J=\{1,...,m\}\atop
{I\cap J=\emptyset\atop
|I|=m_1, |J|=m_2}}\sum_{{\bf l}_1,{\bf l}_2, {\bf a_1,a_2} } $$
$$\int_{X_{\d,1}^{p_1}\times X_{\d,2}^{p_2}}d{\bf x}_{1,p_1}d{\bf x}_{2,p_2}
D^{m_1}K_{1,p_1}^{{\bf l}_1,a_1}(X_{\d,1},\f,{\bf x}_{1,p_1};f^{\times |I|})
D^{m_2}K_{2,p_2}^{{\bf l_2,a_2}}(X_{\d,2},\f,{\bf x}_{2,p_2};f^{\times |J|})
\partial_{\d}^{{\bf l}_1}\partial_{\d}^{{\bf l}_2}g_p({\bf x}_{1,p_1},{\bf x}_{2,p_2})
$$
$$=\sum_{p_1+p_2=p}
\sum_{m_1+m_2=m}\sum_{I\cup J=\{1,...,m\}\atop
{I\cap J=\emptyset\atop
|I|=m_1, |J|=m_2}}\sum_{{\bf l}_1,{\bf a}_1} $$
$$\int_{X_{\d,1}^{p_1}}d{\bf x}_{1,p_1}
D^{m_1}K_{1,p_1}^{{\bf l}_1, {\bf a}_1}(X_{\d,1},\f,{\bf x}_{1,p_1};f^{\times |I|})
\partial_{\d}^{{\bf l}_1} D^{p_2,m_2}K_{2}(X_{\d,2},\f,0;g_p({\bf x}_{1,p_1},\cdot),f^{\times |J|})
\Eq(2.616)$$
Define
$$F_{p_1}({\bf x}_{1,p_1})=D^{p_2,m_2}K_{2}(X_{\d,2},\f,0;g_p({\bf x}_{1,p_1},\cdot),f^{\times |J|})
\Eq(2.617)$$
as a function on $({\tilde X}^{(2)}_{\d,1})^{p_1}$, ( recall the definition of
$g_{p}$). We have suppressed the dependence of $F_{p_1}$ on the other
variables. Then for $0\le k\le 2$
$$\partial_{\d}^k F_{p_1}({\bf x}_{1,p_1})=
D^{p_2,m_2}K_{2}(X_{\d,2},\f,0;\partial_{\d}^k g_p({\bf x}_{1,p_1},\cdot),f^{\times |J|})
$$
where $\partial_{\d}^k$ is the lattice partial derivative of degree $k$ with
respect to
$(x_{1,1},...,x_{1,p_1})$ in multi-index notation,
whence
$$\vert \partial_{\d}^k F_{p_1}({\bf x}_{1,p_1})\vert=
\Vert D^{p_2,m_2}K_{2}(X_{\d,2},\f,0)\Vert\prod_{j\in J}\Vert f_j\Vert_{C^2(X_{\d,2})}
\Vert \partial_{\d}^k g_p({\bf x}_{1,p_1},\cdot)\Vert_{C^2( X_{\d,2}^{p_2})}
$$
Therefore
$$\Vert F_{p_1}\Vert_{C^2(X_{\d,1}^{p_1})}\le
\Vert D^{p_2,m_2}K_{2}(X_{\d,2},\f,0)\Vert\prod_{j\in J}\Vert f_j\Vert_{C^2(X_{\d,2})}
\Vert g_p\Vert_{C^2( X_{\d,2}^{p_2}\times X_{\d,1}^{p_1})}
$$
Since $X_{\d,2}^{p_2}\times X_{\d,1}^{p_1}\subset (X_{\d,1}\cup X_{\d,2})^p$ we have
$$\Vert g_p\Vert_{C^2( X_{\d,2}^{p_2}\times X_{\d,1}^{p_1})}\le
\Vert g_p\Vert_{C^2((X_{\d,1}\cup X_{\d,2})^p)}$$
and
$$\Vert f_j\Vert_{C^2(X_{\d,2})}\le \Vert f_j\Vert_{C^2(X_{\d,1}\cup X_{\d,2})}$$
Hence
$$\Vert F_{p_1}\Vert_{C^2(X_{\d,1}^{p_1})}\le
\Vert D^{p_2,m_2}K_{2}(X_{\d,2},\f,0)\Vert\prod_{j\in J}\Vert f_j\Vert_{C^2(X_{\d,1}\cup X_{\d,2})}
\Vert g_p\Vert_{C^2((X_{\d,1}\cup X_{\d,2})^p)}\Eq(2.618)
$$
Now from \equ(2.616), and the definition \equ(2.617) of $F_{p_1}$
$$D^{ p,m}{\bf K}(X_{\d,1}\cup X_{\d,2},\f,0;f^{\times m},g_p)=\sum_{p_1+p_2=p\atop m_1+m_2=m}
\sum_{I\cup J=\{1,...,m\}\atop
{I\cap J=\emptyset\atop
|I|=m_1, |J|=m_2}} D^{p_1,m_1}K_{1}(X_{\d,1},\f,0;f^{\times |I|}, F_{p_1})
$$
Therefore
$$\vert D^{ p,m}{\bf K}(X_{\d,1}\cup X_{\d,2},\f,0;f^{\times m},g_p)\vert\le
\sum_{p_1+p_2=p\atop m_1+m_2=m}
\sum_{I\cup J=\{1,...,m\}\atop
{I\cap J=\emptyset\atop
|I|=m_1, |J|=m_2}}
\vert D^{p_1,m_1}K_{1}(X_{\d,1},\f,0;f^{\times |I|}, F_{p_1})\vert\le
$$
$$\le \sum_{p_1+p_2=p\atop m_1+m_2=m}
\sum_{I\cup J=\{1,...,m\}\atop
{I\cap J=\emptyset\atop
|I|=m_1, |J|=m_2}}
\Vert D^{p_1,m_1}K_{1}(X_{\d,1},\f,0)\Vert\prod_{i\in I}\Vert f_i\Vert_{C^2(X_{\d,1})}
\Vert F_{p_1}\Vert_{C^2(X_{\d,1}^{p_1})}\Eq(2.619)
$$
\\From \equ(2.618), \equ(2.619) and observing
$\Vert f_i\Vert_{C^2(X_{\d,1})}\le \Vert f_i\Vert_{C^2(X_{\d,1}\cup X_{\d,2})}$
we get
$$\vert D^{ p,m}{\bf K}(X_{\d,1}\cup X_{\d,2},\f,0;f^{\times m},g_p)\vert
\le\sum_{p_1+p_2=p\atop m_1+m_2=m}
\sum_{I\cup J=\{1,...,m\}\atop
{I\cap J=\emptyset\atop
|I|=m_1, |J|=m_2}} \Vert D^{p_1,m_1}K_{1}(X_{\d,1},\f,0)\Vert
\Vert D^{p_2,m_2}K_{2}(X_{\d,2},\f,0)\Vert \cdot $$
$$\cdot \prod_{i=1}^m
\Vert f_i\Vert_{C^2(X_{\d,1}\cup X_{\d,2})}\Vert g_p\Vert_{C^2((X_{\d,1}\cup X_{\d,2})^p)} $$
We perform the sum over $I$ and $J$ using
$$\sum_{I\cup J=\{1,...,m\}\atop
{I\cap J=\emptyset\atop
|I|=m_1, |J|=m_2}}1={m!\over m_1!m_2!}$$
Therefore
$$\Vert D^{ p,m}{\bf K}(X_{\d,1}\cup X_{\d,2},\f,0)\Vert\le
\sum_{p_1+p_2=p\atop m_1+m_2=m}{m!\over m_1!m_2!}\Vert D^{p_1,m_1}K_{1}(X_{\d,1},\f,0)\Vert
\Vert D^{p_2,m_2}K_{2}(X_{\d,2},\f,0)\Vert
$$
Multiply both sides of the previous inequality by $h_B^m/m!$ and $h_F^p$.
Sum over integers $m$, $0\le m\le m_0$, and over all integers
$p\ge 0$ to obtain
$$\Vert{\bf K}(X_{\d,1}\cup X_{\d,2},\f,0)\Vert_{\bf h}\le\Vert K_1(X_{\d,1},\f,0)\Vert_{\bf h}
\Vert K_2(X_{\d,2},\f,0)\Vert_{\bf h}$$
\\This proves the first inequality of Proposition 2.1. The second inequality
follows from the first because for union of disjoint sets
$$G_{\k}(X_{\d}\cup Y_{\d},\f)=G_{\k}(X_{\d},\f)G_{\k}(Y_{\d},\f)$$
\bull
\vskip0.5cm
\\{\bf 3. THE RG MAP }
\numsec=3\numfor=1
\vskip.3truecm
In this section we describe the RG map applied to the generic density
in the polymer representation given in \equ(1.22). This is a lattice
transcription of the continuum RG map described in [BMS], ( see also [M]).
This goes in several steps. First we must perform the
fluctuation integration and rescaling (see \equ(1.48))
$${\cal Z}'(L^{-1}\L_{L^{-1}\d},\f)=S_{L}\mu_{\G}*{\cal Z}(\L_{\d},\Phi)
\Eq(3.0) $$
\\where $\L_{\d}\subset (\d\math{Z})^{3}$ is the volume arrived at after a
certain number of previous RG steps and $\G$ is the fluctuation covariance for
the next step. $\G$ is one of the covariances $\G_{n}$ of Theorem~1.1 and
has the finite range property stated in that theorem. Thus after $n$ RG steps
(see \equ(1.45a)-\equ(1.48a)) $\d=\d_{n}$, $\G= \G_{n}$,
$\L_{\d}=\L_{N-n,n}$ and $L^{-1}\L_{L^{-1}\d}=\L_{N-n-1,n+1}$.
\\The polymer representation \equ(1.22) for ${\cal Z}(\L_{\d})$
is parametrized by the {\it coordinates}
$(V,K)$ on the scale $\d$ where $V$ is a local functional (potential):
$$V(X_{\d})=\sum_{\D_{\d}\subset X_{\d}}V(\D_{\d}) \Eq(3.01)$$
\\Let ${\tilde V}(X_{\d},\Phi)$ be an arbitrary local supersymmetric
functional with ${\tilde V}(X_{\d},0)=0$. We will see
that the polymer representation is preserved under the RG transformation
\equ(3.0) with new coordinates ${\tilde V}_{L},{\cal F}( K)$ on the
next scale $L^{-1}\d$. ${\cal F}$ depends on ${\tilde V}$.
The finite range property of $\G$ leads to
a simple description of this map :
$$V\rightarrow {\tilde V}_{L},\quad
{\tilde V}_{L}(\D_{L^{-1}\d})=S_{L}{\tilde V}(L\D_{\d}) $$
$$K\rightarrow {\cal F}( K) :\> {\cal F}( K)(X_{L^{-1}\d},\Phi)
=\int d\m_{\G}(\xi){\cal B}K(LX_{\d},\xi, S_{L}\Phi) \Eq(3.02) $$
\\where ${\cal B}K$ is a ${\tilde V}$ dependent nonlinear functional of $K$
to be presently described. We call this map the {\it fluctuation map}.
\\We can take advantage of the arbitrariness of the local potential
$\tilde V$ in the above map so as to remove the
expanding ( relevant) parts $F$ in the polymer activity ${\cal F}(K)$
and compensate by a change ${\tilde V}_{L}(F)$
in the local potential ${\tilde V}_{L}$ in such a way that the evolved
density ${\cal Z}'(L^{-1}\L_{L^{-1}\d})$ on the left hand side of \equ(3.0)
remains unchanged. This operation gives rise to the
{\it extraction} map, [BDH-est]
$${\tilde V}_{L}\rightarrow V'(F)={\tilde V}_{L}-{\tilde V}_{L}(F),
\quad {\cal F}(K)\rightarrow K'={\cal E}({\cal F}(K),F)
\Eq(3.03) $$
\\where the image is on the same scale $L^{-1}\d$.
$V'(F)$ and the nonlinear map $\cal E$ have simple expressions which are
lattice transcriptions of those given in [BDH-est].
The composition of the fluctation map \equ(3.02) and
the extraction map \equ(3.03) gives the {\it RG map}
$$f : \quad f(V,K)=(f_{V}(V,K), f_{K}(V,K))$$
\\where
$$f_{V} : V\rightarrow {\tilde V}_{L}\rightarrow V'(F) $$
$$f_{K} : K\rightarrow {\cal F}(K) \rightarrow K'={\cal E}({\cal F}(K),F)
\Eq(3.04) $$
\\The operation of
extraction leads in particular to a discrete flow of the coupling
constants in $V$ on scale $L^{-1}\d$ provided we choose $F,{\tilde V_{L}}(F)$
appropriately. The expanding functionals will be gathered in the local
potential $V'(F)$ whereas the polymer activity ${\cal E}({\cal F}(K),F)$
will be a contracting (irrelevant) error term.
\vskip.3truecm
\\{\it 3.1 The Fluctuation Map}
We now construct the map \equ(3.02) starting from \equ(3.0) with the density
in the polymer representation \equ(1.22). In performing the fluctuation
integration
$$\mu_{\G}*{\cal Z}(\L_{\d},\Phi)=\int d\m_{\G}(\xi) \sum_{N} {1\over N!}
e^{-V(X_{\d}^{(c)},\Phi +\xi)} \sum_{X_{\d,1},..,X_{\d,N}}
\prod_{j=1}^N K(X_{\d,j},\Phi+\xi) \Eq(3.501) $$
\\we will exploit
the independence of $\xi (x)$ and $\xi (y)$ when $|x-y|\ge L$. To
this end we construct an $L$-paving of $\L_{\d}$ and the $L$-closure of
$\bar{X_{\d}}^{(L)}$ of a connected $1$-polymer $X_{\d}$ as in the paragraph
preceding Lemma~2.1. The $1$-polymers will be
combined into larger connected $L-$polymers which by definition are
connected unions of $L$-blocks, ( for the relevant definitions intervening here
and in the following see the paragraph on {\it $L$-polymers and $L$-closures}
before Lemma~2.1). The combination is performed in
such a way that the new polymers are associated to independent
functionals of $\xi$. This is the lattice adaptation of Section~3.1 of
[BMS].
\vskip0.2cm
\\Define the polymer activity P, supported on unit blocks, by:
$$P(\Delta_{\d} ,\xi ,\Phi) =e^{-V(\Delta_{\d} ,\xi + \Phi)} - e^{-\tilde{V}
(\Delta_{\d},\Phi )} \Eq(3.5.v)$$
\\with $\tilde{V}$ , to be chosen. $\tilde{V}(\Delta_{\d},\Phi )$ is required
to satisfy $\tilde{V}(\Delta_{\d},0)=0$.
{\it In the following $V,K$ has field
argument $\xi + \Phi $ whereas $\tilde{V}$ depends only on $\Phi$ }.
The dependence of P on $\xi ,\Phi$ is as defined above.
\\$X_{\d}^{c}=\L_{\d}\setminus \cup_{j=1}^{N}X_{\d,j}$ is a
union of disjoint $1$-blocks $\D_{\d}$. Therefore
$$e^{-V(X_{\d}^{c})}= {\prod}_{{\Delta_{\d}}\subset
{X}_{\d}^{c}}[e^{-\tilde{V}(\Delta_{\d})} + P(\Delta_{\d})] $$
\\Expand the product and insert the expansion into the integrand of
in \equ(3.501) which gives
$${\rm integrand} =\sum_{N} {1\over N!}
{\sum}_{(X_{\d,j}),(\Delta_{\d,i})}
e^{-\tilde{V}(X_{\d,0})}
{\prod}_{j=1}^{N} {K}(X_{\d,j}){\prod}_{i=1}^{M} {P}(\Delta_{\d,i})
\Eq(3.4.v)$$
\\where $X_{\d,0} = \Lambda_{\d} \setminus (\cup X_{\d,j}) \cup (\cup
\Delta_{\d,i})$. Let $Y_{\d}$ be the $L-$closure of $(\cup X_{\d,j})
\cup (\cup \Delta_{\d,i})$ and let $Y_{\d,1},\dots , Y_{\d,P}$ be the
connected components
of $Y_{\d}$. These are $L-$ polymers. Let $f$ be the function that maps
$\pi: = (X_{\d,j}),(\Delta_{\d,i})$ into $\{Y_{\d,1},\dots ,Y_{\d,P} \}$.
Now we
perform the sum over $(X_{\d,j}),(\Delta_{\d,i})$ in \equ(3.4.v) by summing
over $\pi \in f^{-1} (\{Y_{\d,1},\dots ,Y_{\d,P} \})$ and then
$\{Y_{\d,1},\dots ,Y_{\d,P} \}$. The result is:
$${\rm integrand} =\sum_{N} {1\over N!}
{\sum}_{(Y_{\d,j})}
e^{-\tilde{V}(Y_{\d}^{c})}
{\prod}_{j=1}^{N} {\cal B}K(Y_{\d,j})
\Eq(3.12.v)$$
\\where the sum is over strictly disjoint connected $L$ polymers and
$$({\cal B}K)(Y_{\d}) = \sum_{N+M\ge 1}{1\over N !M!}
{\sum}_{(X_{\d,j}),(\Delta_{\d,i})\rightarrow \{Y \}}
e^{-\tilde{V}(X_{\d,0})}
{\prod}_{j=1}^{N}K(X_{\d,j})
{\prod}_{i=1}^{M} {P}(\Delta_{\d,i}) \Eq(3.13)$$
\\where $X_{\d,0}=Y\setminus (\cup X_{\d,j})\cup (\cup\Delta_{\d,i})$
and the $\rightarrow $ is the map $f$. In other words the sum in \equ(3.13)
is over distinct $\Delta_{\d,i}$ and disjoint $1$-polymers $ X_{\d,j}$
such that their $L$-closure is the connected $L$-polymer $Y_{\d}$.
\\We now perform the fluctuation integration of \equ(3.12.v) over $\xi$
followed by rescaling.
Since $\tilde{V}(Y_{\d}^{c})$ is independent of $\xi$ the $\xi$ integration
factors through and acts on the product of polymer activities
$\prod_{j}({\cal B}K)(Y_{\d,j})$. A polymer activity $({\cal B}K)(Y_{\d,j})$
belongs to $\Omega^{0}({\tilde Y}^{(1)}_{\d,j})$ where
${\tilde Y}^{(1)}_{\d,j}$ is $Y_{\d,j}$ to which a $1$-collar has been
attached (see \equ(1.22c), \equ(1.22d)). The $Y_{\d,j}$ are strictly disjoint
connected $L$-polymers and thus necessarily separated from each other by a
distance $\ge L$. The collar attached $L$-polymers ${\tilde Y}^{(1)}_{\d,j}$
are therefore separated from each other by a distance $\ge L-2$.
The fluctation covariance $\G$ has finite range $L/2$ and for $L$ sufficiently
large $L-2\ge L/2$. Therefore the fluctuation integration over the product of
polymer activities factorizes. We now follow this up
by applying the rescaling operator to both sides. This has the effect of
bringing us back to $1$-polymers but on the scale $L^{-1}\d$.
Therefore we obtain
$$S_{L}\mu_{\G}*{\cal Z}(\L_{\d},\Phi)=
\sum_{N} {1\over N!}
{\sum}_{(X_{L^{-1}\d,j})}
e^{-\tilde{V_{L}}(X_{L^{-1}\d}^{c},\Phi)}
{\prod}_{j=1}^{N} \int d\mu_{\G}(\xi){\cal B}K(LX_{\d,j},S_{L}\Phi,\xi)
\Eq(3.14)$$
\\where $X_{L^{-1}\d,j}=X_{j}\cap (L^{-1}\d\math{Z})^{3}$ ( as well as
$X_{\d,j}=X_{j}\cap (\d\math{Z})^{3}$ ) are disjoint $1$-polymers,
$X_{L^{-1}\d}^{c}=L^{-1}\L_{L^{-1}\d}\setminus \cup_{j}X_{L^{-1}\d,j}$.
and $\tilde{V_{L}}(\D_{L^{-1}\d})= S_{L}{\tilde V}(L\D_{\d})$.
This gives the fluctuation map \equ(3.02) : $V\rightarrow {\tilde V}_{L}$,
$K\rightarrow {\cal F}(K)$ with ${\cal B}K$ defined as above.
At the same time we have
shown that the polymer representaion is stable with respect to the RG
transformation.
\\Consider
$${\cal F}( K)(X_{L^{-1}\d},\Phi)
=\int d\m_{\G}(\xi){\cal B}K(LX_{\d},\xi, S_{L}\Phi)$$
\\By construction ${\cal B}K$ is supersymmetric. Therefore since the
supersymmetry operator commutes with the measure ${\cal F}( K)$ is also
supersymmetric. Now since $P(\Delta_{\d} ,\xi ,0)$ and $K(X_{\d},\xi)$
vanish for $\xi=0$ ( the latter by hypothesis, see \equ(2.111)) it follows that
${\cal B}K(LX_{\d},\xi, 0)$ also vanishes for $\xi=0$. Therefore by Lemma~1.1
$${\cal F}( K)(X_{L^{-1}\d},0)
=\int d\m_{\G}(\xi){\cal B}K(LX_{\d},\xi, 0)={\cal B}K(LX_{\d},0,0)=0
\Eq(3.141) $$
\\Thus the condition \equ(2.111) is satisfied by the new polymer activities.
This implies in particular that no field independent relevant parts are
generated by the fluctuation integration as a consequence of supersymmetry.
\vskip.3truecm
\\{\it 3.2 Extraction }
\vskip0.2cm
\\Let $\d'=L^{-1}\d$ and let $\L'=L^{-1}\L_{\d'}$.
The fluctuation map gave us ${\tilde V}_{L}, {\cal F}(K)$ as the coordinates
of the evolved density ${\cal Z}(\L')$. We want to change the
local potential ${\tilde V}_{L}$ and the polymer activity ${\cal F}(K)$
simultaneously such that ${\cal Z}(\L')$ remains invariant.
To this end let $P (\Phi (x))$ be a {\it local} polynomial, which means
that it is a polynomial in $\Phi (x)$ for $x\in \L'$.
Furthemore we require that $P(0)=0$, i.e. $P$ has no field independent part.
\\Given $\D_{\d'}$ a unit block in $\L'$
we consider a change in ${\tilde V}_{L}(\D_{\d'})$ of the form
$${\tilde V}_{L}(F)(\D_{\d'})= \sum_{P}\int_{\D_{\d'}} dx \ \alpha_{P} (x)
P(\Phi (x)) \Eq(33.2d) $$
\\where the sum ranges over finitely many local polynomials and,
for each such $P$, $\alpha _{P}(x)$ has the form
$$ \alpha _{P}(x) = \sum_{X_{\d'}\supset x}\alpha _{P}(X_{\d'},x)
\Eq(33.2cc) $$
\\such that $\alpha _{P}(X_{\d'},x)=0$ if $x\notin X_{\d'}$,
$\alpha_{P} (X_{\d'},x)=0$ if $X_{\d'} \not \subset \Lambda' $ and
$\alpha_{P} (X_{\d'},x)=0$ if $X_{\d'}$ is not a small set (see definition
after \equ(2.4reg)).
The corresponding change in ${\cal F}(K)$ is given in terms of the
{\it relevant parts}
$$ F(X_{\d'},\Phi)= \sum_{P}\int_{X_{\d'}} dx \ \alpha _{P} (X_{\d'},x)
P(\Phi (x)), \quad
F(X_{\d'},\D_{\d'})=
\sum_{P} \int_{\D_{\d'}}dx \ \alpha _{P} (X_{\d'},x)
P(\Phi (x)) \Eq(33.2aa) $$
\\Note that $F(X_{\d'},0)=0$.
\vskip0.2cm
\\{\it Extraction Map: }
\vskip0.2cm
\\{\it Theorem 3.1 ( after Brydges,Dimock and Hurd, [BDH-est])} :
Given $F,\> {\tilde V}_{L}(F)$ as above
there exists a polymer activity which is a non-linear
functional ${\cal E}({\cal F}(K),F)$ of ${\cal F}(K),\> F$ such that
$$V_{L}\rightarrow V'(F)={\tilde V}_{L}-{\tilde V}_{L}(F),
\quad \quad {\cal F}(K)\rightarrow K'={\cal E}({\cal F}(K),F) \Eq(33.17b)$$
\\preserves the polymer representation for the density ${\cal Z}(\L')$
with new coordinates coordinates $(V',\> K')$ satisfing
$V'(F)(\D_{\d'},0)=K'(X_{\d'},0)=0$. Let
${\cal E}_{1}$ denote the linearization of ${\cal E}$. Then the linearization
of the extraction map is given by
$${\cal E}_{1}({\cal F}(K),F) = {\cal F}(K)-Fe^{-{\tilde V}_{L}},
\quad \quad V'(F) ={\tilde V}_{L}- {\tilde V}_{L}(F)
\Eq(33.17c)$$
\\Assume ${\tilde V}_{L}$ to be stable with respect to perturbation $F$:
There are positive numbers $f(X)$ such that
$$\Vert e^{-{\tilde V}_{L}(\D_{\d'})-
\sum_{X_{\d'}\supset \D_{\d'}}z(X)F(X_{\d'},\D_{\d'})}
\Vert_{{\bf h},G_{\k}} \le 2 \Eq(33.17d) $$
\\for all complex numbers $z(X_{\d'})$ with $|z(X_{\d'})|f(X_{\d'})\le 2$.
Then ${\cal E}({\cal F}(K),F)$ is norm analytic and satisfies the bounds
$$\Vert {\cal E}({\cal F}(K),F)\Vert_{{\bf h},G_{\k},\AA,\d'}
\le O(1)(\Vert {\cal F}(K) \Vert_{{\bf h},G_{\k},\AA_{1},\d'}
+\Vert f\Vert_{\AA_{3},\d'} \Eq(33.17e) $$
$$\vert {\cal E}({\cal F}(K),F)\vert_{{\bf h},\AA,\d'}
\le O(1)(\Vert {\cal F}(K) \Vert_{{\bf h},\AA_{1},\d'}
+\Vert f\Vert_{\AA_{3},\d'} \Eq(33.17f) $$
\\ {\it Proof} :
This is a restatement of Theorem 5 in Sec. 4.2 of [BDH-est] with the
substitution $(\tilde V_{L},{\cal F}(K))$ for $(V,K)$, adapted to the
lattice. The proof of Theorem 5 exploited Lemmas~10, 11, 12, 13 the last of
them
providing the extraction formula in equation (121), page 781 of [BDH-est].
In [BDH-est] the continuum unit blocks are open. Our unit lattice
blocks are lattice restrictions of continuum half open cubes. The
complement of a union of ( strictly) disjoint connected lattice polymers in
$\L'$ is given by a disjoint union of lattice unit blocks. Overlap
connectedness is replaced by connectedness. With this in mind
the proofs of Lemmas~10, 11, 12, 13 go through intact on the lattice providing
the extraction map above.
The estimates in Theorem 5 on the norms of
${\cal E}(K,F)$ together with norm analyticity remain valid on the
lattice. \bull.
\vskip0.2cm
\\{\it Remark} : The stability property \equ(33.17d) is proved in Section 5
once we have chosen $\tilde V$ appropriately. The estimate \equ(33.17e) on the
extraction operator $\cal E$ plays an essential role and is exploited in
Section 5.
\vskip0.2cm
\\{\it Formal infinite volume limit}:
We reestablish the notations leading to \equ(1.48a). Choose $\d=\d_{n}$,
$\G=\G_{n}$, $\d'=L^{-1}\d=\d_{n+1}$. $\L_{\d}= \L_{N-n,n},\>
\L_{\d'}= \L_{N-n-1,n+1}$ and ${\cal F}= {\cal F}_{n+1}$ in \equ(3.02).
The RG transformation $T_{N-n-1,n+1}$ of \equ(1.48a)
induces the RG map $f_{N-n-1,n+1}(V,K)$ of \equ(3.04) for the
coordinates of the density ${\cal Z}_{n-1}(\L_{N-n,n})$
in the polymer representation.
$\alpha _{P}(X_{\d_{n+1}},x)$ in \equ({33.2cc}) is chosen later in
Section 4. This choice will be local, in the sense that it is determined
by ${\tilde V}_{L}(\D_{\d_{n+1}}), \ \Delta_{\d_{n+1}} \subset X_{\d_{n+1}}$
and by
${\cal F}_{n+1}(K) (X_{\d_{n+1}})$. Lemma~13 and
equation (112) of [BDH-est] imply that
${\cal E}({\cal F}_{n+1}(K),F))(X_{\d_{n+1}})$
also is local: it is determined by ${\cal F}_{n+1}(K) (Y_{\d_{n+1}}),
\ Y_{\d_{n+1}} \subset X_{\d_{n+1}}$ and ${\tilde V}_{L}(\D_{\d_{n+1}}),
\ \D_{\d_{n+1}} \subset \tilde{X_{\d_{n+1}}}$, where $\tilde{X_{\d_{n+1}}}$
is a neighbourhood of
$X_{\d_{n+1}}$, namely the union of $X_{\d_{n+1}}$ with all small sets that
intersect $X_{\d_{n}}$. Therefore the $K$ component of the map $f_{N-n-1,n+1}$
representing the action of the $n+1$th step of RG, namely
$f_{N-n-1,n+1,K}(K,V)(X_{\d_{n+1}},\Phi )$ is
independent of $N$ for all $N$ large enough so that $\L_{N-n-1,n+1}$
contains $\tilde{X_{\d_{n+1}}}$. Thus $\lim_{N\rightarrow
\infty}f_{N-n-1,n+1,K}(K,V)(X_{\d_{n+1}},\Phi )$ exists pointwise in
$X_{\d_{n+1}}$.
In this paper we are studying the action of this pointwise infinite volume
limit called the {\it formal infinite volume limit}.
\vskip.3truecm \\{\it 3.3 Appendix}:
\\We introduce some notations which will be used later.
We define for $x,y\in (L^{-1}\d\math{Z})^{3}$ and any covariance $u$ on
$(\d\math{Z})^{3}$
$$u_{L}(x-y)= S_{L^{-1}}u(x-y)= L^{2d_{s}}u(L(x-y)) \Eq(3.151)$$
\\Since the fluctuation covariance $\G$ defined on $(\d\math{Z})^{3}$
has finite range $L/2$ we have that $\G_{L}$ defined on $(L^{-1}\d\math{Z})^{3}$
has finite range $1/2$. We recall from section~1.3 that a polymer $X_{\d}$
is defined by $X_{\d}=X\cap (\d\math{Z})^{3}$ where $X$ is a continuum polymer.
We define the rescaling of polymer activities by
$$ S_{L}K(X_{L^{-1}\d},\Phi)=K_{L}(X_{L^{-1}\d},\Phi)
=K(LX_{\d},S_{L}\Phi) \Eq(3.152) $$
\\We write the fluctuation integration of the polymer activity
$K(X_{\d},\Phi,\xi)$ with respect to $\mu_{\G}$ as
$$K^{\sharp}(X_{\d},\Phi) = \int d\mu_{\G}(\xi)
K(X_{\d},\Phi,\xi) \Eq(3.153) $$.
\\We write the fluctuation integration of the polymer activity
$K(X_{L^{-1}\d},\Phi,\xi)$ with respect to $\mu_{\G_{L}}$ as
$$K^{\natural}(X_{L^{-1}\d},\Phi) = \int d\mu_{\G_{L}}(\xi)
K(X_{L^{-1}\d},\Phi,\xi) \Eq(3.154) $$.
\\We define
$$\SS_{L}=S_{L}{\cal B} \Eq(3.155) $$.
\\With these notations it is easy to see that
the fluctuation map can be written as
$$ {\cal F}(K)(X_{L^{-1}\d}) = S_{L}({\cal B}K)^{\sharp}(LX_{\d})
=(\SS_{L}K)^{\natural}K(X_{L^{-1}\d}) \Eq(3.15)$$
\vskip.5truecm
\\{\bf 4. THE RENORMALIZATION GROUP MAP APPLIED }
\numsec=4\numfor=1
\vskip.5truecm
\\In this section we specify the RG map of Section~3 by making choices for
the local potential $\tilde{V}$, and relevant parts $ F$. $\tilde{V}$ is chosen
via first order perturbation theory. $F$ is chosen so as to remove the
expanding part of the fluctuation map. This is the extraction step. This will
be done in second order perturbation theory as well as in the error term.
We will follow closely the strategy in Section~4 of [BMS]. We will use the
notations established in the Appendix to section~3.3, \equ(3.151)-\equ(3.15).
We take $\d=\d_{n}$, $\G=\G_n$. Recall that, see \equ(1.41a),
$\a= {3+\e\over 2}$ where we take $0<\e<1$. The field scaling dimension
is $d_s={3-\e\over 4}$, see \equ(1.41),\equ(1.42).
\vglue0.2cm
\\We assume that starting
from the unit lattice where only the local potential \equ(0.21) is present
$n$ steps of the renormalization group map has been carried out. This produces
a new local potential
$$V_{n}(\D_{\d_n},\Phi)=V(\D_{\d_n},\Phi,C_{n},g_{n},\m_{n})=
g_{n}\int_{\D_{\d_n}} dx:(\Phi\bar\Phi)^2(x):_{C_{n}}+
\m_{n}\int_{\D_{\d_n}} dx(\Phi\bar\Phi)(x)\Eq(4.1)$$
\\together with a polymer activity $K_{n}$ supported on connected polymers in
$(\d_{n}\math{Z})^3$. Note that $(\Phi\bar\Phi)(x)=:\Phi\bar\Phi:_{C_{n}}(x):$
by virtue of \equ(0.19).
We write $K_{n}$ in the form
$$K_{n}=Q_{n}e^{-V_{n}}+R_{n} \Eq(4.12.5)$$
\\where $Q_{n}$ is a polymer activity which is given by
second order perturbation theory in $g$ assuming that $\m$ is $O(g^2)$.
$Q_{n}$ is specified below. $R_{n}$ is the remainder which is formally of
$O(g^3)$. $Q_{n},R_{n}$ vanish when $\Phi=0$ by hypothesis. The RG map will
preserve this property.
In order to carry through the next step of the RG map as described
in Section~3 we must also specify $ \tilde V(\D_{\d_n},\Phi)$. We define
$$\tilde V_n(\D_{\d_n},\Phi)=V(\D_{\d_n},\Phi,C_{n+1,L^{-1}},g_{n},\m_{n})=
g_{n}\int_{\D_{\d_n}} dx:(\Phi\bar\Phi)^2(x):_{C_{n+1,L^{-1}}}+
\m_{n}\int_{\D_{\d_n}} d^3x(\Phi\bar\Phi)(x)\Eq(4.1.1)$$
\\where we have used the notation $C_{n+1,L^{-1}}=S_{L}C_{n+1}$.
Here and in what
follows we adopt the notations introduced in the Appendix of Section~3.3. Thus
$\sharp$ denotes fluctuation integration with
respect to the measure $d\m_{\G_n}(\xi)$ and $\natural$ denotes
fluctuation integration with
respect to the measure $d\m_{\G_{n,L}}(\xi)$, with $\G_{n,L}=S_{L^{-1}}\G_n$.
We recall (see section 3.1) that when we perform the
fluctuation integration the fluctuation field $\xi$ enters $V$ through
$V(\D_{\d_n},\Phi + \xi)$ but $\tilde V$ will remain independent of $\xi$.
\\We now define $Q_n$: $Q_n$ is supported on connected polymers
$X_{\d_{n}} $ such that
$|X_{\d_{n}} |\le 2$. We assume it can be written in the form
$$Q_{n}(X_{\d_n},\Phi)=Q(X_{\d_n},\Phi;C_{n},{\bf w}_{n},g_n) =
g_{n}^{2}\sum_{j=1}^{3}Q^{(j,j)}(\hat X_{\d_n},\Phi;C_n,w_{n}^{(4-j)})
\Eq(4.19.1)$$
\\where
${\bf w}_{n}=(w_{n}^{(1)},w_{n}^{(2)},w_{n}^{(3)})$ is a triple of
integral kernels to be obtained inductively and
$$\hat X_{\d_n}=\cases{\D_{\d_n}\times\D_{\d_n}&if $X_{\d_n}=\D_{\d_n}$\cr
(\D_{\d_n,1}\times\D_{\d_n,2})\cup(\D_{\d_n,2}\times\D_{\d_n,1})
&if $X_{\d_n}=\D_{\d_n,1}\cup\D_{\d_n,2}$\cr
0&otherwise}\Eq(4.17)
$$
$$\eqalign{Q^{(1,1)}(\hat X_{\d_n},\Phi;C_n,w_{n}^{(3)})&=-2
\int_{\hat X_{\d_n}}dxdy(\Phi(x)-\Phi(y))(\bar\Phi(x)-\bar\Phi(y))
w_{n}^{(3)}(x-y)\cr
Q^{(2,2)}(\hat X_{\d_n},\Phi;C_n,w_{n}^{(2)})&=
-\int_{\hat
X_{\d_n}}dxdy\left[:(\Phi(x)-\Phi(y))(\bar\Phi(x)-\bar\Phi(y))
(\Phi(x)+\Phi(y))(\bar\Phi(x)+\bar\Phi(y)):_{C_n}
+\right.\cr
&\left.+3:[(\Phi\bar\Phi)(x)-(\Phi\bar\Phi)(y)]^2:_{C_n}
\right]w_{n}^{(2)}(x-y)\cr
Q^{(3,3)}(\hat X_{\d_n},\Phi;C_n,w_{n}^{(1)})&=4
\int_{\hat X_{\d_n}}dxdy\
:\Phi(x)\bar\Phi(x)\Phi(x)\bar\Phi(y)\Phi(y)\bar\Phi(y):_{C_n}
w_{n}^{(1)}(x-y)\cr}\Eq(4.18)$$
\\Note that in the expression for $Q^{(1,1)}$ is equal to its $C_n$ Wick
ordered form because of \equ(0.19).
\\Next we define the second order approximation to the RG map.
