ctrlcst symbol=κ csts symbol=C normal symbol=c

# Low temperature asymptotic expansion for classical $O(N)$ vector models

Alessandro Giuliani  Dipartimento di Matematica e Fisica, Università degli
Studi Roma Tre, L.go S. L. Murialdo 1, 00146 Roma, Italy &
Centro Linceo Interdisciplinare Beniamino Segre, Accademia Nazionale dei
Lincei, Via della Lungara 10, 00165 Roma, Italy
<EMAIL_ADDRESS>and  Sébastien Ott  Département de
Mathématiques, Université de Fribourg, Chemin du Musée 23, 1700 Fribourg,
Switzerland<EMAIL_ADDRESS>
###### Abstract.

We consider classical $O(N)$ vector models in dimension three and higher and
investigate the nature of the low-temperature expansions for their multipoint
spin correlations. We prove that such expansions define asymptotic series, and
derive explicit estimates on the error terms associated with their finite
order truncations. The result applies, in particular, to the spontaneous
magnetization of the 3D Heisenberg model. The proof combines a priori bounds
on the moments of the local spin observables, following from reflection
positivity and the infrared bound, with an integration-by-parts method applied
systematically to a suitable integral representation of the correlation
functions. Our method generalizes an approach, proposed originally by Bricmont
and collaborators [6] in the context of the rotator model, to the case of non-
abelian symmetry and non-gradient observables.

##  1\. Introduction

In 1976, in one of the most influential papers in mathematical statistical
mechanics, Fröhlich-Simon-Spencer (FSS) [11] proved the existence of
orientational long range order for classical $O(N)$ vector models in three or
more dimensions. In 1981, in the special case of the $O(2)$ model, also known
as the rotator or XY model, Bricmont-Fontaine-Lebowitz-Lieb-Spencer (BFLLS)
[6] extended the FSS result, proving, in particular, that the formal low-
temperature expansion for its magnetization defines an asymptotic series; that
is, the difference between the magnetization at temperature $T$ and the
truncation of its low-temperature series at any finite order $n$ is $o(T^{n})$
as $T\to 0$. A direct generalization of this result to $O(N)$ models with
$N\geq 3$, including the physically relevant case of the Heisenberg model,
remained open since, due to the difficulty in extending the BFLLS method to
the case of non-abelian rotational symmetry. By ‘direct’, here, we mean by
similar elementary methods, combining a priori bounds on the moments of local
spin observables, based on reflection positivity and the related infrared
bound [11, 12], with systematic integration-by-parts applied to a suitable
integral representation of the spin correlations. In this paper we provide
such a direct, self-contained, proof of the asymptotic nature of the low-
temperature expansion for the magnetization and other spin correlation
functions of classical $O(N)$ vector models, for any $N\geq 3$. Technically,
our proof is based on an extension of the BFLLS proof, capable of handling
non-abelian symmetry and non-gradient observables.

Let us recall that there are other, more involved, approaches that can be used
to investigate the low-temperature properties of classical $O(N)$ vector
models: we refer here to Balaban’s multiscale analysis [1, 2, 3] of its
correlation functions [4]. Balaban’s construction implies, in particular, that
the low-temperature expansion for the magnetization is an asymptotic series;
it actually provides much more detailed informations on the correlations at
low temperatures, including their large distance asymptotics and decay
exponents. However, the construction is extremely involved and it is hard to
extract from it explicit informations in a simple and direct way.

For this reason, it is important to develop simpler approaches to the problem,
particularly in cases, such as the classical $O(N)$ vector model, where the
system displays reflection positivity, which in turn implies several strong a
priori bounds on correlation functions. The proof described below shows that,
even in models with non-abelian continuous symmetry, there is no intrinsic
obstacle in using the infrared bound to derive bounds on the low-temperature
expansion for correlations. However, even if our method allows us to control
multipoint spin correlation at any fixed coordinates, it does not provide
informations on their asymptotic behavior as the relative distance between the
spins diverges to infinity. We believe that, in order to compute the critical
exponents and the dominant asymptotics of spin correlations, multiscale
analysis is inevitable. It would be extremely nice to be able to use
systematically reflection positivity within a multiscale scheme, thus
simplifying Balaban’s approach, particularly in connection with the
technically involved procedure related to the small/large field decomposition.
It would also be nice to develop methods, inspired by the BFLLS approach,
capable of controlling the low-temperature series for the magnetization of
reflection positive quantum spin systems, such as the 3D quantum XY model or
quantum Hesienberg anti-ferromagnet. These long term goals are behind the very
motivations of the current work. It remains to be seen whether such
challenging problems can be solved or at least attacked via extensions of the
current work, and we hope to come back to these problems in future
publications.

###  1.1. Definition of the model and finite-volume infrared bound.

Let $\Lambda_{L}$ be the $d$-dimensional discrete torus of side $L$, in
dimension $d\geq 3$. Each site of $\Lambda_{L}$ carries an $N$-component unit
vector, called ‘spin’; we assume that $N\geq 2$, and denote by
$\Omega_{L}=(\mathbb{S}^{N-1})^{\Lambda_{L}}$ the corresponding space of spin
configurations. For $f:\Omega_{L}\to\mathbb{R}$, $\beta\geq 0$, and $h\geq 0$,
we define the un-normalized expectations as follows:

$\begin{split}&Z_{L;\beta}(f)=\int_{\Omega_{L}}\prod_{x\in\Lambda_{L}}d\nu_{N}(S_{x})f(S)e^{\beta\sum_{x\in\Lambda_{L}}\sum_{e}S_{x}\cdot
S_{x+e}},\\\ &Z_{L;\beta,h}(f)\equiv
Z_{L;\beta}\big{(}fe^{h\sum_{x\in\Lambda_{L}}S_{x}^{N}}\big{)},\end{split}$
(1)

where $\nu_{N}$ is the Lebesgue (area) measure on $\mathbb{S}^{N-1}$. Their
normalized counterparts are:

$\langle f\rangle_{L;\beta}=\frac{Z_{L;\beta}(f)}{Z_{L;\beta}(1)},\quad\langle
f\rangle_{L;\beta,h}=\frac{Z_{L;\beta,h}(f)}{Z_{L;\beta,h}(1)}$ (2)

We will write $Z_{L;\beta,h}\equiv Z_{L;\beta,h}(1)$, and often omit $\beta$
from the notation. $\mu_{L;h}$ will denote the probability measure with
expectation $\langle\ \rangle_{L;h}$.

Reflection positivity of $\mu_{L;h}$ implies the following finite-volume
version of the _infrared bound_ [11, Theorem 2.1], which is one of the basic
ingredients of our analysis. We denote by $\hat{\mathrm{e}}_{k}$ with
$k=1,\ldots,d$ the elements of the canonical basis of unit vectors of
$\mathbb{R}^{d}$, and
$\nabla^{\hat{\mathrm{e}}_{k}}_{x}f=f(x+\hat{\mathrm{e}}_{k})-f(x)=(\nabla^{\hat{\mathrm{e}}_{k}}f)(x)$.
We also denote by $\Delta_{L}$ the Laplacian in the torus $\Lambda_{L}$ and by
$G_{L}=(-\Delta_{L})^{-1}$ its finite-volume Green’s function.

######  Proposition 1.1.

For any $h,\beta\geq 0$, any $L$, and any
$f_{1},\cdots,f_{d}:\Lambda_{L}\to\mathbb{R}^{N}$,

$\langle
e^{\sum_{k=1}^{d}\sum_{x\in\Lambda_{L}}S_{x}\cdot\nabla^{\hat{\mathrm{e}}_{k}}_{x}f_{k}}\rangle_{L;h}\leq
e^{\frac{1}{2\beta}\sum_{k=1}^{d}\sum_{x\in\Lambda_{L}}(f_{k}(x))^{2}}.$

In particular, for $f$ such that $\sum_{x}f(x)=0$, letting
$f_{k}(x)=(-\Delta_{L})^{-1}\nabla^{\hat{\mathrm{e}}_{k}}_{x}f$ in the
previous equation, one has:

$\langle e^{\sum_{x\in\Lambda_{L}}S_{x}\cdot f(x)}\rangle_{L;h}\leq
e^{\frac{1}{2\beta}(f,G_{L}f)}.$ (3)

###  1.2. Construction of the infinite volume measure and infrared bound.

We will work with the successive limits $L\to\infty$ (thermodynamic limit),
and $h\searrow 0$ of $\mu_{L;h}$. We denote by $\mu_{h}$ a (arbitrarily
chosen) cluster point of $\mu_{L;h}$ as $L\to\infty$, which exists by
compactness and inherits translation invariance from the finite volume
measures, and by $\langle\cdot\rangle_{h}$ the corresponding expectation. It
is believed (but, to the best of the authors knowledge, not yet proved) that
the limit of $\mu_{L;h}$ as $L\to\infty$ always exists and is exponentially
mixing when $h>0$.

A crucial ingredient for our construction is the following infinite volume
version of the infrared bound:

$\langle e^{\sum_{x}(S_{x}-\langle S_{x}\rangle_{h})\cdot f(x)}\rangle_{h}\leq
e^{\frac{1}{2\beta}(f,Gf)},$ (4)

for any $f$ of finite support, with $G=(-\Delta)^{-1}$ the Green’s function
associated with the lattice Laplacian $\Delta$ on $\mathbb{Z}^{d}$. As proved
in [11, Corollary 2.5], if $\mu_{h}$ is ergodic, then (4) readily follows from
(3). When $N=2,3$, one can extract from Lee-Yang theory (see [9, 10]) that the
limit state $\mu_{h}=\lim_{L\to\infty}\mu_{L;h}$ (with $h>0$) is well defined
and exponentially mixing; in particular, it is ergodic and, therefore, (4)
follows. When $N>3$, Lee-Yang theorem is not available. However, the day can
still be saved. In fact, for the purpose of proving (4), ergodicity is not
required: as shown in Appendix A, the property that
$L^{-d}\sum_{n\in\mathbb{Z}^{d}:|n_{i}|\leq L/2}S_{x+n}$ converges to $\langle
S_{x}\rangle_{h}$ in $L^{p}(\mu_{h})$ for $p\geq 1$ suffices.