Let $p_n$ be the activity supported on unit blocks
defined by
$$p_n(\D_{\d_n},\xi ,\Phi)=V_n(\D_{\d_n},\xi + \Phi)-\tilde V_n
(\D_{\d_n}, \Phi )
= p_{n,g} + p_{n,\mu}
\Eq(4.8a)$$
\\where
$$p_{n,g}=g\int_{\D_{\d_n}} dx\Bigl(:(\xi\bar\xi)^2:_{\G_n}(x)+
2\sum_\a\bigl[\Phi_\a(x):\bar\x_\a(\x\bar\x):_{\G_ n }(x) +
:(\x\bar\x)\x_\a:_{\G_n}(x)\Phi_a(x)\bigr] + $$
$$+2(\Phi\bar\Phi)(x)(\x\bar\x)(x)+
(\Phi\bar\x)^2(x)+
(\x\bar\Phi)^2(x)+ 2\sum_{\a,\b}:(\x_\a\bar\x_\b):_{\G_n} (x)
:(\bar\Phi_\a\Phi_\b)(x):_{C_{n+1,L^{-1}}}+ $$
$$ +2\sum_\a\bigl[\x_\a(x):\bar\Phi_\a(\Phi\bar\Phi):_{C_{n+1,L^{-1}}}(x)+
:(\Phi\bar\Phi)\Phi_\a:_{C_{n+1,L^{-1}}}(x)\bar\x_\a(x)\bigr]
\Bigr)
$$
$$p_{n,\mu}=\m\int_{\D_{\d_n}} dx\left((\x\bar\Phi)(x)+
(\Phi\bar\x)(x)+(\x\bar\x)(x)\right)
\Eq(4.8b)$$
\\In \equ(4.8b) we have used a component notation. Thus $\Phi_1=\f,\>
\Phi_2=\psi$. Similarly for
the fluctuation field $\xi$, $\xi_1=\zeta,\> \xi_2=\eta$.
$\zeta$ is bosonic (degree $0$)and $\eta$ fermionic (degree $1$). In deriving
\equ(4.8b) from \equ(4.8a) have used $C_n=\G_n + C_{n+1,L^{-1}}$ ( see
\equ(0.28)), the independence of $\Phi,\> \xi$ in the sense that their
components are independent and distributed
with covariances $ C_{n+1,L^{-1}},\> \G_n$ respectively. The unordered objects
in \equ(4.8b) are equal to their Wick ordered form.
\vglue0.1cm
\\We will effectuate the RG map of Section~3 following closely the strategy
in [BMS].
Namely, we insert a complex parameter $\lambda$ into our previous definitions
in
such a way that (i) at $\lambda =1$ our $\lambda$ dependent objects
correspond with the previous definitions. (ii) The expansion through
order $\lambda^{2}$ is second order perturbation theory in $g_n$
counting $\mu_n =O (g_n^{2})$. (iii) Powers of $\lambda$ are determined so
as to correspond with leading powers of $g_n$ buried inside polymer
activities. (iv) All functions will turn out to be norm analytic in
$\lambda$ and this will enable us in section~5 to profit from Cauchy estimates.
\\We define
$$
P_n (\lambda) = e^{-\tilde{V_n}} \big(
-\lambda p_{n,g}-\lambda^{2}p_{n,\mu}+{1\over 2}\lambda^{2}p_{n,g}^{2}
\big) + \lambda^{3}r_{n,1}
\Eq(4.9)$$
\\where $r_{n,1}$ is defined by the condition $P_n (\lambda = 1) = P_n =
e^{-V_n} - e^{-\tilde{V_n}}$. Similarly, we define
$$
K_n (\lambda) =
\lambda^{2}e^{-\tilde{V_n}}Q_n +\lambda^{3} \big(
[e^{-V_n} - e^{-\tilde{V_n}}]Q_n + R_n
\big)
\Eq(4.10)$$
\\which, for $\lambda =1$ coincides with $K_n = e^{-V_n}Q_n+R_n$, where $Q_n$
is an explicit polymer activity called the second order avtivity (second
order in the coupling constant) motivated by perturbation theory.
$R_n$ is the remainder. Corresponding to \equ({3.13}) we define
$${\cal B}(\lambda ,K_n) (Y_{\d_n}) = \sum_{N+M\ge 1}{1\over N !M!}
{\sum}_{(X_{\d_n,j}),(\Delta_{\d_n,i})\rightarrow \{Y_{\d_n} \}}
e^{-\tilde{V_n}(X_{\d_n,0})}
{\prod}_{j=1}^{N} K_n (\lambda ,X_{\d_n,j})
{\prod}_{i=1}^{M}P_n (\lambda ,\Delta_{\d_n,i}) \Eq(4.13)$$
where $X_{\d_n,0}=Y_{\d_n}\setminus (\cup X_{\d_n,j})
\cup (\cup\Delta_{\d_n,i})$.
Let $\SS(\lambda,K_n) = S_{L}{\cal B} (\lambda,K_n)$,
where $S_{L}$ is the rescaling defined in the last section.
\\The RG map ( see section~3)
for $K_n$ with parameter $\lambda$ is $K_n\mapsto f_{n+1,K}(\l,K_n)=
{\cal E} (\SS (\lambda,K_n)^{\natural},F_{n} (\lambda ))$, where the
superscript
$\natural$ denotes integration over the fluctuation field $\xi=(\z,\eta)$
with the measure $d\m_{\G_{n,L}}$ and $\G_{n,L}$ is the rescaled covariance
$S_{L^{-1}}\G_n$ as in the Appendix to Section~3. The relevant part
$F_{n} (\lambda ))$ is defined on polymers in $(\d_{n+1}\math{Z})^{3}$
and will be written as
$$F_{n} (\lambda) = \lambda^{2}F_{Q_n}+ \lambda^{3}F_{R_n} \Eq(4.12)$$
\\and $F_{n} (\lambda)= F_{n}$, when $\lambda =1$.
\vglue0.2cm
\\{\it Perturbative contribution to $f_{n+1}$. }
\\Given a function $f (\lambda)$ let
$$T_{\lambda}f = f (0)+f' (0)+{1\over2}f'' (0) \Eq(4.13.5)$$
\\be the Taylor expansion to second order evaluated at $\lambda
=1$. Then the second order approximation to the RG map is
$f_{n+1}^{(\le 2)} =
(f^{(\le 2)}_{n+1,K},f^{(\le 2)}_{n+1,V})$ with
$$
f^{(\le 2)}_{n+1,K} (K_n,V_n)
= T_{\lambda} {\cal E} (\SS(\lambda,K_n)^{\natural},F_{n}(\lambda ))
= {\cal E}_{1} ( T_{\lambda} \SS(\lambda,K_n)^{\natural},F_{Q_n}), \>
\qquad f^{(\le 2)}_{n+1,V}(K_n,V_n) = V_{n+1}^{(\le 2)}\Eq(4.14.1)$$
\\where
$$V_{n+1}^{(\le 2)}= {\tilde V}_{n,L}-{\tilde V}_{n,L}(F_{Q_n})$$
\\Note also that only the linearized ${\cal E}_{1}$ intervenes, because
it will turn out that the nonlinear part of extraction generates terms
only at order $\lambda ^{3}$ or higher.
\vskip.5truecm
\\{\it Proposition 4.1:
\\There is a choice of $F_{Q}$ such that
the form of $Q$ remains invariant under the RG evolution at second
order. In more detail, $f_{n+1}^{(\le 2)}(V_n,Q_n e^{-V_n}) =
(V_{n+1}^{(\le 2)},Q_{n+1}^{(\le 2)} e^{-\tilde V_{n,L}})$ where the
parameters in
$$V_{n+1}^{(\le 2)}(\D_{\d_{n+1}})
=V(\D_{\d_{n+1}},C_{n+1},g'_{n+1,(\le 2)},\m'_{n+1,(\le 2)})$$
\\evolved according to
$$g_{n+1,(\le 2)}=L^\e g_{n}(1-L^\e a_{n}g_{n})\qquad\qquad
\m'_{n+1,(\le 2)}=L^{3+\e\over 2}\m_{n}-L^{2\e}b_{n}g_{n}^{2}
\Eq(4.47)$$
\\The parameters in $Q_{n+1}^{(\le 2)}=Q(C_{n+1},{\bf w_{n+1}},g_{n,L})$, where
$g_{n,L}=L^{\e}g_{n}$, evolved
according to
$${\bf w_{n+1}}={\bf v_{n+1}}+{\bf w}_{n,L}\qquad v_{n+1}^{(1)}=\G_{n,L}
\qquad v_{n+1}^{(p)}=(C_{n,L})^p-(C_{n+1})^p\qquad p=2,3\Eq(4.54)
$$
The constants $a_n, b_n$ are given by
$$a_n = 4\int_{(\d_{n+1}\bf Z)^3} dy\> v_{n+1}^{(2)}(y), \quad\quad
b_n = 2\int_{(\d_{n+1}\bf Z)^3} dy \> v_{n+1}^{(3)}(y) \Eq(4.54.1) $$
}
\\{\it Proof:} We define a polymer activity $\hat Q_{n,L}$ supported on
connected polymers $X_{\d_{n+1}}$ with $|X_{\d_{n+1}}|\le 2$ as follows:
if $|X_{\d_{n+1}}|=1$, say $X_{\d_{n+1}}=\D_{\d_{n+1}}$, then
$$\hat Q_{n,L}(\D_{\d_{n+1}},\xi,\Phi)=
{1\over 2}(p_{n,L }(\D_{\d_{n+1}},\xi,\Phi))^2 $$
\\If $|X_{\d_{n+1}}|=2$ then
$$\hat Q_{n,L}(X_{\d_{n+1}},\xi,\Phi)=
{1\over 2}
\sum_{\D_{n+1,1},\D_{n+1,2}\atop\D_{n+1,1}\cup\D_{n+1,2}=X_{\d_{n+1}}}
p_{n,L,g}(\D_{n+1,1},\xi,\Phi)p_{n,L,g}(\D_{n+1,2},\xi,\Phi) \Eq(4.9.1) $$
\\where $p_{n,L,g}$ is defined by replacing in \equ(4.8a) and \equ(4.8b)
$(g_n,\m_n,\G_n,C_{n+1,L^{-1}})$ by $(g_{n,L},\m_{n,L},\G_{n,L},C_{n+1})$
with $g_{n,L}=L^{\e}g_n$ and $\m_{n,L}=L^{3+\e\over 2}\m_n$.
\\It is easy to check that
$$T_{\lambda }\SS (K_n,\lambda ) = - p_{n,L}e^{-\tilde{V}_{n,L}} + (
e^{-{\tilde V}_{n,L}}\hat Q_{n,L}
+e^{-{\tilde V}_{n,L}}Q_{n,L} ) \Eq(4.21.1)$$
\\where
$$Q_{n,L}(X_{\d_{n+1}},\xi+\Phi)= Q(X_{\d_{n+1}},\xi+\Phi, C_{n,L},
{\bf w_{n,L}},g_{n,L})$$
\\Using $C_{n,L}=\G_{n,L} + C_{n+1}$ and remembering that $\tilde V$
depends only on $\Phi$ we get
$$(e^{-\tilde{V}_{n,L}}Q_{n,L})^{\natural}=e^{-{\tilde V}_{n,L}}
Q(X_{\d_{n+1}},\Phi, C_{n+1},{\bf w_{n,L}},g_{n,L})$$
\\Therefore
$$T_{\lambda }\SS (K_n,\lambda )^{\natural}(X_{\d_{n+1}},\Phi)=
e^{-{\tilde V}_{n,L}}\Bigl(
Q(X_{\d_{n+1}},\Phi, C_{n+1},{\bf w_{n,L}},g_{n,L}) +
{\tilde Q}_n(X_{\d_{n+1}},\Phi,{\bf v}_{n+1}, C_{n+1},g_{n,L} )\Bigr)
\Eq(4.30) $$
\\where ${\tilde Q}_n= \hat Q_{n,L}^{\natural}$ is given after a
straightforward but lengthy computation by
$${\tilde Q}_n(X_{\d_{n+1}},\Phi,{\bf v}_{n+1}, C_{n+1},g_{n,L} )
= g_{n,L}^{2}\sum_{j=1}^{3}
\tilde Q^{(j,j)}(\hat X_{\d_{n+1}} ,\Phi;v^{(4-j)}_{n+1},C_{n+1})
\Eq(4.14.2)$$
\\where
$$\eqalign{\tilde Q^{(1,1)}(\hat X_{\d_{n+1}},\Phi;u)=&2
\int_{\hat X_{\d_{n+1}}}dxdy[\Phi(x)\bar\Phi(y)+\Phi(y)\bar\Phi(x)]
u(x-y)\cr
\tilde Q^{(2,2)}(\hat X_{\d_{n+1}},\Phi;u)=&
\int_{\hat
X_{\d_{n+1}}}dxdy\
\Bigl \{:[\Phi(x)\bar\Phi(y)+\Phi(y)\bar\Phi(x) ]^2:_{C_{n+1}}\cr
&+4:(\Phi(x)\bar\Phi(x))(\Phi(y)\bar\Phi(y)):_{C_{n+1}}\Bigr \}
u(x-y)\cr
\tilde Q^{(3,3)}(\hat X_{\d_{n+1}},\Phi;u)=&4
\int_{\hat X_{\d_{n+1}}}dxdy
:\Phi(x)\bar\Phi(x)\Phi(x)\bar\Phi(y)\Phi(y)\bar\Phi(y):_{C_{n+1}}
u(x-y)\cr}\Eq(4.25)$$
\\Define
$$F_{Q_n} =
\tilde Q(C_{n+1},{\bf v}_{n+1},g_{n,L}) - Q(C_{n+1},{\bf v}_{n+1},g_L)
\Eq(4.52) $$
\\evaluated on $X_{\d_{n+1}},\Phi$.
\\Then we have from \equ({4.30}) and \equ({4.52})
$$ {\cal E}_{1} \bigg(T_{\lambda }\SS
(\lambda,K_n)^{\natural},F_n\bigg)
= T_{\lambda }\SS(\lambda,K_n)^{\natural} - F_{Q_n}e^{-\tilde{V}_{n,L}} =
e^{-\tilde{V}_{n,L}}Q(C_{n+1},{\bf w}_{n+1},g_{n,L})
\Eq(4.15)$$
\\which shows that $Q$ is stable under RG evolution and verifies
\equ(4.54). It remains to show that the chosen perturbative relevant part
$F_{Q_n}$ given by \equ(4.52) is of the form \equ(33.2aa) and thus suitable
for extraction.
\\To compute the difference in \equ(4.52) we will make use of the
following {\it localization formulae}
$$\Phi(x)\bar\Phi(y)+\Phi(y)\bar\Phi(x)=\Phi(x)\bar\Phi(x)+\Phi(y)\bar\Phi(y)
-(\Phi(x)-\Phi(y))(\bar\Phi(x)-\bar\Phi(y))\Eq(4.52.1)$$
$$(\Phi(x)\bar\Phi(y)+\Phi(y)\bar\Phi(x))^2+4(\Phi(x)\bar\Phi(x))(\Phi(y)\bar\Phi(y))=
4[(\Phi\bar\Phi)^2(x)+(\Phi\bar\Phi)^2(y)] -$$
$$-(\Phi(x)-\Phi(y))(\bar\Phi(x)-\bar\Phi(y))
(\Phi(x)+\Phi(y))(\bar\Phi(x)+\bar\Phi(y))-3[(\Phi\bar\Phi)(x)-(\Phi\bar\Phi)(y)]^2
\Eq(4.52.2)$$
\\that are immediate to check. We get
$$F_{Q_n}(X_{\d_{n+1}}) =2
\int_{\hat X_{\d_{n+1}}}dxdy[(\Phi\bar\Phi)(x)+
(\Phi\bar\Phi)(y)]
v^{(3)}(x-y)+ $$
$$+4\int_{\hat X_{\d_{n+1}}} dxdy[:(\Phi\bar\Phi)^2:_{C_{n+1}}(x)+
:(\Phi\bar\Phi)^2:_{C_{n+1}}(y)]v^{(2)}(x-y)
\Eq(4.33)$$
\\Note that due to supersymmetry
there is no field independent part in $F_{Q}$.
\\$F_{Q_n}(X_{\d_{n+1}})$ can be written as:
$$F_{Q_n}(X_{\d_{n+1}})=\sum_{\D_{\d_{n+1}}\subset X_{\d_{n+1}}}F_{Q_n}(X_{\d_{n+1}},\D_{\d_{n+1}})\Eq(4.34)$$
\\where
$$
F_{Q_n}(X_{\d_{n+1}},\D_{\d_{n+1}})=4g_L^2F^{(2)}_{Q_n}(X_{\d_{n+1}},\D_{\d_{n+1}})
+2g_L^2F^{(1)}_{Q_n}(X_{\d_{n+1}},\D_{\d_{n+1}})\Eq(4.35)$$
and
$$F^{(m)}_{Q_n}(X_{\d_{n+1}},\D_{\d_{n+1}})=\int_{\D_{\d_{n+1}}} dx:(\Phi\bar\Phi)^m(x):_{C_{n+1}}
f^{(m)}_{Q_n}(x,X_{\d_{n+1}},\D_{\d_{n+1}})
\Eq(4.36)$$
\\with
$$
f^{(m)}_{Q_n}(x,X_{\d_{n+1}},\D_{\d_{n+1}})=\cases{\int_{\D_{\d_{n+1}}} dy v^{(m')}(x-y)&$X_{\d_{n+1}}=\D_{\d_{n+1}}$\cr
\cr
\int_{\D_{\d_{n+1}}'}dy v^{(m')}(x-y)&$X_{\d_{n+1}}=\D_{\d_{n+1}}\cup\D_{\d_{n+1}}'$, connected\cr}\Eq(4.36a)$$
and $m' = 4- m$.
\eject
$$V(F_{Q_n},\D_{\d_{n+1}})=
\sum_{X_{\d_{n+1}}\supset\D_{\d_{n+1}}}F_{Q_n}(X_{\d_{n+1}},\D_{\d_{n+1}})
=4g_L^2\sum_{X_{\d_{n+1}}\supset\D_{\d_{n+1}}}
F^{(2)}_{Q_n}(X_{\d_{n+1}},\D_{\d_{n+1}}) +$$
$$+2g_L^2\sum_{X_{\d_{n+1}}\supset\D_{\d_{n+1}}}F^{(1)}_{Q_n}(X_{\d_{n+1}},\D_{\d_{n+1}})\Eq(4.37)$$
\\In the following we will use the fact that the
$v^{(j)}_{n+1}(x-y),\ 1\le j\le 3$ vanish for $|x-y|\ge 1/2$. This follows
from the fact that $\G_{n,L}(x-y)$ appears as a factor in the expression
\equ(4.54) for $v^{(j)}_{\d_{n+1}}(x-y)$ and $\G_{n,L}$ has range
$1\over 2$. Returning to \equ(4.37) we have
$$\sum_{X_{\d_{n+1}}\supset\D_{\d_{n+1}}}F^{(m)}_{Q_n}(X_{\d_{n+1}},\D_{\d_{n+1}})=
\int_{\D_{\d_{n+1}}} dx
:(\Phi\bar\Phi)^m(x):_{C_{n+1}}
\Big[\int_{\D_{\d_{n+1}}} dy v^{(m')}_{n+1}(x-y) +$$
$$+\sum_{\D_{\d_{n+1}}'\ne\D_{\d_{n+1}}\atop(\D_{\d_{n+1}},\D_{\d_{n+1}}')\ {\rm connected}}
\int_{\D_{\d_{n+1}}'} dy v^{(m')}_{n+1}(x-y)\Big]$$
\\On the r.h.s. use $v^{(m')}_{n+1}(x-y)=0$ for $|x-y|\ge 1/2$
to extend the sum
on $\D_{\d_{n+1}}'$ to {\it all} the $\D_{\d_{n+1}}'\ne\D_{\d_{n+1}}$.
We then get
$$\sum_{X_{\d_{n+1}}\supset\D_{\d_{n+1}}}F^{(m)}_{Q_n}(X_{\d_{n+1}},\D_{\d_{n+1}})=
\int_{\D_{\d_{n+1}}} dx
:(\Phi\bar\Phi)^m(x):_{C_{n+1}} \int dy v^{(m')}(x-y)$$
\\Hence from \equ(4.37) and above we get
$$V(F_{Q_n},\D_{\d_{n+1}})=
a_n g_{n,L}^2\int_{\D_{\d_{n+1}}} dx:(\Phi\bar\Phi)^2(x):_{C_{n+1}}
+b_n g_{n,L}^2\int_{\D_{\d_{n+1}}} dx(\Phi\bar\Phi)(x)\Eq(4.38)$$
\\where
$$a_n=4\int_{(\d_{n+1}\math{Z})^{3}} dy\ v^{(2)}_{n+1}(y)\qquad\qquad
b=2\int_{(\d_{n+1}\math{Z})^{3}} dy\ v^{(3)}_{n+1}(y)\Eq(4.39)$$
\bull
\\{\it Remark} : $a_n$ and $b_N$ are well defined since the
$v^{j}_{n+1}$ have compact support. They are positive and their properties
are discussed in Lemma~5.12 of Section~5.
\vskip.5truecm
\\{\it The exact RG map $f_{n+1}$ for $K_n=Q_ne^{-V_n}+R_n$}.
\vskip.2truecm
$$ K_n\mapsto K_{n+1} = f_{n+1,K} (\lambda,K_n,V_n)\vert_{\lambda =1}
= {\cal E} (\SS(\lambda,K_n)^{\natural},F_n (\lambda ))\vert_{\lambda =1}
\Eq(4.16)$$
\\induces an evolution of the remainder $R_n$ which is studied by Taylor
series around $\lambda =0$ with remainder written using the Cauchy
formula:
$$ f_{n+1,K} (\lambda =1) = \sum_{j=0}^{3} {f_{n+1,K}^{(j)} (0)\over j!} +
{1\over 2\pi i}
\oint_{\gamma} {d\lambda \over \lambda^{4} (\lambda -1)}
f_{n+1,K} (\lambda)
$$
\\The terms $j=0,1,2$ are the second order part $f^{(\le 2)}_{n+1,K}$. In
the $j=3$ term there are no terms mixing $R_n$ with $Q_n,P_n$ because of the
$\lambda^{3}$ in front of $R_n$. Therefore it splits
$
{f_{K}^{(3)} (0)\over 3!} = R_{n+1,1} + R_{n+1,2}
$ into the third order derivative at $R_n=0$, which we write
using the Cauchy formula as
$$
R_{n+1,1} \equiv R_{n+1,\rm main}
= {1\over 2\pi i}
\oint_{\gamma} {d\lambda \over \lambda^{4}}
{\cal E} \bigg(
\SS(\lambda ,Q_n e^{-V_n})^{\natural},F_{Q_n} (\lambda )
\bigg)
\Eq(4.188)$$
\\and terms linear in $R_n$:
$$\eqalign{
&R_{n+1,2} \equiv R_{n+1,\rm linear}
= \big(\SS_1 R_n\big)^{\natural} - F_{R_n}e^{-\tilde V_{L,n}}\cr
&\SS_1 R_n(Z_{\d_{n+1}}) =
\sum_{X_{\d_{n+1}}:L^{-1}{\bar X}_{\d_{n+1}}^{L}=Z_{\d_{n+1}} }
e^{-\tilde V_{n,L}(Z_{\d_{n+1}}\backslash
L^{-1}X_{\d_{n+1}})} R_{n,L}(L^{-1}X_{\d_{n+1}})\cr
} \Eq(4.19)$$
\\The remainder term in the Taylor expansion is
$$
R_{n+1,3} =
{1\over 2\pi i}
\oint_{\gamma} {d\lambda \over \lambda^{4} (\lambda -1)}
{\cal E} (\SS (\lambda,K_n)^{\natural},F_n (\lambda
))
\Eq(4.21)$$
In Proposition~4.1 the coupling constant in $Q_{n+1}^{(\le 2)}$ is not the
same as
the coupling constant in $V_{n+1}^{(\le 2)}$. Furthermore, the coupling
constant in $V_{n+1}^{(\le 2)}$ will further change because of the
contribution from $F_{R}$. To take this into account we introduce
$$\eqalign{
&V_{n+1}(\D_{\d_{n+1}})=V(\D_{\d_{n+1}} ,C_{n+1} ,g_{n+1} ,\m_{n+1})\cr
&g_{n+1} =L^\e g_n(1-L^\e a_n g_n)+\xi_n(u_n)\cr
&\m_{n+1} =L^{3+\e\over 2}\m_n-L^{2\e}b_n g_n^2+\r_n(u_n)\cr}
\Eq(4.47A)$$
\\where $u_n=(g_n,\m_n,R_n)$ and
the remainders $\xi_n(u_n), \r_n(u_n)$ anticipate the effects of a
yet-to-be-specified $F_{R_n}$. Then we set
$$
R_{n+1,4} = e^{-V_{n+1}}Q(C_{n+1},{\bf w}_{n+1},g_{n+1})
- e^{-\tilde{V}_{n,L}}Q(C_{n+1},{\bf w}_{n+1},g_{n,L})
\Eq(4.22)
$$
\\and define
$$\eqalign{ Q_{n+1}&=Q(C_{n+1},{\bf w}_{n+1},g_{n+1})\cr
R_{n+1}&=R_{n+1,{\rm main}}+R_{n+1,{\rm linear}}+R_{n+1,3}+R_{n+1,4} \cr
K_{n+1}&=Q_{n+1} e^{-V_{n+1}}+R_{n+1} \cr }
\Eq(4.221)$$
\\With these definitions we have obtained the RG map
$$ f_{n+1,V}(V_n,K_n)=V_{n+1},\quad f_{n+1,K}(V_n,K_n) = K_{n+1} \Eq(4.222)$$
\vskip.2truecm
\\{\it Definition of $F_{R_n}$}
\vglue.3truecm
\\To complete the RG step we must specify the relevant part $F_{R_n}$
from the remainder $R_n$.
The goal is to choose $F_{R_n}$ so that the map $R_n
\rightarrow R_{n+1,\rm linear}$ will be contractive in the following sense.
$R_n$ is measured in the norm \equ(2.611), and the kernel norm \equ(2.612),
with $\d=\d_{n}$, with a choice of $\bf h$ and $\bf h'$
(to be made in section~5). $R_{n+1}$ is measured in the same norms but on the
lattice scale $\d_{n+1}$. We will say that the map is contractive if the
size of $R_{n+1,\rm linear}$ is less than the size of $R_n$.
\vglue0.1cm
\\$F_{R_n}$ will have the form given in \equ(33.2aa) with $P$ a
supersymmetric polynomial vanishing at $\Phi=0$. The coefficients $\a_P$
will be
identified via {\it normalization conditions} on the small set part of
$R_{n+1,\rm linear}$. This means that certain derivatives with respect to
$\Phi=(\f,\psi)$ vanish when $\Phi=0$. That the map in question is contractive
when $R_{n+1,\rm linear}$ is suitably normalized is proven in Section~5.
\vglue0.2cm
\\For given coefficients ${\tilde \alpha}_{n,P} (X)$,
we define
$$\tilde F_{R_n}(X_{\d_n},\Phi)= \sum_{P} \int_{X_{\d_n}} dx\
{\tilde \alpha}_{n,P} (X_{\d_n})P (\Phi (x),\dpr_{\d_n} \Phi (x))
\Eq(4.57) $$
$$\tilde F_{R_n}(X_{\d_n},\Phi)=0\> : X_{\d_n} {\rm is\ not\ a\ small\ set}
\Eq(4.571) $$
\\$P$ runs over the {\it relevant} monomials which in this
model are
$P = \Phi\bar\Phi,(\Phi\bar\Phi)^{2},\Phi \dpr_{\d_n,\m}\bar\Phi,
\dpr_{\d_n,\m}\Phi \bar\Phi $ with $\mu =1,2,3
$ with the corresponding coefficients
${\tilde\alpha}_{P}(X_{\d_n})={\tilde\alpha}_{n,2,0}(X_{\d_n}),
{\tilde\alpha}_{n,4}(X_{\d_n}),
{\tilde\alpha}_{n,2,\bar 1}(X_{\d_n},\m),$
\\$ {\tilde\alpha}_{n,2,1}(X_{\d_n},\m) $.
Note that $P=1$ is not a relevant monomial in this model: $R_n$ vanishes when
$\Phi=0$ vanishes by hypothesis. Then
$R_n^{\sharp}(X_{\d_n},\Phi)$ vanishes when $\Phi=0$ by supersymmetry,
(Lemma~1.1) so that no subtraction is necessary at $\Phi=0$.
\\Choose the coefficients ${\tilde \alpha}_{n,P}$ so that
$$J_n= R_n^{\sharp}-\tilde F_{R_n}e^{-\tilde{V_n}}
\Eq(4.59) $$
\\is normalized (details are given below). Note that $J(X_{\d_n},0)=0$.
We define the relevant
part, supported on small sets, by
$$F_{R_n}(Z_{\d_{n+1}},\Phi)=
\sum_{X_{\d_{n+1}}: {\rm small\ sets}\atop L^{-1}{\bar X}_{\d_{n+1}}^{L}
=Z_{\d_{n+1}}} \tilde F_{R_n,L}(L^{-1}X_{\d_{n+1}} ,\Phi)
=\sum_{X_{\d_{n}}: {\rm small\ sets}\atop L^{-1}{\bar X}_{\d_{n}}^{L}
=Z_{\d_{n}}}\tilde F_{R_n}(X_{\d_{n}} ,S_L\Phi)
\Eq(4.58)$$
\\$F_{R_n}$ is supported on small sets by construction.
From the definition of $R_{n+1,\rm linear}$ in \equ(4.19) we get
$$ R_{n+1,\rm linear} (Z_{\d_{n+1}})
= \sum_{X_{\d_{n+1}}: {\rm small\ sets}
\atop L^{-1}{\bar X_{\d_{n+1}}}^{L}=Z_{\d_{n+1}}}
e^{-\tilde{V}_{L} (Z_{\d_{n+1}}\setminus L^{-1}X_{\d_{n+1}})}
J_{n,L} (L^{-1}X_{\d_{n+1}}) + $$
$$+\sum_{X_{\d_{n+1}}: {\rm large\ sets}
\atop L^{-1}{\bar X_{\d_{n+1}}}^{L}=Z_{\d_{n+1}}}
e^{-\tilde{V}_{L} (Z_{\d_{n+1}}\setminus L^{-1}X_{\d_{n+1}})}
J_{n,L} (L^{-1}X_{\d_{n+1}}) \Eq(4.59b)$$
\\Therefore the first sum in $R_{\rm linear}$ is also normalized
because normalization as defined below is preserved under multiplication by
smooth functionals of $\Phi$ and rescaling.
\\Substitution of \equ(4.57) in \equ(4.58) shows that $F_{R_n}$ is of
the form required in \equ({33.2aa}). We have
$$F_{R_n}(Z_{\d_{n+1}},\Phi)= \sum_{P}\int_{Z_{\d_{n+1}}} dx\
\alpha_{n,P} (Z_{\d_{n+1}},x)P (\Phi (x),\dpr_{\d_{n+1}}\Phi (x))
\Eq(4.57b)
$$
\\where
$$ \alpha_{n,P}(Z_{\d_{n+1}},x) =
\sum_{X_{\d_{n+1}}\ {\rm small\ set}\atop L^{-1}{\bar X}_{\d_{n+1}}^{L}
=Z_{\d_{n+1}}} {\tilde \alpha}_{n,P}(X_{\d_{n}} )
L^{-[P]+3}1_{L^{-1}X_{\d_{n+1}} } (x) \Eq(4.67a)$$
\\In \equ(4.67a)
$1_{X}$ is the characteristic function of the set $X$. Note that
$X_{\d_{n+1}}$ fixes $X_{\d_{n}}$ by restriction by our construction of
polymers in section~1.3. $[P]$ is the dimension of the monomial
$P$, ($2kd_s$ for
$(\Phi\bar\Phi)^{k}$ and $2d_s+1$ for $\Phi \dpr_{\d_{n+1}}\bar\Phi $).
$\alpha_{n,P}(Z_{\d_{n+1}},x)$ is supported on small sets $Z_{\d_{n+1}}$
and vanishes if $x\notin Z_{\d_{n+1}}$.
\vglue0.1cm
\\We now compute $V_{F_{R}}$ following \equ({33.2d}). Define
$$
\alpha_{n,P} : = \sum_{Z_{\d_{n+1}}\supset x}\alpha_{n,P}(Z_{\d_{n+1}},x)
\Eq(4.57c)$$
\\This is independent of $x$ by translation invariance. In fact
given an $x$ it belongs uniquely to a block $\D_{\d_{n+1}}$, since our blocks
which are restrictions of half open continuum cubes are always disjoint
(see section~1.3). The sum over all polymers containing a block
$\D_{\d_{n+1}}$ is independent of $\D_{\d_{n+1}}$
by translation invariance.
\\From \equ(4.67a) and \equ(4.57c) we get
$$\alpha_{n,P} = L^{-[P]+3} \sum_{X_{\d_{n+1}} \ {\rm small\ set}:
L^{-1}X_{\d_{n+1}}\supset x }
\tilde{\alpha}_{n,P}(X_{\d_{n}} ) \Eq({4.70})$$
\\$\alpha_{n,P}=0$ for $P=\Phi\dpr_{\d_{n+1}}\bar\Phi\> {\rm or}\>
\dpr_{\d_{n+1}}\Phi\bar\Phi$ by reflection invariance of polymer activities.
\\Therefore
$$\eqalign{ \tilde V_{L}(F_{R_n},\D_{\d_{n+1}})
&=\int_{\D_{\d_{n+1}}} dx\bigg\{\alpha_{n,2,0} \Phi\bar\Phi +
\alpha_{n,4,0} (\Phi\bar\Phi)^{2}
\bigg\}\cr
&=\int_{\D_{\d_{n+1}}} dx\bigg\{\r_n(u_n):\Phi\bar\Phi:_{C_{n+1}} +
\xi_n(u_n):(\Phi\bar\Phi)^{2}:_{C_{n+1}}\bigg\}\cr}\Eq(4.71) $$
\\where $u_n=(g_n,\m_n,R_n)$ and
$$
\r_{n}=\alpha_{n,2,0}+ 2C_{n+1}(0)\a_{n,4,0}\qquad\qquad
\xi_{n}=\alpha_{4,0}
\Eq(4.64)$$
\\which are formulas for the error terms in \equ(4.47A).
\vglue0.3cm
\\{\it Normalization conditions}
\vglue0.3cm
\\By an abuse of notation let $1$ denote the constant function in
$C^{2}(X_{\d_n})$
equal to $1$. Similarly let $1^{2p}$ denote the constant
function in $C^{2}(X_{\d_n}^{2p})$ equal to $1$. We will identify the
$C^{2}(X_{\d_n})$ function
$f(x)=x_{\m}$ with $x_{\m}$. Similarly we will identify $C^{2}(X_{\d_n}^{2})$
functions $g_2(x_{1}, x_{2})=x_{1,\m}$, $g_2(x_{1}, x_{2})= x_{2,\m}$ with
$ x_{1,\m},\>x_{2,\m}$ respectively.
\\Suppose the polymer activity $J(X_{\d_n},\Phi)= J(X_{\d_n},\f,\psi)$ is of
degree $0$, gauge invariant and supersymmetric. We have the following
identities:
$$D^{2,0}J(X_{\d_n},0,0;1^{2})=D^{0,2}J(X_{\d_n},0,0,;1,1) \Eq(4.40)$$
$$D^{2,0}J(X_{\d_n},0,0;x_{1,\m})=D^{0,2}J(X_{\d_n},0,0,;x_{\m},1)
\Eq(4.402)$$
$$D^{2,0}J(X_{\d_n},0,0;x_{2,\m})=D^{0,2}J(X_{\d_n},0,0,;1,x_{\m})
\Eq(4.403)$$
$$D^{2,2}J(X_{\d_n},0,0;1,1,1^{2})=2D^{0,4}J(X_{\d_n},0,0;1,1,1,1) \Eq(4.41)$$
$$D^{4,0}J(X_{\d_n},0,0;1^{4})=0 \Eq(4.401)$$
\\where the field derivatives are taken according to \equ(2.2). The
identities \equ(4.40)-\equ(4.41)
follow by expanding $J(X_{\d_n},\Phi)$ in the fields, retaining a degree 4
supersymmetric polynomial in $\Phi$ and $\dpr_{\d_n}\Phi$ which is all that
enters into the computation. Then express it in the Grassmann
representation \equ(2.1).
\equ(4.401) is trivial. Because $J$ is of degree $0$ the only term that
survives for the computation of \equ(4.401) is of the form
$\int_{X_{\d_n}^{4}}dx\>a(x_1,x_2,x_3,x_4)\psi(x_1)\bar\psi(x_2)\psi(x_3)
\bar\psi(x_4)$ where the kernel $a$ is antisymmetric in $x_1,x_3$ and in
$x_2,x_4$. The integral vanishes if we replace the grassmann piece by $1^4$.
Derivatives on the grassmann fields annihilate $1^4$.
\vglue0.2cm
\\We say that a degree $0$, gauge invariant, supersymmetric polymer activity
$J(X_{\d_n},\Phi)= J(X_{\d_n},\f,\psi)$ with $J(X_{\d_n},0)=0$
is {\it normalized} if, for all small sets $X_{\d_n}$,
$$\eqalign{D^{2,0}J(X_{\d_n},0,0;1^{2})=&
D^{0,2}J(X_{\d_n},0,0,;1,1)=0\cr
D^{2,0}J(X_{\d_n},0,0;x_{1,\m})=& D^{2,0}J(X_{\d_n},0,0;x_{2,\m})=0\cr
D^{0,2}J(X_{\d_n},0,0;1,x_\m)=&
D^{0,2}J(X_{\d_n},0,0;x_\m,1)=0\cr
2D^{0,4}J(X_{\d_n},0,0;1,1,1,1)=&
D^{2,2}J(X_{\d_n},0,0;1,1,1^{2})=0\cr}
\Eq(4.42)$$
\vskip.5truecm
\\{\it Determining coefficients from \equ({4.42})}
\vskip.3truecm
\\We will apply the normalization conditions to $J=J_n$
defined in \equ(4.59).
This will determine the dependence of the error terms
$\xi_n,\>\r_n$ on $R_n$. Lemma~5.17 will show that these terms are bounded
by the kernel norm of $R_n$.
\\In doing the following computations
note that $J_n(X_{\d_n},0,0)=0$ as shown earlier.
Moreover
the odd derivatives $D^{0,j}J_n(X_{\d_n},0;f^{\times j})$,
$j$=odd integer, vanish identically by gauge invariance. It is enough
to take derivatives with respect to the bosonic fields $\f$ because of the
identities stated above, \equ(4.40) et seq.
Taking derivatives of \equ(4.59) and
remembering that $\tilde F_{R_n}(X_{\d_n},0)=0,\> {\tilde V_n}(X_{\d_n},0)=0 $
we get
$$\eqalign{
D^{0,2}J_n(X_{\d_n},0,0;f,\bar f)&=D^{0,2}R_n^{\sharp}(X_{\d_n},0,0;f,\bar f)-
D^{0,2}\tilde F_{R_n}(X_{\d_n},0,0;f,\bar f)\cr
D^{0,4}J_n(X_{\d_n},0,0;f_1,\bar f_1,f_2,\bar f_2)&=D^{0,4}R_n^{\sharp}(X_{\d_n},0,0;f_1,\bar f_1,f_2,\bar f_2)+
D^{0,4}\tilde F_{R_n}(X_{\d_n},0,0;f_1,\bar f_1,f_2,\bar f_2)+\cr
&+4D^{0,2}\tilde F_{R_n}(X_{\d_n},0,0;f,\bar f)D^{0,2}{\tilde V_n}(X_{\d_n},0,0;f,\bar f)
}\Eq(4.64.1)$$
\\where the $f$ are complex valued functions in $C^{2}(X_{\d_n})$. A variation
of $\f$ along $f$ implies that we vary $\bar\f$ along $\bar f$.
Note that from \equ(4.57)
$$\eqalign{
D^{0,2}F_{R_n}(X_{\d_n},0,0;1,1)=&\ |X_{\d_n}|
{\tilde \a}_{n,2,0}(X_{\d_n})\cr
D^{0,2}F_{R_n}(X_{\d_n},0,0;1,x_\m)=&\ |X_{\d_n}|
{\tilde \a}_{n,2,\bar 1}(X_{\d_n},\m)+ {\tilde \a}_{n,2,0}(X_{\d_n})\int_{X_{\d_n}} dx\> x_{\m} \cr
D^{0,2}F_{R_n}(X_{\d_n},0,0;x_{\m},1)=&\ |X_{\d_n}|
{\tilde \a}_{n,2,1}(X_{\d_n},\m)+ {\tilde \a}_{n,2,0}(X_{\d_n})\int_{X_{\d_n}} dx\> x_{\m} \cr
D^{0,4}F_{R_n}(X_{\d_n},0,0;1,1,1,1)=&\ 4|X_{\d_n}|
{\tilde \a}_{n,4}(X_{\d_n})\cr}$$
\\Now imposing successively the conditions \equ(4.42) we get
$$\eqalign{
{\tilde \a}_{n,2,0}(X_{\d_n})&= {1\over |X_{\d_n}|}D^{0,2}R_n^{\sharp}(X_{\d_n},0,0;1,1)\cr
{\tilde \a}_{n,2,\bar 1}(X_{\d_n},\m)&={1\over |X_{\d_n}|}
D^{0,2}R_n^{\sharp}(X_{\d_n},0,0;1,x_\m) -
{1\over |X_{\d_n}|}{\tilde \a}_{2,0}(X_{\d_n})\int_{X_{\d_n}} dx\> x_{\m} \cr
{\tilde \a}_{n,2,1}(X_{\d_n},\m)&={1\over |X_{\d_n}|}
D^{0,2}R_n^{\sharp}(X_{\d_n},0,0;,x_{\m},1) -
{1\over |X_{\d_n}|}{\tilde \a}_{n,2,0}(X_{\d_n})\int_{X_{\d_n}} dx\> x_{\m} \cr
{\tilde \a}_{n,4}(X_{\d_n})&={1\over 4}{1\over |X_{\d_n}|}\Bigl(
D^{0,4}R_n^{\sharp}(X_{\d_n},0,0;1,1,1,1) \cr
&+D^{0,2}{\tilde V_n}(X_{\d_n},0,0;1,1)D^{0,2}
R_n^{\sharp}(X_{\d_n},0,0;1,1)\Bigr)\cr} \Eq(4.62)$$
\textBlack
\\The leading contributions to the ${\tilde \a}_{P}(X_{\d_n})$ are
obtained by setting ${\tilde V_n}=0$ in the above formulae.
\eject
\vskip.5truecm
\\{\bf 5. ESTIMATES}
\numsec=5\numfor=1
\vskip.5truecm
\\Let $u_{n}=(g_{n},\m_{n},R_{n})$. Then $({\bf w}_{n},u_n)$
are the coordinates of the
measure density in the polymer representation after $n$ successive applications
of the RG map $f_j$, $1\le j\le n$, of Section~4. The ${\bf w}_{n}$ evolve
according to ${\bf w}_{n+1}=f_{n+1,{\bf w}}({\bf w}_{n})
={\bf v}_{n+1}+{\bf w}_{n,L}$
as given in \equ(4.54). This evolution is independent of
$u_n$ and is solved in Lemma~5.9 below.
The sequence $\{{\bf w}_{n},u_{n}\}$ with $u_{n+1}= f_{n+1}(u_{n})$,
where the solution for ${\bf w}_{n}$ is incorporated in the map $f_n$, is the
RG trajectory. The index $n$ in $R_{n}$ also indicates that $R_{n}$ is
supported on polymers in $(\d_{n}\math{Z})^{3}$.
Correspondingly the norms for Banach spaces of polymer activities given in
Section~2 are indexed
by the lattice spacing $\d_{n}$. In this section we first set up a uniformly
bounded domain ${\cal D}_{n}$ for $u_{n}$. The rest of this section
is then devoted to the proof of Theorem~5.1 below. This theorem
controls the remainders $(\xi_n,\r_n)$ in the flow equations \equ(4.47A)
together with $R_{n+1}$ in \equ(4.221) when $u_{n}$
belongs $ {\cal D}_n$. It also gives bounds on $g_{n+1}$ and
$\m_{n+1}$.
Theorem~5.1 will provide essential ingredients for
the proof (in Section~6,) of existence of an initial choice of the
mass parameter such that there is a uniformly bounded RG trajectory
at all scales labelled by $n$.
\vglue0.1cm
The aforementioned domain will be a ball defined with Banach space norms
with the
center of the ball fixed i.e. independent of $n$. To this end we first
obtain an approximate discrete flow of the coupling constant $g_{n}$ from
the first equation in \equ(4.47A) by ignoring the remainder
$\xi_{n}(g_n,\m_n,R_n)$. The approximate flow equation has $n$-dependent
coefficients. However
we show below (Lemma~5.12), with no assumption about the domain
${\cal D}_{n}$
given below, that the the positive coefficients $a_{n}$
converge geometrically as $n\rightarrow \infty$ to a constant $a_{c,*}>0$.
This leads us to set up a reference approximate
discrete flow of the coupling constant
$$ g_{c,n+1}= L^{\e}g_{c,n}(1-L^{\e}a_{c,*}g_{c,n})\Eq(55.0)$$
\\This may be thought of as an approximate flow in an underlying continuum
theory. This approximate flow has a nontrivial fixed point, namely
$${\bar g}= {L^{\e}-1\over L^{2\e}a_{c,*}}>0 \Eq(55.4) $$
\\The constant $a_{c,*}=a_{c,*} (L,\e)$
depends on $L,\e$ in such a way that when $\e\rightarrow 0$ with $L$ fixed
$a_{c,*} (L,\e) \rightarrow \bar a_{c,*}(L)$ which depends only on $L$.
We will assume $L$ large but fixed for the rest of the paper. We then choose
$\e$ sufficiently small depending on $L$.
\\We have
$$0< {\bar g}< C_{L}\e \Eq(55.01)$$
\\where $C_{L}$ is a constant which depends only on $L$. $\e$ is then a measure
of smallness of $\bar g$.
\\In the following $O(1)$ denotes a constant {\it independent of $L$, $\e$ and
$n$}. Constants C are {\it independent of $\e$ and $n$ but may
depend on $L$}. These constants may change from line to line. It will not be
necessary to keep track of these changes.
\vskip0.3cm
\\{\it The Domain ${\cal D}_{n}$} :
\vskip0.1cm
\\{\it We will say that $u_{n}= (g_n,\m_n,R_n) $ belongs to the
domain ${\cal D}_{n}$ if
$$\vert g_n -{\bar g}\vert < \n{\bar g},
\ \ \vert\mu_n\vert < {\bar g}^{2-\delta}
\Eq(55.1)$$
$$||| R_n |||_{n}< {\bar g}^{11/4-\eta} \Eq(55.02)$$
\\where the constant $\n$ is held in the range $0<\n<1$, and
$$||| R_n |||_{n}=\max \{\vert R_n\vert_{{\bf h}_{*},\AA,\d_{n}},\>
{\bar g}^{2}\Vert R_n\Vert_{{\bf h},G_{\k},\AA,\d_{n}}\} \Eq(55.03)$$
\\We choose $\k=\k(L)$ as in Lemma~2.1 and $\r=\r(L)$ as in
Lemma~5.3 (independent of the domain hypothesis). $\k,\r$ will be held
fixed after $L$ has been chosen sufficiently large.
$\delta,\eta =O(1)> 0$ are very small fixed numbers, say $1/64$, and
$h_{B}=c{\bar g}^{-1/4}$ with $c=O(1) > 0$ a very small number.
Furthermore we take
$h_{B*}=(\r\k)^{-1/2}$ and choose $m_{0}=9$. $h_{F}=h_{F}(L)$ is an $\e$
independent constant which depends on $L$ and is taken to be sufficiently
large. (The dependence of $h_{F}$ on $L$ enters in the proofs of Lemma~5.15 and
Lemma~5.16 below). We recall that
${\bf h} = (h_{B}, h_{F})$ , ${\bf h_*} = (h_{B*}, h_{F})$.}
\vskip0.1cm
\\{\it Remark} : Note that condition \equ(55.02) is equivalent to having both
$$\Vert R_n\Vert_{{\bf h},G_{\k},\AA,\d_{n}}< {\bar g}^{3/4-\eta}
\Eq(55.2)$$
$$\vert R_n\vert_{{\bf h}_{*},\AA,\d_{n}} < {\bar g}^{11/4-\eta}
\Eq(55.3)$$
\vglue0.2cm
\\Recall the definitions of $\rho_{n}(g_n,\m_n, R_n)$ and
$\xi_{n}(g_n,\m_n, R_n)$ from
\equ({4.64}). We will prove in this section
\vglue.3truecm
\\{\it Theorem~5.1
\\Let $u_{n}=(g_n,\m_n,R_n)\in {\cal D}_{n}$. Let $L$ be
large but fixed followed by $\e$ sufficiently small depending on $L$.
${\bar g}$ is thus sufficiently small. Let $u_{n+1}=f_{n+1}(u_n)$ where
$f_{n+1}$ is the RG map of section~4. Then there exist constants $C_{L}$
independent of $n$ and $\e$ such that}
$$\vert\xi_{n} \vert \le C_L{\bar g}^{11/4-\eta} \Eq(555.3)$$
$$\vert\rho_{n}\vert \le C_L{\bar g}^{11/4-\eta} \Eq(555.4)$$
$$|||R_{n+1}|||_{n+1}\le L^{-1/4}{\bar g}^{11/4-\eta} \Eq(55.7)$$
$$|g_{n+1}-\bar g| < 2\n{\bar g}^{3/2}, \quad
|\m_{n+1}|< O(1)L^{3+\e\over 2} {\bar g}^{2-\d}
\Eq(55.5) $$
\vskip.2truecm
\\{\it Remark } :
The lemmas which follow will serve to prove Theorem~5.1. They are
organized as in Section~5 of [BMS]. We
remark that Lemmas~5.1-5.4, 5.9, and Lemma~ 5.12 are independent of the domain
hypothesis. {\it All the other lemmas are under the assumption that
$(g_n,\m_n,R_n)$ belong to the domain ${\cal D}_{n}$}.
Lemmas~5.21, 5.22, 5.23 and 5.27 are the major
parts of the program. $R_{n+1,\rm main}$ is bounded in Lemma~5.21 and
this result determines the qualitative form of the bound on the
remainder. $R_{n+1,3}$ and $R_{n+1,4}$ are seen, in Lemmas~5.22, 5.23 to be
negligible in comparison. $R_{n+1,\rm linear}$ is the crux of the program
and it is bounded in Lemma~5.27. The remaining Lemmas are auxiliary
results on the way to these Lemmas.
These auxiliary lemmas implement some of the following principles: Wick
constants $C_{n}(0)$ are uniformly bounded by constants $C=C_{L}$. In
bounds by $G_{\k},{\bf h},{\cal A}$ norms, a fluctuation field $\zeta$
contributes a constant $C= O(1)(\r(L)\k(L))^{-1/2}$ and a field
$\f$ contributes a constant $O (1){\bar g}^{-1/4}$.
The contributions of these fields as well as of the Grassmann fields
$\psi, \h$ are controlled by the structure
of the norms defined in section 2 (with above choice of ${\bf h},{\bf h}_*$)
and later in this section ((5.20) et seq).
Integration over the Grassman fluctuation fields $\h$ is controlled by the
Gramm inequality.
In bounds by the ${\bf h_{*}},{\cal A}$ norms, fluctuation fields $\z$
contribute a constant $C= O(1)(\r(L)\k(L))^{-1/2}$ and
fields $\f$ contribute $O (1)$. $h_{B*}$ has been adjusted to take care
of the constant $C$ above in the contribution of the fluctuation field.
\vskip0.3cm
\\{\it Lattice Taylor expansions}
\\In the following we will have occasion to estimate the difference of lattice
fields at two different points of a hypercubic lattice $(\d_n\math{Z})^{d}$.
Let $f$ be a lattice function and $x,y$ be two points in the lattice. We
write $y-x=\sum_{j=1}^{d}\d_n h_j e_j$ where $h_j\in\bf Z$ and the $e_j$ are
the unit vectors of the lattice. We will express the difference
$f(y)-f(x)$ as a sum of forward lattice derivatives of $f$ along segments of a
specified lattice path. Given $j\in \{1,2,..,d\}$, $s\in \bf Z$ and
$v\in (\d_n\math{Z})^{d}$ we define $p_j(v,s)\in (\d_n\math{Z})^{d}$ by
$$p_j(v,s)=\sum_{i=1}^{j-1}(v,e_i)e_i +\d_n s e_j \Eq(5.001) $$
\\with the convention that if $j=1$ the sum is empty.
Then it is a simple matter to verify that
$$f(y)-f(x)=\d_n\sum_{j=1}^{d}\sum_{s_j=0}^{h_j-1}
\dpr_{\d_n,e_j}f(x+p_j(y-x,s_j))\Eq(5.002) $$
\\By iterating \equ(5.002) we get the second order lattice Taylor expansion
$$f(y)-f(x)=\sum_{j=1}^{d} ((y-x),e_{j})
\dpr_{\d_n,e_j}f(x) +\d_n^{2}\sum_{j,k=1}^{d}\sum_{s_j=0}^{h_j-1}
\sum_{s_k=0}^{h_k-1}\dpr_{\d_n,e_j}\dpr_{\d_n,e_k}
f(x+ p_k(p_j(y-x,s_j),s_k)) \Eq(5.003) $$
\vskip.3truecm
\\{\it Lemma 5.1
\\Let $Z_{\d_{n}},X_{\d_{n}}$ be connected 1-polymers in
$(\d_{n}\math{Z})^{3}$. Let $Y_{\d_{n}}=\emptyset$ or
$ Y_{\d_{n}} =L^{-1}X_{\d_{n}} \subset Z_{\d_{n}}$ such that
vol$(Z_{\d_{n}}\setminus Y_{\d_{n}})$
\\$\ge {1\over 2}$. Choose
any $\g=O(1)>0$ and $\k=\k(L)>0$ as in Lemma~2.1. Let
$\f :{\tilde Z}_{\d_{n}}^{(5)}\rightarrow \math{C}$ where
${\tilde Z}_{\d_{n}}^{(5)}$ is $Z_{\d_{n}}$ with $5$-collar attached
(see \equ(1.22c), \equ(1.22d)). Then
there exists a constant $C=\k^{-j/2}O(1) $ such
that
$$\Vert\f\Vert_{C^{2}(Z_{\d_{n}})}^j\le C\>2^{|Z|}{\bar g}^{-{j\over 4}}
e^{\g {\bar g}\int_{Z_{\d_{n}}\backslash Y_{\d_{n}}}dy
|\f(y)|^4}G_\k(Z_{\d_{n}},\f)\Eq(5.1)$$
\\For $Y_{\d_{n}}=\emptyset$ the above bound holds without the factor
$2^{|Z|}$.
}
\\{\it Proof}: This is the lattice analogue of Lemma~5.1 in [BMS].
The proof reposes on the H\"older inequality and
the lattice Sobolev inequality ( see [BGM], Appendix B for an
elementary proof). Let $x\in Z_{\d_{n}}$. Write
$$\f(x)= {1\over vol( Z_{\d_{n}}\backslash Y_{\d_{n}})}
\int_{ Z_{\d_{n}}\backslash Y_{\d_{n}} }dy (\f(y)+\f(x) -\f(y))$$
\\and bound
$$|\f(x)|\le {1\over vol(Z_{\d_{n}}\backslash Y_{\d_{n}} )}
\int_{Z_{\d_{n}}\backslash Y_{\d_{n}} }dy |\f(y)|+
{1\over vol( Z_{\d_{n}}\backslash Y_{\d_{n}} )}
\int_{Z_{\d_{n}}\backslash Y_{\d_{n}} }dy | \f(x) -\f(y)| $$
\\We bound the first term on the right hand side by
$ O(1)\Vert \f\Vert_{L^{4}(Z_{\d_{n}}\backslash Y_{\d_{n}} )}$.
To bound the second term we
write the difference $\f(y) -\f(x)$ as a sum of lattice derivatives along the
segments of the path as in \equ(5.002). Any connected polymer
$Z_{\d_n}$ as defined in
section~1.3 can be represented as $Z_{\d_n}=Z\cap (\d_n\math{Z})^{3}$ where
$Z$ is a connected continuum polymer. Recall from section~1.3 that
$Z_{\d_n}$ is said to be convex if
$Z$ is a convex set. If $Z_{\d_n}$ is convex then the path $p_j(y-x,s_j)$
in \equ(5.002)
lies entirely in $Z_{\d_n}$. If $Z_{\d_n}$ is not convex then it decomposes
as a connected union of connected convex polymers. If $x,y$ are not in the same
convex component then it suffices to consider the case when they are in
adjacent components. We pick a point $z_0$ in the intersection of the closures,
write $f(x)-f(y)=(f(x)-f(z_0))+(f(z_0)-f(y))$ and use the above first order
taylor expansion for each summand. The estimates below remain valid. Therefore
it is sufficient to consider the case $Z_{\d_n}$ is convex. From
$$\f(y)-\f(x)=\d_n\sum_{j=1}^{3}\sum_{s_j=0}^{h_j-1}
\dpr_{\d_n,e_j}\f(x+p_j(y-x,s_j))\Eq(5.1a) $$
\\we get the bound
$$|\f(y) -\f(x)|\le \sum_{j=1}^{3}\d_n|h_j|
\sup_{z\in Z_{\d_{n}}}|\dpr_{\d_{n},e_j}\f(z)|
\le 3\d_n(\max_{j}|h_j|) \max_{j}\sup_{z\in Z_{\d_{n}}}
|\dpr_{\d_{n},e_j}\f(z)| $$
$$\le O(1)|y-x|\> \Vert \f \Vert_{Z_{\d_{n}},1,5}\Eq(5.1b)$$
\\where in the last step we have used the lattice Sobolev embedding theorem.
We have $|y-x|\le |Z|$. Putting the above bounds together we get
$$|\f(x)|\le 0(1)|Z|(\Vert \f\Vert_{L^{4}(Z_{\d_{n}}\setminus Y_{\d_{n}})} +
\Vert \f \Vert_{Z_{\d_{n}},1,5})$$
\\We also have for $k=1,2$,
$|\dpr_{\d_{n},\m_1,..,\m_k}^{k}\f(x)|\le \Vert \f \Vert_{Z_{\d_{n}},1,5}$
by Sobolev embedding. Therefore combining with the previous inequality we get
$$\Vert\f\Vert_{C^{2}(Z_{\d_{n}})}\le
0(1)|Z|(\Vert \f\Vert_{L^{4}(Z_{\d_{n}}\setminus Y_{\d_{n}})} +
\Vert \f\Vert_{Z_{\d_{n}},1,5})$$
\\Hence
$$\Vert\f\Vert_{C^{2}(Z_{\d_{n}})}^{j}\le
0(1)^{j}|Z|^{j}(\Vert \f\Vert_{L^{4}(Z_{\d_{n}}\setminus Y_{\d_{n}})}^{j} +
\Vert \f\Vert_{Z_{\d_{n}},1,5}^{j})\le C\> 2^{|Z|}
{\bar g}^{-j/4} e^{\g {\bar g}\int_{Z_{\d_{n}}\setminus Y_{\d_{n}}}dy\> \f^{4}(y)}
G_{\k}(Z_{\d_{n}},\f) $$
\\where $C$ depends on $\k, \g, j$. This proves the bound \equ(5.1). We now
prove the statement following \equ(5.1). Let
$Y_{\d_{n}}=\emptyset$. For $x\in Z_{\d_{n}}$ pick the unit
block $\D_{\d_{n}}\subset Z_{\d_{n}},\> \D_{\d_{n}}\ni x $. We have
$$|\f(x)|\le \int_{\D_{\d_{n}}}dy\>|f(y)|+\int_{\D_{\d_{n}}}dy\>|\f(x)-\f(y|$$
\\Proceeding as before the first term is bounded by the $L^{4}(\D_{\d_{n}})$
norm which is less than the $L^{4}(Z_{\d_{n}})$ norm. The second term is
bounded as before except that since $x,y\in \D_{\d_{n}}$ we have
$|x-y| \le O(1)$. The rest is as before. \bull
\vglue.3truecm
\\In effecting the fluctuation map in Section~3.1 we created polymer activities
which depended separately on $\Phi$ and the fluctuation field $\xi$. The
following lemmas will enable us to estimate the contributions of the bosonic
fluctuation field $\z$ at various steps. Define a large field regulator for
the bosonic fluctuation field $\z : \tilde {X}_{\d_{n}}^{5}\rightarrow \bf C$
$$\tilde G_{\k,\r}(X_{\d_{n}},\z)=e^{\r\Vert\z\Vert^{2}_{L^{2}(X_{\d_{n}})}}
G_{\k}(X_{\d_{n}} ,\z) ,\quad \r,\k>0\Eq(5.2)$$
\\$\k$ is chosen as in Lemma~2.1 and is held sufficiently small. The choice
of $\r >0$ is dictated by Lemma 5.3 below.
\vglue.3truecm
\\{\it Lemma 5.2
\\For any $x\in X_{\d_{n}}$
$$|\z(x)|^j\le C_{\r,j,\k}\tilde G_{\k,\r}(X_{\d_{n}},\z)\Eq(5.3)$$
\\where $C_{\r,j}=(\r\k)^{-(j/2)}O(1) $
and $O(1)$ depends on $j$. We have isolated out the $\r,\k$
dependence in the bound.
}
\\{\it Proof} : The proof follows the lines of the proof of Lemma~5.1 for the
case $Y_{\d_{n}}=\emptyset$. Take the unit block $\D_{\d_{n}}
\subset X_{\d_{n}}$ such that $\D_{\d_{n}}\ni x$. We replace
the $L^{4}$ norm by the $L^{2}$ norm in the appropriate place and estimate
$|\z(x)-\z(y)|$ with
$x,y\in \D_{\d_{n}}$ as before now using the regulator $ \tilde G_{\k,\r}$.
\bull
\vglue.3truecm
\\The parameter $\r > 0$ is chosen such that the following
Lemma~5.3 holds. $\r$ depends on $L$.
\vglue.1truecm
\\{\it Lemma 5.3 : Let $\k >0$ be chosen as in Lemma~2.1. Then there exists
$\r_{0}=\r_{0}(L) >0$ independent of $n$ such that for all
$\r$, $0 <\r\le \r_{0}$
$$\int d\m_{\G_{n}}(\z)\tilde G_{\k,\r}(X_{\d_{n}},\z)\le 2^{|X_{\d_{n}}|}
\Eq(5.4)$$
}
\\Lemma~5.3 is proved in the same way as Lemma~2.1.
\vglue.3truecm
\\We introduce a new
intermediate large field regulator which combines the ones introduced earlier
$$\hat G_{\k,\r}(X_{\d_{n}},\z,\f)
=G_\k(X_{\d_{n}},\z+\f)G_\k(X_{\d_{n}},\f)\tilde G_{\k,\r}(X_{\d_{n}},\z)
\Eq(5.9)$$
\vglue.2truecm
\\{\it Lemma 5.4 : Let $\k,\>\r$ be chosen as in Lemma~2.1 and Lemma~5.3
respectively. Then we have
$$\int d\m_{\G_{n}}(\z)\hat G_{\k,\r}(X_{\d_{n}},\z,\f)\le
2^{|X_{\d_{n}}|}G_{3\k}(X_{\d_{n}},\f)
\Eq(5.12)$$
}
\vglue.3truecm
\\{\it Proof}: The proof follows from an application of the Cauchy-Schwarz
inequality and Lemmas~2.1, 5.3.
\vglue.3truecm
\\In effecting the fluctuation map we have introduced polymer activities
which are functions
of the four separate fields $\f,\z,\psi,\h$ and not just of $\f+\z,\psi +\h$.
The polymer representation will have the form
$$\tilde K(X_{\d_{n}},\f,\z,\psi,\tilde\h)=\sum_{p\ge 0}
\sum_{I\subset\{1,...,2p\}} \sum_{{\bf l},{\bf a}}\int_{X_{\d_{n}}^{2p}}
d{\bf x}\tilde K^{{\bf l},{\bf a},I}_{2p}
(X_{\d_{n}},\f,\z,{\bf x})\ \dpr^{\bf l}_{\d_{n}}
\prod_{j\in I}\psi_{a_j}(x_{j})
\prod_{j\in I^c}\h_{a_j}(x_{j})\Eq(2.1.2)$$
where we have kept the same conventions as in \equ(2.1).
\\Note that if $\tilde K(X_{\d_{n}},\f,\z,\psi,\h)$ is actually
$\tilde K_{n}(X_{\d_{n}},\f+\z,\psi+\h)$ then
$$\tilde K^{{\bf l},{\bf a},I}_{2p}
=\tilde K^{{\bf l},{\bf a}}_{n,2p},\quad
\forall\> I\subset\{1,...,2p\}\Eq(2.1.3)$$
\\The field derivatives of polymer activities represented in \equ(2.1.2)
are defined by :
$$D^{2p,m}\tilde K(X_{\d_{n}},\f,\z,0,0; f^{\times m},
g_{2p})= \sum_{I\subset\{1,...,2p\}}
\sum_{{\bf l},{\bf a}}\int_{X_{\d_{n}}^{2p}} d{\bf x}D^m
\tilde K^{{\bf l},{\bf a},I}_{2p}
(X_{\d_{n}},\f,\z,{\bf x};f^{\times m})\ \dpr^{\bf l}_{\d_n}g_{2p}({\bf x})
\Eq(2.2.2)$$
\\where the derivative $D^m$ of the bosonic coefficient
is with respect to the field $\f$ (and not the fluctuation field $\z$).
The fermionic field derivatives ( at the origin) are being
computed with respect to both $\psi$ {\it and}
the fermionic fluctuation field $\h$.
\\The norm of the multilinear functional \equ(2.2.2) is defined as in
\equ(2.3), namely
$$\Vert D^{2p,m}\tilde K(X_{\d_{n}},\f,\z,0,0)\Vert =
\sum_{I\subset\{1,...,2p\}}
\sup_{\Vert f_j\Vert_{C^{2}(X_{\d_{n}})}\le 1
\atop{ \Vert g_{2p}\Vert_{C^{2}(X_{\d_{n}}^{2p})}
\le 1\atop \> \forall 1\le j\le m}} \left\vert\sum_{{\bf l},{\bf a}}
\int_{X_{\d_{n}}^{2p}} d{\bf x} D^m\tilde K^{{\bf l},{\bf a},I}_{2p}
(X_{\d_{n}},\f,\z,{\bf x};f^{\times m})\>
\dpr^{\bf l}g_{2p}({\bf x})\right\vert\Eq(2.3.2)$$
\\For $I\subset\{1,...,2p\}$ define a polymer activity
$$\tilde K_{2p}^{I}(X_{\d_{n}},\f,\z,\psi,\tilde\h)=
\sum_{{\bf l},{\bf a}}\int_{X_{\d_{n}}^{2p}}
d{\bf x}\tilde K^{{\bf l},{\bf a},I}_{2p}
(X_{\d_{n}},\f,\z,{\bf x})\ \dpr^{\bf l}_{\d_{n}}
\prod_{j\in I}\psi_{a_j}(x_{j})
\prod_{j\in I^c}\h_{a_j}(x_{j})\Eq(2.33.2)$$
\\Then
$$D^{2p,m}\tilde K(X_{\d_{n}},\f,\z,0,0; f^{\times m},
g_{2p})=\sum_{I\subset\{1,...,2p\}}
D^{2p,m}\tilde K_{2p}^{I}(X_{\d_{n}},\f,\z,0,0; f^{\times m},
g_{2p})\Eq(2.33.2a) $$
\\and observe that
$$\Vert D^{2p,m}\tilde K(X_{\d_{n}},\f,\z,0,0)\Vert =
\sum_{I\subset\{1,...,2p\}}
\Vert D^{2p,m}\tilde K_{2p}^{I}(X_{\d_{n}},\f,\z,0,0)\Vert \Eq(2.3.2a) $$
\\In the beginning of this section we specified
${\bf h}=(h_F,h_B)$ and ${\bf h}_{*}=(h_F,h_{B*})$. We define
in addition
$${\bf \hat h}=(\hat h_F,\hat h_B)=({h_F\over 2},h_B) \Eq(2.2.2.1)$$
$${\bf \hat h}_{*}=(\hat h_F,\hat h_{B*})=({h_F\over 2},h_{B*})
\Eq(2.2.2.2) $$
\\We define the norms
$$\Vert\tilde K(X_{\d_{n}},\f,\z,0,0)\Vert_{\bf \hat h}=
\sum_{p=0}^\io\sum_{m=0}^{m_0}\hat h_F^{2p}{h_B^m\over m!}
\Vert D^{2p,m}\tilde K(X_{\d_{n}},\f,\z,0,0)\Vert \Eq(2.4.2) $$
$$\Vert\tilde K(X_{\d_{n}})\Vert_{{\bf \hat h},\hat G_{\k,\r}}=
\sup_{\f,\z\in {\cal F}_{\tilde X_{\d}^{(5)}} }
\Vert\tilde K(X_{\d_{n}},\f,\z,0,0)
\Vert_{\bf \hat h}\hat G^{-1}_{\k,\r}(X_{\d_{n}},\f,\z)\Eq(2.6.2)$$
$$\Vert\tilde K(X_{\d_{n}},0,\z,0,0)\Vert_{\bf h_{*}}=
\sum_{p=0}^\io\sum_{m=0}^{m_0} h_F^{2p}{ h_{B*}^m\over m!}
\Vert D^{2p,m}\tilde K(X_{\d_{n}},0,\z,0,0)\Vert \Eq(2.6.2.1)$$
$$\Vert\tilde K(X_{\d_{n}})\Vert_{\bf h_{*},\tilde G_{\k,\r}}
= \sup_{\z\in {\cal F}_{\tilde X_{\d}^{(5)}}}
\Vert\tilde K(X_{\d_{n}},0,\z,0,0)\Vert_{\bf h_{*}}
G^{-1}_{\k,\r}(X_{\d_{n}},\z) \Eq(2.6.2.2) $$
$$\vert\tilde K(X_{\d_{n}})\vert_{\bf\hat {h_{*}}}=
\sum_{p=0}^\io\sum_{m=0}^{m_0}\hat {h_{F}}^{p}{ {h_{B}}_{*}^{m}\over m!}
\Vert D^{p,m}\tilde K(X_{\d_{n}},0,0,0,0)\Vert\Eq(2.5.2)$$
\\Note that if we represent a polymer activity
$\tilde K(X_{\d_{n}},\f+\z,\psi+\eta)$ in the form \equ(2.1.2) then
$$\Vert\tilde K(X_{\d_{n}},\f+\z,0)\Vert_{\bf h}=
\Vert\tilde K(X_{\d_{n}},\f,\z,0,0)\Vert_{\bf \hat h} \Eq(2.7)$$
\\where we have used \equ(2.1.3) and the fact that
$$\sum_{I\subset\{1,...,2p\}}1=2^{2p} \Eq(2.8)$$
\\It is straightforward to prove that the above norms satisfy the
multiplicative property of Proposition~2.3 (with $G_{\k}$ replaced by
$\hat G_{\k}$).
\vglue.3truecm
\\{\it Lemma 5.5
\\Let $g_n, \m_{n}$ belong to ${\cal D}_{n}$. Let $Y_{\d_{n}}$ be a
1-polymer. Then
for $V_{n} (Y_{\d_{n}},\Phi,\xi)=
V(Y_{\d_{n}},\Phi+\xi,C_{n},g_n,\m_n)$ or
$V(Y_{\d_{n}},\Phi,S_{L}C_{n+1} ,g_n,\m_n)$, we have
$$\Vert e^{- V_{n}(Y_{\d_{n}} ,\Phi,\xi )}\Vert_{\bf h}\le 2^{|Y_{\d_{n}}|}
e^{-{\bar g}/4\int_{Y_{\d_{n}}} dx |\f +\z|^4(x)}\Eq(5.5)$$
$$
\Vert e^{-V_{n}(Y_{\d_{n}},\Phi,\xi)}\Vert_{{\bf h}_{*}}
\le 2^{|Y_{\d_{n}}|}\Eq(5.lem5.1) $$
\\for $\e >0$ sufficiently small depending
on $L$. In the above norms ${\bf h}=(h_{B},h_{F})$ and
${{\bf h}_{*}}=(h_{B*},h_{F})$ are chosen as in the hypothesis for the
domain $\cal D_{\d_{n}}$. Thus $h_{F}=h_{F}(L)$,
$h_{B}=c{\bar g}^{-{1\over 4}}$ with $c=O(1)$ sufficiently small, and
$h_{B*}=(\r\k)^{-1}$. Note that $ h_{B*}$ depends on $L$.
}
\vglue.3truecm
\\{\it Proof} : It is sufficient to prove this when $Y_{\d_n}$ is a 1-block
$\D_{\d_{n}}$. Because otherwise we can write $Y_{\D_n}$ as a disjoint union
of 1-blocks and write the left hand side as a product over 1-block
contributions. Then the multiplicative property of the $\bf h$ norm (
Proposition~2.3) gives the lemma.
\\From the definition of $V$ in \equ(4.1) we get on undoing
the Wick ordering
$$V_{n}(\Delta_{\d_{n}},\Phi)=V_{n,u}(\Delta_{\d_{n}},\f)+
2g_{n}\int_{\D_{\d_{n}}}dx\f\bar\f(x)\psi\bar\psi(x)+
{\tilde\m_{n}}\int_{\D_{\d_{n}}
}dx\psi\bar\psi(x) \Eq(5.12.1)$$
\\where
$$V_{u,n}(\D_{\d_{n}},\f)=
g_{n}\int_{\D_{\d_{n}}}dx [(\f\bar\f(x))^{2}+{\tilde\m_{n}}\f\bar\f(x)]
\Eq(5.12.2)$$
\\and ${\tilde\m_{n}}=\m_{n} -2g_{n} C_{n}(0)$.
\\By the multiplicative property of the $\bf h$ norm,
$$\Vert e^{- V(\D_{\d_{n}},\Phi )}\Vert_{\bf h}\le
\Vert e^{- V_{u,n}(\D_{\d_{n}},\f )}\Vert_{h_{B}}
\Vert e^{-2g_{n}\int_{\D_{\d_{n}}}dx\f\bar\f(x)\psi\bar\psi(x)}\Vert_{\bf h}
\Vert e^{-{\tilde\m_{n}}\int_{\D_{\d_{n}}}dx\psi\bar\psi(x)}
\Vert_{h_{F}} \Eq(5.12.3)$$
\\We estimate each of the factors on the right hand side in turn. We observe
that by taking $\e$ sufficiently small we can make $\bar g$ as small as
necessary since $0<{\bar g}\le C\e$. Since
$g_{n}, \m_{n}$ belong to $ {\cal D}_{n}$
we have $g_{n}=O(\bar g)$ and $\m =O({\bar g}^{2-\d})$.
Moreover from from (5a) of Theorem~1.1 and \equ(0.25) we have the uniform bound
$|C_{n}(0)|\le C_{L}$.
\\For the first factor we have the bound
$$\Vert e^{- V_{u,n}(\D_{\d_{n}},\f )}\Vert_{h_{B}}\le
2^{{1\over 2}|\D_{\d_{n}}|}e^{-g_{n}/2\int_{\D_{\d_{n}}}dx (\f\bar\f)^2(x)}
\Eq(5.12.4) $$
\\This can be proved on the lines of the proof of Lemma~ 5.5 of [BMS]
by substituting there $\bar g$ for $\e$ and
taking account of the previous observations. Now consider the second factor.