Let $\psi:\mathbb{R}^{N}\to\mathbb{R}$ be the pressure of the $O(N)$ model,
defined by
$\psi(v)\equiv\psi_{\beta}(v)=\lim_{L\to\infty}\frac{1}{|\Lambda_{L}|}\log
Z_{L;\beta}(e^{\sum_{x\in\Lambda_{L}}v\cdot S_{x}})$. Let

$D^{*}:=\\{h>0:\ \psi\textnormal{ is differentiable at }hs\ \forall
s\in\mathbb{S}^{N-1}\\}$

(note that the definition is well-posed, thanks to $O(N)$ invariance: if
$\psi$ is differentaible at $hs_{0}$ for some $s_{0}\in\mathbb{S}^{N-1}$, then
it is automatically differentiable at $hs$ for all $s\in\mathbb{S}^{N-1}$).
$D^{*}$ has full Lebesgue measure on $(0,+\infty)$, thanks to the convexity of
$\psi$. Moreover, let $\mu$ be a (arbitrarily chosen) limit point of
$\mu_{h}$, as $h\in D^{*}$ tends to $0^{+}$, and let $\langle\cdot\rangle$ be
the corresponding expectation. Then the following generalization of [11,
Corollary 2.5 and Theorem 3.1] holds (for the proof, see Appendix A).

######  Proposition 1.2.

If $h\in D^{*}$, for any $f$ with finite support,

$\langle e^{\sum_{x}(S_{x}-\langle S_{x}\rangle_{h})\cdot f(x)}\rangle_{h}\leq
e^{\frac{1}{2\beta}(f,Gf)}.$ (5)

In particular,

$\langle S_{0}\rangle_{h}\cdot\langle S_{0}\rangle_{h}\geq 1-{NT}G(0,0),$ (6)

where
$G(0,0)=(2\pi)^{-d}\int_{p\in[-\pi,\pi]^{d}}dp\big{(}2\sum_{k=1}^{d}(1-\cos(p_{k}))\big{)}^{-1}$.
The bounds (5) and (6) also hold with $\langle\cdot\rangle$ replacing
$\langle\cdot\rangle_{h}$.

###  1.3. Main results

We are now ready to state the main result of this work.

######  Theorem 1.3.

For any $n,K\in\mathbb{Z}_{+}:=\\{0,1,2,\cdots\\}$, and $\epsilon>0$ there
exists $C<\infty$ such that for any
$\alpha^{1},\cdots,\alpha^{N}:\mathbb{Z}^{d}\to\mathbb{Z}_{+}$ with
$\sum_{k=1}^{N}\sum_{x\in\mathbb{Z}^{d}}\alpha^{k}_{x}\leq K$, there are
coefficients $a_{0},\cdots,a_{n}\in\mathbb{R}$ such that

$\Big{|}\langle\prod_{x}\prod_{k}(S_{x}^{k})^{\alpha_{x}^{k}}\rangle-\sum_{i=0}^{n}a_{i}T^{i}\Big{|}\leq
CT^{n+1-\epsilon}$

where the $a_{i}$s are given by an explicit inductive algorithm.

A Corollary of Theorem 1.3 and of our inductive algorithm is the following
second order approximation of the magnetization. Recall that $\Delta$ denotes
the Laplacian on $\mathbb{Z}^{d}$ and $G=(-\Delta)^{-1}$ its Green function.
Recall also that $\\{\hat{\mathrm{e}}_{k}\\}_{k=1,\ldots,d}$ denotes the
canonical basis of unit vectors of $\mathbb{R}^{d}$. The discrete derivative
of $G$ acting (say) on its second coordinate,
$\nabla^{\hat{\mathrm{e}}_{k}}_{x}G(x_{0},\cdot)$, was defined right before
the statement of Proposition 1.1. Similarly, we let
$\nabla^{\hat{\mathrm{e}}_{k},\hat{\mathrm{e}}_{k^{\prime}}}_{x,y}G=G(x+\hat{\mathrm{e}}_{k},y+\hat{\mathrm{e}}_{k^{\prime}})-G(x,y+\hat{\mathrm{e}}_{k^{\prime}})-G(x+\hat{\mathrm{e}}_{k},y)+G(x,y)$.

######  Corollary 1.4.

For $d\geq 3$ and $N\geq 2$, the following second order expansion for the
spontaneous magnetization of the $O(N)$ vector model holds, as $T\to 0^{+}$:

$\langle
S_{0}^{N}\rangle=1-T\frac{(N-1)}{2}G(0,0)+T^{2}\frac{(N-1)}{2}\Big{[}\frac{(3N-5)}{4}G(0,0)^{2}\\\
-\sum_{x_{1}\in\mathbb{Z}^{d}}\sum_{e\in\\{\hat{\mathrm{e}}_{i},i=1,\cdots
d\\}}\big{(}\nabla_{x_{1}}^{e}G(0,\cdot)\big{)}^{2}\Big{(}\frac{1}{2}\nabla_{x_{1},x_{1}}^{e,e}G+(N-2)G(0,0)\Big{)}\Big{]}+o(T^{2}).$
(7)

Of course, similar formulas can be derived for third or higher order
truncations, but we decided to spell out for illustrative purposes only the
simplest non-trivial one.

A second immediate consequence of the proof of our main result is the
following.

######  Corollary 1.5.

Let $S=(S_{x})_{x\in\mathbb{Z}^{d}}$ be distributed with respect to the
measure $\mu$. Then the first $N-1$ components of the rescaled field
$\sqrt{\beta}S$, that is, $\sqrt{\beta}(S^{1},S^{2},\dots,S^{N-1})$, converge
in law, as $\beta\to\infty$, to $N-1$ independent Gaussian Free Fields on
$\mathbb{Z}^{d}$.

###  1.4. Organization of the paper

We will prove Theorem 1.3 by introducing an inductive procedure to compute the
coefficients in the Taylor series. This expansion follows the same scheme as
in [6]: a regularized version of the Gaussian integration by part and a priori
bounds on the moments of suitable observables, obtained by using the infrared
bound. The main difference lies in the fact that for $N>2$, the $O(N)$ model
can not be written as a gradient perturbation of a Gaussian free field. This
substantially complicates the picture and, as a result, we need to introduce a
more involved procedure than the one of [6], particularly in regards to the
procedure required for summing over the coordinates the remainder terms. The
paper is organized as follows: after having summarized the notations used in
the paper in Section 2, in Section 3 we introduce the coordinates we use to
parametrize the spin space; then, using these coordinates, we express the
Gibbs measure as a perturbation of a Gaussian one; finally, we use this re-
writing of the measure to derive the formal low-temperature expansion, whose
structure will be used below, in the description of the inductive procedure.
In Section 4, we explain how to use the infrared bound to derive a priori
bound on the spin moments (this is an adaptation of the method in [6]). In
Section 5 we describe the version of the Gaussian integration by parts to be
used in the following. Finally, in Section 6, we describe the inductive
procedure: we first illustrate the procedure for the computation of the
spontaneous magnetization at order $T^{2}$, thus proving Corollary 1.4, and
then discuss the general scheme at all orders, thus proving Theorem 1.3 and
Corollary 1.5. Appendix A contains the proof of Proposition 1.2. The proofs of
the priori bounds on the decay of correlations (which use reflection
positivity in a relatively standard manner) are collected in Appendix B, while
Appendix C discusses some standard Gaussian estimates, and their implications
for the summation over the coordinates of certain connected Gaussian
expectations, which systematically appear in the evaluation of the remainder
terms.

##  2\. Notations

Before starting the proof of our main results, let us summarize here the
notations and conventions used in this paper. For ease of reference, we
include here also those already introduced in the previous section.

We write $\mathbb{Z}_{+}=\\{0,1,2,\cdots\\}$ and $[n]=\\{1,2,\cdots,n\\}$.

We will work in dimensions $d\geq 3$, on either $\mathbb{Z}^{d}$ or
$\Lambda_{L}=\\{0,\cdots,L-1\\}^{d}$ the $d$-dimensional discrete torus of
side $L$. We denote $E_{L},E$ the set of nearest neighbour pairs in
$\Lambda_{L},\mathbb{Z}^{d}$. We write $i\sim j$ when $i$ and $j$ are nearest
neighbours. When writing $\sum_{x}$, we mean summing over
$x\in\mathbb{Z}^{d}$.

The symbols $\hat{\mathrm{e}}_{i}$, with $i\in\\{1,\ldots,d\\}$, will denote
the canonical basis unit vectors of $\mathbb{R}^{d}$ and
$\mathrm{B}=\\{-\hat{\mathrm{e}}_{1},\hat{\mathrm{e}}_{1},\ldots,-\hat{\mathrm{e}}_{d},\hat{\mathrm{e}}_{d}\\}$.
Sometimes we will also use the notation $\hat{\mathrm{e}}_{i}$, with
$i\in\\{1,\ldots,N\\}$, for the base vectors of $\mathbb{R}^{N}$ and hope that
it will not confuse the reader. We attribute a canonical orientation to each
edge: any edge can be written $\\{x,x+e\\}$ with
$e\in\mathrm{B}_{+}=\\{\hat{\mathrm{e}}_{i},i=1,\ldots,d\\}$, and we orient it
from $x$ to $x+e$. When writing $\sum_{e}$, we mean summing over
$e\in\mathrm{B}_{+}$. Sums over $e\in\mathrm{B}$ will be explicitly mentioned.
Let $\mathbb{E}_{d}=\big{\\{}(x,e):\
x\in\mathbb{Z}^{d},e\in\mathrm{B}_{+}\big{\\}}$. There is a canonical
bijection between $E$ and $\mathbb{E}_{d}$.