From
$$e^{-2g_{n}\int_{\D_{\d_{n}}}dx\f\bar\f(x)\psi\bar\psi(x)}
=\sum_{p\ge 0}{(-2g_{n})^{p}\over p!}
\int_{\D_{\d_{n}}^{p}}d{\bf x}K_{2p}(\f,\D,{\bf x})\prod_{j=1}^{p}
\psi\bar\psi(x_{j})
$$
where
$$K_{2p}(\f,\D,{\bf x})=\prod_{j=1}^{p}\f\bar\f(x_{j})$$
\\we obtain
$$\Vert e^{-2g_{n}\int_{\D_{\d_{n}}}dx\f\bar\f(x)\psi\bar\psi(x)}\Vert_{\bf h}\le
\sum_{p\ge 0}{(2g_{n})^{p}\over
p!}h_{F}^{2p}\sum_{m=0}^{m_0}{h_B^m\over m!}$$
$$\sup_{{\Vert f_i\Vert_{ C^{2}(\D_{\d_{n}})}\le 1,\> \forall i}
\atop\Vert g_{2p}\Vert_{ C^{2}(\D_{\d_{n}}^{2p})}\le 1}\left\vert
\int_{\D_{\d_{n}}^{p}} d{\bf x}D^m K_{2p}(\f,\D_{\d_n},{\bf x};f^{\times m})
g_{2p}(x_1,x_1,...,x_p,x_p)\right\vert\le
$$
$$\le \sum_{p\ge 0}{(2g_{n})^{p}\over p!}h_{F}^{2p}\int_{\D_{\d_{n}}^{p}} d{\bf x}
\prod_{j=1}^{p}\Vert \f\bar\f(x_{j})\Vert_{h_B}\le
e^{2g_{n} h_{F}^{2}\int_{\D}dx\Vert\f\bar\f(x)\Vert_{h_B}}
$$
\\It is straightforward to show that
$$2g_{n} h_{F}^{2}\int_{\D}dx\Vert\f\bar\f(x)\Vert_{h_B} \le C_{L}
{\bar g}^{1\over 2} \bigl ({{\bar g}\over 4}\int_{\D_{n}}dx
(\f\bar\f)^{2}(x) +O(1)\bigr )$$
\\and therefore for the second factor we have the bound
$$\Vert e^{2g_{n}\int_{\D_{\d_{n}}}dx\f\bar\f(x)\psi\bar\psi(x)}\Vert_{\bf h}\le
2^{C_{L}{\bar g}^{1\over 2}|\D|}
e^{C_{L}{\bar g}^{1\over 2} {{\bar g}\over 4}\int_{\D}dx (\f\bar\f)^{2}(x)}
\Eq(5.12.5) $$
\\Finally for the third factor we have straightforwardly the bound
$$\Vert e^{{\tilde\m_{n}}\int_{\D_{\d_{n}}}dx\psi\bar\psi(x)}
\Vert_{h_{F}} \le 2^{C_{L}{\bar g}|\D|} \Eq(5.12.6) $$
\\Putting together the bounds for the three factors, we obtain for $\e$
sufficiently small depending on $L$
$$\Vert e^{- V(\D_{\d_{n}},\Phi )}\Vert_{\bf h} \le
2^{|\D|}e^{-{\bar g}/4\int_{\D}dx (\f{\bar \f})^2(x)}$$
\\which proves the first part of the lemma follows.
The proof of the second part follows the same lines. \bull
\vglue0.3cm
\\{\it Lemma 5.6
\\Let $p_{n,g}(\D_{\d_n},\xi,\Phi)$ and $p_{n,\m} (\D_{d_n}, \xi, \Phi)$
be as given in
\equ(4.8b). Let $g_n, \m_n$ belong to ${\cal D}_{n}$. Let
$h_{B}=c{\bar g}^{-1/4}$ and $h_{B*}$ be as in the definition of
${\cal D}_{n}$. Recall that ${\bf h}=(h_{B},h_{F})$,
and ${\bf h_{*}}=(h_{B*},h_{F})$, where $h_{F}=h_{F}(L)$.
Let $\k=\k(L)>0$ and $\r=\r(L)>0$, be as specified in Lemmas~2.1 and 5.3.
Then for
any $\g=O(1)>0$, $0\le s < 1$ we have constants $C_{L}$ independent of $n$ and
$\e$ but depending on $L$ such that
$$\Vert p_{n,g}(\D_{\d_n},\f,\z,0,0)\Vert_{\bf h}\le C_{L} {\bar g}^{1/4}(1-s)^{-3/4}
\tilde G_{\k,\r}(\D_{\d_n},\z)G_{\k}(\D_{\d_n},\f)e^{{\bar g}(1-s)
\g\int_{\D_{\d_n}}
dx\ (\f{\bar \f})^2(x)}\Eq(5.6)$$
$$\Vert p_{n,\m}(\D_{\d_n},\f,\z,0,0)\Vert_{\bf h}\le C_{L} {\bar g}^{7/4-\delta
}(1-s)^{-1/2} \tilde G_{\k,\r}(\D_{\d_n},\z)G_{\k}(\D_{\d_n},\f)
e^{{\bar g}(1-s)\g\int_{\D_{\d_n}}
dx\ (\f{\bar \f})^2(x)}\Eq(5.6b)$$
$$\Vert p_{n,g}(\D_{\d_n},0,\z,0,0)\Vert_{{\bf h}_{*}}\le C_{L}\> {\bar g}
\tilde G_{\k,\r}(\D_{\d_n},\z)
\Eq(5.6.1)
$$
$$\Vert p_{n,\m}(\D_{\d_n},0,\z,0,0)\Vert_{{\bf h}_{*}}\le C_{L}
{\bar g}^{2-\delta }\tilde G_{\k,\r}(\D_{\d_n},\z) \Eq(5.6.2)
$$
}
\\{\it Proof} :
$p_{n,g}(\D_{\d_n},\xi,\Phi)$ is given in \equ(4.8a). We undo the Wick ordering
which produces constants $C_{n}(0)$ uniformly bounded by constant
$C_{L}$ from Corollary~1.2. We can then write it in the form
\equ(2.1.2) by expanding out in the Grassmann fields. Since it is a local
polynomial of degree four we get,
$$p_{n,g}(\D_{\d_n},\f,\z,\psi,\h)=\sum_{p=0}^{2}
\sum_{{\bf a}}\sum_{I\subset\{1,...,2p\}}\int_{\D_{\d_n}}
dx{\tilde p}_{n,g,2p}^{{\bf 0},{\bf a},I}(\D_{\d_n},\f,\z,x)
\prod_{i\in I}\psi_{a_i}(x)
\prod_{i\in I^c}\h_{a_i}(x)\Eq(5.6.3)$$
\\where ${\bf 0}$ means that
$l_{i}=0\ \forall i$. We have following the definition of the norm
in \equ(2.4.2) with $\hat{\bf h}$ replaced by ${\bf h}$
$$\Vert p_{n,g}(\D_{\d_n},\f,\z,0,0)\Vert_{\bf h} \le
\sum_{p=0}^{2} h_{F}^{2p} \sup_{||g_{2p}||\le 1}
\sum_{{\bf a}}\sum_{I\subset\{1,...,2p\}}
\int_{\D_{\d_n}}dx\Vert {\tilde p}_{n,g,2p}^{{\bf 0},{\bf a},I}(\D_{\d_n},\f,\z,x)
\Vert_{h_{B}}
|g_{2p}({\bf x})|$$
$$\le \sum_{p=0}^{2} h_{F}^{2p}
\sum_{{\bf a}}\sum_{I\subset\{1,...,2n\}\atop |I|\ {\rm even}}
\int_{\D_{\d_n}}dx\Vert {\tilde p}_{n,g,2p}^{{\bf 0},{\bf a},I}(\D_{\d_n},\f,\z,x)
\Vert_{h_{B}}
\Eq(5.6.4)$$
\\where $g_{2p}({\bf x})=g_{2p}(x,x,...,x)$ and
$\Vert g_{2p}\Vert$ is the $C^2(\D_{\d_{n}}^{2p})$ norm of $g_{2p}$.
${\tilde p}_{n,g,2p}^{{\bf 0},{\bf a},I}(\D_{\d_n},\f,\z,x)$ is a polynomial
in $\f,\z$
and every term in the $h_B$ norm of ${\tilde p}_{n,g,2p}$
can be estimated as in the proof of Lemma 5.6
of [BMS]. Each term carries a factor $g_{n}=O({\bar g})$.
The fluctuation fields $\z$ are estimated via Lemma 5.2 and the
fields $\f$ via Lemma 5.1. For each field $\f$ we loose
${\bar g}^{-{1\over 4}}$.
In the $p=0$ term the maximum power of $\f$ in ${\tilde p}_{n,g,p}$
is $3$, for $p=1$ the maximum power is $2$, and for $p=2$ it is $0$.
The bound \equ(5.6) now follows as in Lemma 5.6, [BMS]. The bound \equ(5.6b)
for $p_{n,\mu}(\D_{\d_n},\xi,\Phi)$ is proved in the same way. We just have
to
remember that $\m_{n} =O({\bar g}^{2-\d})$ from the domain hypothesis, and
that the
maximum power of the field $\f$ in the ${\tilde p}_{n,\m,p}$ is $1$. The
remaining parts are proved in the same way. \bull
\vglue0.3cm
Define $p_n (s) = p_n (s,\D_{\d_{n}} ,\Phi,\xi )$ by
$p_n (s) = sp_{n,g}+s^{2}p_{n,\mu} $.
Then $r_{n,1} = r_{n,1} (\D_{\d_{n}} ,\Phi ,\xi)$ defined by \equ({4.9}) is
given by
$$r_{n,1} =
{1\over 2}\int_0^1ds(1-s)^{2}e^{-p_n (s)-\tilde{V_n}}
\big(-p'_n (s)^{3}+6p'_n (s)p_{n,\mu }\big)\Eq(5.7)$$
\\with $p'_n (s)={d\over ds}p_n (s)=p_{n,g}+2sp_{n,\mu }$ and
$p''_n (s)=2p_{n,\mu}$.
\vglue.3truecm
\\{\it Lemma 5.7
\\Under the conditions of the domain ${\cal D}_{n}$ there exists
a constant $C_L$ independent of $n$ and $\e$ but dependent on $L$ such that
$$\Vert r_{n,1}(\D_{\d_n})\Vert_{h,\hat G_{\k,\r}} \le C_{L}{\bar g}^{3/4}
\Eq(5.8)$$
$$\Vert r_{n,1}(\D_{\d_n})\Vert_{h_{*},\tilde G_{\k,\r}}\le
C_{L}{\bar g}^{3-\delta } \Eq(5.8.1)$$
}
\\{\it Proof} : Follow the proof of the corresponding Lemma 5.7 of [BMS].
Write ${\tilde V}_{n} +p_{n}(s)= V_{n,1}(s) + V_{n,2}(s)$ where
$$ V_{n,1}(s)=V(\D_{\d_n}, \Phi +\xi, C_{n}, sg_{n}, s^{2}\m_{n}),\quad
V_{n,2}(s)=V(\D_{\d_n},\Phi,S_{L}C_{n},(1- s)g_{n}, (1-s^{2})\m_{n})$$
\\We have
$$\Vert r_{n,1}(\D_{\d_n},\f,\z,0,0)\Vert_{\bf h}
\le {1\over 2} \int_{0}^{1} ds(1-s)^{2}\Vert e^{-V_{n,1}(s)}\Vert_{\bf h}
\Vert e^{-V_{n,2}(s)}\Vert_{\bf h}\Bigl(\Vert p'(s)\Vert_{\bf h}^{3} +
6\Vert p'(s)\Vert_{\bf h} \Vert p_{\m}\Vert_{\bf h}\Bigr) $$
\\$g_n,\m_n$ belong to ${\cal D}_n$. Lemmas~ 5.5 and 5.6 continue to hold
with $g_n,\m_n$ replaced by $sg_n,s^{2}\m_n$ or $(1-s)g_n,(1-s^{2})\m_n$.
We bound
$\Vert e^{-V_{n,1}(s)}\Vert_{\bf h}\le 2$ and
$\Vert e^{-V_{n,2}(s)}\Vert_{\bf h} \le 2e^{-(1-s){{\bar g}\over 4}
\int_{\D_{\d_n}}dx |\f|^{4}(x)} $. We bound the remaining factor (using
Lemma~5.6) by $C_{L}{\bar g}^{3\over 4}{\hat G}_{\r,\k}(\D_{\d_n},\f,\z)
e^{(1-s){\bar g}3\g \int_{\D_{\d_n}}dx |\f|^{4}(x)}$. We put the three bounds
together and choose $0<\g< {1\over 12}$. This gives the bound \equ(5.8).
The proof of \equ(5.8.1) is similar. \bull
\vglue0.3cm
\\{\it Lemma 5.8
\\Under the conditions for the domain ${\cal D}_{\d_n}$ there exists a
constant $C_L$, independent of $n$ and $\e$ but dependent on $L$ such that
$$\Vert P_{n}(\l)\Vert_{{\bf h},\hat G_{\k,\r},\AA,\d_n}\le
C_{L}\vert\l {\bar g}^{1/4}\vert
\qquad {\rm for } \ |\l {\bar g}^{1/4}| \le 1
\Eq(5.13)
$$
$$\Vert P_{n}(\l)\Vert_{{\bf h}_{*},\tilde G_{\k,\r},\AA,\d_n}
\le C_{L}\vert\l {\bar g}^{1-\delta /2}\vert
\qquad {\rm for } \ |\l {\bar g}^{1-\delta /2}| \le 1
\Eq(5.13.1)$$
}
\vglue.3truecm
\\{\it Proof}:
This follows on applying Lemmas 5.5, 5.6 and 5.7 to $ P_{n}(\l)$ defined in
\equ(4.9). \bull
\vglue0.3cm
\\{\it Estimates for $Q_ne^{-V_n}$}
\\We now turn to the estimate of $Q_n e^{-V_n}$. From \equ(4.19.1)
$$ Q_{n}(X_{\d_n},\Phi)=Q(X_{\d_n},\Phi;C_{n},{\bf w}_{n},g) =
g_{n}^{2}\sum_{j=1}^{3}Q^{(j,j)}(\hat X_{\d_n},\Phi;C_n,w_{n}^{(4-j)})
\Eq(5.18)$$
where the $Q^{(m,m)}$ are given
in \equ(4.18). Under an iteration, see Proposition~4.1, we have
$$w_{n}^{(p)}\rightarrow w_{n+1}^{(p)}=v_{n+1}^{(p)}+w_{n,L}^{p}$$,
\\where $p=1,2,3$ and the $v_{n}^{(p)}$ are given in Proposition~4.1.
Starting with $w_0^{(p)}=0$ we get by iterating
$$w_{n}^{(p)}=\sum_{j=0}^{n-1}v_{n-j,L^{j}}^{(p)}\Eq(5.19)$$
\\For every integer $n\ge 0$
we consider the Banach spaces ${\cal W}_{p,\d_n}$ of functions
$f:\> (\d_n\math{Z})^3\mapsto \bf R$ with norms
$\Vert\cdot\Vert_{p,n}$, $p=1,2,3$:
$$\Vert f\Vert_{p,n}= \sup_{x\in (\d_n\math{Z})^3}
\left((|x|+\d_n)^{6p+1\over
4}|f(x)|\right)\Eq(5.20)$$
\\We define the Banach space ${\cal W}_{n}= {\cal
W}_{1,n}\times {\cal W}_{2,n}\times{\cal W}_{3,n}$ consisting of
vectors
${\bf f}=(f^{(1)},f^{(2)},f^{(3)})$, $f^{(p)}:\>(\d_n\math{Z})^3\mapsto \bf R$
with the norm
$$\Vert{\bf f}\Vert_{n} = \max_{p} \Vert f^{(p)}\Vert_{n}\Eq(5.20a) $$
\\Let ${\bf w}_n=(w_n^{(1)},w_n^{(2)},w_n^{(3)})$ as above.
\vglue.3truecm
\\{\it Lemma~5.9
\\1. For $L$ sufficiently large and $\e>0$ sufficiently small there
exists a constant $k_{L}$ independent of $n$ and $\e$ such that
for all $n\ge 1$,
$$\Vert {\bf w}_n\Vert_{n}\le k_{L}/2 \Eq(5.20b) $$
\\If we start the sequence $\{{\bf w}_n\}_{n\ge 0}$ with
$ {\bf w}_0\not = 0$, with $\Vert {\bf w}_n\Vert_{\d_0}\le k_{L}/2 $, then
$$\Vert {\bf w}_n\Vert_{n}\le k_{L},\> \forall\> n\ge 0 \Eq(5.20c) $$
\\2. There exists a function ${\bf w}_{*}$ defined on
$\cup_{n\ge 0}(\d_n\math{Z})^{3}\subset \math{Q}^{3}$
such that for every integer $l\ge 0$ held fixed , the sequence
$\{{\bf w}_n\}_{l\le n} $ converges to ${\bf w}_{*}$ in the
norm $\Vert\cdot\Vert_{l}$ as $n\rightarrow \io$. The convergence rate
is given by
$$\Vert {\bf w}_n-{\bf w}_{*}\Vert_{l}\le \tilde c_L L^{-qn} \Eq(5.20d) $$
\\where $q>0$ is the constant in Theoerem~1.1 and Corollary~1.2..
We have ${\bf w}_{*}={\bf v}_{c*}+{\bf w}_{*,L}$ in ${\cal W}_l$
for every $l\ge 0$.
}
\vglue.3truecm
\\{\it Proof}:
\\1. Let $m=n-j$ with $0\le j\le n-1$. By definition
$v_{m}^{(p)}=C_{m,L}^{p}- C_{m+1}^{p}$ with pointwise multiplication. Since
$C_{m,L}=\G_{m,L}+ C_{m+1}$, it follows that $v_{m}^{(p)}$ has $\G_{m,L}$ as
factor. From the finite range property of $\G_{m,L}$ it follows that
$$v_{m}^{(p)}(x)=0:\> |x|\ge 1$$
\\Theorem~1.1, part (5a), and Corollary~1.2
give uniform bounds on the $\G_{m}$ and $C_{m}$. Therefore there exists a
constant $c_{L,p}$ independent of $n$ such that
$$\Vert v_{m}^{(p)}\Vert_{L^{\infty}((\d_m\math{Z})^{3})} \le c_{L,p}$$
\\2. By definition
$$\eqalign{\Vert v_{n-j,L^{j}}^{(p)}\Vert_{p,n}& =
\sup_{x\in (\d_n\math{Z})^{3}}
\left(L^{2d_{s}j}(|x|+\d_n)^{6p+1\over 4}|v_{n-j}^{(p)}(L^{j}x)|\right)\cr
&= L^{-j({6p+1\over 4}- 2d_{s})}\sup_{y\in (\d_{n-j}\math{Z})^{3}}
\left((|y|+\d_{n-j})^{6p+1\over 4}|v_{n-j}^{(p)}(y)|\right) \cr}
$$
\\Because of the finite range property of $v_{n-j}^{(p)}$ of paragraph 1,
we can bound $|y|\le 1$ in the weight factor on the right. Because
$n-j\ge 1$ we can bound in the weight factor $\d_{n-j}\le \d_1=L^{-1}$.
Therefore on using the bound
on $v_{n-j}^{(p)}$ of paragraph 1 we get
$$\Vert v_{n-j,L^{j}}^{(p)}\Vert_{p,n}
\le L^{-j({6p+1\over 4}- 2d_{s})}(1+L^{-1})^{6p+1\over 4}c_{L,p}$$
\\We bound the first geometric factor by taking $p=1$ and
$\e>0$ very small in $d_{s}=(3-\e)/4$. This gives $L^{-j/5}$.
This gives the bound
$$\Vert v_{n-j,L^{j}}^{(p)}\Vert_{p,n}\le L^{-j/5}\>c_{L,p} $$
\\with a new constant $c_{L,p}$ independent of $n$. Using the above bound
we get from \equ(5.19) the bound
$$\Vert w_n^{(p)}\Vert_{p,n} \le c_{L,p}
\sum_{j=0}^{\infty}L^{-j/5}\le 2c_{L,p} $$
\\for $L$ sufficiently large. Therefore setting $k_{L}=4\max_{p}c_{L,p}$ we get
$$\Vert {\bf w}_n\Vert_{n} \le k_{L}/2$$
\\which proves \equ(5.20b). \equ(5.20c) is a trivial consequence of the above.
This proves the first part of the lemma.
\\3. Let $v_{c*}^{(p)}=C_{c*,L}^p-C_{c*}^p$, with pointwise multiplication,
where $C_{c*}$ is the smooth continuum covariance in $\math{R}^3$ of
Corollary~1.2. By factoring out $\G_{c*,l}$, Theorem~1.1 and Corollary~1.2
we have that $v_{c*}^{(p)}$ exists in $L^{\io}(\math{R}^3)$ and has finite
range: $v_{c*}^{(p)}(x)=0 : |x|\ge 1$. Moreover by Theorem~1.1 and
Corollary~1.2 we have, see the proof of Lemma~5.12 for the detailed argument,
$$\Vert v_{m}^{(p)}- v_{c*}^{(p)}\Vert_{L^{\infty}((\d_{m}\math{Z})^3)}
\le c_{L,p}L^{-q(m-1)} $$
\\Define
$$w_{*}^{(p)}=\sum_{j=0}^{n-1}v_{c*,L^{j}}^{(p)} $$
\\Fix any integer $l\ge 0$. Then for $n\ge l$ the $(p,l)$ norm is dominated by
the $(p,n)$ norm. Then proceeding as in the first part and using the previous
inequality we get
$$\eqalign{\Vert w_n^{p}-w_{*}^{p}\Vert_{p,l} &\le
\sum_{j=0}^{n-1}\Vert v_{n-j,L^j}^{p}-v_{c*,L^j}^{p}\Vert_{p,n}\le
c_{L,p} \sum_{j=0}^{n-1}L^{-j({6p+1\over 4}- 2d_{s})}
\Vert v_{n-j}^{p}-v_{c*}^{p}\Vert_{L^{\infty}((\d_{n-j}\math{Z})^3)} \cr
&\le c_{L,p}\sum_{j=0}^{n-1} L^{-j/5}L^{-q(n-j-1)} \le c'_{L,p} L^{-qn}
}$$
\\Now take the maximum over $p$. This proves \equ(5.20) and at the same time
the convergence statement of part 2 of the lemma. The last statement of
part 2 is trivial to prove. This completes the proof of the second part of
the lemma. \bull
\vglue.3cm
\\{\it Lemma 5.10
\\Under the conditions of the domain ${\cal D}_n$ there exists
constants $C_{p,L}$ independent of $n$ such that
$$\Vert Q_{n}e^{-V_{n}}\Vert_{{\bf h},G_{\k},\AA_p,\d_n}\le
C_{p,L}\>{\bar g}^{1/2}
\Eq(5.55.1)
$$
$$\vert Q_{n}e^{-V_{n}}\vert_{{\bf h}_{*},\AA_p,\d_n}
\le C_{p,L}\>{\bar g}^{2}
\Eq(5.55.2)$$
}
\vglue.3truecm
\\{\it Proof}
$$Q_{n}(X_{\d_n})e^{-V_{n}(X_{\d_n})}=
g_{n}^2\sum_{m=1}^3 Q^{(m,m)}(\hat X_{\d_n},\Phi;C_{n},w_{n}^{(4-m)})
e^{-V_{n}(X_{\d_n})}\Eq(5.55.3)$$
$$\left\Vert Q_{n}(X_{\d_n},\f)e^{-V_{n}(X_{\d_n},\f)}\right\Vert_{\bf h} \le
g_{n}^{2}\sum_{m=1}^3
\left\Vert Q^{(m,m)}(\hat X_{\d_n},\Phi;C_{n},w_{n}^{(4-m)})\right\Vert_{\bf h}
\left\Vert e^{-V_{n}(X_{\d_n})}\right\Vert_{\bf h}\Eq(5.55.4)$$
Here $X_{\d_n}$ is a small set because of the support properties of $Q_{n}$. The last
factor will be estimated by Lemma 5.5. From \equ(4.18) we have
$$Q^{(3,3)}(\hat X_{\d_n},\Phi;w_{n}^{(1)})=4
\int_{\hat X_{\d_n}}dxdy\
:\Phi(x)\bar\Phi(x)\Phi(x)\bar\Phi(y)\Phi(y)\bar\Phi(y):_{C_{n}}
w_{n}^{(1)}(x-y)\Eq(5.55.5)$$
\\We exhibit \equ(5.55.5) as an element of the Grassmann algebra:
$$Q^{(3,3)}(\hat X_{\d_n},\Phi;w_{n}^{(1)})=
Q_0^{(3,3)}(\hat X_{\d_n},\f;w_{n}^{(1)})+
\int_{\hat X_{\d_n}}dxdy\
Q_1^{(3,3)}(\hat X_{\d_n},\f,x,y;w_{n}^{(1)}):\psi(x)\bar\psi(y):_{C_n}+$$
$$+\int_{X_{\d_n}}dx\
Q_2^{(3,3)}(\hat X_{\d_n},\f,x;w_{n}^{(1)}):\psi(x)\bar\psi(x):_{C_n}+$$
$$+\int_{\hat X_{\d_n}}dxdy\
Q_3^{(3,3)}(\hat X_{\d_n},\f,x,y;w_{n}^{(1)}):\psi(x)\bar\psi(x)\psi(y)\bar\psi(vy):_{C_n}
\Eq(5.55.6)$$
\\where
$$\eqalign{Q_0^{(3,3)}(\hat X_{\d_n},\f;w_{n}^{(1)})&=
4\int_{\hat X_{\d_n}}dxdy\
:\f(x)\bar\f(x)\f(x)\bar\f(y)\f(y)\bar\f(y):_{C_{n}}
w_{n}^{(1)}(x-y)\cr
Q_1^{(3,3)}(\hat X_{\d_n},\f,x,y;w_{n}^{(1)})&=
4:\f(x)\bar\f(x)\f(y)\bar\f(y):_{C_{n}}
w_{n}^{(1)}(x-y)\cr
Q_2^{(3,3)}(\hat X_{\d_n},\f,x;w_{n}^{(1)})&=
4\int_{X_{\d_n}^{\prime}}dy:(\f(x)\bar\f(y)+\f(y)\bar\f(x))
\f(y)\bar\f(y):_{C_{n}}
w_{n}^{(1)}(x-y)\cr
Q_3^{(3,3)}(\hat X_{\d_n},\f,x,y;w_{n}^{(1)})&=
4:\f(x)\bar\f(y):_{C_{n}}
w_{n}^{(1)}(x-y)\cr}\Eq(5.55.7)$$
where, denoting with $\D(x)$ the block $\D$ such that
$x\in\D$, we have $X_{\d_n}^{\prime}=\cases{\D&if $X_{\d_n}=\D$\cr X_{\d_n}\setminus\D(x)&if
$X_{\d_n}=\D_1\cup\D_2$\cr}$.
\\Undo the Wick ordering, which produces lower order terms with coefficients
which are uniformly bounded independent of $n$ by Corollary~1.2. It is
therefore enough to estimate with the Wick ordering taken off. We get
$$\Vert Q^{(3,3)}(\hat X_{\d_n},\Phi;w_{n}^{(1)})\Vert_{\bf h}\le
\Vert Q_0^{(3,3)}(\hat X_{\d_n},\f;w_{n}^{(1)})\Vert_{h_B}+$$
$$+
h_F^2\sup_{\Vert g_2\Vert_{C^{2}(\hat X_{\d_n}^2)} \le 1}
\int_{\hat X_{\d_n}}dxdy\
\Vert Q_1^{(3,3)}(\hat X_{\d_n},\f,x,y;w_{n}^{(1)})\Vert_{h_B}
|g_2(x,y)|+$$
$$+h_F^2\sup_{\Vert g_2\Vert_{C^{2}(\hat X_{\d_n}^2)}\le 1}\int_{X_{\d_n}}dx\
\Vert Q_2^{(3,3)}(\hat X_{\d_n},\f,x;w_{n}^{(1)})\Vert_{h_B}
|g_2(x,x)|+$$
$$+h_F^4\sup_{\Vert g_4 \Vert_{C^{2}(\hat X_{\d_n}^4)}\le 1}
\int_{\hat X_{\d_n}}dxdy\
\Vert Q_3^{(3,3)}(\hat X_{\d_n},\f,x,y;w_{n}^{(1)})\Vert_{h_B}
|g_4(x,x,y,y)|$$
\\To estimate the $h_B$ norm of the $Q_j^{(3,3)}$ we apply to \equ(5.55.7)
$h_B^k D^{k}$, with $D$ the bosonic field derivative, $h_B=c{\bar g}^{-1/4}$
and use Lemma~5.1. Contributions for $k>4$ vanish.
We use $g_{n}=O({\bar g})$ from the domain hypothesis of
Theorem~5.1. We estimate the kernel $w_{n}^{(1)}(x-y)$ using Lemma~5.9.
As a result we get
$$\Vert Q^{(3,3)}(\hat X_{\d_n},\Phi;w_{n}^{(1)})\Vert_{\bf h}\le
C_L {\bar g}^{-3/2}\int_{\hat X_{\d_n}}dxdy\
{1\over(|x-y| +\d_n)^{7/4}}
e^{ {\bar g}/4\int_{X_{\d_n}}dx|\f\bar\f(x)|^2}G_\k(X_{\d_n},\f)
\Eq(5.55.8)$$
\\The integral over $\hat X_{\d_n}$ exists and is of $O(1)$ since
$X_{\d_n}$ is a small set in $(\d_n\math{Z})^{3}$. Therefore
$$\Vert Q^{(3,3)}(\hat X_{\d_n},\Phi;w_{n}^{(1)})\Vert_{\bf h}\le
C_L {\bar g}^{-3/2}
e^{ {\bar g}/4\int_{X_{\d_n}}dx|\f\bar\f(x)|^2}G_\k(X_{\d_n},\f)
\Eq(5.55.8a)$$
\\Next turn to $Q^{(m,m)}$, $m=1,2$. From \equ(4.18)
$$Q^{(2,2)}(\hat X_{\d_n},\Phi;C,w_{n}^{(2)})=
-\int_{\hat
X_{\d_n}}dxdy\left[:(\Phi(x)-\Phi(y))(\bar\Phi(x)-\bar\Phi(y))
(\Phi(x)+\Phi(y))(\bar\Phi(x)+\bar\Phi(y)):_{C_{n}}\right.
+$$
$$\left.+:[(\Phi\bar\Phi)(x)-(\Phi\bar\Phi)(y)]^2:_{C_{n}}
\right]w_{n}^{(2)}(x-y)$$
\\We exhibit this as an element of the Grassmann algebra. This gives
$$Q^{(2,2)}(\hat X_{\d_n},\Phi;C_n,w_{n}^{(2)})=
-Q^{(2,2)}_{0}(\hat X_{\d_n},\f;C_n,w_{n}^{(2)})-\int_{\hat
X_{\d_n}}dxdy $$
$$\left[Q^{(2,2)}_{1}(\hat X_{\d_n},\f,x,y;C_n,w_{n}^{(2)})
:(\psi(x)+\psi(y))(\bar\psi(x)+\bar\psi(y)):_{C_{n}}+\right.$$
$$Q^{(2,2)}_{2}(\hat X_{\d_n},\f,x,y;C_n,w_{n}^{(2)},)
:(\psi(x)-\psi(y))(\bar\psi(x)-\bar\psi(y)):_{C_{n}}+$$
$$+Q^{(2,2)}_{3}(\hat X_{\d_n},\f,x,y;C_n,w_{n}^{(2)})
:(\psi\bar\psi(x)-\psi\bar\psi(y)):_{C_{n}}+$$
$$+Q^{(2,2)}_{4}(\hat X_{\d_n},x,y,w_{n}^{(2)})
\{(:\psi(x)-\psi(y))(\bar\psi(x)-\bar\psi(y))
(\psi(x)+\psi(y))(\bar\psi(x)+\bar\psi(y)):_{C_{n}}+$$
$$\left. :(\psi\bar\psi(x)-\psi\bar\psi(y))^{2}):_{C_{n}})\}
\right]\Eq(5.55.10)
$$
\\where
$$\eqalign{Q^{(2,2)}_{0}(\hat X_{\d_n},\f;C_n,w_{n}^{(2)})=&\int_{\hat
X_{\d_n}}dxdy\>w_{n}^{(2)}(x-y)(:|\f(x)-\f(y)|^{2}|\f(x)+\f(y)|^{2}:_{C_{n}}\cr
&+:(|\f|^{2}(x)-|\f|^{2}(y))^{2}:_{C_{n}})\cr
Q^{(2,2)}_{1}(\hat X_{\d_n},\f,x,y;C_n,w_{n}^{(2)})=&
w_{n}^{(2)}(x-y):|\f(x)- \f(y)|^{2}_{C_n} \cr
Q^{(2,2)}_{2}(\hat X_{\d_n},\f,x,y;C_n,w_{n}^{(2)})=&
w_{n}^{(2)}(x-y) :|\f(x)+\f(y)|^{2}:_{C_{n}} \cr
Q^{(2,2)}_{3}(\hat X_{\d_n},\f,x,y;C_n,w_{n}^{(2)} )=&
2w_{n}^{(2)}(x-y):(|\f|^{2}(x)-|\f|^{2}(y)) :_{C_{n}}\cr
Q^{(2,2)}_{4}(\hat X_{\d_n},x,y,w_{n}^{(2)} )=&
w_{n}^{(2)}(x-y) \cr}\Eq(5.55.11)$$
\\ The $\Vert \cdot\Vert_{{\bf h},G_{\k},\AA_{p}}$ norm
estimate for $Q^{(2,2)}(\hat X_{\d_n},\Phi;C,w_{n}^{(2)})$ reposes on
the following principles:
\\1. Undoing the Wick ordering produces lower order terms
with Wick coefficients which are uniformly bounded independent of $n$ by
Corollary~1.2. Moreover by the domain hypothesis $g_{n}=O({\bar g})$
\\2. By Lemma~5.9, the kernel $w_{n}^{(2)}$ has the bound
$|w_{n}^{(2)}(x-y)|\le k_{L}(|x-y|+\d_n)^{-13/4}$ where the constant $k_L$ is
independent of $n$.
\\3. The fields $\f(x)$ are estimated by Lemma~5.1. Differences of fields
$|\f(x)-\f(y)|$ are estimated by \equ(5.1b). This produces a factor $|x-y|$
which we retain, and majorise the Sobolev factor by the large field regulator.
Diffferences of fields $\f\bar\f(x)-\f\bar\f(y)$ can also be expressed as
in \equ(5.1a), substituting $\f\bar\f$ for $\f$. This requires estimating
$(\dpr_{\d_n,e_{j}}\f\bar\f)(x+\cdot\cdot)$. We apply the lattice Leibniz which
modifies
the continuum rule by producing an extra term
$\d_{n} |\dpr_{\d_n,e_{j}}\f(x+\cdot\cdot)|^{2}$,
(see equation (5.2),page 432 of [BGM]).
We estimate the $\f$ by Lemma~5.1, with $\k/2$ in the large field regulator.
We estimate the gradient pieces by the Sobolev inequality as in
\equ(5.1b), and then by the large field regulator with $\k/2$. We have also
produced a factor $|x-y|$ as in \equ(5.1b).
\\Invoking the above principles we get the following bounds for the bosonic
coefficients:
$${h_{B}^{k}\over k!}
\Vert D^{k}Q^{(2,2)}_{0}(\hat X_{\d_n},\f;C_n,w_{n}^{(2)})\Vert
\le c_{L}{\bar g}^{-1/2}\int_{\hat X_{\d_n}}dxdy\>
(|x-y|+\d_n) ^{-({13\over 4} +2)}
e^{ {\bar g}/4\int_{X_{\d_n}}dx|\f\bar\f(x)|^2}G_\k(X_{\d_n},\f) \Eq(5.55.81)$$
\\and for $j=1,2,3$
$${h_{B}^{k}\over k!}
\Vert D^{k}Q^{(2,2)}_{j}(\hat X_{\d_n},\f;x,y,C_n,w_{n}^{(2)})\Vert
\le c_{L}{\bar g}^{-1/2}\>(|x-y|+\d_n)^{-13/4}f_{j}(x-y)
e^{ {\bar g}/4\int_{X_{\d_n}}dx|\f\bar\f(x)|^2}G_\k(X_{\d_n},\f) \Eq(5.55.82)$$
\\where the maximum value of $k$ which gives a nonvanising contribution
is $4$ and
$$f_{1}(x-y)=|x-y|^{2},\>f_{2}(x-y)=1, \>
f_{3}(x-y)=|x-y|,f_{4}(x-y)=1 \Eq(5.55.83)$$
\\4. We must estimate the contribution of the fermionic pieces to the
$\bf h$ norm. To this end
denote by $F_j(\psi)$ the fermionic factor multiplying $Q^{(2,2)}_{j}$
in \equ(5.55.10). Express the differences $\psi(x)-\psi(y)$ by the fermionic
analogue of \equ(5.002). We do the same also for
$\psi\bar\psi(x)-\psi\bar\psi(y)$ and then apply the lattice Leibnitz rule
to $\dpr_{\d_{n},\e_j}\psi\bar\psi(x+\cdot)$. We replace the fermionic
pieces by the functions $g_{2p}$ on $\hat X_{\d_n}\cup\dpr_{2}\hat X_{\d_n}$.
and their lattice derivatives. Corresponding to $F_j(\psi)$ we get the
contribution $G_{ j}$ which is a linear form on $g_{2p_j}$, where
$p_j=1$ for $j=1,2,3$ and $p_4=2$.
Let $\d_n h_j$, $h_j\in \bf Z$ be the component of $y-x$ along the unit vector
$e_j$. We have
$$G_{1}=g_2(x,x)+g_2(x,y)+g_2(y,x)+g_2(y,y)$$
$$G_{2}=\d_n^{2}\sum_{i_1,i_2=1}^{3}\sum_{0\le s_{i_l}\le h_{i_l}-1,\>l=1,2}
\dpr_{\d_n,e_{i_1}}^{(1)}\dpr_{\d_n,e_{i_2}}^{(2)}
g_2(x+ p_{i_1}(y-x, s_{i_1}),x+ p_{i_2}(y-x, s_{i_2}) )$$
$$G_{3}=\d_n\sum_{i=1}^{3}\sum_{s_i=0}^{h_i-1}
\Bigl[\dpr_{\d_n,e_{i}}^{(1)}g_2(x+ p_{i}(y-x,s_{i}),
x+p_{i}(y-x,s_{i})) +\dpr_{\d_n,e_{i}}^{(2)}g_2(x+p_{i}(y-x,s_{i}),
x+p_{i}(y-x,s_{i}))+$$
$$+\d_n\dpr_{\d_n,e_{i}}^{(1)}\dpr_{\d_n,e_{i}}^{(2)}
g_2(x+ p_{i}(y-x,s_{i}),x+p_{i}(y-x,s_{i}))\Bigr] $$
$$G_{4}=\d_n^{2}\sum_{i_1,i_2=1}^{3}\sum_{0\le s_{i_l}\le h_{i_l}-1,
\>l=1,2}\dpr_{\d_n,e_{i_1}}^{(1)}\dpr_{\d_n,e_{i_2}}^{(2)}
\Bigl(g_4(x+p_{i_1}(y-x,s_{i_i}),x+ p_{i_2}(y-x,s_{i_2}),x,x)+$$
$$+g_4(x+\cdot\cdot,x+\cdot\cdot,x,y)
+g_4(x+p_{i_1}(y-x,s_{i_1}),x+p_{i_2}(y-x,s_{i_2}),y,x)+$$
$$+g_4(x+p_{i_1}(y-x,s_{i_1}),x+ p_{i_2}(y-x,s_{i_2}) ,y,y)\Bigr)
+\cdot\cdot $$
\\where the superscript on the lattice derivative denotes the argument on which
it acts. The omitted terms $\cdot\cdot$ in $G_4$ comes from the square of the
(first order) lattice Taylor expansion of $\psi\bar\psi(x)-\psi\bar\psi(y)$
and then replacing the product of $4$ Grassmann fields by the test function
$g_4$. For $j=1,2,3$ we have the bounds
$$|G_{j}|
\le O(1){\tilde f}_j(x-y)\Vert g_2\Vert_{C^{2}({\hat X}_{\d_n}^{2}) }
\Eq(5.55.84) $$
\\and for $j=4$ we have
$$|G_{4}|
\le O(1){\tilde f}_4(x-y)\Vert g_4\Vert_{C^{2}({\hat X}_{\d_n}^{4}) }
\Eq(5.55.85) $$
\\where
$$\tilde f_1=|x-y|,\>\tilde f_2=|x-y|^2,\>\tilde f_3=|x-y|,
\>\tilde f_4=|x-y|^2 \Eq(5.55.86) $$.
\\On using the bounds \equ(5.55.81)-\equ(5.55.86) we get for $0\le k\le 4$
and $0\le p\le 2$
$$h_{F}^{2p}{h_B^k\over k!}
\vert D^{2p,n}Q^{(2,2)}({\hat X}_{\d_n},\f,0;f^{\times k},g_{2p})\vert
\le c_{L}{\bar g}^{-1/2}\int_{\hat X_{\d_n}}dxdy\>
(|x-y|+\d_n)^{-({13\over 4} -2)}$$
$$\times e^{ {\bar g}/4\int_{\hat X_{\d_n}}dx|\f\bar\f(x)|^2}G_\k(X_{\d_n},\f)
\Vert f\Vert_{C^{2}({\hat X}_{\d_n})}^{\times k}
\Vert g_{2p}\Vert_{C^{2}({\hat X}_{\d_n}^{2p}) } \Eq(5.55.87)$$
\\For $k>4$ or $p>2$ we have vanishing contribution. The integral over
$\hat X_{\d_n}$ exists and gives a contribution of $O(1)$ since
$X_{\d_n}$ is a small set in $ (\d_n\math{Z})^3$. Therefore we obtain
from the previous inequality
$$\Vert Q^{(2,2)}({\hat X}_{\d_n},\f,0)\Vert_{\bf h}
\le c_{L}{\bar g}^{-1/2}
e^{ {\bar g}/4\int_{\hat X_{\d_n}}dx|\f\bar\f(x)|^2}G_\k(X_{\d_n},\f)
\Eq(5.55.88) $$
\\We can estimate in the same way the case $m=1$.
We have
$$\Vert Q^{(1,1)}(\hat X_{\d_n},\Phi;C,w^{(3)})\Vert_{\bf h}\le
c_{L}{\bar g}^{-1/2} \int_{\hat X_{\d_n}}dxdy\
(|x-y|+\d_n)^{-({19\over 4}-2)}
e^{ {\bar g}/4\int_{X_{\d_n}}dx|\f\bar\f(x)|^2}G_\k(X_{\d_n},\f) \Eq(5.55.13)
$$
\\The integral over $\hat X_{\d_n}$ exists and gives a contribution of
$O(1)$. Therefore
$$\Vert Q^{(1,1)}(\hat X_{\d_n},\Phi;C,w^{(3)})\Vert_{\bf h}\le
c_{L}{\bar g}^{-1/2}
e^{{\bar g}/4\int_{X_{\d_n}}dx|\f\bar\f(x)|^2}G_\k(X_{\d_n},\f) \Eq(5.55.13a)
$$
\\Therefore from \equ(5.55.4), Lemma 5.5 and the bounds \equ(5.55.8a),
\equ(5.55.88),\equ(5.55.13a) we get
$$\left\Vert Q_n(X_{\d_n})e^{-V_n(X_{\d_n})}\right\Vert_{\bf h,G_\k}
\le c_{L}{\bar g}^{1/2}$$
and since $Q_n$ is supported on small sets we get
$$\Vert Q_ne^{-V_n}\Vert_{{\bf h},G_{\k},\AA_p}\le
C_{L,p}{\bar g}^{1/2}
\Eq(5.55.14)
$$
which is \equ(5.55.1). To prove \equ(5.55.2) we estimate the r.h.s of
\equ(5.55.4) at $\Phi=0$ after undoing the Wick ordering, set
${\bf h}={\bf h}_*$, and use Lemma~5.5. \bull
\vglue0.1cm
\\In the following lemma we consider $Q_n(\Phi+\xi)e^{-V_n(\Phi+\xi)}$ as
a function of $\f,\z,\psi,\eta$.
\vglue.3truecm
\\{\it Lemma 5.11
\\Under the conditions of the domain ${\cal D}_n$ there exists constants
$C_{L,p}$ independent of $n$ such that
$$\Vert Q_ne^{-V_n}\Vert_{{\bf h},\hat G_{\k,\r},\AA_p,\d_n}\le
C_{L,p}{\bar g}^{1/2}
\Eq(5.55.15)
$$
$$\Vert Q_ne^{-V_n}\Vert_{{\bf h}_{*},\tilde G_{\k,\r},\AA_p,\d_n}
\le C_{L,p}{\bar g}^{2}
\Eq(5.55.16)$$
} \vglue.3truecm
\\{\it Proof}
\\The bound \equ(5.55.15) follows from \equ(5.55.1) since
$\hat G_{\k,\r}>G_{\k}$. To prove \equ(5.55.16) we first express
$Q_n^{(m,m)}(X_{\d_n},\f+\z,\psi+\eta) $ in the Grassmann representation as in the proof of Lemma~5.10, substituting in the expressions there
$\f\rightarrow\f+\z$, $\psi\rightarrow\psi+\h$. Field derivatives are
defined as in \equ(2.2.2). For the bosonic
coefficients we take derivatives at $\f=0$.
The resulting dependence on $\z$ is estimated by Lemma~5.2. The rest of the
proof follows that of Lemma~5.10. We use Lemma~5.5 which implies that
$\Vert e^{-V_n(X_{\d_n},\z,\eta)}\Vert_{{\bf h}_*}\le 2^{|X_{\d_n}|}$,
and $X_{\d_n}$ is a small set. \bull
\vglue.3truecm
\\We now prove a lemma to
control the perturbative flow coefficients $a_n,b_n$ given in \equ(4.47) and
\equ(4.54.1). This lemma is independent of the domain ${\cal D}_n$.
\vglue.3truecm
\\{\it Lemma~5.12
\\Let $v_{c*}^{(p)}=C_{c*,L}^p-C_{c*}^p$, with pointwise multiplication,
where $C_{c*}$ is the smooth continuum covariance in $\math{R}^3$ of
Corollary~1.2. Define
$$a_{c*}=2\int_{\math{R}^3}dy\> v_{c*}^{(2)}(y),\quad
b_{c*}=4\int_{\math{R}^3}dy\> v_{c*}^{(3)}(y)$$
\\We have that $a_n,b_n,a_{c*},b_{c*}$ are strictly positive. Moreover
there exist constants $c_{L}$ independent of $n$ such that
$$|a_n|\le c_L, \quad |b_n|\le c_L , \quad |a_{c*}|\le c_L,
\quad |b_{c*}| \le c_L \Eq(5.361) $$
\\and
$$|a_n-a_{c*}|\le c_{L}L^{-qn},\quad |b_n-b_{c*}|\le c_{L}L^{-qn} \Eq(5.362) $$
\\where $q>0$ is as in Therem~1.1.
}
\\{\it Remark} : The convergence rate estimates \equ(5.362) play no role
in the estimates of the present section. They are used in Section~6 for the
existence proof of a global renormalization group trajectory.
\vglue.3truecm
\\{\it Proof}
\\From \equ(4.54), for $p=2,3$, using $C_{n,L}=\G_{n,L}+C_{n+1}$
$$v_{n+1}^{(p)}=C_{n,L}^p-C_{n+1}^p
=\G_{n,L}(\G_{n,L}^{p-1}+p\G_{n,L}^{p-2}C_{n+1}+\d_{p,3}3C_{n+1}^2)
\Eq(5.36)$$
\\with pointwise multiplication. The positivity in Fourier space of the
integral kernels on the right hand side implies that $a_n>0,\> b_n>0$
as claimed. The common factor of $\G_{n,L}(x)$ which has finite range $1$
implies that $v_{n+1}^{(p)}(x)$ has support in the unit ball in
$(\d_{n+1}\math{Z})^3$. From Theorem.1.1 and Corollary~1.2 we have that
$v_{n+1}^{p}$ above are uniformly bounded in
$L^{\infty}((\d_{n+1}\bf Z)^{3})$ by constants $c_L$. By the same arguments
$v_{c*}$ has finite range and belongs to $L^{\infty}(\math{R}^{3})$.
The uniform bounds in the first part of the lemma now follow.
\\By the same arguments using
$C_{*,L}=\G_{c*}+C_{c*}$ we have that $a_{c*}>0,\>b_{c*}>0$ and
$v_{c*}^{(p)}(x)$ has support in the unit ball in $\math{R}^3$. Moreover
using Corollary~1.2 we have $\Vert v_{c*}^{(p)}\Vert_{C^{k}(\math{R}^{3})}
\le c_{k,L}$ for all $\k\ge 0$.
\\Define
$$a_n^{(p)}=\int_{(\d_{n+1}\math{Z})^3}dy\>v_{n+1}^{(p)}(y),\quad
a_{c*}^{(p)}=\int_{\math{R}^3}dy\>v_{c*}^{(p)}(y) \Eq(5.36.1)$$
\\Then using the compact support property of $v_{n+1}^{(p)}$ and
$v_{c*}^{(p)}$ we get
$$|a_n^{(p)}-a_{c*}^{(p)}|\le \Vert v_{n+1}^{(p)}- v_{c*}^{(p)}
\Vert_{L^{\infty}((\d_{n+1}\math{Z})^3)}
+\left\vert \int_{(\d_{n+1}\math{Z})^3}dy\>v_{c*}^{(p)}(y)-
\int_{\math{R}^3}dy\>v_{c*}^{(p)}(y)\right\vert \Eq(5.36.2)
$$
\\We estimate the first term on the right hand side of \equ(5.36.2). We have
$$\Vert v_{n+1}^{(p)}- v_{c*}^{(p)}\Vert_{L^{\infty}((\d_{n+1}\math{Z})^3)}
\le \Vert C_{c*,L}^{p}-C_{n,L}^{p}\Vert_{L^{\infty}((\d_{n+1}\math{Z})^3)}+
\Vert C_{c*}^{p}-C_{n+1}^{p}\Vert_{L^{\infty}((\d_{n+1}\math{Z})^3)}\Eq(5.36.3)
$$
\\In the first term on the right in \equ(5.36.3)
we factor out $C_{c*,L}-C_{n,L}$ and
in the second term we factor out $C_{c*}-C_{n+1}$. Then use of the bounds
in Corollary~1.2 gives
$$\Vert v_{n+1}^{(p)}- v_{c*}^{(p)}\Vert_{L^{\infty}((\d_{n+1}\math{Z})^3)}
\le c_{L,p}L^{-qn} \Eq(5.36.4) $$
\\We estimate the second term on the right in \equ(5.36.2) using Lemma~6.6
of [BGM] and the compact support of $v_{c*}$. This gives
$$\left\vert \int_{(\d_{n+1}\math{Z})^3}dy\>v_{c*}^{(p)}(y)-
\int_{\math{R}^3}dy\>v_{c*}^{(p)}(y)\right\vert \le O(1)\d_{n+1}
\Vert v_{c*}^{(p)}\Vert_{C^{1}(\math{R}^3)} \le c_{L,p}L^{-(n+1)}\Eq(5.36.5)$$
\\From \equ(5.36.2), \equ(5.36.4) and \equ(5.36.5) we get with $q$ that of
Theorem~1.1
$$|a_n^{(p)}-a_{c*}^{(p)}|\le c_{L,p}L^{-qn} \Eq(5.36.6)$$
\\which completes the proof of the lemma. \bull
\vglue.3truecm
\\{\it Lemma 5.13
\\Under the conditions of the domain ${\cal D}_n$ there exist constants
$C_{p,L}$ independent of $n$ such that
$$\Vert Q_n(e^{-V_n}-e^{-\tilde V_n})\Vert_{{\bf h},\hat G_{\k,\r},\AA_p,\d_n}
\le C_{L,p}{\bar g}^{3/4}
\Eq(5.55.18)
$$
$$\Vert Q_n(e^{-V_n}-e^{-\tilde V_n})\Vert_{{\bf h}_{*},\tilde G_{\k,\r},\AA_p
,\d_n}
\le C_{L,p}{\bar g}^{3}
\Eq(5.55.19)$$
}
\\{\it Proof}: The proof is on the same lines as that of the corresponding
Lemma 5.13 in [BMS]. It follows from Lemmas 5.11, 5.6, 5.10, and 5.5 which
are lattice equivalents of the corresponding lemmas in [BMS].
\vglue.3truecm
\\{\it Lemma 5.14
\\Under the conditions of the domain ${\cal D}_n$ there exists constants
$C_L$ independent of $n$ such that
$K_n(\l)$ given by \equ(4.10) satisfies the bounds
$$\Vert K_n(\l)\Vert_{{\bf h},\hat G_{\k,\r},\AA,\d_n}\le
C_{L}|\l {\bar g}^{1/4-\h/3}|^2\quad {\rm for}\ |\l{\bar g}^{1/4-\h/3}|<1
\Eq(5.55.20)
$$
$$\Vert K(\l)\Vert_{{\bf h}_{*},\tilde G_{\k,\r},\AA,\d_n }
\le C_{L}|\l{\bar g}^{11/12-\h/3}|^2\quad {\rm for}\
|\l{\bar g}^{11/12-\h/3}|<1
\Eq(5.55.21)$$
}
\\{\it Proof}: This follows from Lemmas 5.11 and 5.13 and the hypothesis
\equ(55.02) on $R_n$.
\vglue.1truecm
\\The following lemma generalizes Lemma~5.15 of [BMS] to the lattice and
the additional presence of Grassmann fields. It will play a key role later
in obtaining contractive estimates.
\vglue0.2cm
\\{\it Lemma 5.15
\\For any polymer activity $\tilde K(X_{\d_n},\phi+\z,\psi+\h)$:
$$\Vert \tilde K(X_{\d_n},\z,0)\Vert_{{\bf h}_{*}}\le
O(1) \tilde G_{\r,\k}(X_{\d_n},\z)\left[
\vert \tilde K(X_{\d_n})\vert_{{\bf h}_{*}}+h_B^{-m_0}h_{B*}^{m_{0}}\Vert
\tilde K(X_{\d_n})\Vert_{{\bf h},G_\k}\right]\Eq(5.45)$$
$$\Vert \tilde K(Y_{\d_n},\phi,0)\Vert_{{\bf h}}\le O(1)
e^{\g {\bar g}\int_{Z_{\d_n}\backslash Y_{\d_n}}dy(\phi\bar\phi(y))^2}G_\k(Z_{\d_n},\phi)
\left[
\vert \tilde K(Y_{\d_n})\vert_{{\bf h}}+L^{-m_0 d_s}\Vert
\tilde K(Y_{\d_n})\Vert_{h_F,L^{d_s}h_B,G_\k}\right]\Eq(5.45b)$$
$$\vert \tilde K^\sharp(X_{\d_n})\vert_{\hat {\bf h}_{*}}\le O(1) 2^{|X_{\d_n}|}\left[ \vert
\tilde K(X_{\d_n})\vert_{{\bf h}_{*}}+h_B^{-m_0}h_{B*}^{m_{0}}\Vert
vv\tilde K(X_{\d_n})\Vert_{{\bf h},G_\k}\right]\Eq(5.46)$$
$$\Vert \tilde K^\sharp(X_{\d_n})\Vert_{\hat {\bf h}, G_{2\k}} \le 2^{|X_{\d_n}|}
\Vert \tilde K(X_{\d_n})\Vert_{{\bf h}, G_{\k}} \Eq(5.4666) $$
\\where $\tilde G_{\r,\k}$ is as defined in \equ(5.2), and $m_0=9$ is
the maximum number of derivatives appearing in the definition of
Kernel and $h$ norms. In \equ({5.45b}), $Y_{\d_n},Z_{\d_n},\gamma$ are as described
in Lemma~5.1. Moreover in the above norms ${\bf h}_{*}=(h_{F},h_{B*})$,
and ${\hat {\bf h}}_{*}=({h_{F}\over 2},h_{B*})$ where
$h_{B*}=(\r\k)^{-1/2}$, ${\bf h}=(h_F,h_B)$,
${\hat {\bf h}}=({h_{F}\over 2},h_{B})$, $h_B=c{\bar g}^{-1/4}$, c=O(1) very
small, and $h_{F}$ is taken to be
sufficiently large depending on $L$. The last condition plays a role in the
proofs of \equ(5.46) and \equ(5.4666). }
\vglue.2truecm
\\The superscript $\sharp$ stands for
$d\m_{\G_n}(\z)$ integration. $\r$ is chosen as in Lemma~5.3, and $\k$ as in
Lemma~2.1.
Note that we have that the constant $C(\r,\k,j)$ appearing in
Lemma~5.2 ( this bounds $\z^{j}$) satisfies
$$C(\r,\k,j)=h_{B*}^{j}O(1)^{j} \Eq(5.45a) $$.
\vglue.3truecm
\\{\it Proof}
\\We will first prove \equ(5.45)
following the lines of the proof of lemma 5.15 of [BMS] where the Grassman
fields were absent. Recall from the definition in \equ(2.6.2.1) that
$$\Vert\tilde K(X_{\d_n},0,0,\z,0,0)\Vert_{\bf h_{*}}
=\sum_{m=0}^{m_0}{h_{B*}^m\over m!} A_m\Eq(2.6.2.3)$$
\\where
$$A_m =\sum_{p\ge 0} h_{F}^{2p}{h_{B*}^{m}\over m!}
\Vert D^{2p,m}\tilde K(X_{\d_n},0,0,\z,0,0)\Vert \Eq(2.6.2.3a)$$
\\First conside the case $m=m_0$. Then
$$A_{m_{0}}\le h_{B*}^{m_0}h_{B}^{-m_0}\Vert\tilde K(X_{\d_n}\Vert_{{\bf h},G_{\k}}
\tilde G_{\r,\k} \Eq(5.46.2) $$
\\since $G_\k \le \tilde G_{\r,\k}$. Now let $mm$. Hence:
$$\eqalign{A_m &\le O(1)\tilde G_{\r,\k}(X,\z)\Bigr[
\sum_{j=0}^{m_0-m-1}{(j+m)!\over j!m!}
\Vert\tilde K(X_{\d_n})\Vert_{{\bf h}_*} +
{m_{0}!h_{B*}^{m_0}h_{B}^{-m_0}\over m!(m_0-m-1)!}
\Vert \tilde K(X_{\d_n})\Vert_{{\bf h},G_{\k}}\Bigr] \cr
&\le O(1)\tilde G_{\r,\k}(X,\z)\Bigr[\Vert\tilde K(X_{\d_n})\Vert_{{\bf h}_*}
+h_{B*}^{m_0}h_{B}^{-m_0}\Vert\tilde K(X_{\d_n})\Vert_{{\bf h},G_{\k}}\Bigr]
\cr } \Eq(5.46.3) $$
\\Summing \equ(5.46.3) over $0\le m\le m_0-1$ and adding
\equ(5.46.2) proves \equ(5.45).
\vglue0.2truecm
\\Inequality \equ(5.45b) is also proved in the same way as \equ(5.45).
We are estimating the $\bf h$ norm which is given by \equ(2.6.2.3) with
$\bf h_{*}$ replaced by $\bf h$, $h_B*$ by $h_B$ and $\z$ by $\f$. We replace
$\tilde G_{\r,\k}$ by $G_\k$. Then
\equ(5.46.2) remains true with $\z$ replaced by $\f$ and $\tilde G_{\r,\k}$
replaced by $G_k$. Subsequently for $m < m_0$ we expand in Taylor series
as above but now in $\f$. We do the norm estimate as above but now using
Lemma 5.1 in place of Lemma 5.2. For $\e$ sufficiently small depending on
$L$ we have $\bar g$ sufficiently small and therefore $h_{B}^{-1}$ is
sufficiently small. Hence $h_{B}^{-j}C\le O(1)$ where $C=\k^{-j/2}O(1)$ is the
constant appearing in Lemma~5.1. In the Taylor remainder term we replace $h_B$
by $L^{d_s}h_B$. which leads to the factor $L^{-m_0 d_s}$.
\vglue.2cm
\\Next we will prove \equ(5.46) and \equ(5.4666). Recall that
${\bf \hat h}_*=({h_F\over 2},h_{B*})$ and ${\bf \hat h}=({h_F\over 2},h_B)$
. We consider the fluctuation integration
on $\tilde K(X,\f,\psi)$ carrying it out explicitly for the Grassman fields.
Then we have
$$\tilde K^\sharp(X_{\d_n},\f,\psi)= \int d\m_{\G_{n}}(\z) \Bigl[
\sum_{p\ge 0}\sum_{{\bf l},{\bf a}} \sum_{I\subset\{1,...,2n\}\atop|I|
{\rm even} }
\int_{X_{\d_n}^{2p}} d {\bf x}{\tilde K^{{\bf l} ,{\bf a}}}_{2p}
(X_{\d_n},\f+\z,{\bf x})\dpr_{\d_n}^{\bf l} \prod_{i\in I}\psi_{a_i}(x_{i})
\det{\G_{n}^{\bf a}}_{I^c}({\bf x}_{I^c}) \Bigr] \Eq(5.53)$$
\\where $I^c=\{1,...,2p\}\setminus I $.
The matrix ${\G_{n}^{\bf a}}_{I^c}$ is an $|I^c|/2\times |I^c|/2 $
square matrix whose entry $({\G_{n}^{\bf a}}_{I^c})_{rs}$ is defined for
$r,s \in I^c$ when $a_r=1$ and $a_s=-1$ and is then given by
$$({\G_{n}^{\bf a}}_{I^c})_{r,s}=\G_{n}(x_{r}-x_{s}) $$
Therefore,
$$D^{2j,m}\tilde K^\sharp(X_{\d_n},\f,0; f^{\times m},g_{2j}) =
\int d\m_{\G_{n}}(\z) \Bigl[
\sum_{p\ge 0}\sum_{I\subset\{1,...,2p\}\atop|I|
{\rm even} } \d_{2j,|I|}
\sum_{{\bf l},{\bf a}}
\int_{X_{\d_n}^{2p}} d {\bf x} D^{m}\tilde K^{{\bf l},{\bf a}}_{2p}
(X_{\d_n},\f+\z,{\bf x};f^{\times m} )$$
$$ \dpr_{\d_n}^{\bf l} g_{|I|}(x_{i_{1}},...,x_{i_{|I|}})
\det{\G_{n}^{a}}_{I^c}({\bf x})\Bigr] \Eq(5.53.1)$$
\\Define the function $\tilde g_{2p,I}$ on $X_{\d_n}^{2p}$ by
$$\tilde g_{2p,I}(x_1,...,x_{2p})= g_{|I|}(x_{i_{1}},...,x_{i_{|I|}})
\det{\G_{n}^{\bf a}}_{I^c}({\bf x}_{I^c})$$
\\Then we can write \equ(5.53.1) as
$$D^{2j,m}\tilde K^\sharp(X_{\d_n},\f,0; f^{\times m},g_{2j}) =
\int d\m_{\G_{n}}(\z) \Bigl[
\sum_{p\ge 0}\sum_{I\subset\{1,...,2p\}\atop|I|
{\rm even} } \d_{2j,|I|}
D^{2p,m}\tilde K(X_{\d_n},\f+\z,0; f^{\times m},\tilde g_{2p,I})\Bigr] $$
\\Let $\dpr_{\d_n}^{k}$ be the lattice forward derivative of order $k$ in
multi-index notation. By part (5a) of
Theorem~1.1 we have
$$\max_{0\le k\le 4}\Vert\dpr_{\d_n}^{k}\G_{n}
\Vert_{L^{\infty}((\d_n\math{Z})^{3})}\le C_{L} \Eq(5.61.11) $$
\\where $C_{L}$ is a constant independent of $n$.
Let now $\dpr_{\d_n}^{k_r}$, $0\le k\le 2$, be the lattice forward derivative
of order $k_r$, $0\le k_r\le 2$
with respect to the points $x_{r}\in I^{c}$. Let ${\bf k}=(k_1,...,k_{|I^c|})$
and define $\dpr_{\d_n}^{\bf k}=\prod_{r=1}^{|I^c|}\dpr_{\d_n}^{k_r}$.
Let $\dpr_{\d_n}^{\bf k}$ act on the determinant. This
produces another determinant with derivatives acting on the
matrix elements $\G_{n} (x_{r}-x_{s})$. Since $\G_n$ is positive definite
these matrices can be written
as Gram matrices by a standard argument. Thus the matrix
$a_{ij}=\dpr_{\d_n}^{k_i}\dpr_{\d_n}^{k_j}\G_n(x_i-x_j)$ with
$i,j\in I^c$ can be written as $a_{i,j}=(f_i,f_j)_{L^{2}((\d_n\math{Z})^3)}$
where $f_i(\cdot)=\dpr_{\d_n}^{k_i}\G_n^{1\over 2}(x_i,\cdot)$.
Gram's inequality says $|{\rm det}\>a_{ij}|\le
\prod_{i=1}^{|I^c|}\Vert f_i\Vert_{L^{2}((\d_n\math{Z})^3)}^{2}$. We have
$\Vert f_i\Vert_{L^{2}((\d_n\math{Z})^3)}^{2}=\dpr_{\d_n}^{* k_i}
\dpr_{\d_n}^{k_i}\G_n(x_i-y)$ for $y= x_i$. $\dpr_{\d_n}^{*}$, the
$L^{2}$ adjoint of $\dpr_{\d_n}$, is the backward derivative. Since
$2k_i\le 4$, we have by \equ(5.61.11) the bound
$\Vert f_i\Vert_{L^{2}((\d_n\math{Z})^3)}^{2}\le C_{L}$.
We therefore get
$$|\dpr_{\d_n}^{\bf k}\det {\G_{n}^{\bf a}}_{I^c}({\bf x}_{I^c})|
\le C_{L}^{|I^c|}
\Eq(5.61.1)$$
\\Hence
$\det {\G_{n}^{\bf a}}_{I^c}({\bf x}_{I^c}))$ belongs to
$C^{2}(X_{\d_n}^{I_{c}})$ and its $C^{2}$ norm is
bounded by $C_{L}^{|I^c|}$. Since $g_{|I|}$ belongs to
$C^{2}(X_{\d_n}^{|I|})$ we have that the product function
$\tilde g_{2p,I}= g_{2|I|}\det {\G_{n}^{\bf a}}_{I^c}$ belongs
to $C^{2}(X_{\d_n}^{2p})$ and its norm is bounded by the product of the two
norms. Therefore on using the definition of the norm in \equ(2.3) we get
$$\Bigl |D^{2p,m}\tilde K(X_{\d_n},\f+\z,0; f^{\times m},\tilde g_{2p,I})\Bigr|
\le \Vert D^{2p,m}\tilde K(X_{\d_n},\f+z,0)\Vert C_{L}^{|I_c|}.
\Vert g_{2j}\Vert_{C^{2}(X_{\d_n}^{2j})}\Vert f \Vert_{C^{2}(X_{\d_n})}^{m} $$
\\where we have used the constraint $|I|=2j$. Hence
$$|D^{2j,m}\tilde K^\sharp(X_{\d_n},\f,0; f^{\times m},g_{2j})|
\le \int d\m_{\G_{n}}(\z) \sum_{p\ge 0}\sum_{I\subset\{1,...,2p\}}\d_{2j,|I|}
\Vert D^{2p,m}\tilde K(X_{\d_n},\f+\z,0)\Vert \times $$
$$ \times C_{L}^{|I_c|}
\Vert g_{2j}\Vert_{C^{2}(X_{\d_n}^{2j})}\Vert f \Vert_{C^{2}(X_{\d_n})}^{m} $$
whence
$$\Vert D^{2j,m}\tilde K^\sharp(X_{\d_n},\f,0)\Vert \le
\int d\m_{\G_{n}}(\z) \sum_{p\ge 0}\sum_{I\subset\{1,...,2p\}}\d_{2j,|I|}
C_{L}^{|I_c|} \Vert D^{2p,m}\tilde K(X_{\d_n},\f+\z,0)\Vert $$
\\Multiplying both sides by ${\hat h_{F}}^{2j}$ and summing over $j$ gives
$$\sum_{j\ge 0} {\hat h_{F}}^{2j}
\Vert D^{2j,m}\tilde K^\sharp(X_{\d_n},\f,0)\Vert
\le \int d\m_{\G_{n}}(\z) \sum_{p\ge 0}\sum_{I\subset\{1,...,2p\}}
{\hat h_{F}}^{|I|}C_{L}^{|I_c|} \Vert D^{2p,m}\tilde K(X_{\d_n},\f+\z,0)\Vert $$
\\Observe that since
$\hat h_F = h_F/2$ and $2p=|I|+|I_c|$
$$\hat h_F^{|I|}=2^{-2p} h_F^{2p} \hat h_F^{-|I_c|}$$
\\Choose $h_F > 2C_{L}$, so that $\hat h_F > C_{L}$.
We then obtain
$$ \sum_{j\ge 0}{\hat h_{F}}^{2j}
\Vert D^{2j,m}\tilde K^\sharp(X_{\d_n},\f,0)\Vert \le
\int d\m_{\G_{n}}(\z) \sum_{p\ge 0}
2^{-2p} h_F^{2p} \sum_{I\subset\{1,...,2p\}}
\Vert D^{2p,m}\tilde K(X_{\d_n},\f+\z,0)\Vert $$
\\Now $\sum_{I\subset\{1,...,2p\}}1 =2^{2p}$. Therefore
$$\sum_{j\ge 0} {\hat h_{F}}^{2j}
\Vert D^{2j,m}\tilde K^\sharp(X_{\d_n},\f,0)\Vert\le
\int d\m_{\G_{n}}(\z)\sum_{p\ge 0} h_F^{2p}\Vert D^{2p,m}\tilde K(X_{\d_n},\f +\z,0)\Vert
\Eq(5.333)$$
Set $\f=0$ in \equ(5.333), multiply both sides of the above inequality by
$h_B*^{m}/m!$ and sum over $0\le m\le m_0$ which gives
$$\vert \tilde K^\sharp(X_{\d_n},0,0)\vert_{\bf {\hat h_*}} \le
\int d\m_{\G_{n}}(\z)\Vert \tilde K(X_{\d_n},\z,0)\Vert_{\bf h_*} \Eq(5.61.2) $$
\\The inequality \equ(5.46) now follows on using \equ(5.45) followed by
Lemma 5.3. To prove \equ(5.4666) we multiply \equ(5.333) by $h_{B}^{m}/m!$ and
sum over $m$ as before to obtain
$$\Vert \tilde K^\sharp(X_{\d_n},\f,0)\Vert_{\bf{\hat h}} \le
\int d\m_{\G_{n}}(\z)\Vert \tilde K(X_{\d_n},\f +\z,0)\Vert_{\bf h} $$
\\ \equ(5.4666) now follows on using the stability of the large field
regulator $G_\k$. \bull
\vglue0.5truecm
\noindent The next lemma extends lemma~5.16 of [BMS] to the case when
Grassmann fields are also present.
\\{\it Lemma 5.16
\\For any $q>0$, there exists constants $c_L$ independent of $n$
such that for $L$ sufficiently large, $\e$ sufficiently small and
$h_F$ sufficiently large depending on $L$,
$$\Vert\SS(\l, K_n)^\natural\Vert_{{\bf h},G_{\k},\AA_{p},\d_{n+1}}\le q
\qquad {\rm when} \ |\l {\bar g}^{1/4-\eta/3} |\le c_L
\Eq(5.57) $$
$$\vert\SS(\l, K_n)^\natural\vert_{{\bf h}_{*},\AA_{p},\d_{n+1}}\le q
\qquad {\rm when} \ |\l {\bar g}^{11/12-\eta/3} |\le c_L
\Eq(5.58)$$
where ${\bf h}_{*}=(h_{B*},h_{F})$ and $\natural$ denotes integration with
respect to $d\m_{\G_{n,L}}(\xi)$, $\G_{n,L}=S_{L^{-1}}\G_n$ being the
rescaled fluctuation covariance.
\\When $R_n=0$ we may set $\eta =0$ in \equ(5.57) and replace
$\l {\bar g}^{11/12-\eta/3}$ by $\l {\bar g}^{1/2-\delta/2}$ in \equ(5.58)
}
\vglue.3truecm
\\{\it Proof}
\\We suppress the dependence on $\l$ which plays a passive role in most of the
following and make the dependence explicit towards the end when necessary.
By definition
$$ (\SS(K_n))^{\natural}(Z_{\d_{n+1}},\Phi) =
\int d\m_{\G_n}(\xi){\cal B}K_n(LZ_{\d_n},S_L\Phi,\xi) $$
\\where ${\cal B}$ is the reblocking operator defined in \equ(3.13) and after
the introduction of the $\l$ parameter in \equ(4.13).
We write the above in the Grassmann representation \equ(2.1.2)
$${\cal B}K_n(LZ_{\d_n},S_L\Phi,\xi)
=\sum_{p\ge 0}\sum_{{\bf l},{\bf a}}
\sum_{I\subset \{1,..,2p\}}
\int_{(LZ_{\d_n})^{2p}} d {\bf x}\>{\cal B K}^{{\bf l} ,{\bf a},I}_{n,2p}
(LZ_{\d_n},S_L\f,\z,{\bf x})\dpr_{\d_n}^{\bf l}\prod_{k\in I}S_L\psi_{a_k}(x_k)
\prod_{k\in I^c}\eta_{a_k}(x_k) $$
\\Therefore
$$(\SS(K_n))^{\natural}(Z_{\d_{n+1}},\f,\psi)=\int d\m_{\G_n}(\z) \Bigl[
\sum_{p\ge 0}\sum_{{\bf l},{\bf a}}\sum_{I\subset \{1,..,2p\}\atop|I|
{\rm even} }
\int_{(LZ_{\d_n})^{2p}} d {\bf x}\>{\cal B K}^{{\bf l} ,{\bf a},I}_{n,2p}
(LZ_{\d_n},S_L\f,\z,{\bf x}) \times $$
$$\times \dpr_{\d_n}^{\bf l}\prod_{k\in I}S_L\psi_{a_k}(x_k)
{\rm det}\G_{I^c}({\bf x}_{I^c})\Bigr] $$
\\where we have carried out the Grassmann integration explicitly as in
\equ(5.53) with the same notations for the determinant. We follow the arguments
after \equ(5.53). Thus we define
the function $\tilde g_{2p,I}$ on $(LZ_{\d_n})^{2p}$ by
$$\tilde g_{2p,I}(x_1,...,x_{2p})= S_L g_{|I|}(x_{i_{1}},...,x_{i_{|I|}})
\det{\G_{n}^{\bf a}}_{I^c}({\bf x}_{I^c})$$
\\Note that $g_{|I|}$ scales in the same way as the product of $|I|$
Grassmann fields. Observe that
$$D^{2j,m}(\SS(K_n))^{\natural}(Z_{\d_{n+1}},\f,0; f^{\times m},g_{2j}) =
\int d\m_{\G_{n}}(\z) \Bigl[
\sum_{p\ge 0}\sum_{I\subset\{1,...,2p\}\atop|I|
{\rm even} } \d_{2j,|I|} \times $$
$$\times D^{2p,m}({\cal B}K_n)_{2p}^{I}(LZ_{\d_n},\f,\z,0; f^{\times m},
\tilde g_{2p,I})\Bigr] \Eq(5.58.1)$$
\\where $({\cal B}K_n)_{2p}^{I}$ is defined as after \equ(2.3.2).
By \equ(5.61.1),
the arguments following it, and the constraint $|I|=2j$
$$\Vert \tilde g_{2p,I}\Vert_{C^{2}((LZ_{\d_n})^{2p})}
\le C_L^{|I^c|}
\Vert S_L g_{2j}\Vert_{C^{2}((LZ_{\d_n})^{2j})}
\Vert S_L f\Vert_{C^{2}(LZ_{\d_n})}^{\times m} $$
$$\le C_L^{|I^c|} L^{-|I|d_s}L^{-md_s}
\Vert g_{2j}\Vert_{C^{2}(Z_{\d_{n+1}}^{2j})}
\Vert f\Vert_{C^{2}(Z_{\d_{n+1}})}^{\times m} $$
\\Therefore
$$\left|D^{2p,m}({\cal B}K_n)_{2p}^{I}(LZ_{\d_n},\f,\z,0; f^{\times m},
\tilde g_{2p,I})\right| \le C_L^{|I^c|}
L^{-|I|d_s}L^{-md_s}\Vert D^{2p,m}{\cal B}K_n(LZ_{\d_n},S_L\f,\z,0)\Vert
\times $$
$$\times\Vert g_{2j}\Vert_{C^{2}(Z_{\d_{n+1}}^{2j})}
\Vert f\Vert_{C^{2}(Z_{\d_{n+1}})}^{\times m} \Eq(5.58.1.2)$$
\\where we have used $\Vert D^{2p,m}({\cal B}K_n)_{2p}^{I}\Vert
\le \Vert D^{2p,m}{\cal B}K_n\Vert$ as follows from \equ(2.3.2a).