For $p:\mathbb{Z}^{d}\to\mathbb{Z}_{+}$, $q:\mathbb{E}_{d}\to\mathbb{Z}_{+}$,
and $\tilde{p}:\mathbb{Z}^{d}\times[N-2]\to\mathbb{Z}_{+}$ we write

$\begin{split}&\phi^{p}=\prod_{x\in\mathbb{Z}^{d}}\phi_{x}^{p_{x}},\quad(\nabla\phi)^{q}=\prod_{(x,e)\in\mathbb{E}_{d}}(\nabla_{x}^{e}\phi)^{q_{x,e}},\quad
u^{\tilde{p}}=\prod_{x}\prod_{k=1}^{N-2}(u_{x}^{k})^{\tilde{p}_{x}^{k}},\\\
&\left\|p\right\|_{\scriptscriptstyle
1}=\sum_{x\in\mathbb{Z}^{d}}|p_{x}|,\quad\left\|q\right\|_{\scriptscriptstyle
1}=\sum_{(x,e)\in\mathbb{E}_{d}}|q_{x,e}|,\quad\left\|\tilde{p}\right\|_{\scriptscriptstyle
1}=\sum_{x}\sum_{k}|\tilde{p}_{x}^{k}|,\end{split}$ (8)

where: $\tilde{p}^{k}:\mathbb{Z}^{d}\to\mathbb{Z}_{+}$ is given by
$\tilde{p}^{k}_{x}=\tilde{p}(x,k)$, and, for
$f:\mathbb{Z}^{d}\to\mathbb{R}^{N}$,

$\nabla^{e}_{x}f=f(x+e)-f(x)=(\nabla^{e}f)(x),\ e\in\mathrm{B}.$

We say that $p$ is odd if $\left\|p\right\|_{\scriptscriptstyle 1}$ is, same
for $q$. We say that $\tilde{p}$ is even if each $\tilde{p}^{k}$ is, we say
that it is odd if there is $k\in\\{1,\cdots,N-2\\}$ such that $\tilde{p}^{k}$
is. We define the _support_ of $p,q,\tilde{p}$ as above:

$\begin{split}&\mathrm{supp}_{p}=\\{x\in\mathbb{Z}^{d}:\ p_{x}\neq
0\\},\quad\mathrm{supp}_{\tilde{p}}=\bigcup_{k=1}^{N-2}\mathrm{supp}_{\tilde{p}^{k}},\\\
&\mathrm{supp}_{q}=\bigcup_{(x,e)\in\mathbb{E}_{d}:q_{x,e}\neq
0}\\{x,x+e\\}.\end{split}$ (9)

We also let $\mathfrak{P},\mathfrak{Q},\tilde{\mathfrak{P}}$ be the sets of
tuples $p,q,\tilde{p}$ of finite support of the form described above,
respectively. Analogous definitions hold for tuples
$p:\Lambda_{L}\to\mathbb{Z}_{+}$, $q:\big{\\{}(x,e):\
x\in\Lambda_{L},e\in\mathrm{B}_{+}\big{\\}}\to\mathbb{Z}_{+}$, and
$\tilde{p}:\Lambda_{L}\times[N-2]\to\mathbb{Z}_{+}$, whose sets will be
denoted $\mathfrak{P}_{L},\mathfrak{Q}_{L},\tilde{\mathfrak{P}}_{L}$,
respectively.

For $f:\mathbb{Z}^{d}\to\mathbb{R}^{N}$ and
$g:\mathbb{Z}^{d}\times\mathbb{Z}^{d}\to\mathbb{R}$, let

$\displaystyle(-\Delta
f)(x)=2df(x)-\sum_{e\in\mathrm{B}}f(x+e)=-\sum_{e\in\mathrm{B}}\nabla^{e}_{x}f,$
$\displaystyle\nabla^{e,e^{\prime}}_{x,y}g=g(x+e,y+e^{\prime})-g(x,y+e^{\prime})-g(x+e,y)+g(x,y),\quad
e,e^{\prime}\in\mathrm{B}.$

$\Delta$ is the Laplacian on $\mathbb{Z}^{d}$, we let $G=(-\Delta)^{-1}$ be
its Green function. $G^{m}$ will denote the Green function of the massive
Laplacian: $G^{m}=(-\Delta+m^{2}\mathrm{Id})^{-1}$. We write $\Phi_{m}$ for
the law (and expectation) of the (massive) Gaussian Free Field on
$\mathbb{Z}^{d}$ (the centred Gaussian field with covariance $G^{m}$).
$\Delta_{L}$ will denote the Laplacian in the torus $\Lambda_{L}$ and
$G^{m}_{L}=(-\Delta_{L}+m^{2}\mathrm{Id})^{-1}$. $\Phi_{m;L}$ will denote the
law (and expectation) of the centred Gaussian vector with covariance matrix
$G^{m}_{L}$. When $m$ is omitted, it is set to $0$. We will write
$G^{m}(\cdot,y)$ the function $x\mapsto G^{m}(x,y)$, and define similarly
$G^{m}(y,\cdot)$. We will sometimes use the notation $G_{xy}^{m}\equiv
G^{m}(x,y)$.

Scalar product will be denoted $\cdot$ or $(\ ,\ )$. By convention, if $f,g$
have value in $\mathbb{R}^{N}$, $(f,g)=\sum_{x\in\mathbb{Z}^{d}}f(x)\cdot
g(x)$.

We will denote $\mathbb{S}^{k}$ the unit sphere in $\mathbb{R}^{k+1}$ and
$\nu_{k+1}$ the Lebesgue (un-normalized uniform) measure on $\mathbb{S}^{k}$.
We also denote $\mathbb{B}^{k}=\\{x\in\mathbb{R}^{k}:\ \|x\|\leq 1\\}$.
Moreover, $\Omega_{L}=(\mathbb{S}^{N-1})^{\Lambda_{L}}$.

Finally, we recall that $\mu_{L;\beta,h}$, or simply $\mu_{L;h}$ (we often
omit $\beta$ from the notation), denotes the probability measure associated
with the average $\langle\cdot\rangle_{L;\beta,h}$, or simply
$\langle\cdot\rangle_{L;\beta}$, in (2). $Z_{L;\beta,h}\equiv
Z_{L;\beta,h}(1)$, or simply $Z_{L;h}$ denotes the partition function (see
(1)), and $\psi:\mathbb{R}^{N}\to\mathbb{R}$ the pressure in the thermodynamic
limit, defined as $\psi(v)=\lim_{L\to\infty}|\Lambda_{L}|^{-1}\log
Z_{L;\beta}(e^{\sum_{x\in\Lambda_{L}}v\cdot S_{x}})$. $\mu_{h}$ denotes an
arbitrarily chosen limit point of $\mu_{L;h}$ as $L\to\infty$, and
$\langle\cdot\rangle_{h}$ denotes the corresponding average. $\mu$ denotes an
arbitrarily chosen limit point of $\mu_{h}$ as $h\searrow 0$ taken along a
sequence of $h$s in $D^{*}:=\\{h>0:\ \psi\textnormal{ is differentiable at
}hs\ \forall s\in\mathbb{S}^{N-1},\\}$, and $\langle\cdot\rangle$ the
corresponding average.

In the following sections, $C,C^{\prime},\ldots,c,c^{\prime},\ldots$ denote
constants independent of $T,h,m$, whose specific values may change from line
to line; dependence upon additionally auxiliary parameters will usually be
specified explicitly, with the exception of $N\geq 2$, which we shall not
track the dependence of the various constants on. We will explicitly focus on
the case $N>2$, the case $N=2$ being significantly simpler, in that the $u$
variables introduced at the beginning of the next section are not present; the
interpretation and adaptation of the proof to the case $N=2$ is immediate and
left to the reader (moreover, the case $N=2$ is already covered by [6]).

##  3\. Gaussian coordinates and the formal low-temperature expansion

###  3.1. The Gibbs measure as a perturbed Gaussian

We will use a particular parametrization of $\mathbb{S}^{N-1}$: for
$u\in\mathbb{R}^{N-2}$ with $\|u\|\leq 1$, and $-\pi<\theta\leq\pi$, we let

$\Pi(u,\theta)=\big{(}u_{1},\cdots,u_{N-2},\rho\sin(\theta),\rho\cos(\theta)\big{)}$
(10)

with $\rho=\rho(u)=\sqrt{1-\|u\|^{2}}$, which provides a parametrization of
$\mathbb{S}^{N-1}$. Note that if $N=2$ there is no variable $u$ and $\rho$
should be replaced by $1$ (in this case, many of the formulas and equations
below simplify significantly). One has the change of coordinate formula: for
any integrable $f:\mathbb{S}^{N-1}\to\mathbb{R}$,

$\int_{\mathbb{S}^{N-1}}d\nu_{N}(S)f(S)=\int_{\mathbb{B}^{N-2}}du\,\int_{-\pi}^{\pi}d\theta
f(u_{1},\cdots,u_{N-2},\rho\sin\theta,\rho\cos\theta).$ (11)

In light of (11), given an observable $a:\Omega_{L}\to\mathbb{R}$ of the spin
configuration in $\Lambda_{L}$, letting $A$ be the composition of $a$ with the
transformation (10), to be applied to each spin $S_{x}\in\mathbb{S}^{N-1}$,
$x\in\Lambda_{L}$, we find that the un-normalized expectation of $a$ can be
rewritten as follows:

$\begin{split}Z_{L;\beta,h}(a)&=\int_{\Omega_{L}}\prod_{x\in\Lambda_{L}}d\nu_{N}(S_{x})\,a(S)e^{\sum_{x\in\Lambda_{L}}(\beta\sum_{\mathrm{e}}S_{x+\mathrm{e}}\cdot
S_{x}+hS^{N}_{x})}\\\
&=\int\limits_{(\mathbb{B}^{N-2})^{\Lambda_{L}}}\\!d{u}\\!\int\limits_{[-\pi,\pi]^{\Lambda_{L}}}\\!\\!\\!d\theta\,A(u,\theta)e^{H_{L;h}(u,\theta)},\end{split}$
(12)

where, in the first line, $S=(S_{x})_{x\in\Lambda_{L}}$, and

$\begin{split}H_{L;h}(u,\theta)&=\sum_{x\in\Lambda_{L}}\Big{[}\beta\sum_{e}\Big{(}u_{x}\cdot
u_{x+\mathrm{e}}+\sqrt{(1-\|u_{x}\|^{2})(1-\|u_{x+e}\|^{2})}\cos(\nabla^{e}_{x}\theta)\Big{)}\\\
&+h\sqrt{(1-\|u_{x}\|^{2})}\cos(\theta_{x})\Big{]}.\end{split}$ (13)

If we now rescale the variables as follows:

$\phi_{x}=\sqrt{\beta}\theta_{x},\quad\tilde{u}_{x}=\sqrt{\beta}u_{x},$

the expectation in (12) becomes, letting $T=\beta^{-1}$ and
$\tilde{A}(\tilde{u},\phi)=A(\sqrt{T}\tilde{u},\sqrt{T}\phi)$:

$\begin{split}Z_{L;\beta,h}(a)&=T^{L^{d}(N-1)/2}\\!\\!\\!\\!\int\limits_{(\sqrt{\beta}\mathbb{B}^{N-2})^{\Lambda_{L}}}\\!\\!\\!\\!d{\tilde{u}}\,e^{\mathcal{H}_{L;h}(\tilde{u})}\\!\\!\\!\\!\int\limits_{[-\sqrt{\beta}\pi,\sqrt{\beta}\pi]^{\Lambda_{L}}}\\!\\!\\!\\!d\phi\,\tilde{A}(\tilde{u},\phi)e^{-\frac{1}{2}\|\nabla\phi\|^{2}}e^{W_{L;h}(\tilde{u},\phi)}\end{split}$
(14)

where
$\|\nabla\phi\|^{2}=\sum_{x\in\Lambda_{L}}\sum_{e}(\nabla^{e}_{x}\phi)^{2}$,
$\|\phi\|^{2}=\sum_{x\in\Lambda_{L}}(\phi_{x})^{2}$,

$\begin{split}\mathcal{H}_{L;h}(\tilde{u})&=\sum_{x\in\Lambda_{L}}\Big{[}\sum_{e}\Big{(}\tilde{u}_{x}\cdot\tilde{u}_{x+e}+\beta\sqrt{(1-T\|\tilde{u}_{x}\|^{2})(1-T\|\tilde{u}_{x+e}\|^{2})}\Big{)}\\\
&+h\sqrt{1-T\|\tilde{u}_{x}\|^{2}}\Big{]}\end{split}$ (15)

and

$\begin{split}W_{L;h}(\tilde{u},\phi)&=\sum_{x\in\Lambda_{L}}\Big{\\{}\sum_{e}\Big{[}\beta\sqrt{(1-T\|\tilde{u}_{x}\|^{2})(1-T\|\tilde{u}_{x+e}\|^{2})}\Big{(}\cos(\sqrt{T}\nabla^{e}_{x}\phi)-1\Big{)}\\\
&+\frac{1}{2}(\nabla^{e}_{x}\phi)^{2}\Big{]}+h\sqrt{1-T\|\tilde{u}_{x}\|^{2}}\big{(}\cos(\sqrt{T}\phi_{x})-1\big{)}\Big{\\}}.\end{split}$
(16)

Note that $W_{L;h}$ vanishes linearly in $T$ as $T\to 0$. More precisely, by
expanding in Taylor series in $T$ the right side of (16), we find that
$W_{L;h}$ admits the following low-temperature expansion:

$W_{L;h}(\tilde{u},\phi)=\sum_{s\geq
1}T^{s}\sum_{x\in\Lambda_{L}}\sum_{\begin{subarray}{c}r,p\geq 0:\\\
r+p=s\end{subarray}}\Big{[}c_{r}\sum_{e}(\nabla^{e}_{x}\phi)^{2r+2}\mathcal{G}^{p}_{x,e}(\tilde{u})+c_{r-1}\mathds{1}_{r\geq
1}h(\phi_{x})^{2r}\tilde{c}_{p}\|\tilde{u}_{x}\|^{2p}\Big{]},$ (17)

where $c_{r}=\tfrac{(-1)^{r+1}}{(2r+2)!}$, $\tilde{c}_{p}=(-1)^{p}{1/2\choose
p}$ (recall that $\binom{1/2}{0}=1$ and
$\binom{1/2}{p}=\frac{1}{p!}\frac{1}{2}(\frac{1}{2}-1)\cdots(\frac{1}{2}-p+1)$
for integer $p>0$) and
$\mathcal{G}^{p}_{x,e}(\tilde{u})=\sum_{l+m=p}\tilde{c}_{l}\tilde{c}_{m}\|\tilde{u}_{x}\|^{2l}\|\tilde{u}_{x+e}\|^{2m}$.

Therefore, (14) expresses the $O(N)$ model as a perturbation of a Gaussian
measure, as far as the integration over the $\phi$ variables is concerned.
Moreover, as a consequence of the $O(N-1)$ symmetry of the model in the hyper-
plane orthogonal to the magnetic field, for any $k\in\\{1,\cdots,N-2\\}$ one
has the equality in law (with respect to the probability measure with density
$\propto
e^{\mathcal{H}_{L;h}(\tilde{u})-\frac{1}{2}\|\nabla\phi\|^{2}+W_{L;h}(\tilde{u},\phi)}$):

$\big{(}\tilde{u}^{1},\cdots,\tilde{u}^{N-2}\big{)}\stackrel{{\scriptstyle\textnormal{law}}}{{=}}\Big{(}\tilde{u}^{1},\cdots,\tilde{u}^{k-1},\mathfrak{F}(\tilde{u},\phi),\tilde{u}^{k+1},\cdots,\tilde{u}^{N-2}\Big{)}.$
(18)

where
$\mathfrak{F}(\tilde{u},\phi)=\big{(}\sqrt{\beta(1-T\|\tilde{u}_{x}\|^{2})}\sin\big(\sqrt{T}\phi_{x}\big{missing})\big{)}_{x\in\Lambda_{L}}$,

###  3.2. The formal low temperature expansion

Eqs.(14),(17),(18) allow us to recursively define the coefficients of the (a
priori formal) low-temperature expansion for the average of any observable
$a:\Omega_{L}\to\mathbb{R}$. Let $\tilde{A}(\tilde{u},\phi)$ be, as before,
the form that the observable takes, once it is re-expressed in terms of the
variables $\tilde{u},\phi$. We denote its formal Taylor expansion in $T$ by

$\tilde{A}(\tilde{u},\phi)\stackrel{{\scriptstyle(\infty)}}{{=}}\sum_{s\geq
0}T^{s}\sum_{p,\tilde{p}}a_{s}^{p,\tilde{p}}\phi^{p}\tilde{u}^{\tilde{p}},$
(19)

where $\stackrel{{\scriptstyle(\infty)}}{{=}}$ denotes identity between formal
power series, or between a function and its formal Taylor series, the sums
over $p,\tilde{p}$ run over $\mathfrak{P}_{L},\tilde{\mathfrak{P}}_{L}$,
respectively (see definition after (9)), and $a_{s}^{p,\tilde{p}}$ are
suitable real coefficients, independent of $T$. The formal low-$T$ expansion
for

$\langle
a\rangle_{L;h}=\frac{\int\limits_{(\sqrt{\beta}\mathbb{B}^{N-2})^{\Lambda_{L}}}\\!\\!\\!\\!d{\tilde{u}}\,e^{\mathcal{H}_{L;h}(\tilde{u})}\\!\\!\\!\\!\int\limits_{[-\sqrt{\beta}\pi,\sqrt{\beta}\pi]^{\Lambda_{L}}}\\!\\!\\!\\!d\phi\,\tilde{A}(\tilde{u},\phi)e^{-\frac{1}{2}\|\nabla\phi\|^{2}}e^{W_{L;h}(\tilde{u},\phi)}}{\int\limits_{(\sqrt{\beta}\mathbb{B}^{N-2})^{\Lambda_{L}}}\\!\\!\\!\\!d{\tilde{u}}\,e^{\mathcal{H}_{L;h}(\tilde{u})}\\!\\!\\!\\!\int\limits_{[-\sqrt{\beta}\pi,\sqrt{\beta}\pi]^{\Lambda_{L}}}\\!\\!\\!\\!d\phi\,e^{-\frac{1}{2}\|\nabla\phi\|^{2}}e^{W_{L;h}(\tilde{u},\phi)}}$
(20)

is obtained by neglecting the constraints $|\tilde{u}_{x}|\leq\sqrt{\beta}$,
$|\phi_{x}|\leq\pi\sqrt{\beta}$ in the integration domain (as we will show in
the next section, this approximation produces an exponentially small error in
$\beta$, as $\beta\to\infty$), and by replacing (17) and (19) in (20). In view
of this, we write