>From \equ(5.58.1) and \equ(5.58.1.2) we have
$$\Vert D^{2j,m}(\SS(K_n))^{\natural}(Z_{\d_{n+1}},\f,0)\Vert
\le \int d\m_{\G_{n}}(\z) \Bigl[
\sum_{p\ge 0}\sum_{I\subset\{1,...,2p\}} \d_{2j,|I|}
C_L^{|I^c|} L^{-|I|d_s}L^{-md_s} \times $$
$$\times \Vert D^{2p,m}{\cal B}K_n(LZ_{\d_n},S_L\f,\z,0)\Vert \Bigr]$$
\\Multipling the previous bound by $h_F^{2j}$ and summing over $j$ gives
$$\sum_{j\ge 0}h_F^{2j}
\Vert D^{2j,m}(\SS(K_n))^{\natural}(Z_{\d_{n+1}},\f,0)\Vert
\le \int d\m_{\G_{n}}(\z) \Bigl[
\sum_{p\ge 0}\sum_{I\subset\{1,...,2p\}} h_F^{|I|}C_L^{|I^c|}L^{-|I|d_s}
\times $$
$$\times \Vert D^{2p,m}{\cal B}K_n(LZ_{\d_n},S_L\f,\z,0)\Vert \Bigr]$$
\\Observe that
$$C_L^{|I^c|}L^{-|I|d_s}h_F^{|I|}= ({C_L L^{d_s}\over h_{F}})^{|I^c|}
L^{-2pd_s}h_F^{2p} \le L^{-2pd_s}h_F^{2p} $$
\\for $h_F$ sufficiently large depending on $L$. Moreover for $L$ sufficiently
large and $\e$ small we have
$$L^{-2pd_s}\sum_{I\subset\{1,...,2p\}}1\> = ({2\over L^{d_s}})^{2p} \le 1$$.
\\Therefore
$$\sum_{j\ge 0}h_F^{2j}
\Vert D^{2j,m}(\SS(K_n))^{\natural}(Z_{\d_{n+1}},\f,0)\Vert
\le \int d\m_{\G_{n}}(\z)
\sum_{p\ge 0}h_F^{2p}\Vert D^{2p,m}{\cal B}K_n(LZ_{\d_n},S_L\f,\z,0)\Vert
\Eq(5.58.4.3) $$
\\Multiplying \equ(5.58.4.3)
by ${h_B^{m}\over m!}$ and summing over $m:\> 0\le m\le m_0$
gives
$$\Vert \SS(K_n))^{\natural}(Z_{\d_{n+1}},\f,0)\Vert_{\bf h}
\le \int d\m_{\G_{n}}(\z)\Vert {\cal B}K_n(LZ_{\d_n},S_L\f,\z,0)\Vert_{\bf h}
\Eq(5.58.4.4) $$
\\Evaluate \equ(5.58.4.3) at $\f=0$. Then multiplying by
${h_{B_*}^{m}\over m!}$ and summing over $m:\> 0\le m\le m_0$
gives
$$\vert\SS(K_n))^{\natural}(Z_{\d_{n+1}})\vert_{\bf h_*} \le
\int d\m_{\G_{n}}(\z)\Vert {\cal B}K_n(LZ_{\d_n},0,\z,0)\Vert_{\bf h_*}
\Eq(5.58.4.5) $$
\\We now prove \equ(5.57) starting from \equ(5.58.4.4). Inserting
the definition \equ(4.13) in \equ(5.58.4.4) and using the multiplicative
property of the $\bf h$ norm we get
$$ \Vert (\SS (K_n)^{\natural})(Z_{\d_{n+1}},\f,0)\Vert_{\bf h}
\le \sum_{N+M\ge 1}{1\over N !M!}
{\sum}_{(X_{\d_n,j}),(\Delta_{\d_n,i})\rightarrow LZ_{\d_{n+1}}}
\Vert e^{-\tilde V_{n,L}(Z_{\d_{n+1}}\setminus L^{-1}({\bf X}_{\d_{n+1}}
\cup {\bf \Delta_{\d_{n+1}}}),\f)}\Vert_{\bf h}\times $$
$$\times \int d\m_{\G_n}(\z)
{\prod}_{j=1}^{N}\Vert K_n(X_{\d_n,j},S_L\f,\z,0)\Vert_{\bf h}
{\prod}_{i=1}^{M}\Vert P_n(\Delta_{\d_n,i},S_L\f,\z,0)\Vert_{\bf h}$$
$$\le 2^{|Z_{\d_{n+1}}|}\sum_{N+M\ge 1}{1\over N !M!}
{\sum}_{(X_{j}),(\Delta_{\d_n,i})\rightarrow LZ_{\d_n}}\int d\m_{\G_L}(\z)
{\hat G}_{\k,\r}({\bf X}_{\d_n}\cup {\bf \Delta_{\d_n}}, S_L\f,\z)
{\prod}_{j=1}^{N}\Vert K_n(X_{j})\Vert_{{\bf h},{\hat G}_{\k,\r}} $$
$$\times {\prod}_{i=1}^{M}\Vert P_n(\Delta_{\d_n,i})\Vert_{{\bf h},{\hat G}_{\k,\r}}
\Eq(5.58.9) $$
\\where ${\bf X}_{\d_n}=\cup X_{\d_n,j}$,
${\bf\Delta}_{\d_n}=\cup\Delta_{\d_n,i}$ and we have bounded
$e^{-\tilde V_{n,L}}$ using lemma~5.5 which continues to apply.
\\Lemma~5.4 bounds the $\z$ integral by
$$ 2^{|{\bf X}_{\d_n}\cup{\bf\Delta_{\d_n}}|}
G_{3\k}({\bf X}_{\d_n}\cup {\bf\Delta_{\d_n}},S_L\f)
\le 2^{|{\bf X}_{\d_n}\cup{\bf\Delta_{\d_n}}|}
G_{\k}(L^{-1}({\bf X}_{\d_{n+1}}\cup{\bf\Delta}_{\d_{n+1}}), \phi) $$
$$\le \prod_{j=1}^{M} 2^{|X_{\d_n,j}|}\prod_{i=1}^{N} 2^{|\Delta_{\d_n,i}|}
G_{\k}(Z_{\d_{n+1}},\phi)$$
\\since $L^{-1}({\bf X}_{\d_{n+1}}\cup{\bf\Delta_{\d_{n+1}}})
\subset Z_{\d_{n+1}}$. Therefore
$$\Vert (\SS (K_n)^{\natural})(Z_{\d_{n+1}})\Vert_{\bf h, G_{\k}}
\le 2^{|Z_{\d_{n+1}}|}\sum_{N+M\ge 1}{1\over N !M!}
{\sum}_{(X_{\d_n,j}),(\Delta_{\d_n,i})\rightarrow LZ_{\d_n}}
{\prod}_{j=1}^{M}2^{|X_{\d_n,j}|}\Vert K_n(X_{j})\Vert_{{\bf h},{\hat G}_{\k,\r}}
\times $$
$${\prod}_{i=1}^{N}2^{|\Delta_{\d_n,i}|}
\Vert P_n(\Delta_{\d_n,i})\Vert_{{\bf h},{\hat G}_{\k,\r}}
$$
\\The condition on the sum over the polymers above implies that
$Z_{\d_n}=(\cup L^{-1}\bar X_{\d_n,j}^{L})
\cup (\cup L^{-1}\bar \Delta_{\d_n,i}^{L})$.
This also implies that
$Z_{\d_{n+1}}=(\cup L^{-1}\bar X_{\d_{n+1},j}^{L})
\cup (\cup L^{-1}\bar\Delta_{\d_{n+1},i}^{L})$.
\\Multiply both sides by $\AA_p(Z_{\d_{n+1}})$ and observe on the right hand
side
$$\AA_{p+1}(Z_{\d_{n+1}})
\le {\prod}_{j=1}^{M} \AA_{p+1}(L^{-1}\bar X_{j,\d_{n+1}})
{\prod}_{i=1}^{N} \AA_{p+1}(L^{-1}\bar \Delta_{\d_{n+1},i})$$
$$\le O(1)^{N+M} {\prod}_{j=1}^{M} \AA_{-2}(X_{j,\d_{n+1}})
{\prod}_{i=1}^{N} \AA_{-2}(\Delta_{\d_{n+1},i})$$
\\where we have first used the fact that the $L$-closures of the polymers
are connected by definition of the reblocking operation,
then Lemma~2.2 together with $|X_{j,\d_{n+1}}|=|X_{j,\d_{n}}|$ and
$|\Delta_{\d_{n+1},i}|=|\Delta_{\d_{n},i}|$ . The last observation follows
from our definition of polymers in section~1.3 and \equ(1.22f). Therefore
$$\Vert (\SS (K_n)^{\natural})(Z_{\d_{n+1}})\Vert_{\bf h, G_{\k}}
\AA_{p}(Z_{\d_{n+1}})
\le \sum_{N+M\ge 1}{1\over N !M!} O(1)^{N+M}
{\sum}_{(X_{j}),(\Delta_{\d_n,i})\rightarrow LZ } $$
$${\prod}_{j=1}^{M}\Vert K_n(X_{j,\d_n})\Vert_{{\bf h},{\hat G}_{\k,\r}}
\AA_{-1}(X_{j,\d_n}){\prod}_{i=1}^{N}
\Vert P_n(\Delta_{\d_n,i})\Vert_{{\bf h},{\hat G}_{\k,\r}}
\AA_{-1}(\Delta_{\d_n,i}) $$
\\Fix any $\D_{\d_{n+1}}$ and sum over $Z_{\d_{n+1}}\ni \Delta_{\d_{n+1}}$.
This fixes on the right hand side the sum over $Z_{\d_{n}}\ni \Delta_{\d_{n}}$
with $\D_{\d_{n}}$ fixed by restriction. The spanning tree
argument of Lemma~7.1 of [BY] controls the sums over
$N, M, Z_{\d_n}, (X_{j,\d_n} ), (\Delta_{i,\d_n})\rightarrow LZ_{\d_n} $
with the result ( we have now made
explicit the dependence on $\l$)
$$\Vert (\SS (\l, K_n)^{\natural})\Vert_{\bf h, G_{\k},\AA_{p},\d_{n+1}}
\le O(1)\sum_{N\ge 1} O(1)^{N} L^{3N}
\Bigl (\Vert K_n(\l)\Vert_{{\bf h},{\hat G}_{\k,\r},\AA,\d_n} +
\Vert P_n(\l)\Vert_{{\bf h},{\hat G}_{\k,\r},\AA,\d_n} \Bigr )^{N} $$
\\The proof of \equ(5.57) is completed by Lemmas~5.8 and 5.14. When $R=0$
we can use Lemma~5.8 and replace Lemma~5.14 by Lemma~5.11.
\\To prove \equ(5.58) we start from \equ(5.58.4.5) and proceed as before. We
replace ${\hat G}_{\k,\r}$ by
${\tilde G}_{\k,\r}$ and then use Lemma~5.3 to estimate the $\z$ integral.
Proceeding as before now leads to
$$\vert (\SS (\l, K_n)^{\natural})\vert_{{\bf h}_{*}, \AA_{p},\d_{n+1}}
\le O(1)\sum_{N\ge 1} O(1)^{N} L^{3N}
\Bigl (\Vert K_n(\l)\Vert_{{\bf h}_{*},{\tilde G}_{\k,\r},\AA,\d_n} +
\Vert P(\l)\Vert_{{\bf h}_{*},{\tilde G}_{\k,\r},\AA,\d_n} \Bigr )^{N} $$
\\Now use Lemmas~5.8 and 5.14 to complete the proof of \equ(5.58). Finally
when $R = 0$ use Lemmas~5.8 and 5.11 as before. \bull
\vglue.3truecm
\\{\it Estimates on relevant parts and flow coefficients
from the remainder}
\vglue.3truecm
\\Let $({\tilde \a}_{n,P})$ be the coefficients $({\tilde
\a}_{n,2,0}, {\tilde \a}_{n,2,1}, {\tilde \a}_{n,2,\bar 1}
{\tilde \a}_{n,4})$ defined in \equ(4.57)
and \equ({4.62}). The flow coefficients $\xi_{n, R}$, $\r_{n, R}$
are given in \equ({4.64}).
\vglue.3truecm
\\{\it Lemma 5.17 : Under the conditions of the domain ${\cal D}_n$ we have
$$\Vert R_n^\sharp\Vert_{{\bf \hat h},G_{3\k},\AA_{-1},\d_n}\le
{\bar g}^{3/4-\eta} \Eq(5.65)$$
$$\vert R_n^\sharp\vert_{{\bf {\hat h}}_*,\AA_{-1},\d_n}
\le O(1){\bar g}^{11/4-\eta} \Eq(5.66)$$
$$\vert{\tilde \a}_{n,P}\vert_{\AA,\d_n}
\le O(1){\bar g}^{11/4-\eta} \Eq(5.67)$$
$$|\xi_{n}|\le C_L{\bar g}^{11/4-\eta}
\Eq(5.69)$$
$$|\r_{n}|\le C_L{\bar g}^{11/4-\eta} \Eq(5.70)$$
where the constants $C_L$ are independent of $n$ and $\e$ }
\vglue.3truecm
\\{\it Proof}
\\\equ(5.65) follows from \equ({55.2}) and Lemma~5.15,
\equ(5.4666). \equ(5.66) follow from
\equ({55.3}) and lemma 5.15 with $m_0 =9$
and $\e$ sufficiently small depending on L so that $\bar g$ is
sufficiently small. In fact in lemma~5.15 (with $\tilde K=R_n$) the first term
has the desired bound by \equ(55.3). By \equ({55.2}) rogether with
$h_B^{-1}=c{\bar g}^{1\over 4}$ and $h_{B*}=h_{B*}(L)$ we see that
the second term is bounded by
$O(1){\bar g}^{1\over 4}h_{B*}^{9}{\bar g}^{11/4 -\eta}
\le {\bar g}^{11/4 -\eta}$ for $\bar g$ sufficiently small.
\\Recall that ${\tilde \a}_{n,P}(X_{\d_n})$, are supported on small
sets. Then \equ(5.67) follows from \equ(4.62) and
\equ(5.66). In fact the dominant contribution
comes by setting ${\tilde V_n}=0$ because the difference gives
additional powers of $\bar g$. Then we have
$$\Vert {\tilde \a}_{n,P}\Vert_{\AA,\d_n} \le O(1) n(P)!
{\bf {\hat h}}_{*}^{-n(P)}
\vert 1_{S}R_n^\sharp\vert_{{\bf {\hat h}}_{*},\AA,\d_n}$$
where $n(P)$ is the number of fields in the monomial $P$,
we have used the shorthand notation
${\bf {\hat h}}_{*}^{-n(P)}=\max_{n(P)_{F}+n(P)_{B}=n(P)}(h_{*B}^{-n(P)_{B}}
{\hat h}_F^{-n(P)_{F}})$
and $1_{S}$ is the
indicator function on small sets. Now use \equ(5.66) to get \equ(5.67).
\equ(5.69), \equ(5.70) follow from \equ(5.67), the
definitions \equ(4.64), \equ(4.70) and Wick
coefficients $C_n(0)$ are uniformly bounded by a $L$ dependent constant by
Corollary~1.2. \bull
\vglue.3truecm
\\{\it Lemma~5.18
\\Under the conditions of ${\cal D}_n$ and
$\e$ sufficiently small depending on $L$, there exists a constant
$C_L$ independent of $\e$ and $n$ such that
$$\vert g_{n+1}-\bar g\vert < 2\n{\bar g}^{3/2},
\quad \vert\mu_{n+1}\vert < C_L {\bar g}^{2-\d} \Eq(55.9) $$
}
\\{\it Proof} :
\\It is convenient to define
$$\tilde g_n = g_n -\bar g $$
\\Then from the flow equation \equ(4.47A) for
$g_n$ and the definition of ${\bar g}$ in \equ(55.4) we get
$$\tilde g_{n+1}=(2-L^{\e})\tilde g_{n} + \tilde\xi_n \Eq(55.9a) $$
\\where
$$\tilde\xi_n=-L^{2\e}a_{c*}{\tilde g_n}^{2}-L^{2\e}(a_n-a_{c*}) g_n^{2}+
\xi_{n} \Eq(55.9b) $$
\\From Lemma~5.12, Lemma~5.17 and $g_n\in {\cal D}_n$
we get for $\e$ sufficiently small depending on $L$
the bound
$$|\tilde\xi_n|\le C_L{\bar g}^{2} \Eq(55.9c) $$
\\Therefore
$$|\tilde g_{n+1}|< \n{\bar g}((2-L^{\e}) +
{C_L\over \n} {\bar g}) \Eq(55.9d) $$
\\For $\e$ sufficiently small depending on $L$ we get
$$|(1-L^{\e}) + {C_L\over \n} {\bar g}|\le 1 \Eq(55.9e) $$
\\Therefore $|\tilde g_{n+1}|\le 2\n{\bar g}$ which proves the first
inequality of \equ(55.9).
\\The bound on $\mu_{n+1}$
follows from the second of the flow equations \equ({4.47}), on using
$\mu_{n}$, $g_n$ belong to ${\cal D}_n$, Lemma~5.12
and the bound \equ(5.70) on $\rho_{n}$. \bull
\vglue0.2cm
\\As stated in Theorem~3.1 borrowed from [BDH] the assumption
of stability of the local potential with respect to perturbation by
relevant parts (see \equ(33.17d) ensures the extraction estimate of
\equ(33.17e). The following lemma proves the stability for the case at hand,
namely that of $\tilde V_{n,L}(\D_{n+1})$ with respect to the relevant
part $F_n$ defined in section~4.
\vglue0.2cm
\\Recall from \equ(4.12) that $F_n (\lambda) = \lambda^{2}F_{Q_n}+
\lambda^{3}F_{R_n}$ and from \equ({33.2aa}) that (each part of) $F_n$
decomposes: $F_n (X_{\d_{n+1}})=\sum_{\Delta_{\d_{n+1}} \subset X_{\d_{n+1}}}
F_n(X_{\d_{n+1}},\Delta_{\d_{n+1}})$.
\vglue.3truecm
\\{\it Lemma 5.19
\\For any $R>0$ and $\xi :=R \max(|\lambda^{2}|\bar g,
|\lambda^{3}|\bar g^{7/4-\eta})$ sufficiently small,
$$
\Vert e^{-\tilde V_{n,L}(\D_{n+1})-\sum_{X_{\d_{n+1}}\supset\D_{n+1}} z(X_{\d_{n+1}})F(\lambda,
X_{\d_{n+1}},\D_{n+1})}\Vert_{{\bf h},G_\k}\le 2^2 \Eq(5.88)$$
\\where $z(X_{\d_{n+1}})$ are complex parameters with $|z(X_{\d_{n+1}})|
\le R$.
}
\vglue.3truecm
\\{\it Proof}
\\It is easy to see that Lemma~5.5 still holds if we replace $\tilde V_n$ by
$\tilde V_{n,L}$ provided $\e$ is sufficiently small. This implies
that $\bar g$ is sufficiently small. We then have
$$
\Vert e^{-\tilde V_{n,L}(\D_{n+1})-\sum_{X_{\d_{n+1}}\supset\D_{n+1}}
z(X_{\d_{n+1}})F(\lambda, X_{\d_{n+1}},\D_{n+1})}\Vert_{\bf h}\le $$
$$ 2\ e^{-{\bar g}/4\int_{\D_{n+1}}
dx(|\f(x)|^{2})^2+\sum_{X_{\d_{n+1}}\supset\D_{n+1}}R\Vert F(\lambda
,X_{\d_{n+1}},\D_{n+1})\Vert_{\bf h}} \Eq(55.88) $$
\\Recall that the relevant parts $F(X_{\d_{n+1}},\D_{n+1})$ are supported on
small sets $X_{\d_{n+1}}$. The proof now follows easily from the following
\vglue.3truecm
\\{\it Claim:} For $\e$ sufficiently small
$$|z(X_{\d_{n+1}})|\Vert F(\lambda,X_{\d_{n+1}},\D_{n+1})\Vert_{\bf h}
\le C_{L}\xi\left(\bar g \int_{\D_{n+1}}
d^{3}x(|\f(x)|^{2})^{2} + \bar g^{1/2}\Vert|\f|^{2}
\Vert_{\D_{n+1},1,5}+1\right)\Eq(5.91)
$$
\\where $\Vert\phi\Vert^{2}_{\D_{n+1},1,5}$ is the square of the lattice
Sobolev norm defined in \equ(2.10reg).
\vglue0.1cm
\\{\it Proof of the Claim:}
\\We have
$\Vert F(\lambda,X_{\d_{n+1}},\D_{n+1})\Vert_{\bf h}\le |\lambda |^{2}\Vert
F_Q(X_{\d_{n+1}},\D_{n+1})\Vert_{\bf h}+|\lambda |^{3}
\Vert F_{ R}(X_{\d_{n+1}},\D_{n+1})\Vert_{\bf h}$.
\\Consider \equ(4.33)-\equ(4.36a). Undo the Wick ordering on the superfield
field
and note that, by virtue of supersymmetry, no field independent terms arise.
The $m=1$
term in \equ(4.36) remains unchanged and in the $m=2$ case there results
an additional contribution $-2C_{n+1}(0)\Phi\bar\Phi$. The Wick
constant $C_{n+1}(0)$ has a uniform bound $C$ which depends only on $L$
by Corollary~1.2. We write this in the Grassmann representation and notice that
for the $m=2$ case the $\psi\bar\psi(x)^{2}$ contribution vanishes by
statistics. From the definition of the $\bf h$ norm with
$h_B=c{\bar g}^{-1/4}$ and $h_F=h_F(L)$ we get the bound
$$\Vert F_Q(X_{\d_{n+1}},\D_{n+1})\Vert_{\bf h}\le C_{L}{\bar g}^2\left(
\int_{\D_{n+1}} d^{3}x\Vert(|\f|^{2})^2(x)\Vert_{{h_{B}}}
\sup_{x\in \D_{n+1}}|f^{(2)}_{Q}(X_{\d_{n+1}},x,\D_{n+1})|+\right.$$
$$\left.\left(\int_{\D_{n+1}} d^{3}x\Vert(|\f|^{2})(x)\Vert_{h_{B}}+1\right)
\sum_{m=1}^{2} \sup_{x\in \D_{n+1}}|f^{(m)}_{Q}(X_{\d_{n+1}},x,\D_{n+1})|
\right)$$
\\Now for $m=1,2$
$$
\bar g\int_{\D_{n+1}} d^{3}x\Vert(|\f|^{2})^m(x)\Vert_{h_{B}}\le O(1)\left(
\bar g\int_{\D_{n+1}} d^{3}x(|\f|^{2})^{2}(x)+1\right)$$
\\From the definition $\equ(4.36a)$ and the estimates obtained
in the course of proving Lemma~5.12, we have
$$\sup_{x\in \D_{n+1}}|f^{(m)}_{Q}(X_{\d_{n+1}},x,\D_{n+1})|\le C_{L}$$
Therefore
$$|\lambda^{2}||z(X_{\d_{n+1}})| \ \Vert F_Q(X_{\d_{n+1}},\D_{n+1})\Vert_h
\le C_{L} R |\lambda |^{2}\bar g \left(\bar g\int_{\D_{n+1}} dx(|\f|^{2})^2(x)
+1\right)\Eq(5.96)$$
\\Next consider $F_{ R_n}$, supported on small sets, defined in
\equ(4.57), \equ(4.58). Recall \equ({4.57b}),
$$F_{R}(X_{\d_{n+1}},{\bf\Phi})= \sum_{P}\int_{\D_{n+1} } dx\
\alpha_{P} (X_{\d_{n+1}},x)P ({\bf\Phi} (x),\dpr_{\d_{n+1}}{\bf\Phi} (x))
$$
\\By Lemma~5.17 and \equ(4.67a) we have
$|\a_P(X_{\d_{n+1}},x)|\le C_{L} \bar g^{11/4-\eta}$,
so that
$$\eqalign{
|\lambda |^{3}\vert z(X_{\d_{n+1}})\vert \Vert F_{ R}(X_{\d_{n+1}},\D_{n+1})\Vert_{\bf h}
\le &C_{L} R|\lambda |^{3}\bar g^{11/4-\eta}\sum_{P}
\int_{\D_{n+1}} dx\ \Vert
P(\Phi (x),\dpr_{\d_{n+1}}\Phi (x))
\Vert_{\bf h}\cr
\le &C_{L} R|\lambda |^{3}\bar g^{7/4-\eta}\bigg(
\bar g\int_{\D_{n+1}} dx \, (|\f|^{2})^2(x)+
\bar g^{1/2}\Vert|\f|^{2}\Vert_{\D_{n+1},1,5}+1
\bigg)\cr}$$
\\The claim follows by combining this with \equ({5.96}). In
the above inequality the Sobolev norm when estimating the
term giving arise to $\phi\dpr_{\d_n,\m}\bar\phi$. We bound
$\vert\phi\dpr_{\d_n,\m}\bar\phi\vert \le 1/2(\vert|\f|^{2}\vert
+\vert \dpr_{\d_n,\m}\phi\vert^{2})$
and then use the lattice Sobolev embedding inequality. \bull
\vglue.3truecm
\\{\it Lemma 5.20
\\For any $R>0$ and $\xi :=R \max(|\lambda^{2}|{\bar g}^{2},
|\lambda^{3}|{\bar g}^{11/4-\eta})$ sufficiently small,
$$
\vert e^{-\tilde V_{n,L}(\D_{n+1})-\sum_{X_{\d_{n+1}}\supset\D_{n+1}}
z(X_{\d_{n+1}})F(\lambda,X_{\d_{n+1}},\D_{n+1})}
\vert_{{\bf h}_{*}}\le 2^2
\Eq(5.88.1)$$
\\where $z(X)$ are complex parameters with $|z(X)|\le R$.
}
\vglue.3truecm
\\{\it Proof}
\\The proof is similar to the previous one except that we can use the estimate
$\vert F(\lambda,X,\D)\vert_{{\bf h}_{*}}
\le C_{L}\xi$ in place of \equ({5.91}) since the ${\bf h}_{*}$ norm is computed
with field derivatives at $\Phi=0$. \bull
\vglue0.2cm
We will now bound the remainder $R_{n+1}$ given in \equ(4.221). It consist
of a sum of four contributions, namely $R_{n+1,\rm main}$, $R_{n+1,linear}$,
$R_{n+1,3}$ and $R_{n+1,4}$ which we will estimate in turn. These estimates
parallel those obtained for the continuum bosonic theory in [BMS].
\vglue0.2cm
Recall from \equ({4.188}) that
$$
R_{n+1,\rm main}
= {1\over 2\pi i}
\oint_{\gamma} {d\lambda \over \lambda^{4}}
{\cal E} \bigg(
\SS(\lambda ,Q_ne^{-V_n})^{\natural},F_{Q_n} (\lambda )
\bigg)
\Eq(5.lem35-0)
$$
\vglue.2truecm
\\{\it Lemma 5.21
$$\Vert R_{n+1,\rm main}\Vert_{{\bf h},G_{\k},\AA,\d_{n+1}}
\le C_{L}{\bar g}^{3/4} \Eq(5.lem35-1)$$
$$\vert R_{n+1,\rm main}\vert_{{\bf h}_{*},\AA,\d_{n+1}}
\le C_{L}{\bar g}^{3-3\d/2} \Eq(5.lem35-2)$$
}
\\{\it Proof}
\\Choose the contour $\g$ to be a circle of radius $|\l| =c_L{\bar g}^{-1/4}$.
Then from Lemma~5.16, we have that for any $q>0$ ,
$\Vert\SS(\lambda ,Q_ne^{-V_n})^{\natural}\Vert_{{\bf h},G_{\k},\AA_1,\d_{n+1}}
\le q$. For $\l\in \g$, $R^{-1}=O(1)$ and $|z(X_{\d_{n+1}})|=R$ the stability
estimate of Lemma~5.19 holds since for $\e$ sufficiently small,
$\bar g$ is sufficiently small and the $\xi$ in Lemma~5.16 is sufficiently
small. The $f(X_{\d_{n+1}})$ which figures in the extraction estimate below
is supported on small sets and $f(X_{\d_{n+1}})\le 2|z(X_{\d_{n+1}})|^{-1}$.
Therefore by the extraction estimate \equ(33.17e) of Theorem~3.1 we have
$$\Vert {\cal E}(\SS(\lambda ,Q_ne^{-V_n})^{\natural},F_{Q_n}(\l))
\Vert_{{\bf h},G_{\k},\AA,\d_{n+1}} \le
\Vert\SS(\lambda ,Q_ne^{-V_n})^{\natural}\Vert_{{\bf h},G_{\k},\AA_1,\d_{n+1}}
+\Vert f\Vert_{\AA_3,\d_{n+1}} $$
$$\le q +c'_L R^{-1} \le c_L $$
\\The bound \equ(5.lem35-1) now follows from \equ(5.lem35-0) by
a Cauchy estimate since the
contour is of radius $c_L{\bar g}^{-1/4}$.
\\To prove \equ({5.lem35-2}) we choose the
contour $\gamma$ to be a circle of radius $c_{L}{\bar g}^{-1+\d/2}$
so that $\xi$ in Lemma~5.20 is small and the hypothesis of
Lemma~5.16 is satisfied. $z(X_{\d_{n+1}})$ and $f(X_{\d_{n+1}})$ are chosen
as bedore. Then \equ({5.lem35-2}) follows from the extraction estimate
\equ(33.17f), \equ(5.58) of Lemma~5.16 and a Cauchy bound. \bull
Recall from \equ({4.21}) that
$$
R_{n+1,3} =
{1\over 2\pi i}
\oint_{\gamma} {d\lambda \over \lambda^{4} (\lambda -1)}
{\cal E}\Bigl (\SS (\lambda,K_n)^{\natural},F_n(\lambda )\Bigr)
\Eq(5.lem36-3)$$
\vglue.3truecm
\\{\it Lemma 5.22
$$\Vert R_{n+1,3}\Vert_{{\bf h},G_{\k},\AA,\d_{n+1}}\le C_{L}
{\bar g}^{1-4\eta /3} \Eq(5.lem36-4)$$
$$\vert R_{n+1,3}\vert_{{\bf h}_{*},\AA,\d_{n+1}}
\le C_{L}{\bar g}^{10/3-4\eta /3} \Eq(5.lem36-2)$$
}
\\{\it Proof}
\\The proof follows the lines of that of Lemma~5.21. To prove \equ(5.lem36-4)
we take the contour $\g$ to of radius
$|\lambda|= c_{L}{\bar g}^{-(1/4-\eta/3)}$.
This ensures that the hypothesis of Lemma~5.16, \equ(5.58) is satisfied
and $\xi$ of Lemma~5.19 is sufficiently small so that stability holds.
\equ(5.lem36-4) now follows from the extraction estimate \equ(33.17e)
and the Cauchy bound as before. To prove \equ(5.lem36-4) we take
$\g$ to be of radius $|\lambda|= c_{L}{\bar g}^{-(5/6-\eta/3)}$. Then for
$\bar g$ sufficiently small the hypothesis for \equ(5.58) of Lemma~5.16
is satisfied. Moreover then $\xi$ of Lemma~5.20 is sufficiently small and
and Lemma~5.20 holds. \equ(5.lem36-4) now follows from the extraction
estimate \equ(33.17f) and the Cauchy bound as before. \bull
\vglue0.2cm
From the definition of $R_{n+1,4}$ in \equ(4.22) we have
$$R_{n+1,4} = \bigg(e^{-V_{n+1}} - e^{-\tilde{V}_{n,L}}\bigg)
Q(C_{n+1},{\bf w}_{n+1},g_{n+1}) + e^{-\tilde{V}_{n,L}}
Q(C_{n+1},{\bf w}_{n+1},(g_{n+1}^{2}-g_{n,L}^{2}))
$$
\\{\it Lemma 5.23
$$
\Vert R_{n+1,4} \Vert_{{\bf h},G_{\kappa},\AA,\d_{n+1}}
\le C_L {\bar g}^{3/2}
\qquad\qquad\vert R_{n+1,4}\vert_{{\bf h}_{*},\AA,\d_{n+1}}
\le C_L {\bar g}^{3}
$$
}
\vglue.2truecm
\\{\it Proof}
\\We follow the lines of the proof of Lemma~5.23 in [BMS].
From the flow equations \equ(4.47A), together with the domain hypothesis
${\cal D}_n$ and Lemmas~5.12, 5.17, 5.9 we obtain the bounds
$$\vert g_{n+1}-g_{n,L}\vert \le C_L {\bar g}^2 \Eq(5.lemm37-1)$$
$$\vert \m_{n+1}-\m_{n,L}\vert \le C_L {\bar g}^2 \Eq(5.lemm37-2)$$
$$\Vert {\bf w}_{n+1} \Vert_{\d_{n+1}} \le k_l \Eq(5.lemm37-3)$$
\\From \equ(5.lemm37-1) and Lemma~5.18 we get
$$\vert g_{n+1}^{2}-g_{n,L}^{2}\vert \le C_L {\bar g}^3 \Eq(5.lemm37-1a)$$
\\We estimate the two terms in the expression for $R_{n+1,4}$
above. Because of Q each term is supported on small sets.
We write the first term as
$$\bigg(
e^{-V_{n+1}} - e^{-\tilde{V}_{n,L}}
\bigg)
Q(C_{n+1},{\bf w_{n+1}},g_{n+1})=Q(C_{n+1},{\bf w}_{n+1},g_{n+1})
e^{-{1\over 2}V_{n+1}}\times $$
$$\times \int_0^1 ds (\tilde{V}_{n,L} -
V_{n+1})e^{-({1\over 2}-s)V_{n+1}-s{V}_{n,L}}$$
\\evaluated on a small set $X_{\d_{n+1}}$. We bound
$$\Vert \bigg(e^{-V_{n+1}} - e^{-\tilde{V}_{n,L}}\bigg)Q(C_{n+1},{\bf
w_{n+1}},g_{n+1})\Vert_{{\bf h},G_{\k}}
\le \Vert Q(C_{n+1},{\bf w}_{n+1},g_{n+1})e^{-{1\over
2}V_{n+1}}\Vert_{{\bf h},G_{\k} } \times $$
$$\times \int_0^1 ds \Vert \tilde{V}_{n,L}
-V_{n+1}\Vert_{{\bf h}}\Vert e^{-({1\over 2}-s)V_{n+1}}\Vert_{{\bf h}}\Vert
e^{-s{V}_{n,L}}\Vert_{{\bf h}} \Eq(5.lemm37-4) $$
\\We get by undoing the Wick orderings
$$\tilde{V}_{n,L}(X_{\d_{n+1}})-V_{n+1}(X_{\d_{n+1}})
=\int_{X_{\d_{n+1}}}dx \Bigl((g_{n,L}-g_{n+1})(|\f|^{2})^{2}(x) +
2(g_{n,L}-g_{n+1})\f\bar\f(x)\psi\bar\psi(x) +$$
$$+(\tilde\m_{n,L}-\tilde\m_{n+1})
\psi\bar\psi(x)\Bigr)$$
\\where $\tilde\m_{n+1}=\m_{n+1} -2 C_{n+1}(0) g_{n+1}$ and
$\tilde\m_{n,L}=\m_{n,L} -2 C_{n+1}(0) g_{n,L}$.
Using \equ(5.lemm37-1), \equ(5.lemm37-2) and the uniform boundedness of
the Wick coefficient $C_{n+1}(0)$ we can bound
$$\Vert \tilde{V}_{n,L}(X_{\d_{n+1}}) -V_{n+1}(X_{\d_{n+1}} )\Vert_{{\bf h}}
\le C_L{{\bar g}\over \g}
e^{\g{\bar g}\int_{X_{\d_{n+1}}}dx (\phi\bar\phi)^2(x)}\Eq(5.lemm37-5) $$
for any $\g=O(1) >0$. By Lemma 5.5 we can bound
$$\Vert e^{-({1\over 2}-s)V_{n+1}(X_{\d_{n+1}})}\Vert_{{\bf h}}
\le 2^{|X_{\d_{n+1}}|}
e^{-({1\over 2}-s){{\bar g}\over 4}\int_{X_{\d_{n+1}}}dx
(\phi\bar\phi)^2(x)}\Eq(5.lemm37-6) $$
$$\Vert e^{-s{V}_{n,L}(X_{\d_{n+1}})}\Vert_{{\bf h}} \le 2^{|X_{\d_{n+1}}|}
e^{-s{{\bar g}\over 4}\int_{X_{\d_{n+1}}}dx
(\phi\bar\phi)^2(x)}\Eq(5.lemm37-7) $$
\\We now plug into \equ(5.lemm37-4) the bounds \equ(5.lemm37-6) and
\equ(5.lemm37-7) and choose $\g=1/2$.
Then the $s$-integral in \equ(5.lemm37-4) is bounded by $C_L{\bar g}$.
Lemma~5.10 continues to hold if we replace $n$ by $n+1$ by virtue of Lemma~5.9
and \equ(55.9) of Lemma~5.18. Therefore the first factor in \equ(5.lemm37-4)
is bounded by $C_L{\bar g}^{1\over 2}$. Putting these bounds
together and recalling that the $Q$ are supported on small sets we
obtain
$$\Vert \bigg(e^{-V_{n+1} } - e^{-\tilde{V}_{n,L}}\bigg)Q(C_{n+1},
{\bf w}_{n+1},g_{n+1})\Vert_{{\bf h},G_{\k},\AA,\d_{n+1}}
\le C_L{\bar g}^{3\over 2} \Eq(5.lemm37-8) $$
\\The second term in the expression
for $R_4$ is bounded using \equ(5.lemm37-1a) and Lemma~5.10 with $n$
replaced by $n+1$ and $g_n^{2}$ by $g_{n+1}^{2}-g_{n,L}^{2}$. We get
$$\Vert e^{-\tilde{V}_{n,L}}
Q(C_{n+1},{\bf w}_{n+1},(g_{n+1}^{2}-g_{n,L}^{2}))
\Vert_{{\bf h},G_{\k},\AA,\d_{n+1}}
\le C_L{\bar g}^{3\over 2} \Eq(5.lemm37-9) $$
\\Adding together \equ(5.lemm37-8) and \equ(5.lemm37-9) we get
the first bound of the Lemma.
The second bound is proved similarly taking into account that
all derivatives in the ${\bf h}_*$ norm are at $\Phi =0$. \bull
\vglue.3truecm
\\{\it Lemma 5.24
\\
Let $X_{\d_n}$ be a small set and let $J(X_{\d_n},\Phi)$ be normalized as in
\equ({4.42}). Recall that the rescaled activity $J_{L}$ is defined by
$J_{L}(L^{-1}X_{\d_{n+1}},\f,\psi)=J(X_{\d_n},\f_{L^{-1}},\psi_{L^{-1}})$
where $\f_{L^{-1}}=S_L\f$ and $\psi_{L^{-1}}=S_L\psi$.