$\langle a\rangle_{L;h}\stackrel{{\scriptstyle(\infty)}}{{=}}\sum_{s\geq
0}T^{s}\sum_{p,\tilde{p}}a_{s}^{p,\tilde{p}}\frac{\int\\!d{\tilde{u}}\,e^{\mathcal{H}_{L;h}(\tilde{u})+\mathcal{V}_{L;h}(\tilde{u})}\,\tilde{u}^{\tilde{p}}\langle\phi^{p}\rangle_{L;h,\tilde{u}}}{\int\\!d{\tilde{u}}\,e^{\mathcal{H}_{L;h}(\tilde{u})+\mathcal{V}_{L;h}(\tilde{u})}}$
(21)

where the integral over $\tilde{u}$ is performed over
$(\mathbb{R}^{N-2})^{\Lambda_{L}}$, we defined
$\mathcal{V}_{L;h}(\tilde{u})=\log$ $\int
d\phi\,e^{-\frac{1}{2}\|\nabla\phi\|^{2}+W_{L;h}(\tilde{u},\phi)}$ (where the
integral over $\phi$ is performed over
$(\mathbb{R}^{\Lambda_{L}})^{\Lambda_{L}}$), and

$\langle\phi^{p}\rangle_{L;h,\tilde{u}}=\int
d\phi\,e^{-\frac{1}{2}\|\nabla\phi\|^{2}+W_{L;h}(\tilde{u},\phi)-\mathcal{V}_{L;h}(\tilde{u})}\phi^{p}.$
(22)

By formally expanding in $W_{L;h}$, this expectation can be rewritten as

$\langle\phi^{p}\rangle_{L;h,\tilde{u}}\stackrel{{\scriptstyle(\infty)}}{{=}}\sum_{k\geq
0}\tfrac{1}{k!}\Phi_{L}(\phi^{p};\underbrace{W_{L;h};\cdots;W_{L;h}}_{k\
\text{times}}),$ (23)

where $\Phi_{L}(A_{1};\cdots;A_{k})$ denotes truncated expectation with
respect to the Gaussian law $\Phi_{L}$ with covariance $G_{L}$, that is,

$\Phi_{L}(A_{1};\cdots;A_{k}):=\partial_{\lambda_{1}}\cdots\partial_{\lambda_{k}}\log\Big(\Phi(e^{\sum_{i=1}^{k}\lambda_{i}A_{i}})\Big{missing})\Big{|}_{\lambda_{0}=\cdots=\lambda_{k}=0}$
(24)

or, equivalently,

$\Phi_{L}(A_{1};\cdots;A_{k})=\sum_{\pi\in\mathcal{P}_{k}}(|\pi|-1)!(-1)^{|\pi|-1}\prod_{Y\in\pi}\Phi_{L}(A_{Y}),$
(25)

where $\mathcal{P}_{k}\equiv\mathcal{P}([k])$ is the set of partitions of
$[k]=\\{1,\cdots,k\\}$, and $A_{Y}=\prod_{i\in Y}A_{i}$. Inserting the formal
low-$T$ expansion of $W_{L;h}$, (17), and taking $L\to\infty$, $h\searrow 0$,
we find:

$\begin{split}\langle
a\rangle\stackrel{{\scriptstyle(\infty)}}{{=}}&\sum_{s_{0}\geq
0}T^{s_{0}}\sum_{p,\tilde{p}}a_{s_{0}}^{p,\tilde{p}}\,\Big{[}\Phi(\phi^{p})\langle\tilde{u}^{\tilde{p}}\rangle+\sum_{s\geq
1}T^{s}\sum_{k\geq
1}\tfrac{1}{k!}\sum_{\begin{subarray}{c}p_{1},r_{1},\ldots,p_{k},r_{k}\geq
0:\\\ p_{l}+r_{l}=s_{l}\geq 1\\\ s_{1}+\cdots+s_{k}=s\end{subarray}}\
\sum_{x_{1},e_{1},\ldots,x_{k},e_{k}}\\\ &\hskip-19.91684pt\cdot
c_{r_{1}}\cdots
c_{r_{k}}\Phi\big{(}\phi^{p};(\nabla^{e_{1}}_{\\!x_{1}}\phi)^{2r_{1}+2};\cdots;(\nabla^{e_{k}}_{\\!x_{k}}\phi)^{2r_{k}+2}\big{)}\langle\tilde{u}^{\tilde{p}}\prod_{l=1}^{k}\mathcal{G}^{p_{l}}_{x_{l},e_{l}}\rangle\Big{]},\end{split}$
(26)

where $\Phi$ is the law of the massless Gaussian Free Field on
$\mathbb{Z}^{d}$, the sum over $p,\tilde{p}$ can be freely restricted to even
tuples (terms with $p$ or $\tilde{p}$ odd are zero by parity), and the sums
over $x_{1},\ldots,x_{k}$ are now over $\mathbb{Z}^{d}$.

###  3.3. The finite order truncations of the low temperature expansion

If we denote by $\langle a\rangle^{(n)}$ the truncation at order $n$ of the
formal Taylor series in $T$, (26) implies that

$\begin{split}&\langle
a\rangle^{(n)}=\sum_{s_{0}=0}^{n}T^{s_{0}}\sum_{p,\tilde{p}}a_{s_{0}}^{p,\tilde{p}}\,\Big{[}\Phi(\phi^{p})\langle\tilde{u}^{\tilde{p}}\rangle^{(n-s_{0})}+\mathds{1}_{n>s_{0}}\mathds{1}_{\|p\|_{1}>0}\sum_{s=1}^{n-s_{0}}T^{s}\sum_{k\geq
1}\tfrac{1}{k!}\sum_{\begin{subarray}{c}p_{1},r_{1},\ldots,p_{k},r_{k}\geq
0:\\\ p_{l}+r_{l}=s_{l}\geq 1\\\ s_{1}+\cdots+s_{k}=s\end{subarray}}\cdot\\\
&\qquad\cdot c_{r_{1}}\cdots
c_{r_{k}}\sum_{x_{1},e_{1},\ldots,x_{k},e_{k}}\Phi\big{(}\phi^{p};(\nabla^{e_{1}}_{\\!x_{1}}\phi)^{2r_{1}+2};\cdots;(\nabla^{e_{k}}_{\\!x_{k}}\phi)^{2r_{k}+2}\big{)}\langle\tilde{u}^{\tilde{p}}\prod_{l=1}^{k}\mathcal{G}^{p_{l}}_{x_{l},e_{l}}\rangle^{(n-s-
s_{0})}\Big{]},\end{split}$ (27)

where, again, the sum over $p,\tilde{p}$ can be restricted to even tuples.
Now, the terms $\langle\tilde{u}^{\tilde{p}}\rangle^{(n-s_{0})}$ in the right
side (and similarly
$\langle\tilde{u}^{\tilde{p}}\prod_{l=1}^{k}\mathcal{G}^{p_{l}}_{x_{l},e_{l}}\rangle^{(n-s-
s_{0})}$, which are linear combinations of terms of the same form, i.e., of
$\langle\tilde{u}^{\tilde{p}^{\prime}}\rangle^{(n-s-s_{0})}$ for suitable
$\tilde{p}^{\prime}$), can be computed recursively, by using (18) and (27)
itself for a finite number of times. In fact, let $m+\|\tilde{p}\|_{1}/2\geq
0$ be the order of $\langle\tilde{u}^{\tilde{p}}\rangle^{(m)}$, where
$\tilde{p}$ is an even tuple with non-negative elements. If the order is zero,
then $\langle 1\rangle^{(0)}=1$. More generally, if $\tilde{p}=0$, $\langle
1\rangle^{(m)}=1$, for all $m\geq 0$. Let us then consider a term
$\langle\tilde{u}^{\tilde{p}}\rangle^{(m)}$ of positive order
$k=m+\|\tilde{p}\|_{1}/2$, with $\|\tilde{p}\|_{1}$ positive. Let us assume
without loss of generality that $\|\tilde{p}^{1}\|$ is even and positive, so
that $\tilde{p}^{\geq 2}=(0,\tilde{p}^{2},\ldots,\tilde{p}^{N-2})$ has norm
$\|\tilde{p}^{\geq 2}\|_{1}\leq\|\tilde{p}\|_{1}-2$. Using (18), and expanding
$\prod_{x}\Big{(}\sqrt{\beta(1-T\|\tilde{u}_{x}\|^{2})}\sin\big(\sqrt{T}\phi_{x}\big{missing})\Big{)}^{\tilde{p}^{1}_{x}}$
in Taylor series in $T$ in the form

$\begin{split}&\phi^{\tilde{p}^{1}}\sum_{s\geq
0}(-1)^{s}T^{s}\sum_{\begin{subarray}{c}(s_{x})_{x\in\mathbb{Z}^{d}}:\\\
s_{x}\geq 0,\
\sum_{x}s_{x}=s\end{subarray}}\prod_{x\in\mathbb{Z}^{d}}\sum_{\begin{subarray}{c}i_{x},j_{x}\geq
0:\\\
i_{x}+j_{x}=s_{x}\end{subarray}}\phi_{x}^{2i_{x}}\|\tilde{u}_{x}\|^{2j_{x}}\frac{1}{(2i_{x}+1)!}{1/2\choose
j_{x}}\\\ \equiv\ &\phi^{\tilde{p}^{1}}\sum_{s\geq
0}T^{s}\sum_{n\in\mathfrak{P}}\sum_{\tilde{n}\in\tilde{\mathfrak{P}}}c^{n,\tilde{n}}_{s}\phi^{n}\tilde{u}^{\tilde{n}}\end{split}$
(28)

(note that $c^{n,\tilde{n}}_{s}$ is non zero only if $n,\tilde{n}$ are both
even and $\|n\|_{1}+\|\tilde{n}\|_{1}=2s$), we find

$\langle\tilde{u}^{\tilde{p}}\rangle^{(m)}=\sum_{s=0}^{m}T^{s}\sum_{\begin{subarray}{c}n\in\mathfrak{P}\\\
\tilde{n}\in\tilde{\mathfrak{P}}\end{subarray}}c_{s}^{n,\tilde{n}}\langle\phi^{\tilde{p}^{1}+n}\tilde{u}^{\tilde{p}^{\geq
2}+\tilde{n}}\rangle^{(m-s)}.$ (29)

By applying formula (27) to the terms
$\langle\phi^{\tilde{p}^{1}+n}\tilde{u}^{\tilde{p}^{\geq
2}+\tilde{n}}\rangle^{(m-s)}$ in the right side, we can re-express each of
them as linear combinations of terms
$\langle\tilde{u}^{\tilde{p}^{\prime}}\rangle^{(m^{\prime})}$, whose order,
$k^{\prime}=m^{\prime}+\|\tilde{p}^{\prime}\|_{1}/2$, is strictly smaller than
the original order $k$. Therefore, by proceeding recursively, we can compute
$\langle\tilde{u}^{\tilde{p}}\rangle^{(m)}$ explicitly. By plugging the result
back into (27) we obtain the (truncation at order $n$ of the) low temperature
expansion of $\langle a\rangle$, for any observable $a$.

For later reference, we formalize the definition of $\langle\ \rangle^{(n)}$
as follows. Let $\mathcal{A}$ be the space of formal power series in $T$ of
the form $\sum_{s\geq
0}T^{s}\sum_{p\in\mathfrak{P}}\sum_{\tilde{p}\in\tilde{\mathfrak{P}}}a^{p,\tilde{p}}_{s}\phi^{p}\tilde{u}^{\tilde{p}}$
with real coefficients $a^{p,\tilde{p}}_{s}$ independent of $T$. Let also
$\mathcal{A}_{0}$ be the subspace of $\mathcal{A}$ consisting of formal power
series independent of $\tilde{u},\phi$, of the form $\sum_{s}T^{s}a_{s}$, with
$a_{s}\in\mathbb{R}$.

######  Definition 3.1.

For any $n\geq 0$, $\langle\ \rangle^{(n)}$ is the operator acting on
$\mathcal{A}$ that satisfies the following properties.

  * •

$\langle 1\rangle^{(n)}=1$.

  * •

$\langle\ \rangle^{(n)}$ is translation invariant, that is, for any
$p\in\mathfrak{P}$, $\tilde{p}\in\tilde{\mathfrak{P}}$,
$x_{0}\in\mathbb{Z}^{d}$,
$\langle\phi^{p}\tilde{u}^{\tilde{p}}\rangle^{(n)}=\langle\phi^{\tau_{x_{0}}p}\tilde{u}^{\tau_{x_{0}}\tilde{p}}\rangle^{(n)}$,
where $(\tau_{x_{0}}p)_{x}=p_{x-x_{0}}$, and similarly for
$\tau_{x_{0}}\tilde{p}$.

  * •

For any $Q,Q^{\prime}\in\mathcal{A}$, $\langle
Q+Q^{\prime}\rangle^{(n)}=\langle Q\rangle^{(n)}+\langle
Q^{\prime}\rangle^{(n)}$.

  * •

For any $\lambda\in\mathcal{A}_{0}$ of the form $\lambda=\sum_{s\geq
0}T^{s}\lambda_{s}$, and any $Q\in\mathcal{A}$, $\langle\lambda
Q\rangle^{(n)}=\sum_{s=0}^{n}T^{s}\lambda_{s}\langle Q\rangle^{(n-s)}$.

  * •

For any $p\in\mathfrak{P}$ and any $\tilde{p}\in\tilde{\mathfrak{P}}$, letting
$\mathcal{F}=\mathcal{F}(\tilde{u}^{1},\ldots,\tilde{u}^{N-2})=\tilde{u}^{\tilde{p}}$,

$\langle\phi^{p}\mathcal{F}(\tilde{u}^{1},\cdots,\tilde{u}^{N-2})\rangle^{(n)}=\langle\phi^{p}\mathcal{F}(\tilde{u}^{\pi(1)},\cdots,\tilde{u}^{\pi(N-2)})\rangle^{(n)}$
(30)

for any permutation $\pi$ of $\\{1,\cdots,N-2\\}$.

  * •

For any $\tilde{p}\in\tilde{\mathfrak{P}}$, if
$\tilde{p}^{1}=(\tilde{p}^{1}_{x})_{x\in\mathbb{Z}^{d}}\in\mathfrak{P}$ is not
zero, letting $\tilde{p}^{\geq 2}=(0,\tilde{p}^{2},\ldots,\tilde{p}^{N-2})$,

$\langle\tilde{u}^{\tilde{p}}\rangle^{(n)}=\langle\phi^{\tilde{p}^{1}}\tilde{F}\tilde{u}^{\tilde{p}^{\geq
2}}\rangle^{(n)}$ (31)

where $\tilde{F}=\sum_{s\geq
0}T^{s}\sum_{n,\tilde{n}}c^{n,\tilde{n}}_{s}\phi^{n}\tilde{u}^{\tilde{n}}\in\mathcal{A}$
(the coefficients $c^{n,\tilde{n}}_{s}$ were defined in (28)).

  * •

For any even $p\in\mathfrak{P}$, $\tilde{p}\in\tilde{\mathfrak{P}}$, the
following “extraction” formula holds:

$\begin{split}\langle\phi^{p}\tilde{u}^{\tilde{p}}\rangle^{(n)}&=\Phi(\phi^{p})\langle\tilde{u}^{\tilde{p}}\rangle^{(n)}+\mathds{1}_{n\geq
1}\sum_{k=1}^{n}\frac{1}{k!}\sum_{s=k}^{n}T^{s}\sum_{\begin{subarray}{c}s_{1},\cdots,s_{k}\geq
1\\\ \sum s_{l}=s\end{subarray}}\sum_{\begin{subarray}{c}x_{1},\cdots,x_{k}\\\
e_{1},\cdots,e_{k}\end{subarray}}\sum_{\begin{subarray}{c}r_{1},p^{\prime}_{1},\cdots,r_{k},p^{\prime}_{k}\geq
0\\\ p^{\prime}_{l}+r_{l}=s_{l}\end{subarray}}\times\\\
&\times\big{(}\prod_{l=1}^{k}c_{r_{l}}\big{)}\Phi\big{(}\phi^{p};(\nabla_{x_{1}}^{e_{1}}\phi)^{2r_{1}+2};\cdots;(\nabla_{x_{k}}^{e_{k}}\phi)^{2r_{k}+2}\big{)}\langle\tilde{u}^{\tilde{p}}\prod_{l=1}^{k}\mathcal{G}_{x_{l},e_{l}}^{p_{l}^{\prime}}\rangle^{(n-s)},\end{split}$
(32)

where $c_{r}$ and $\mathcal{G}^{p}_{x,e}$ were defined right after (17).

Note that the presence of the connected correlation
$\Phi\big{(}\phi^{p};(\nabla_{x_{1}}^{e_{1}}\phi)^{2r_{1}+2};\cdots;(\nabla_{x_{k}}^{e_{k}}\phi)^{2r_{k}+2}\big{)}$
ensures absolute summability of the second line of (32), see Lemma C.5, thus
making $\langle\ \rangle^{(n)}$ a well defined object. More precisely, Lemma
C.5 implies the following quantitative bounds, which will be used in the
inductive proof discussed in Section 6.

######  Lemma 3.1.

Let $K>0,n\geq 0$ be integers. Let $\epsilon>0$. Let
$p,p^{\prime}:\mathbb{Z}^{d}\to\mathbb{Z}_{+}$ be odd,
$\tilde{p}:\mathbb{Z}^{d}\times[N-2]\to\mathbb{Z}_{+}$ be even, and such that
$\left\|p\right\|_{\scriptscriptstyle
1}+\left\|p^{\prime}\right\|_{\scriptscriptstyle
1}+\left\|\tilde{p}\right\|_{\scriptscriptstyle 1}\leq 2K$, then,

$|\langle\phi^{p}\phi^{p^{\prime}}\tilde{u}^{\tilde{p}}\rangle^{(n)}|\leq
C\sum_{x\in\mathrm{supp}_{p}}\sum_{y\in\mathrm{supp}_{p^{\prime}}}\frac{1}{(1+|x-y|)^{d-2-\epsilon}},$

where $C=C(K,n,\epsilon)$.

######  Lemma 3.2.

Let $K>0,n\geq 0$ be integers. Let $\epsilon>0$. Let
$p:\mathbb{Z}^{d}\to\mathbb{Z}_{+}$ be odd,
$q:\mathbb{E}_{d}\to\mathbb{Z}_{+}$ be odd,
$\tilde{p}:\mathbb{Z}^{d}\times[N-2]\to\mathbb{Z}_{+}$ be even, and such that
$\left\|p\right\|_{\scriptscriptstyle 1}+\left\|q\right\|_{\scriptscriptstyle
1}+\left\|\tilde{p}\right\|_{\scriptscriptstyle 1}\leq 2K$, then,

$|\langle\phi^{p}(\nabla\phi)^{q}\tilde{u}^{\tilde{p}}\rangle^{(n)}|\leq
C\sum_{x\in\mathrm{supp}_{p}}\sum_{y\in\mathrm{supp}_{q}}\frac{1}{(1+|x-y|)^{d-1-\epsilon}},$

where $C=C(K,n,\epsilon)$.