Then we have
\\1. For $ 2p+m=2 $ so that $(p,m)=(1,0),\> (0,2)$
$$\Vert D^{2p,m}J_{L}(L^{-1}X_{\d_{n+1}},0,0)\Vert
\le O(1)L^{-(7-\e)/2}\Vert
D^{2p,m}J(X_{\d_n},0,0)\Vert \Eq(5.79) $$
\\2. For $ 2p+m=4 $ so that $(p,m)=(2,0),(1,2),(0,4)$
$$\Vert D^{2p,m}J_{L}(L^{-1}X_{\d_{n+1}},0,0)\Vert \le O(1)L^{-(4-\e)}\Vert
D^{2p,m}J(X_{\d_n},0,0)\Vert
\Eq(5.80)$$
}
\vglue.2truecm
\\{\it Proof} :
\\In the proof of this lemma we will need to use the lattice Taylor expansion
introduced in \equ(5.001),\equ(5.002) and \equ(5.003) with a particular
choice of
a lattice path joining two points. The polymer $X_{\d_n}$ being a small set
is connected. It can be represented
as $X_{\d_n}=X\cap (\d_n\math{Z})^{3}$ where $X$ is a continuum connected
polymer which is a small set. As in the proof of Lemma~5.1 we say that
$X_{\d_n}$ is convex if $X$ is a convex set. By the argument in the proof of
Lemma~5.1 it suffices to consider the case when $X_{\d_n}$ is convex.
\\Let $u_{k}$ be a function in $C^{2}(X_{\d_n}^{k})$ for $k\ge 2$.
In the following $u$ will represent
one of the functions $f_j$ in $C^{2}(X)$ giving a direction for a bosonic
derivative or a function $g_{2p}$ in $C^{2}(X^{2n})$ associated with a
fermionic derivative of order $2p$. Let $e_1,e_2,.....,e_{3k}$ be the basis
vectors of $(\d_n\math{Z})^{3k}$. Let $x=\sum_{i=1}^{3k}(x,e_i)e_i $
denote a point in $X_{\d_n}^{k}$.
\\We recall the definition of the rescaled function
$$ u_{k,L^{-1}}(x)= L^{-kd_{s}}u_{k}({x\over L})$$
\\Observe that
$$\Vert u_{k,L^{-1}}\Vert_{C^{2}(X_{\d_n}^{k})} \le L^{-kd_{s}}
\Vert u_{k}\Vert_{C^{2}(L^{-1}X_{\d_{n+1}}^{k})} \Eq(8.303.5) $$
\\Fix a point $x_0\in X_{\d_n}^{k}$. We write
$x-x_0=\d_n\sum_{i=1}^{3k}h_i e_i$ where the $h_i$ are integers.
\\From \equ(5.003) we have
$$u_{k,L^{-1}}(x) = u_{k,L^{-1}}(x_0)+ \sum_{i=1}^{3k} ((x-x_0),e_{i})
\dpr_{\d_n,e_i}u_{k,L^{-1}}(x_0) +\d_n^{2}\sum_{i,j=1}^{3k}\sum_{s_i=0}^{h_i-1}
\sum_{s_j=0}^{h_j-1}$$
$$\dpr_{\d_n,e_i}\dpr_{\d_n,e_j}
u_{k,L^{-1}}(x_0+ p_j(p_i(x-x_0,s_i),s_j)) $$
\\The argument of $u_{k,L^{-1}}$ in the last term lies entirely in
$X_{\d_n}^{k}$ since $X_{\d_n}$ is convex.
\\Define
$$ \d u_{k,L^{-1}}(x)=u_{k,L^{-1}}(x)-
u_{k,L^{-1}}(x_0)= \d_n\sum_{j=1}^{3k}\sum_{s_j=0}^{h_j-1}
\dpr_{\d_n,e_j}u_{k,L^{-1}}(x_0+p_j(x-x_0,s_j)) \Eq(5.803) $$
\\and
$$\d^2u_{k,L^{-1}}( x)=h_{k,L^{-1}}(x)- u_{k,L^{-1}}(x_0)-\sum_{i=1}^{3k}
(( x- x_0),e_i)\dpr_{\d_n,e_i}u_{k,L^{-1}}(x_0)$$
$$=\d_n^{2}\sum_{i,j=1}^{3k}\sum_{s_i=0}^{h_i-1}
\sum_{s_j=0}^{h_j-1}\dpr_{\d_n,e_i}\dpr_{\d_n,e_j}
u_{k,L^{-1}}(x_0+ p_j(p_i(x-x_0,s_i),s_j))\Eq(5.803.2) $$
\\Now from \equ(5.803), \equ(5.803.2) we have
using the definition of the rescaled function $u_{k,L^{-1}}$
$$\d u_{k,L^{-1}}(x)=L^{-(1+k{3-\e\over 4})}
\d_n\sum_{j=1}^{3k}\sum_{s_j=0}^{h_j-1}
(\dpr_{\d_n,e_j}u_k)_{L^{-1}}(x_0+p_j(x-x_0,s_j)) $$
\\and for $l\ge 1$
$${\dpr_{\d_n}}^{l}\d u_{k,L^{-1}}(x)=L^{-(l+k{3-\e\over 4})}
({\dpr_{\d_n}}^{l}u_{k})(L^{-1}x)$$
\\where for $ {\dpr_{\d_n}}^{l}$ a multi-index convention is implicit.
This implies that
$$\Vert \d u_{k,L^{-1}}\Vert_{C^{2}(X_{\d_n}^{k})} \le c_{1} L^{-(1+k{3-\e\over 4})}
\Vert h_k \Vert_{C^{2}(L^{-1}X_{\d_{n+1}} ^{k})} \Eq(8.303.3)$$
where $c_{1}=O(1)$ since $X$ is a small set.
In the same way starting from \equ(5.803.2)
a little bit of work shows that
$$\Vert \d^{2}h_{k,L^{-1}}\Vert_{C^{2}(X_{\d_n}^{k})}
\le c_{2} L^{-(2+k{3-\e\over 4})}
\Vert h_k\Vert_{C^{2}(L^{-1}X_{\d_{n+1}^{k}})} \Eq(8.303.4) $$
where $c_{2}= O(1)$.
\vskip0.2cm
\vskip0.2cm
\\We will first prove the bounds of \equ(5.79).
\\Consider first the case $(p,m)=(1,0)$. We have
$$ D^{2,0}J_{L}(L^{-1}X_{\d_{n+1}},0,0;g_{2}) =
D^{2,0}J(X_{\d_{n}},0,0;g_{2,L^{-1}})
= D^{2,0}J(X_{\d_{n}},0,0;\d^2 g_{2,L^{-1}}) \Eq(5.809.1) $$
where we have Taylor expanded the function $ g_{2,L^{-1}}$ as in \equ(5.803.2)
and then used the first and second normalization conditions in \equ(4.42).
Therefore
$$\vert D^{2,0}J_{L}(L^{-1}X_{\d_{n+1}},0,0;g_2 )\vert
\le \Vert D^{2,0}J(X_{\d_{n}},,0,0)\Vert \>
\Vert \d^{2}g_{2,L^{-1}}\Vert_{C^{2}(X_{\d_{n}}^{2})}$$
$$\le O(1)L^{-(7-\e)/2}\Vert D^{2,0}J(X_{\d_{n}},0)\Vert\>
\Vert g_2\Vert_{C^{2}(L^{-1}X_{\d_{n+1}}^{2})} $$
where in the last step we have used the bound in \equ(8.303.4) for $k=2$
This proves the case $(p,m)=(1,0)$ of the lemma.
\\To prove the case $(p,m)=(0,2)$ we Taylor expand the function $f_{L^{-1}}(x)$
to second order, then use the first and the third
conditions in \equ(4.42) to get
$$\eqalign{ D^{0,2}J(X_{\d_{n}},0,0;f_{L^{-1}}^{\times 2}) =
D^{0,2}J(X_{\d_{n}},0,0;f_{1,L^{-1}}(x_0),\d^2 f_{2,L^{-1}})
+& D^{0,2}J(X_{\d_{n}},0,0;\d^2 f_{1,L^{-1}}, f_{2,L^{-1}}(x_0))\cr
+& D^{0,2}J(X_{\d_{n}},0,0;\d f_{1,L^{-1}},\d f_{2,L^{-1}}) \cr }\Eq(5.809) $$
\\Therefore
$$\eqalign{
\vert D^{0,2}J(X_{\d_{n}},0,0;f_{L^{-1}}^{\times 2})\vert
&\le \Vert D^{0,2}J(X_{\d_{n}},0,0)\Vert
\Bigl (\Vert f_{1,L^{-1}}\Vert_{C^{2}(X_{\d_{n}})}
\Vert \d^{2}f_{2,L^{-1}}\Vert_{C^{2}(X_{\d_{n}})} \cr
&+\Vert f_{2,L^{-1}}\Vert_{C^{2}(X_{\d_{n}})}\Vert \d^{2}f_{1,L^{-1}}\Vert_{C^{2}(X_{\d_{n}})}\cr
&+\Vert \d f_{1,L^{-1}}\Vert_{C^{2}(X_{\d_{n}})}\Vert \d f_{2,L^{-1}}\Vert_{C^{2}(X_{\d_{n}})})
\Bigr ) \cr
&\le O(1)L^{-(7-\e)/2}\Vert D^{0,2}J(X_{\d_{n}},0,0)\Vert \prod_{j=1}^{2}
\Vert f_{j}\Vert_{C^{2}(L^{-1}X_{\d_{n+1}})} \cr } $$
\\where we have used the bounds \equ(8.303.5),
\equ(8.303.3) and \equ(8.303.4) for the case
$k=1$. This proves the case $(p,m)=(0,2)$.
\\ Next we prove the bounds \equ(5.80) . For this case
$(p,m)=(2,0),(1,2),(0,4)$ so that $2p+m=4$. Taylor expand test functions
around the fixed point $x_{0} \in X_{\d_{n}}$ to first order with remainder.
We get for $(p,m)=(2,0)$
$$ D^{4,0}J_{L}(L^{-1}X_{\d_{n+1}},0;g_{4}) = D^{4,0}J(X_{\d_{n}},0;g_{4,L^{-1}})
= D^{4,0}J(X_{\d_{n}},0;\d g_{4,L^{-1}}) \Eq(5.810) $$
where we have Taylor expanded the function $ g_{4,L^{-1}}$ as in \equ(5.803)
and then used \equ(4.401).
\\Therefore exploiting the bound \equ(8.303.3) for $k=4$ we get
$$\vert D^{4,0}J_{L}(L^{-1}X_{\d_{n+1}},0,0;g_{4} )\vert
\le O(1)L^{-(4-\e)}\Vert D^{4,0}J(X_{\d_{n}},0)\Vert
\Vert g_{4}\Vert_{C^{2}(L^{-1}X_{\d_{n+1}}^4)} $$
which proves the case $(p,m)=(2,0)$.
\\Next we turn to the case $(p,m)=(1,2)$. We have
$$ D^{2,2}J_{L}(L^{-1}X_{\d_{n+1}},0;f^{\times 2},g_{2})=D^{2,2}J(X_{\d_{n}},0;f_{L^{-1}}^{\times 2},g_{2,L^{-1}})
= D^{2,2}J(X_{\d_{n}},0;\d f_{L^{-1}}^{(1)},f_{L^{-1}}^{(2)},g_{2,L^{-1}})+$$
$$+ D^{2,2}J(X_{\d_{n}},0;f_{L^{-1}}^{(1)}(x_0),\d f_{L^{-1}}^{(2)},g_{2,L^{-1}})+
D^{2,2}J(X_{\d_{n}},0;f_{L^{-1}}^{(1)}(x_0),f_{L^{-1}}^{(2)}(x_0),\d g_{2,L^{-1}})\Eq(5.812)$$
where we have used the fourth condition in \equ(4.42). Therefore
$$\vert D^{2,2}J_{L}(L^{-1}X_{\d_{n+1}},0,0;f^{\times 2},g_{2} )\vert
\le \Vert D^{2,2}J(X_{\d_{n}},0,0) \Vert
\Bigl (\Vert \d f^{(1)}_{L^{-1}}
\Vert_{C^{2}(X_{\d_{n}})}\Vert f^{(2)}_{L^{-1}}\Vert_{C^{2}(X_{\d_{n}})}
\Vert g_{2,L^{-1}}\Vert_{C^{2}(X_{\d_{n}}^{2})} + $$
$$+\Vert \d f^{(2)}_{L^{-1}}\Vert_{C^{2}(X_{\d_{n}})}
\Vert f^{(1)}_{L^{-1}}\Vert_{C^{2}(X_{\d_{n}})}
\Vert g_{2,L^{-1}}\Vert_{C^{2}(X_{\d_{n}}^{2})} +
\Vert \d g_{2,L^{-1}}\Vert_{C^{2}(X_{\d_{n}}^{2})}
\Vert f^{(1)}_{L^{-1}}\Vert_{C^{2}(X_{\d_{n}})}
\Vert f^{(2)}_{L^{-1}}\Vert_{C^{2}(X_{\d_{n}})} \Bigr ) $$
Then using the bounds \equ(8.303.5), \equ(8.303.3) for $k=1,2$ and
\equ(8.303.4) for $k=1$ we get
$$\vert D^{2,2}J_{L}(L^{-1}X_{\d_{n+1}},0,0;f^{\times 2},g_{2} )\vert
\le O(1)L^{-(4-\e)}\Vert D^{2,2}J(X_{\d_{n}},0)\Vert
\prod_{j=1}^{2}\Vert f^{(j)}\Vert_{C^{2}(L^{-1}X_{\d_{n}})}
\Vert g_2\Vert_{C^{2}(L^{-1}X_{\d_{n+1}}^{2})}
$$
\\which proves the case $(p,m)=(1,2)$.
\\Finally we treat the case $(n,m)=(0,4)$.
Let ${\cal N}_{4} =(1,2,3,4)$. Then
using the fourth condition of \equ(4.42) we get
$$D^{0,4}J_{L}(L^{-1}X_{\d_{n+1}},0,0;f^{\times 4})=
D^{0,4}J(X_{\d_{n}},0;f_{L^{-1}}^{\times 4})=
\sum_{I\subset {\cal N}_{4}, |I|\not= 4 }
D^{0,4}J(X_{\d_{n}},0;f_{L^{-1}}(x_0)^{\times |I|},
\d f_{L^{-1}}^{\times |I_{c}|})$$
where $f^{\times |I|}=(f_{j})|_{j\in I}$ .
Therefore
$$\vert D^{0,4}J_{L}(L^{-1}X_{\d_{n}},0,0;f^{\times 4} )\vert
\le \Vert D^{0,4}J(X_{\d_{n}},0,0)\Vert
\sum_{I\subset {\cal N}_{4}, |I|\not= 4 }
\Vert f_{L^{-1}}\Vert_{C^{2}(X_{\d_{n}})}^{|I|}
\Vert \d f_{L^{-1}}\Vert_{C^{2}(X_{\d_{n}})}^{|I_{c}|} $$
\\Because of the condition on $I$ in the above sum $|I_{c}|\ge 1$.
Therefore using the bounds \equ(8.303.5) and \equ(8.303.3)
we get
$$\vert D^{0,4}J_{L}(L^{-1}X_{\d_{n}},0,0;f^{\times 4} )\vert
\le O(1) L^{-(4-\e)}\Vert D^{0,4}J(X_{\d_{n}},0)\Vert \prod_{j=1}^{4}
\Vert f_j\Vert_{C^{2}(L^{-1}X_{\d_{n+1}})} $$
\\which proves the case $(p,m)=(0,4)$ and thus completes the proof of
Lemma~5.24. \bull
\vglue.3truecm
\\{\it Corollary 5.25
\\Let $Y_{\d_{n+1}}=L^{-1}X_{\d_{n+1}}$ where $X$ is a small set,
$Z_{\d_{n+1}}=L^{-1}{{\bar X}_{\d_{n+1}}}^{L}$
and let $J(X_{\d_n},\Phi)$ be normalized as in \equ({4.42}). By definition
$J_{L}(Y_{\d_{n+1}},\Phi)=J(X_{\d_n},S_L\Phi)$. Then
$$
|J_{L} (Y_{\d_{n+1}})|_{\bf h}
\le O (1) L^{-(7-\e)/2}|J(X_{\d_n})|_{\bf \hat h}
\Eq(5.lem39-1)
$$
$$
\Vert
J_{L} (Y_{\d_{n+1}}) e^{-\tilde{V}_{L} (Z_{\d_{n+1}}\setminus Y_{\d_{n+1}})}
\Vert_{{\bf h},G_{\k}} \le O (1) L^{-(7-\e)/2}
\bigg[
|J(X_{\d_n})|_{\bf \hat h} + \Vert J(X_{\d_n})\Vert_{{\bf \hat h},G_{3\k}}
\bigg]
\Eq(5.lem39-2)
$$
$$
|J_{L}(Y_{\d_{n+1}})|_{\bf h_*}
\le O (1) L^{-(7-2\e)/2}|J(X_{\d_n})|_{\bf {\hat h*}}
\Eq(5.lem39-3)
$$
}
\vglue.2truecm
\\{\it Proof}
\\\equ({5.lem39-1}), \equ({5.lem39-3})
follow easily from Lemma~5.24 taking advantage of the scaling present to
shift
from $\bf h$ to $\bf \hat h$.
To see this recall that $ h_{F}= 2 \hat h_{F}$. Hence
$$|J_{L} (Y_{\d_{n+1}})|_{\bf h}= \sum_{n=0}^{\infty}\sum_{m=0}^{m_0}
{\hat h_{F}}^{2n} {h_{B}^{m}\over m !} 2^{2n}
\Vert D^{2n,m}J_{L}(Y_{\d_{n+1}},0)\Vert
\Eq(5.995)$$
\\Only terms with $2n+m$ even contribute. For $2n+m \le 4 $ use Lemma~5.24 and
observe that $2^{2n} \le O(1)$. For $2n+m \ge 6 $, we have for $L$
sufficiently large and $\e$ sufficiently small depending on $L$
$$\eqalign{
2^{2n} \Vert D^{2n,m}J_{L} (Y_{\d_{n+1}},0)\Vert
\le 2^{2n} L^{-(2n+m){(3-\e)\over 4}}
\Vert D^{2n,m}J (X_{\d_n},0)\Vert \cr
\le L^{-(2n+m){(3-{1\over 3} -\e)\over 4}}
\Vert D^{2n,m}J (X_{\d_n},0)\Vert \cr
\le O(1) L^{(7-\e)/2}\Vert D^{2n,m}J (X_{\d_n},0)\Vert
}$$
\\Putting the two case together in \equ(5.995) gives \equ({5.lem39-1}). The
proof of \equ({5.lem39-3}) is the same on replacing $h_B$ by $h_{B*}$.
\\For \equ({5.lem39-2}) we write
$$
\Vert J_{L} (Y_{\d_{n+1}},{\bf \Phi})
e^{-\tilde{V}_{L} (Z_{\d_{n+1}}\setminus Y_{\d_{n+1}},{\bf \Phi})}
\Vert_{\bf h}
\le
\Vert J_{L} (Y_{\d_{n+1}},{\bf \Phi}) \Vert_{\bf h}
\Vert e^{-\tilde{V}_{L} (Z_{\d_{n+1}}\setminus Y_{\d_{n+1}},{\bf \Phi})}\Vert_{\bf h}\le$$
$$\le O (1)G_\k(Z_{\d_{n+1}},{\phi})\bigg[
|J_{L}(Y_{\d_{n+1}})|_{\bf h} +
L^{-m_{0}d_s}\Vert J_{L} (Y_{\d_{n+1}})\Vert_{h_F,L^{[\phi]}h_B,G_{\kappa}}
\bigg]
$$
where we used Lemmas~5.5 and Lemma~5.15. By
\equ({5.lem39-1}), and rewriting the second term by
moving the scaling from $J$ to the norm,
$$
\Vert J_{L} (Y_{\d_{n+1}},{\bf \Phi})
e^{-\tilde{V}_{L} (Z_{\d_{n+1}}\setminus Y_{\d_{n+1}},{\bf \Phi})}
\Vert_{\bf h}\le
O (1)G_\k(Z_{\d_{n+1}},\f)\bigg[
L^{-(7-\e)/2}|J(X_{\d_{n}})|_{\bf \hat h} +
L^{-m_{0}d_s}\Vert J(X_{\d_{n}})\Vert_{\bf \hat h ,G_{3\kappa}}
\bigg]
$$
\\Recall that $m_0=9$ and the scaling dimension $d_s=(3-\e)/4$.
\equ({5.lem39-2}) now follows by multiplying both sides by
$G_\k^{-1}(Z_{\d_{n+1}},\f)$, and taking the supremum over $\f$. \bull
\vglue.2truecm
\\{\it Lemma 5.26
$$\Vert\tilde F_{R_n}e^{-\tilde V_n}\Vert
_{{\bf h},G_{\k},\AA,\d_n}\le O(1)\e^{3/4-\eta}
\Eq(5.71)$$
$$\vert\tilde F_{R_n}e^{-\tilde V_n}\vert
_{{\bf {\hat h_{*}}},\AA,\d_n}\le O (1)\e^{11/4-\eta}
\Eq(5.72)$$
\\and $J_n= R_n^{\sharp}-\tilde F_{R_n}e^{-\tilde{V_n}}$ satisfies on small
sets the bounds}
$$\Vert J_n\Vert_{{\bf{\hat h}},G_{3\k},\AA,\d_n}
\le O(1){\bar g}^{{3\over 4}-\eta}
\qquad \vert J_n \vert_{{\bf{\hat h}},\AA,\d_n }
\le O(1){\bar g}^{{3\over 4}-\eta}
\qquad\qquad\vert J_n\vert_{{\bf{\hat h_{*}}},\AA,\d_n}
\le O(1){\bar g}^{{11\over 4}-\eta} \Eq(5.71a) $$
\vglue.3truecm
\\{\it Proof}
\\First we prove \equ({5.71}). $\tilde F_{R_n}$ is defined
in \equ(4.57) and \equ({4.59}), and is supported on
small sets. We estimate its $\bf h$ norm as in the proof of Lemma~5.19.
$$\eqalign{ \Vert {\tilde F}_{ R_n}(X_{\d_n},{\bf \Phi})\Vert_{\bf h}
&\le \sum_{P} \vert {\tilde \a}_{n,P}(X_{\d_n})\vert
\int_{X_{\d_n} } dx\ \Vert
P ({\bf \Phi}(x),\dpr{\bf \Phi}(x))\Vert_{\bf h} \cr
&\le C_L\sum_{P}\vert {\tilde\a}_{n,P}(X_{\d_n})\vert {\bar g}^{-1}
\bigg(
{\bar g}\int_{X_{\d_n}}dx\> (|\f|^{2})^2(x)
+{\bar g}^{1/2}\Vert\f\Vert^2_{X_{\d_n},1,\s}+1
\bigg) \cr
&\le C_L\sum_{P}\vert {\tilde\a}_{n,P}(X_{\d_n})\vert
{\bar g}^{-1}G_\k(X_{\d_n},\f)e^{\g{\bar g}\int_{X_{\d_n}}dy(|\f|^2)^2(y)}
\cr }
$$
\\for any $\g =O(1)>0$. Hence, using Lemma 5.5
$$\Vert\tilde F_{R_n}(X_{\d_n},{\bf \Phi})e^{-\tilde V_n(X_{\d_n},{\bf \Phi})}\Vert_{\bf h}
\le\Vert\tilde F_{R_n}(X_{\d_n},{\bf \Phi})\Vert_{\bf h}
\Vert e^{-\tilde V_n(X_{\d_n},{\bf \Phi})}\Vert_{\bf h}
\le C_L\sum_{P}2^{|X_{\d_n}|}\vert{\tilde\a}_{n,P}
(X_{\d_n})\vert {\bar g}^{-1}G_\k(X_{\d_n},\f) $$
\\We thus obtain ( remembering that ${\tilde\a}_{n,P}$ are supported on small
sets) on using \equ(5.67) of Lemma~5.17 for $\e$ sufficiently small depending
on $L$, implying $\bar g$ sufficiently small,
$$\Vert\tilde F_{R_n}(X_{\d_n},{\bf \Phi})e^{-\tilde V_n(X_{\d_n},{\bf \Phi})}
\Vert_{{\bf h},G_{\k},\AA,\d_n}
\le C_L{\bar g}^{-1}\sum_{P}\Vert \tilde{\a}_{n,P}\Vert_{\AA,\d_n}
\le O(1){\bar g}^{3/4 -\eta}$$
\\This proves \equ({5.71}).
Now we turn to the proof of \equ({5.72}).
As observed in the proof of Lemma~5.17,
${\bf {\hat h_{*}}}^{n_P}\vert{\tilde\a}_{n,P}\vert_{\AA,\d_n} \le n(P)!\>
\vert 1_{S}R_n^\sharp\vert_{\bf {\hat h_{*}},\AA,\d_n}
\le O(1) {\bar g}^{11/4-\eta}$.
We have from the definition of $\tilde F_{R_n}$ given in \equ(4.57)
$\vert \tilde F_{R_n}(X_{\d_n})\vert_{\bf {\hat h_{*}}} \le O(1)\sum_{P}
\vert {\tilde\a}_{n,P}(X_{\d_n})\vert {\bf {\hat h_{*}}}^{n_P}$, whence
$$\vert \tilde F_{R_n}\vert_{{\bf {\hat h_{*}}},\AA,\d_n} \le
\sum_{P}\vert{\tilde\a}_{n,P}\vert_{\AA,\d_n}
{\bf {\hat h_{*}}}^{n_P}\le O(1){\bar g}^{11/4-\eta}$$
which proves \equ({5.72}).
\\To get these bounds for $J_n= R_n^{\sharp}-\tilde F_{R_n}e^{-\tilde{V_n}}$
we use \equ({5.71}) and \equ({5.72}) to bound $\tilde
F_{R_n}e^{-\tilde{V_n}}$ part. We can substitute ${\bf{\hat h}}$ for $\bf h$
in \equ({5.71}) since the ${\bf{\hat h}}$ norm is smaller than the $\bf h$
norm. We bound $R_n^{\sharp}$ by
Lemma~5.17. We have also used the trivial bound
$\vert J \vert_{{\bf{\hat h}},\AA,\d_n }
\le\Vert J\Vert_{{\bf{\hat h}},G_{3\k},\AA,\d_n}$ to obtain the second
inequality for $J$ from the first. \bull
\vglue.3truecm
\\{\it Lemma 5.27}
$$\Vert R_{n+1,\rm linear}\Vert
_{{\bf h},G_{\k},\AA, \d_{n+1}}\le O(1)L^{- (1-\e)/2}{\bar g}^{3/4-\eta}
\Eq(5.73)$$
$$\vert R_{n+1,\rm linear}\vert
_{{\bf h}_{*},\AA, \d_{n+1}}\le O(1)L^{- (1-\e)/2}{\bar g}^{11/4-\eta}
\Eq(5.74)$$
\\{\it Proof}
\\Let $R_{n+1,\rm linear}$, given in \equ(4.59b) is the sum of two
terms which represent contributions from small/large sets respectively. Let
$R_{n+1,\rm linear, s.s}$ denote the first term:
$$R_{n+1,\rm linear, s.s}(Z_{\d_{n+1}}) =\sum_{X_{\d_{n+1}}: {\rm small\ sets}
\atop L^{-1}{\bar X_{\d_{n+1}}}^{L}=Z_{\d_{n+1}}}
e^{-\tilde{V}_{L} (Z_{\d_{n+1}}\setminus Y_{\d_{n+1}})}
J_{n,L} (Y_{\d_{n+1}}) \Eq(4.59b3) $$
\\where $Y_{\d_{n+1}}=L^{-1}X_{\d_{n+1}}$ and
$J_n= R_n^{\sharp}-\tilde F_{R_n}e^{-\tilde{V_n}}$. By Corollary~5.25 we get
$$ \Vert R_{n+1,\rm linear, s.s}(Z_{\d_{n+1}})\Vert_{{\bf h},G_{\k}}\le
O (1)L^{-(7-\e)/2}\sum_{X_{\d_{n+1}}: {\rm small\ sets}
\atop L^{-1}{\bar X_{\d_{n+1}}}^{L}=Z_{\d_{n+1}}}
\bigg[
|J_n(X_{\d_n})|_{\hat{\bf h}} +
\Vert J(X_{\d_{n}})\Vert_{\hat{\bf h},G_{3\k}} \bigg]
\Eq(4.59b1)
$$
\\Note that $Z_{\d_{n+1}}$ fixes $Z_{\d_{n}}$ by restriction and the sum
on the right hand side is the same as the sum over $X_{\d_{n}}$ such that
$L^{-1}{\bar X_{\d_{n}}}^{L}=Z_{\d_{n}}$. We
multiply both sides by $\AA(Z_{\d_{n+1}})$. On the right hand side
we have $\AA(Z_{\d_{n+1}})=\AA(Z_{\d_{n}})=\AA({\bar X_{\d_{n}}}^{L})
\le O(1)\AA(X_{\d_{n}})$ by \equ(2.5reg). We fix a unit block $\D_{n+1}$
and sum over $Z_{\d_{n+1}}\supset\D_{n+1}$. This fixes by restriction to the
over $Z_{\d_{n}}\supset\D_{n}$ on the right hand side. The argument on
p.790 of [BDH-est] controls the constrained sum on $X_{\d_{n}}$ such that
$L^{-1}{\bar X_{\d_{n}}}^{L}=Z_{\d_{n}}\supset\D_{n} $ by $L^{3}$ times the
sum over $X_{\d_{n}}\supset\D_{n}$. Taking then the supremum over the
fixed unit block gives
$$\eqalign{\Vert R_{n+1,\rm linear, s.s}\Vert_{{\bf h},G_{\k},\AA,\d_{n+1}}
&\le
O (1)L^{-(7-\e)/2}L^{3} \bigg[
|J_n|_{\hat{\bf h},\AA,\d_{n}} +
\Vert J_n\Vert_{\hat{\bf h},G_{3\k},\AA,\d_{n}}
\bigg]\cr
&\le O(1)L^{- (1-\e)/2}{\bar g}^{3/4-\eta}}
\Eq(4.59b3) $$
\\where for the second inequality we used Lemma~5.26.
\\The second term in \equ(4.59b)for $R_{n+1,\rm linear}$ which gets
contributions only from large sets is
$$R_{n+1,\rm linear,l.s.}(Z_{\d_{n+1}}) =\sum_{X_{\d_{n+1}}: {\rm large\ sets}
\atop L^{-1}{\bar X_{\d_{n+1}}}^{L}=Z_{\d_{n+1}}}
e^{-\tilde{V}_{L} (Z_{\d_{n+1}}\setminus Y_{\d_{n+1}})}
R_{n,L}^{\natural} (L^{-1}X_{\d_{n+1}}) \Eq(4.59b2) $$
\\where we have used $J_n=R_n^{\sharp}$ since the relevant part
${\tilde F}_{R_n}$ is supported on small sets. We first bound
in the ${\bf h}$ norm and observe that because of the rescaling
involved and Lemma~5.17
$$\Vert R_{n,L}^{\natural} (L^{-1}X_{\d_{n+1}})\Vert_{{\bf h}}
\le \Vert R_{n}^{\sharp} (X_{\d_{n}})\Vert_{\hat{\bf h},G_{3\k}}
G_{\k}(X_{\d_{n+1}})
\le O(1) 2^{|X_{\d_n}|} {\bar g}^{3/4-\eta} G_{\k}(Z_{\d_{n+1}}) $$
\\so that on using Lemma~5.5 for $e^{-\tilde{V}_{L}}$
$$\Vert R_{n+1,\rm linear,l.s.}(Z_{\d_{n+1}})\Vert_{{\bf h}, G_{\k}}
\le O(1){\bar g}^{3/4-\eta} \sum_{X_{\d_{n+1}}: {\rm large\ sets}
\atop L^{-1}{\bar X_{\d_{n+1}}}^{L}=Z_{\d_{n+1}}} 2^{2|X_{\d_{n+1}}|} $$
\\We estimate the $\AA$ norm as before except that for large sets we use
from \equ(2.6reg)
\\$\AA(L^{-1}{\bar X_{\d_{n+1}}}^{L})
\le c_{p}L^{-4}\AA_{-p}(X_{\d_{n+1}}) $ for any positive integer
$p$ with $c_p =O(1)$. Choose $p=2$. Therefore
$$\Vert R_{n+1,\rm linear,l.s.}\Vert_{{\bf h}, G_{\k}, \AA, \d_{n+1}}
\le O(1) L^{-1}{\bar g}^{3/4-\eta} \Eq(4.59b4) $$
\\Adding the contributions \equ(4.59b3) and \equ(4.59b4) we get \equ(5.73).
\equ(5.74) can be proved in the same way.
For the small set contribution we use the kernel bounds in Corollary~5.25
and Lemma~5.26. For the large set contribution we first use the rescaling
involved to shift on the right hand side the ${\bf h}_*$ norm to the
$\hat{\bf h}_*$ norm followed by the kernel bound in Lemma~5.17. \bull
\vglue.3truecm
\\{\it Proof of Theorem~5.1 }
\vglue.2truecm
\\From \equ(4.221), $R_{n+1}$ is the sum of $R_{n+1,\rm main}$,
$R_{n+1,\rm linear}$, $R_{n+1,3}$ and $R_{n+1,4}$. $R_{n+1,\rm main}$
satisfies the bound given in Lemmas~5.21. For $L$ large and
$\e$ small depending on $L$ implying $\bar g$ sufficiently small
$C_L{\bar g}^{3/4}\le L^{-1/2}{\bar g}^{3/4 -\eta}$ with $\eta=1/64$. Similarly
$C_L{\bar g}^{3-3\d/2}\le L^{-1/2}{\bar g}^{11/4 -\eta}$ for $\d=\eta$.
Therefore
$|||R_{n+1,\rm main}|||_{n+1}\le $
\\$L^{-1/2}{\bar g}^{11/4 -\eta}$. Similarly from Lemmas~5.22 and 5.23 we
get $ |||R_{n+1,j}|||_{n+1}\le L^{-1/2}{\bar g}^{11/4 -\eta}$, for
$j=3,4$. Adding these bounds to that provided for $R_{n+1,\rm main}$
by Lemma~5.27 we have that the
sum satisfies the bound \equ(55.7) for $L$ sufficiently large. The bounds
\equ(555.3), \equ(555.4) and \equ(55.5) have been proved in Lemmas~5.17 and
5.18. \bull
\vglue0.5truecm
\\{\bf 6. EXISTENCE OF THE GLOBAL RENORMALIZATION GROUP TRAJECTORY
AND THE STABLE MANIFOLD }
\numsec=6\numfor=1
\vskip.5cm
\\In this section we will prove the existence of a global RG
trajectory. The global trajectory is one that exists on every scale
$n$ no matter how large, in other words on every lattice $(\d_n\math{Z})^{3}$
no matter how fine, and the entire trajectory lies in a uniformly bounded
domain. Our strategy will be to hold the variables of the initial unit
lattice theory in a suitable domain. Then apply the RG map
$n_0$ times where $n_0\ge 0$ is any finite integer.
The relevant variable (mass) will have expanded. Then restart the
trajectory at this scale ($n\ge n_{0}$). We will show that for all variables
at scale $n_0\ge 0$ held in a suitable domain ( arrived at by $n_0$
iterations from
the unit lattice domain) there exists a critical mass $\m_{n_0}$ in this domain
such that the entire trajectory for all $n\ge n_0$ is uniformly bounded. We
will then prove that when $n_0$ is sufficiently large the
critical mass is a Lipshitz continuous function of the other
(contracting) variables in the domain. This proves the existence of the
stable manifold at scale $n_0$. The main theorems are Theorems~6.2 and 6.4.
One consequence is that the coupling constant $g_n$ is bounded away from
$0$ uniformly in $n$.
\\The idea of proving the
existence of the stable manifold after a sufficient number of RG iterations
originated in discussions with David Brydges. Indeed the stable manifold
has been established for various continuum models (see [BDH-eps], [BMS]).
By moving forward sufficiently on the trajectory we move sufficiently
close to the continuum, lattice RG artifacts become less important
and the stable manifold can be estblished.
\vskip0.5truecm
\\{\it 6.1}. Define $\tilde g_{n}= g_{n}-\bar g $ with $\bar g $ defined by
\equ(55.4) and $\tilde{\bf w}_n= {\bf w}_n- {\bf w}_*$. Here ${\bf w}_*$ is
the function on $\cup_{n\ge 0}(\d_n\math{Z})^{3}$ defined in Lemma~5.9.
According to Lemma~5.9, $\tilde{\bf w}_n \rightarrow 0$ geometrically fast
in ${\cal W}_l$ for all $l\ge 0$.
\\We will use as coordinates of the RG trajectory
$$\y_n=(\tilde g_{n},\m_{n},R_{n},\tilde{\bf w }_n)\Eq(6.2) $$
\\The RG map
$$\y_{n+1}=f_{n+1}(\y_n) \Eq(6.3)$$
\\can be written in components
(see \equ(4.47A), \equ({4.222}), \equ(4.54) and Lemma~5.9)
$$\tilde g_{n+1}=f_{n+1,g}(\y_n)=\alpha(\epsilon)\tilde g_n +\tilde\xi_n(\y_n)
\Eq(6.4)$$
$$\m_{n+1}=f_{n+1,\m}(\y_n)=L^{3+\e\over 2}\m_n+\tilde\r_n(\y_n)\Eq(6.5)$$
$$R_{n+1}=f_{n+1,R}(\y_n)=: U_{n+1}(\y_n)\Eq(6.6)$$
$$\tilde{\bf w}_{n+1}= f_{n+1,w}(\y_n)= ({\bf v}_{n+1}-{\bf v}_{c,*})
+\tilde{\bf w}_{n,L} \Eq(6.7)$$
with initial
$$\y_0=(\tilde g_{0},\m_{0},R_0,\tilde{\bf w}_0) \Eq(6.8)$$
\\$\a(\e),\> \xi_n,\> \r_n$ are defined by
$$\alpha(\epsilon)=2 - L^{\e} = 1 - O (\log L)\e$$
\\for $\e$ sufficiently small depending on $L$, and
$$\tilde\xi_n(\y_n)=-L^{2\e}a_{c*}\tilde g_n^2
-L^{2\e}(a_n-a_c*)(\tilde g_n +\bar g)^{2}
+\xi_{n}(\y_n)\Eq(6.9)$$
$$\tilde \r_n(\y_n)=-L^{2\e}b_n(\bar g+\tilde g_n)^2
+\r_{n}(\y_n)\Eq(6.10)$$
\\Let $E_n$ be the Banach space consisting of elements $\y_n$ with the norm
$$\Vert \y_n\Vert_{n} =\max((\n{\bar g})^{-1}|\tilde g_n|,\>
{\bar g}^{-(2-\d)}|\m_n|,\> {\bar g}^{-(11/4-\eta)}|||R_n|||_{n},
{\tilde c}_L^{-1}\Vert \tilde{\bf w}_n\Vert_{n}) \Eq(6.11)$$
\\where the norm $|||R|||_{\d_n}$ of $R_n$ is as defined in \equ(55.03). $\n$
is the $O(1)$ constant which figures in the specification of the domain
${\cal D}_n$, \equ(55.1). We have $0<\n<1$. We will take $\n>0$ sufficiently
small depending on $L$ and this will be specified in the proofs of
Lemmas~6.3 and 6.4 below. The norm $\Vert \tilde{\bf w}_n\Vert_{\d_n}$ and the
constant $\tilde c_L$ are as specified in Lemma~5.9.