These two lemmas follows straightforwardly from Lemma C.5 and the definition
of $\langle\ \rangle^{(n)}$. In fact, notice first of all that eq.(86) of
Lemma C.5 implies that, for any even $\tilde{p}\in\tilde{\mathfrak{P}}$,

$|\langle\tilde{u}^{\tilde{p}}\rangle^{(n)}|\leq C,$ (33)

with $C=C(\|\tilde{p}\|_{1},n)$: this can be easily proved by induction in
$n+\|\tilde{p}\|_{1}/2$, proceeding as described after (27). More in detail:
consider a non zero even $\tilde{p}$ and $n\geq 0$; assume inductively that
$|\langle\tilde{u}^{\tilde{p}^{\prime}}\rangle^{(n^{\prime})}|\leq C$ is known
for all $\tilde{p}^{\prime},n^{\prime}$ such that
$n^{\prime}+\|\tilde{p}^{\prime}\|_{1}/2<n+\|\tilde{p}\|_{1}/2$; use (30) to
reduce to the case of $\tilde{p}^{1}$ non zero; apply (31), then use (32);
write $\tilde{p}^{1}=p+p^{\prime}$ with $p,p^{\prime}$ odd, then use (86) and
the inductive hypothesis to conclude the proof of the induction step. At this
point, the two lemmas follow immediately from (32), the use of (86) and (87),
respectively, and of (33) (details left to the reader).

##  4\. Applications of the infrared bound

In this section we describe an important application of the infrared bound
summarized in Proposition 1.2 above. In particular, generalizing a method in
[6], we derive a priori bounds on the moments of the spin correlation
functions, in the $(u,\theta)$, or equivalently $(\tilde{u},\phi)$,
coordinates. These will be used in the following sections in order to control
the remainder of the low temperature expansion, as produced by systematic
integration by parts.

###  4.1. Large deviation estimates on the spin correlations.

######  Lemma 4.1.

For any $\beta\geq 0$, $h\in D^{*}$ and $a>0$, if $k\in\\{1,\cdots,N-1\\}$,

$\langle e^{a\sqrt{\beta}|S_{0}^{k}|}\rangle_{h}\leq 2e^{a^{2}G_{00}/2}.$ (34)

Moreover,

$\langle e^{a\sqrt{\beta}(1-S_{0}^{N})}\rangle_{h}\leq
e^{(a\sqrt{T}N+a^{2}/2)G_{00}}.$ (35)

###### Proof.

By the infrared bound (5) applied to the function
$f(x)=-a\sqrt{\beta}\delta_{0,x}\mathrm{e}_{k}$, one obtains

$\langle e^{-a\sqrt{\beta}(S_{0}^{k}-\langle
S_{0}^{k}\rangle_{h})}\rangle_{h}\leq e^{a^{2}G_{00}/2}.$ (36)

Now, for $k=1,\cdots,N-1$, by symmetry $S_{0}^{k}$ has the same law as
$-S_{0}^{k}$. In particular, $\langle S_{0}^{k}\rangle_{h}=0$ and

$\langle e^{a\sqrt{\beta}|S_{0}^{k}|}\rangle_{h}=\langle
e^{a\sqrt{\beta}S_{0}^{k}}\mathds{1}_{S_{0}^{k}\geq 0}\rangle_{h}+\langle
e^{-a\sqrt{\beta}S_{0}^{k}}\mathds{1}_{S_{0}^{k}<0}\rangle_{h}\leq 2\langle
e^{-a\sqrt{\beta}S_{0}^{k}}\rangle_{h}.$

Thus (34) holds for $k=1,\cdots,N-1$. Notice that this bound holds for any
value of $T$. For $k=N$, the lower bound (6) on $\langle
S_{0}^{N}\rangle_{h}$, and $\langle S_{0}^{N}\rangle_{h}\geq 0$ implies that

$\langle
S_{0}^{N}\rangle_{h}\geq\sqrt{\max(1-TNG_{00},0)}\geq\max(1-TNG_{00},0).$

Plugging this bound in (36) with $k=N$, we find

$e^{a^{2}G_{00}/2}\geq e^{a\sqrt{\beta}\langle S_{0}^{N}\rangle_{h}}\langle
e^{-a\sqrt{\beta}S_{0}^{N}}\rangle_{h}\geq e^{-a\sqrt{T}NG_{00}}\langle
e^{a\sqrt{\beta}(1-S_{0}^{N})}\rangle_{h}$

so that

$\langle e^{a\sqrt{\beta}(1-S_{0}^{N})}\rangle_{h}\leq
e^{a^{2}G_{00}/2}e^{a\sqrt{T}NG_{00}}\leq e^{(a\sqrt{T}N+a^{2}/2)G_{00}},$

proving (35). ∎

A direct consequence is

######  Corollary 4.2.

Let $\beta\geq 0$, $h\in D^{*}$ and $b>0$. For any $1\leq k_{0}\leq N-1$,

$\mu_{h}\big{(}\sum_{k=1}^{k_{0}}|S_{0}^{k}|^{2}\geq b^{2}\big{)}\leq
2e^{-b^{2}\beta/(2k_{0}^{2}G_{00})}.$ (37)

If, additionally, $T\leq b/(2G_{00}N)$, then

$\mu_{h}(S_{0}^{N}\leq 1-b)\leq e^{-b^{2}\beta/(8G_{00})}.$ (38)

###### Proof.

Using the exponential version of Chebyshev’s inequality, Hölder’s inequality,
and (34), we find that, for any $a>0$,

$\begin{split}\mu_{h}(\sum_{k=1}^{k_{0}}|S_{0}^{k}|^{2}\geq
b^{2})&\leq\mu_{h}(\sum_{k=1}^{k_{0}}|S_{0}^{k}|\geq b)\\\ &\leq
e^{-ab\sqrt{\beta}}\langle
e^{a\sqrt{\beta}\sum_{k=1}^{k_{0}}|S_{0}^{k}|}\rangle_{h}\\\ &\leq
e^{-ab\sqrt{\beta}}\prod_{k=1}^{k_{0}}\langle
e^{ak_{0}\sqrt{\beta}|S_{0}^{k}|}\rangle^{1/k_{0}}_{h}\\\ &\leq
2e^{-ab\sqrt{\beta}}e^{a^{2}k_{0}^{2}G_{00}/2}.\end{split}$

Choosing $a=b\sqrt{\beta}/(k_{0}^{2}G_{00})$ implies (37). Next, using (35)
and, again, the exponential version of Chebyshev’s inequality, for any $a>0$,

$\mu_{h}(1-S_{0}^{N}\geq b)\leq e^{-ab\sqrt{\beta}}\langle
e^{a\sqrt{\beta}(1-S_{0}^{N})}\rangle_{h}\leq
e^{-ab\sqrt{\beta}}e^{(a\sqrt{T}N+a^{2}/2)G_{00}}.$

Choosing $a=b\sqrt{\beta}/G_{00}-\sqrt{T}N$ and $T\leq b/(2G_{00}N)$ implies
(38). ∎

###  4.2. Bounds on the moments of the $(u,\theta)$ coordinates

The next Theorem is the main objective of this section. Its proof is a
generalization of the one of the corresponding result in [6], namely [6,
Eq.(10) and Lemma 2]. Here and below, with some abuse of notation, we denote
by $\langle\ \rangle_{h}$ the average with respect to $\mu_{h}$, even when re-
expressed in terms of the $(u,\theta)$ variables, rather than of $S$ (a
similar convention will be used for $\langle\ \rangle_{L;h}$ and $\langle\
\rangle$).

######  Theorem 4.3.

There exist constants $C,C_{0},c>0$ and $T_{0}>0$ such that, if $T\leq T_{0}$,
the following holds.

  * •

For any $k\in\\{1,\cdots,N-2\\}$ and $a>0$, $\langle
e^{a|\tilde{u}_{0}^{k}|}\rangle_{h}\leq 2e^{a^{2}G_{00}/2}$. Moreover, for any
$n\geq 0$,

$\langle|\tilde{u}^{k}_{0}|^{n}\rangle_{h}\leq C^{n}\sqrt{n!}.$

  * •

$\langle e^{|\phi_{0}|}\rangle_{h}\leq C$. In particular, for any $n\geq 0$,

$|\langle|\phi_{0}|^{n}\rangle_{h}|\leq Cn!.$

  * •

For any $F$ function of $\phi,\tilde{u}$,

$|\langle
F\delta(\theta_{0}\pm\pi)\rangle_{h}|\leq\left\|F\right\|_{\scriptscriptstyle\infty}C_{0}e^{h-c\beta},$

where the sup-norm is computed over
$\phi_{x}\in[-\sqrt{\beta}\pi,\sqrt{\beta}\pi]$ and
$\|\tilde{u}_{x}\|\leq\sqrt{\beta}$ for all $x\in\mathbb{Z}^{d}$.

###### Proof.

The first inequality in the first item, $\langle
e^{a|\tilde{u}_{0}^{k}|}\rangle_{h}\leq 2e^{a^{2}G_{00}/2}$, is a direct
consequence of (34) and the definition of $\tilde{u}$. From such inequality,
it follows that $\langle|\tilde{u}_{0}^{k}|^{n}\rangle_{h}\leq
2e^{a^{2}G_{00}/2}a^{-n}n!$. The choice $a=\sqrt{n/G_{00}}$ implies the stated
bound on $\langle|\tilde{u}_{0}^{k}|^{n}\rangle_{h}$.