\vglue0.3cm
\\{\it 6.2. Domains and bounds}
\vglue0.2cm
Let $E_n(r)\subset E_n$ be the open ball of radius $r$,
centered at the origin:
$$E_n(r)=\{\y_n\in E_n: \Vert \y_n\Vert_n < r\}\Eq(6.13)$$
\\Let ${\cal D}_n$ be the domain of $(g_n,\m_n,R_n)$ defined in
\equ(55.1) and \equ(55.02).
$$\y_n\in E_n(1)\Rightarrow (g_n,\m_n,R_n)\in {\cal D}_n\Eq(6.14)$$
\\and then Theorem 5.1 holds.
\\Let $\y_n\in E_n(1)$. Let $\e>0$ be sufficiently small (depending on $L$).
Then from Theorem~5.1, Lemma~5.12 , \equ(6.9) and \equ(6.10) we get the bounds
$$\eqalign{
|\tilde\xi_{n}(\y_n)|\le & C_L\Bigl((\n^2 + L^{-nq}){\bar g}^{2}
+{\bar g}^{11/4-\eta}\Bigr) \cr
|\tilde\r_{n}(\y)|\le & C_L {\bar g}^{2}\cr
|||U_{n+1}(\y_n)|||_{n+1}\le &L^{-1/4}{\bar g}^{11/4-\eta} \cr}\Eq(6.15)$$
\\We have the following Lipshitz bounds :
\vglue.3truecm
\\{\it Lemma 6.1}
\\Let $\y_n,\y_n'\in E_n(1/4)$. Then we have:
$$\eqalign{
{\rm (i) }&\quad
|\tilde\xi_n(\y_n)-\tilde\xi_n(\y_n')|\le C_L\Bigl((\n^2 + L^{-nq}){\bar g}^{2}
+{\bar g}^{11/4-\eta}\Bigr) \Vert \y_n-\y_n'\Vert_n \cr
{\rm (ii) }&\quad
|\tilde\r_n(\y_n)-\tilde\r_n(\y_n')|\le
C_L {\bar g}^{2}\Vert \y_n-\y_n'\Vert_n \cr
{\rm (iii) }&\quad
|||U_{n+1}(\y_n)-U_{n+1}(\y_n')|||_{n+1}
\le O(1)L^{-1/4}{\bar g}^{11/4-\eta}\Vert \y_n-\y_n'\Vert_n \cr
{\rm (iv) } &\quad
{\tilde c}_L^{-1}
\Vert f_{n+1,\bf w}(\y_n)- f_{n+1,\bf w}(\y_n')\Vert_{n+1}
\le L^{-1/5}\Vert \y_n-\y_n'\Vert_n \cr}
\Eq(6.17)$$
\\{\it Proof}:
\\$\tilde\xi_n,\tilde\r_n,U_{n+1}$ are (norm) analytic functions in
${\cal D}_n$
and thus in $E_n(1)$. The analyticity follows from the algebraic
operations in Section 4, the norm analyticity of the reblocking map
together with the norm analyticity of the extraction map
(Theorem 5 [BDH-est]). Therefore we can use Cauchy estimates exactly as in
in the proof of Lemma~6.1 of [BMS] together with
the bounds \equ(6.15) to get (i), (ii) and (iii). To get (iv) note that
from \equ(6.7) and the definition of the norms in \equ(5.20) we have
$$\eqalign{\Vert f_{n+1,\bf w}(\y_n)- f_{n+1,\bf w}(\y_n')\Vert_{n+1}
&\le L^{2d_s} \max_{1\le p\le 3}
\sup_{x\in(\d_{n+1}{\bf Z})^{3}}
\left((|x|+\d_{n+1}) ^{6p+1\over 4}|{\tilde w}_{n}^{(p)}(Lx)-
{\tilde w}^{\prime (p)}_{n}(Lx)|
\right) \cr
&\le L^{2d_s} \max_{1\le p\le 3}
\sup_{y\in(\d_{n}{\bf Z})^{3}}
\left(({|y|\over L}+\d_{n+1}) ^{6p+1\over 4}
|\tilde w_{n}^{(p)}(y)-\tilde w_{n}^{\prime (p)}(y)|
\right) \cr
&\le L^{2d_s -7/4} \max_{1\le p\le 3}
\sup_{y\in(\d_{n}{\bf Z})^{3}}
\left((|y|+\d_{n}) ^{6p+1\over 4}
|\tilde w_{n}^{(p)}(y)-\tilde w_{n}^{\prime (p)}(y)|
\right) \cr
&\le L^{-1/5}\Vert \tilde{\bf w}_n-\tilde{\bf w}_{n}^{\prime}\Vert_{n}\cr }
$$
\\where we have used $d_s={3-\e\over 4}$ and then $L$ large and $\e$
sufficiently small. Now dividing both sides by $\tilde c_L$ we get (iv).
$\bull$
\vglue0.2cm
\\{\it 6.3. Existence of the global RG trajectory}
\vglue0.1cm
Let ${\cal D}_{n}$ be the domain specified by \equ(55.1),\equ(55.02),
and \equ(55.03).
Let $(\tilde g_0,\m_0, R_0)$ belong to
$\tilde{\cal D}_{0}\subset {\cal D}_{0}$
where $\tilde{\cal D}_{0}$ is specified by
$$\vert \tilde g_0 \vert < 2^{-(n_0 +5)}\n {\bar g},
\ \ \vert\mu_0\vert < 2^{-(n_0 +5)}L^{-{3+e\over 2}n_0} {\bar g}^{2-\delta}$$
$$||| R_0 |||_{0}\le {\bar g}^{11/4-\eta} $$
\\Let $n_0$ be a positive integer. By iterating the RG map $n_0$ times
using Theorem~5.1 and the flow equation \equ(4.47A) recursively we obtain for
$\e$ sufficiently small depending on $L$ and $n_0$,
$(g_{n_0},\m_{n_0},R_{n_0}) \in {\cal D}_{n_0}(1/32)$ where
${\cal D}_{n_0}(1/32)$ is specified by
$$\vert \tilde g_{n_0}\vert < {1\over 32}\n {\bar g},
\ \ \vert \m_{n_0}\vert < {1\over 32} {\bar g}^{2-\delta} $$
$$||| R_{n_0} |||_{n_0}<{1\over 32}{\bar g}^{11/4-\eta} $$
\\We will now prove the existence of a global solution to
the discrete flow map \equ(6.3):
$$\y_{n+1}=f_{n+1}(\y_{n}),\> \forall \> n\ge n_0 $$
with initial condition
$$\y_{n_0}=(\tilde g_{n_0},\m_{n_0}, \tilde{\bf w}_{n_0},R_{n_0} )$$
\\in a bounded domain.
To this end we consider the Banach space ${\bf E}_{n_0}$ of
sequences ${\bf s}_{n_0}=\{\y_n\}_{n\ge n_{0}} $, each $\y_n \in E_n$ ,
with the norm
$$\Vert {\bf s}_{n_0}\Vert = \sup_{n\ge n_{0}}\Vert \y_n \Vert_{n} \Eq(6.111)$$
\\and the open ball ${\bf E}_{n_0}(r)\subset \bf E$
$${\bf E}_{n_0}(r)=\{{\bf s}_{n_0} : \Vert{\bf s}_{n_0}\Vert < r \}
\Eq(6.131)$$
\\We will derive on the space of sequences ${\bf E}_{n_0} $ an equation that a
global RG trajectory must solve and then prove for
$\y_{n_0}\in E_{n_0}(1/32)$
the existence of a unique solution in the ball
${\bf E}_{n_0}(1/4)$,
for a suitable choice of $r$, by the contraction mapping principle.
This adapts a standard method from the theory of hyperbolic dynamical systems
in Banach spaces due to Irwin in [I]. Irwin's analysis is explained by Shub
in Appendix 2, Chapter 5 of [S]. For earlier applications see
section 5 of [BDH-eps] and section 6 of [BMS].
\vglue.3truecm
\\{\it Theorem 6.2
\\Let $L$ be large, $\n$ be sufficiently small depending on $L$, then
$\e$ sufficiently small depending on $L$. Let
$\y_{n_0}\in E_{n_0}(1/32)$ for any integer $n_{0}\ge 0$. Then there is a
$\m_{n_0}$ such that there exists a sequence
${\bf s}_{n_0}=\{\y_n\}_{n\ge n_0} $ in ${\bf E}_{n_0}(1/4)$ satisfying
$\y_{n+1}=f_{n+1}(\y_{n})$ for all $n\ge n_0$.}
\vglue.2truecm
\\{\it Proof}
\\Our initial data will be at scale $n_0$. Let $n_0\le n\le N-1$. We iterate
the map \equ(6.4) forwards $N$ times. We iterate the
map \equ(6.5) backwards $N-n_0$ times starting from a given $\m_{N}$.
We then easily derive
$$\tilde g_{n+1}=\alpha(\epsilon)^{n+1-n_0}\tilde g_{n_0} +
\sum_{j=n_0}^{n}\alpha(\epsilon)^{n-j}\tilde\xi_j(\y_j),
\quad n_0\le n\le N-1$$
$$\m_{n+1}=L^{-{3+\e\over 2}(N-n-1)}\m_{N}-
\sum_{j=n+1}^{N-1}L^{-{3+\e\over 2}(j-n)}\tilde\r_j(\y_j),
\quad n_0-1\le n\le N-2 $$
\\Let us fix $\m_{N}=\m_f$ and take $N\rightarrow\io$. In other words we
assume the $\m_n$ flow is bounded and then must show that such a flow exists.
We have
$$\tilde g_{n+1}= \alpha(\epsilon)^{n+1-n_0}\tilde g_{n_0} +
\sum_{j=n_0}^{n} \alpha(\epsilon)^{n-j}\tilde\xi_j(\y_j),\quad n\ge n_0
\Eq(6.18)$$
$$\m_{n+1}=-
\sum_{j=n+1}^{\io}L^{-{3+\e\over 2}(j-n)}\tilde\r_j(\y_j),
\quad n\ge n_0-1\Eq(6.19)$$
\\together with
$$R_{n+1}=U_{n+1}(\y_{n}), \quad n\ge n_0\Eq(6.20)$$
$$\tilde{\bf w}_{n+1}= ({\bf v}_{n+1}-{\bf v}_{c*})
+\tilde{\bf w}_{n,L} \Eq(6.201) $$
\\The solution of the
autonomous flow $\tilde{\bf w}_n$ given by lemma~5.9 can be incorporated
in the $\y_n$ and $\tilde{\bf w}_n$ is then no longer a flow variable.
\\For $\e$ sufficiently small (depending on $L$)
$$0< \alpha(\epsilon) <1 \Eq(6.21)$$
\\Note that ${\bf s}_{n_0}\in {\bf E}_{n_0}(1/4)$ implies $\y_j\in E_j(1/4)$
for all $j\ge n_0$.
Then the infinite sum of \equ(6.19)
converges by \equ(6.21) and \equ(6.15). So $\m_{n_0}$ has now
been determined provided \equ(6.18)-\equ(6.20) has a solution in the afore
mentioned ball. It is easy to verify that any solution of \equ(6.18)-
\equ(6.19), together with the autonomous $\tilde{\bf w}$ flow, is a solution
of the RG flow $\y_{n+1}=f_{n+1}(\y_{n})$ for $n\ge n_0$.
\vglue0.2cm
\\We write \equ(6.18)-\equ(6.20) in the form
$$\y_{n+1}=F_{n+1}({\bf s}_{n_0}), \quad n\ge n_0 \Eq(6.22)$$
\\where ${\bf s}_{n_0}=(\y_{n_0}, \y_{n_0+1}, \y_{n_0+2},...)$ and
$F_{n+1}$ has components
$(F_{n+1}^{(g)},F_{n+1}^{(\m)},F_{n+1}^{(R)})$ given by the r.h.s. of
\equ(6.18), \equ(6.19), \equ(6.20) respectively.
\\If we write
$${\bf F}({\bf s}_{n_0})=
(F_{n_0}({\bf s}_{n_0}), F_{n_0+1}({\bf s}_{n_0}),...)$$
\\then \equ(6.22) can be written as a fixed point equation
$${\bf s}_{n_0}={\bf F}({\bf s}_{n_0})\Eq(6.23)$$
\\We seek a solution of \equ(6.23) in the open ball ${\bf E}_{n_0}(1/4)$ with
initial data $\y_{n_0}=(\tilde g_{n_0},\m_{n_0},R_{n_0},\tilde{\bf w}_{n_0})$
in $E_{n_0}(1/32)$.
The existence of a unique solution follows by the standard
contraction mapping principle and the next Lemma. \bull
\vglue.3truecm
\\{\it Lemma 6.3}
$${\bf s}_{n_0}\in {\bf E}_{n_0}(1/32)\Rightarrow{\bf F}({\bf s}_{n_0})
\in {\bf E}_{n_0}(1/16)\Eq(6.26)$$
Moreover, for ${\bf s}_{n_0},{\bf s}_{n_0}'\in {\bf E}_{n_0}(1/4)$
$$\Vert{\bf F}({\bf s}_{n_0})-{\bf F}({\bf s}_{n_0}')\Vert\le
{1\over 2}\Vert {\bf s}_{n_0}-{\bf s}_{n_0}'\Vert\Eq(6.27)$$
\vglue.3truecm
\vglue.3truecm
\\{\it Proof}
\\First we prove \equ(6.26), and thus take
${\bf s}_{n_0}\in {\bf E}_{n_0}(1/32)$.
Then $\y_n\in E_n(1/32)$ for every $n\ge n_0$ and we can use the estimates
\equ(6.15). From \equ(6.18) and the estimates in \equ(6.15) we have
$$(\n{\bar g})^{-1}|F_{n+1}^{(g)}({\bf s}_{n_0})| <
\alpha(\epsilon){1\over 32}+ (\n{\bar g})^{-1}
\sum_{j=n_0}^{n} \alpha(\epsilon)^{n-j} C_L\Bigl((\n^2 + L^{-jq}){\bar g}^{2}
+{\bar g}^{11/4-\eta}\Bigr) $$
$$< {1\over 32}+C_L {{\bar g}^{7/4-\eta}\over \n(1- \alpha(\epsilon)})+
C_L{{\bar g}\n\over 1- \alpha(\epsilon)}
+C_L {{\bar g}\over \n(1-\alpha(\e)^{-1}L^{-q})}$$
$$< {1\over 32}+{C_L \over \n\log L}\e^{1/4-\eta}+ {C_L \over \log L}\n +
{C_L \over \n(1-L^{-q})}\e <
{1\over 16}$$
\\for $L$ sufficiently large, $\n$ sufficiently small depending on $L$ so that
$ {C_L\over \log L}\n \le 1/96$ and then $\e$ sufficiently small depending on
$L$ so that $\bar g \le C_L\e$ is sufficiently small,
${C_L \over \n(1-L^{-q})}\e \le 1/96$ and
${C_L \over \n\log L}\e^{1/4-\eta} \le 1/96$
\\Similarly from \equ(6.19) and \equ(6.15) we have
$${\bar g}^{-(2-\d)}|F_{n+1}^{(\m)}({\bf s}_{n_0})|\le c_L{\bar g}^{\d}
\sum_{j=n+1}^{\io} L^{-{3+\e\over 2}(j-n)}
\le L^{-{3+\e\over 2}}(1-L^{-{3+\e\over 2}})^{-1}<
{1\over 16}$$
since $\d=1/64$, $L$ sufficiently large, and $\e$ sufficiently small
depending on $L$.
\\Finally from \equ(6.20) and \equ(6.15)
$${\bar g}^{-(11/4-\eta)}|||F_{n+1}^{(R)}({\bf s}_{n_0})|||_{n+1}
\le L^{-1/4}< {1\over 16}$$
\\for $L$ sufficiently large. This proves \equ(6.26).
\\To prove \equ(6.27), take ${\bf s}_{n_0},{\bf s}_{n_0}'\in {\bf E}_{n_0}(1/4)$.
This implies that $\y_n, \y'_{n}\in E_n(1/4)$ for every $n\ge n_0$ and we can
use the Lipshitz estimates of lemma 6.1.
Note that the initial coupling $g_{n_0}$ is held fixed. Then we have
$$(\n{\bar g})^{-1}|F_{n+1}^{(g)}({\bf s}_{n_0})-F_{n+1}^{(g)}
({\bf s}_{n_0}')|\le
\sum_{j=n_0}^{n} \alpha(\epsilon)^{n-j}(\n{\bar g})^{-1}
|\tilde \xi_j(\y_j)-\tilde\xi_j(\y'_j)|$$
$$\le (\n{\bar g})^{-1}\sum_{j=n_0}^{n} \alpha(\epsilon)^{n-j}
C_L\Bigl((\n^2 + L^{-jq}){\bar g}^{2}
+{\bar g}^{11/4-\eta}\Bigr) \Vert {\bf s}_{n_0}-{\bf s}_{n_0}'\Vert $$
$$\le \Bigl({C_L \over \n\log L}\e^{1/4-\eta}+
C_L{{\bar g}\n\over 1- \alpha(\epsilon)}
+C_L {{\bar g}\over \n(1-\alpha(\e)^{-1}L^{-q}})\Bigr)
\Vert {\bf s}_{n_0}-{\bf s}_{n_0}'\Vert
\le
{1\over 2}\Vert {\bf s}_{n_0}-{\bf s}_{n_0}'\Vert$$
\\by estimating as above in the bound for $F_{n+1}^{(g)}({\bf s}_{n_0})$
with $L$ sufficiently large, $\n$ sufficiently small
depending on $L$ and $\e$ sufficiently small depending on $L$. Similarly,
$${\bar g}^{-(2-\d)}|F_{n+1}^{(\m)}({\bf s}_{n_0})-F_{n+1}^{(\m)}({\bf s}_{n_0}')|\le
\sum_{j=n+1}^{\io}
L^{-{3+\e\over 2}(j-n)}{\bar g}^{-(2-\d)}|\tilde \r_j(\y_j)-\tilde\r_j(\y'_j)| $$
$$\le L^{-{3+\e\over 2}}(1-L^{-{3+\e\over 2}})^{-1}
C_L{\bar g}^{\d}\Vert {\bf s}_{n_0}-{\bf s}_{n_0}'\Vert\le
{1\over 2}\Vert {\bf s}_{n_0}-{\bf s}_{n_0}'\Vert$$
\\for $L$ sufficiently large and $\e$ sufficiently small depending on $L$.
Finally
$${\bar g}^{-(11/4-\eta)}|||F_{n+1}^{(R)}({\bf s}_{n_0})-F_{n+1}^{(R)}({\bf s}_{n_0}')|||_{n+1} =
{\bar g}^{-(11/4-\eta)}|||U_{n+1}(\y_{n})-U_{n+1}(\y'_{n})|||_{n+1} $$
$$\le O(1)L^{-1/4}\Vert {\bf s}_{n_0}-{\bf s}_{n_0}'\Vert
\le {1\over 2}\Vert {\bf s}_{n_0}-{\bf s}_{n_0}'\Vert$$
\\for $L$ sufficiently large. Thus \equ(6.27) has been proved.
This completes the proof of Theorem~6.2. $\bull$
\vglue0.3cm
\\{\it 6.4}. Theorem~6.2 says that
if $\y_{n_0}=(\tilde g_{n_0}, \m_{n_0}, R_{n_0},\tilde{\bf w}_{n_0})
\in E_{n_0}(1/32)$ for any $n_0\ge 0$
then there is an $\m_{n_0}$
such that a uniformly bounded RG trajectory exists. We call this a critical
mass. The next theorem proves the uniqueness of $\m_{n_0}$ for $n_0$
sufficiently large : $\m_{n_0}$ is a norm continuous function of
$(\tilde g_{n_0},R_{n_0},\tilde{\bf w}_{n_0})$.
\vglue0.3cm
\\To this end we represent the Banach space $E_n$ as a
product of two Banach spaces $E_n= E_{n,1}\times E_{n,2}$. We write
$\y_n\in E_n$ as $\y_n=(\y_{n,1},\y_{n,2})$ where
$\y_{n,1}=(\tilde g_n, R_n,\tilde{\bf w}_n )$ and
$\y_{n,2}=\m_n$. $\y_{n,2}$ is the expanding (relevant)variable.
Let $p_i,\ i=1,2$, denote the projector onto $E_{n,i}$
and $f_{n,i}=p_i\circ f_n$. The norm $\Vert\cdot\Vert_n$ on $E_n$ being a
a box norm we have
$$\Vert \y_n\Vert_n =\max(\Vert \y_{n,1}\Vert_n,\Vert \y_{n,2}\Vert_n)$$
\\where $\Vert \y_{n,2}\Vert_n={\bar g}^{-(2-\d)}|\m_n|$ and
$\Vert \y_{n,1}\Vert_n=\max((\n{\bar g})^{-1}|\tilde g_n|,\>
{\bar g}^{-(11/4-\eta)}|||R_n|||_{\d_n},
{\tilde c}_L^{-1}\Vert \tilde{\bf w}_n\Vert_{\d_n})$.
\vglue0.2cm
\\In the following we continue to assume
that $L$ is sufficiently large, followed by $\n$ sufficiently small depending
on $L$, then $\e$ sufficiently small depending on $L$. The last condition also
implies that ${\bar g} \le C_L \e$.
\vglue0.3cm
\\{\it Theorem~6.4 : Let
${\bf s}_{n_0}= \{\y_n : \y_{n+1}=f_{n+1}(\y_n)\}_{n\ge n_0}\in {\bf E}(1/4)$
be the global RG trajectory of Theorem~6.2. Then for $n_0$ sufficiently large
there exists a Lipshitz continuous function
$h\>:\> E_{n_0,1}\rightarrow \math{R}$
with Lipshitz constant $1$ such that the stable manifold of the sequence
of maps $\{f_n\}_{n\ge n_0+1}$,
$W_{n_0}^{s}=\{\y_{n_0}\in E_{n_0}(1/32) :\>{\bf s}_{n_0} \in {\bf E}(1/4)\}$
is the graph
$$W_{n_0}^{s}= \{\y_{n_0,1}, h(\y_{n_0,1}\} $$
}
\vglue0.2cm
\\We will prove the theorem following the analysis of Shub in
[S, Section~5]. Similar results were proved in [Section~5.3, BDH-eps] and
[Section~6, BMS] for continuum models.
The proof rests on the following lemma.
\vglue.3truecm
\\{\it Lemma 6.5}
\\Let $\y_n,\y_n'\in E_n(1/4)$. Then for $n\ge n_0$, $n_0$ sufficiently large
$$\Vert f_{n+1,1}(\y_n)-f_{n+1,1}(\y_n')\Vert_{n+1}
\le (1-\e)\Vert \y_n-\y_n'\Vert_{n} \Eq(6.30)$$
\\and, if
$\Vert \y_{n,2}-\y_{n,2}'\Vert_{n}\ge \Vert \y_{n,1}-\y_{n,1}'\Vert_{n}$
then
$$\Vert f_{n+1,2}(\y_n)-f_{n+1,2}(\y_n')\Vert_{n+1}
\ge (1+\e)\Vert \y_n-\y_n'\Vert_{n}\Eq(6.31)$$
\vglue.3truecm
\\{\it Proof}
\\First we prove
\equ(6.30). $f_{n+1,1}$ has components $f_{n+1,g}$, $f_{n+1,R}$ and
$f_{n+1,w}$. From \equ(6.4)
$$f_{n+1,g}(\y_n)= \alpha(\epsilon) \tilde g_n +\tilde\xi_n(\y_n)$$
\\Since $\y_n,\y_n'\in E_n(1/4)$ we can use lemma 6.1. Therefore for $n\ge n_0$
$$\eqalign{(\n{\bar g})^{-1}|f_{n+1,g}(\y_n)-f_{n+1,g}(\y_n')|
&\le \alpha(\epsilon) \Vert \y_n-\y_n'\Vert_{n} +
(\n{\bar g})^{-1}|\tilde\xi_n(\y_n)-\tilde\xi_n(\y_n')|\cr
&\le \Bigl(1-\e\log(L) +C_L\e(\n + {1\over \n}L^{-n_{0}q}+
{1\over \n} \e^{3/4-\eta}\Bigr)\Vert \y_n-\y_n'\Vert_{n}\cr} $$
\\Let $L$ be large. Let $\n$ be sufficiently small and $n_0$ sufficiently
large
so that $C_L L^{-n_{0}q/2}\le \n\le {1\over C_L}$ and
$\e$ sufficiently small so that $\n\ge C_L\e^{3/8 -\eta}$. Then we have
$$\eqalign{1-\e\log(L) +\e C_L(\n + {1\over \n}L^{-n_0 q}+
{1\over \n} \e^{3/4-\eta}) &\le 1+\e(-\log(L) +1+ L^{-qn_0 /2} + \e^{3/8})\cr
&\le 1+\e(-\log(L) +3) \le 1-\e \cr } $$
\\Therefore
$$(\n{\bar g})^{-1}|f_{n+1,g}(\y_n)-f_{n+1,g}(\y_n')|
\le (1-\e)\Vert \y_n-\y_n'\Vert_{n}$$
\\Since $f_{n+1,R}(\y_n)=U_{n+1}(\y_n)$, we have from lemma 6.1 for $L$
sufficiently large
$${\bar g}^{-(11/4-\eta)}|||f_{n+1,R}(\y_n)-f_{n+1,R}(\y_n')|||_{n+1}
\le(1-\e)\Vert \y_n-\y_n'\Vert_{n}$$
\\as well as
$$\tilde c_L^{-1}
\Vert f_{n+1,\bf w}(\y_n)- f_{n+1,\bf w}(\y_n')\Vert_{n+1}\le (1-\e)
\Vert \y_n-\y_n'\Vert_{n}$$
\\These three inequalities prove \equ(6.30).
\\Next we turn to \equ(6.31). In this case by assumption
$\Vert \y_{n,2}-\y_{n,2}'\Vert\ge \Vert \y_{n,1}-\y_{n,1}'\Vert$ and hence,
since our norms are box norms, we have
$$\Vert \y_n-\y_n'\Vert_n=\Vert \y_{n,2}-\y_{n,2}'\Vert_n
={\bar g}^{-(2-\d)}|\m_n-\m'_n|$$
\\From \equ(6.5)
$$f_{n+1,\m}(\y_n)=L^{3+\e\over 2}\m_n +\tilde\r_n(\y_n)$$
\\Then, using lemma 6.1, we have
$$\eqalign{{\bar g}^{-(2-\d)}|f_\m(\y_n)-f_\m(\y_n')|\ge
& L^{3+\e\over 2}\Vert \y_n-\y_n'\Vert_{n} -
{\bar g}^{-(2-\d)}|\tilde\r(\y_n)-\tilde\r(\y_n')| \cr
& \ge(L^{3+\e\over 2}-C_L{\bar g}^{\d})\Vert \y_n-\y_n'\Vert_{n} \cr
& \ge (1+\e)\Vert \y_n-\y_n'\Vert}_{n}$$
\\for $\e$ sufficiently small depending on $L$. This proves \equ(6.31). $\bull$
\vglue.3truecm
\\{\it Proof of Theorem~6.4 }
\\To prove that $W_{n_0}^{s}$ is given by a graph of a function
$\y_{n_0,2}=h(y_{n_0,1}) $ it is enough to prove that
if in $M_{n,0}$ we take two points $\y_{n_0}=(\y_{n_0,1},\y_{n_0,2})$ and
$\y_{n_0}'=(\y_{n_0,1}',\y_{n_0,2}')$ then
$$\Vert \y_{n_0,2}-\y_{n_0,2}'\Vert_{n_0}\le
\Vert \y_{n_0,1}-\y_{n_0,1}'\Vert_{n_0}\Eq(6.33)$$
\\because then for a given $\y_{n_0,1}$ we would have at most one
$\y_{n_0,2}$, and by theorem 6.2 there exists such a $\y_{n_0,2}$. This means
that $W_{n_0}^{s}$ is the graph of a function $h$, $\y_{n_0,2}=h(\y_{n_0,1})$,
and moreover
$$\Vert h(\y_{n_0,1})-h(\y_{n_0,1}')\Vert_{n_0}\le\Vert \y_{n_0,1}-\y_{n_0,1}'\Vert_{n_0}$$
\\Suppose \equ(6.33) is not true. Then
$$\Vert \y_{n_0,2}-\y_{n_0,2}'\Vert_{n_0}>
\Vert \y_{n_0,1}-\y_{n_0,1}'\Vert_{n_0}\Eq(6.33.1)$$
\\\equ(6.33.1) implies that \equ(6.31) holds. The latter followed by
\equ(6.30) gives
$$\Vert f_{n_0+1,2}(\y_{n_0})-f_{n_0+2}(\y_{n_0}')\Vert_{n_0+1}
\ge (1+\e)\Vert \y_{n_0}-\y_{n_0}'\Vert_{n_0} >
(1-\e)\Vert \y_{n_0}-\y_{n_0}'\Vert_{n_0}$$
$$\ge\Vert f_{n_0+1,1}(\y_{n_0})-f_{n_0+1,1}(\y_{n_0}')\Vert_{n_0+1}
\Eq(6.33.2) $$
\\and hence
$$\Vert f_{n_0+1}(\y_{n_0})-f_{n_0+1}(\y_{n_0}')\Vert_{n_0+1}=
\Vert f_{n_0+1,2}(\y_{n_0})-f_{n_0+2}(\y_{n_0}')\Vert_{n_0+1}
\ge (1+\e)\Vert \y_{n_0}-\y_{n_0}'\Vert_{n_0} \Eq(6.33.3) $$
\\Define the composition of maps
$${\cal P}_{n}^{k}\doteq f_{n+k}\circ .....\circ f_{n+2}\circ f_{n+1}$$
\\Now
$$\Vert {\cal P}_{n_0}^{2}(\y_{n_0})-{\cal P}_{n_0}^{2}(\y_{n_0}')\Vert_{n_0+2}
=\Vert f_{n_0+2}(f_{n_0+1}(\y_{n_0}))-f_{n_0+2}(f_{n_0+1}(\y_{n_0}'))
\Vert_{n_0+2} $$
\\By \equ(6.33.2),the second part of Lemma 6.4 followed by \equ(6.33.3) we get
$$\Vert{\cal P}_{n_0}^{2}(\y_{n_0})-{\cal P}_{n_0}^{2} (\y_{n_0}')\Vert_{n_0+2}
\ge(1+\e)\Vert f_{n_0+1}(\y_{n_0})-f_{n_0+1}(\y_{n_0}')\Vert_{n_0+1}
\ge (1+\e)^2\Vert\y_{n_0}-\y_{n_0}'\Vert_{n_0} $$
\\Repeating this $k$ times we get for all $k\ge 0$
$$\Vert{\cal P}_{n_0}^{k}(\y_{n_0})-{\cal P}_{n_0}^{k}(\y_{n_0}')\Vert_{n_0+k}
\ge(1+\e)^k\Vert \y_{n_0}-\y_{n_0}'\Vert_{n_0} \Eq(6.33.4)$$
\\Now $\y_{n_0},\y_{n_0}'$ belong to $W_{n_0}^{s}$ and
${\cal P}_{n_0}^{k}(\y_{n_0})$ is a member of the sequence
${\bf s}_{n_0}\in {\bf E}(1/4)$. Therefore
\\$\Vert{\cal P}_{n_0}^{k}(\y_{n_0})\Vert_{n_0+k}< 1/4 $. Therefore we have
from \equ(6.33.4) the bound
${1\over 2} >(1+\e)^k\Vert \y_{n_0}-\y_{n_0}'\Vert_{n_0}$.
By making $k$ arbitrarily large we get a
contradiction because $\y_{n_0}\ne \y_{n_0}'$ under \equ(6.33.1).
Hence \equ(6.33) is true and the theorem 6.4 has been proved \bull
\vglue0.3cm
\\As a consequence of theorem~6.4 we have
$\y_{n}\in E_n(1/4)$, $\forall n\ge n_0$. This implies
that $|\tilde g_{n}| < {1\over 4}\n {\bar g}$, $\forall n\ge n_0$. Whence
for all $n\ge 0$
$$(1-{1\over 4}\n) {\bar g}< g_{n} < (1+{1\over 4}\n) {\bar g} \Eq(6.33.5) $$
\\We have $0<\n< 1 $. Therefore the effective coupling constant generated by
the discrete RG flow is uniformly bounded away from $0$ at all RG scales.
\vglue.3truecm
\\{\bf Acknowledgements} : One of us (PKM) is especially grateful to
David Brydges for many fruitful conversations during the course of this work.
He thanks G\'erard Menessier for helpful conversations and Erhard Seiler
for his comments on the manuscript.
\vglue.5truecm
\\{\bf References}
\vglue.3truecm
\\[AR] A. Abdesselam: A Complete Renormalization Group Trajectory Between
Two Fixed Points, Commun. Math. Phys (to be published),
http://arXiv:math-ph/0610018.
\vglue.3truecm
\\[Bal1] Tadeusz Balaban: $({\rm Higgs})_{2,3}$ quantum fields in a
finite volume. I. A lower bound,
Commun. Math. Phys.
{\bf 85},
603--626 (1982).
\vglue.3truecm
\\[Bal2] Tadeusz Balaban: $({\rm Higgs})_{2,3}$ quantum fields in a
finite volume. II. An upper bound,
Commun. Math. Phys.
{\bf 86},
555--594 (1982).
\vglue.3truecm
\\[Bal3] Tadeusz Balaban:
Renormalization group approach to lattice gauge theories. I.
Generation of effective actions in a small field approximation and a
coupling constant renormalization in four dimensions,
Commun. Math. Phys.
{\bf 109},
249--301 (1987).
\vglue.3truecm
\\[BDH-est] D. Brydges, J. Dimock and T.R. Hurd:
Estimates on Renormalization Group Transformation,
Canad. J. Math.
{\bf 50},
756--793 (1998), no. 4.
\vglue.3truecm
\\[BDH-eps] D. Brydges, J. Dimock and T.R. Hurd:
A Non-Gaussian Fixed Point for $\phi^4$ in 4-$\epsilon$ Dimensions,
Commun. Math. Phys.
{\bf 198},
111--156 (1998).
\vglue.3truecm \textRed
\\[BGM] D. Brydges, G. Guadagni and P. K. Mitter: Finite range
Decomposition of Gaussian Processes,
J.Statist.Phys.
{\bf 115},
415--449 (2004)
\textBlack
\vglue.3truecm
\\[BEI] David Brydges, Steven N. Evans, John Z. Imbrie:
Self-Avoiding Walk on a Hierarchical Lattice in Four Dimensions,
The Annals of Probability,
{\bf 20},
82--124 (2003).
\vglue.3truecm
\\[BI1] D. Brydges and J. Imbrie: End-to-End Distance from the Green's
Function for a Hierarchical
Self-Avoiding Walk in Four Dimensions, Commun. Math. Phys.
{\bf 239},
523--547 (2003).
\vglue.3truecm
\\[BI2] D. Brydges and J. Imbrie: Green's Function for a Hierarchical
Self-Avoiding Walk in Four Dimensions, Commun. Math. Phys.
{\bf 239},
549--584 (2003).
\vglue.3truecm
\\[BMS] D.C. Brydges, P.K.Mitter and B.Scoppola :
Critical $(\Phi^{4})_{3,\epsilon}$,
Commun. Math. Phys.
{\bf 240},
281--327 (2003).
\vglue.3truecm
\\[BY] D. Brydges and H.T. Yau:
Grad $\phi$ Perturbations of Massless Gaussian Fields,
Commun. Math. Phys.
{\bf 129},
351--392 (1990).
\vglue.3truecm
\\[F] W. Feller,
An introduction to probability theory and its applications, Vol.2,
John Wiley and Sons, Hoboken, N.J.,
1968.
\vglue.3truecm
\\[GK1] K. Gawedzki and A. Kupiainen:
A rigorous block spin approach to massless field theories,
Commun. Math. Phys.
{\bf 77},
31-64 (1980)
\vglue.3truecm
\\[GK2] K. Gawedzki and A. Kupiainen:
A rigorous block spin approach to massless field theories,
Ann.Phys.
{\bf 147},
198 (1980)
\vglue.3truecm
\\[GK3] K. Gawedzki and A. Kupiainen:
Massless $(\phi)^{4}_4$ theory: Rigorous control of a renormalizable
asmptotically free model,
Commun. Math. Phys.
{\bf 99},
197-252 (1985).
\vglue.3truecm
\\[I] M.C. Irwin:
On the stable manifold theorem,
Bull. London Math. Soc.
{\bf 2},
196-198 (1970).
\vglue.3truecm
\\[KG] A. N. Kolmogoroff and B. V. Gnedenko,
Limit Distributions of sums of independant random variables,
Addison Wesley, Cambridge,Mass
1954.
\vglue.3cm
\\[M] P. K. Mitter:
On the convergence to the continuum of a class of lattice covariances,
(in preparation).
\vglue.3truecm
\\[Mc] A. J. McKane:
Reformulation of $n\rightarrow 0$ models using supersymmetry,
Phys.Lett.
A {\bf 41},
22--44 (1980)
\vglue.3truecm
\\[PS] G. Parisi and N. Sourlas:
Self-avoiding walk and supersymmetry,
J.Phys.Lett.
{\bf 41},
L403--L406 (1980)
\vglue.3truecm
\\[S] M. Shub:
Global Stability of Dynamical Systems,
Springer-Verlag, New York,
1987.
\ciao
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