To get the second item, write

$\displaystyle\langle e^{\sqrt{\beta}|\theta_{0}|}\rangle_{h}$
$\displaystyle=\langle
e^{\sqrt{\beta}|\theta_{0}|}\mathds{1}_{|\theta_{0}|\leq\pi/2}\mathds{1}_{\|u_{0}\|\leq
1/2}\rangle_{h}+\langle
e^{\sqrt{\beta}|\theta_{0}|}\mathds{1}_{|\theta_{0}|\leq\pi/2}\mathds{1}_{\|u_{0}\|>1/2}\rangle_{h}+\langle
e^{\sqrt{\beta}|\theta_{0}|}\mathds{1}_{|\theta_{0}|>\pi/2}\rangle_{h}$
$\displaystyle\leq\langle
e^{\pi^{-1}\sqrt{3\beta}\sqrt{1-\|u_{0}\|^{2}}|\sin(\theta_{0})|}\rangle_{h}+e^{\pi\sqrt{\beta}/2}\mu_{h}(\|u_{0}\|>1/2)+e^{\pi\sqrt{\beta}}\mu_{h}(|\theta_{0}|>\pi/2)$
$\displaystyle=\langle
e^{\pi^{-1}\sqrt{3\beta}|S_{0}^{N-1}|}\rangle_{h}+e^{\pi\sqrt{\beta}/2}\mu_{h}\big{(}{\sum_{k=1}^{N-2}|S^{k}_{0}|^{2}}>1/4\big{)}+e^{\pi\sqrt{\beta}}\mu_{h}(S_{0}^{N}\leq
0)$ $\displaystyle\leq
2e^{3G_{00}/(2\pi^{2})}+2e^{\pi\sqrt{\beta}/2}e^{-\beta/(8G_{00}(N-2)^{2})}+e^{\pi\sqrt{\beta}}e^{-\beta/(8G_{00})},$

where in the last inequality we used (34), (37) and (38). If we now choose
$\beta$ large enough and recall the definition of $\phi$, we obtain, as
desired, that $\langle e^{|\phi_{0}|}\rangle_{h}\leq C$ for a suitable $C>0$.

Finally, we consider the term $\langle F\delta(\theta_{0}\pm\pi)\rangle_{h}$
involving the delta function. Using the DLR equation and letting
$S_{0}(u_{0})=\Pi(u_{0},\pi)=(u_{0}^{1},\cdots,u_{0}^{N-2},0,-\sqrt{1-\|u_{0}\|^{2}})$,

$\langle F\delta(\theta_{0}\pm\pi)\rangle_{h}=\int d\mu_{h}(S_{i},i\neq
0)\frac{\int_{\|u_{0}\|\leq 1}du_{0}\,F(S,S_{0}(u_{0}))e^{\beta\sum_{i\sim
0}S_{0}(u_{0})\cdot S_{i}-\sqrt{1-\|u_{0}\|^{2}}h}}{\int
d\nu_{N}(\tilde{S}_{0})e^{\beta\sum_{i\sim 0}\tilde{S}_{0}\cdot
S_{i}+h\tilde{S}_{0}^{N}}}.$

Define

$J(S_{i},i\sim 0)=\sup_{\|u_{0}\|\leq 1}\Big{(}\int
d\nu_{N}(\tilde{S}_{0})e^{\beta\sum_{i\sim 0}(\tilde{S}_{0}-S_{0}(u_{0}))\cdot
S_{i}+h\tilde{S}_{0}^{N}}\Big{)}^{-1}\equiv\sup_{\|u_{0}\|\leq
1}J_{u_{0}}(S_{i},i\sim 0).$

One has

$|\langle
F\delta(\theta_{0}\pm\pi)\rangle_{h}|\leq\left\|F\right\|_{\scriptscriptstyle\infty}\frac{\pi^{N/2-1}}{\Gamma(N/2)}\int
d\mu_{h}(S_{i},i\neq 0)J(S_{i},i\sim 0).$

Now, for any event $A$ measurable with respect to $\\{S_{i}:i\sim 0\\}$, one
has

$\int d\mu_{h}(S_{i},i\neq 0)J(S_{i},i\sim 0)\leq\sup_{(S_{i})_{i\sim 0}\in
A}J(S_{i},i\sim 0)+\mu_{h}(A^{c})\sup_{(S_{i})_{i\sim 0}\in
A^{c}}J(S_{i},i\sim 0).$ (39)

We choose $A=\\{S_{i}^{N}\geq 7/8,i\sim 0\\}$. Study the first term. For any
$(S_{i})_{i\sim 0}\in A$, any $u_{0}\in\mathbb{B}^{N-2}$ and $h\geq 0$, one
has the bound

$\begin{split}J_{u_{0}}(S_{i},i\sim 0)&\leq\Big{(}\int
d\nu_{N}(\tilde{S}_{0})\mathds{1}_{\tilde{S}_{0}^{N}\geq
7/8}e^{\beta\sum_{i\sim 0}(\tilde{S}_{0}-S_{0}(u_{0}))\cdot
S_{i}}\Big{)}^{-1}\\\ &\leq\Big{(}\int
d\nu_{N}(\tilde{S}_{0})\mathds{1}_{\tilde{S}_{0}^{N}\geq 7/8}e^{\beta
2d/32}\Big{)}^{-1}=Ce^{-\beta 2d/32}.\end{split}$

for some constant $C>0$. Indeed, on the one hand, for $S_{i}^{N}\geq 7/8$ and
$\|u_{0}\|\leq 1$, by Cauchy-Schwartz,

$\begin{split}S_{0}(u_{0})\cdot
S_{i}&=\sum_{k=1}^{N-2}u_{0}^{k}S_{i}^{k}-\sqrt{1-\|u_{0}\|^{2}}S_{i}^{N}\leq\|u_{0}\|\Big{(}\sum_{k=1}^{N-1}(S_{i}^{k})^{2}\Big{)}^{1/2}\\\
&\leq\|u_{0}\|\Big{(}1-\frac{49}{64}\Big{)}^{1/2}\leq\|u_{0}\|\cdot\frac{1}{2}\leq\frac{1}{2}.\end{split}$

On the other hand, if $\tilde{S}^{N}_{0}\geq 7/8$ and $S_{i}^{N}\geq 7/8$,

$\tilde{S}_{0}\cdot S_{i}\geq\cos(2\arccos(7/8))=\frac{17}{32}.$ (40)

We now turn to the second term in (39). First, by a union bound,

$\mu_{h}(A^{c})\leq 2d\mu_{h}(1-S_{0}^{N}>1/8)\leq
2de^{-\beta/(8^{3}G_{00})},$

where we assumed $T$ sufficiently small and used (38). Moreover, for any
$u_{0}\in\mathbb{B}^{N-2}$,

$\begin{split}J_{u_{0}}(S_{i},i\sim 0)&\leq e^{h}\Big{(}\int
d\nu_{N}(\tilde{S}_{0})\mathds{1}_{\|\tilde{S}_{0}-S_{0}(u_{0})\|\leq
T}e^{\beta\sum_{i\sim 0}(\tilde{S}_{0}-S_{0}(u_{0}))\cdot
S_{i}}\Big{)}^{-1}\\\ &\leq e^{h+2d}\Big{(}\int
d\nu_{N}(\tilde{S}_{0})\mathds{1}_{\|\tilde{S}_{0}-S_{0}(u_{0})\|\leq
T}\Big{)}^{-1}\leq C^{\prime}e^{h}\beta^{N-1},\end{split}$

for some $C^{\prime}>0$ independent of $T$. Combining the two previous
equations, we obtain the desired estimate on the second term in (39), which
concludes the proof. ∎

###  4.3. Taylor expansion of the spin correlations

Given $\alpha:\mathbb{Z}^{d}\times[N]\to\mathbb{Z}_{+}$ with
$\|\alpha\|_{1}=\sum_{k=1}^{N}\sum_{x\in\mathbb{Z}^{d}}\alpha^{k}_{x}$ finite,
we let $a_{\alpha}(S)$ be the spin observable

$a_{\alpha}(S)=\prod_{x}\prod_{k}(S_{x}^{k})^{\alpha_{x}^{k}}.$

In order to prove our main result, Theorem 1.3, we are interested in computing
$\langle a_{\alpha}(S)\rangle$ for $\alpha$s such that
$\|\alpha^{k}\|_{1}=\sum_{x}\alpha^{k}_{x}$ are even, for all
$k\in\\{1,\ldots,N-1\\}$ (if any of these $\|\alpha^{k}\|_{1}$ is odd,
$\langle a_{\alpha}(S)\rangle=0$, due to the residual $O(N-1)$ symmetry in the
directions orthogonal to the magnetic field). The main result of this
subsection is the following:

######  Lemma 4.4.

Let $\alpha$ and $a_{\alpha}(S)$ be defined as above. Then there exist
coefficients $a_{\alpha,s}^{p,\tilde{p}}$, with $s\geq 0$ and
$p\in\mathfrak{P}$, $\tilde{p}\in\tilde{\mathfrak{P}}$ both even, such that,
for any $n\geq 0$,

$\langle
a_{\alpha}(S)\rangle=\sum_{s=0}^{n}T^{s}\sum_{p,\tilde{p}}a_{\alpha,s}^{p,\tilde{p}}\langle\phi^{p}\tilde{u}^{\tilde{p}}\rangle+R_{n+1}(\alpha),$
(41)

where the sum over $p,\tilde{p}$ in the right hand side runs over even tuples
in $\mathfrak{P},\tilde{\mathfrak{P}}$, $a_{\alpha,s}^{p,\tilde{p}}=0$ if
$\|p\|_{1}+\|\tilde{p}\|_{1}\neq 2s$, and the remainder satisfies

$|R_{n+1}(\alpha)|\leq C(n,\alpha)T^{n+1}$ (42)

for some $C(n,\alpha)>0$.

######  Remark 4.1.

The coefficients $a_{\alpha,s}^{p,\tilde{p}}$ are explicitly computable in
terms of the Taylor expansions of $\sqrt{1-x}$, $\sin x$ and $\cos x$, see
(45)-(46) below.

###### Proof.

We denote by $A_{\alpha}(\tilde{u},\phi)$ the rewriting of $a_{\alpha}(S)$ in
the $(\tilde{u},\phi)$ variables:

$A_{\alpha}(\tilde{u},\phi)=\prod_{x}\prod_{k}\Big{(}\Pi_{k}(\sqrt{T}\tilde{u}_{x},\sqrt{T}\phi_{x})\Big{)}^{\alpha_{x}^{k}},$
(43)

with $\Pi_{k}$ the components of the function $\Pi$ in (10). Letting
$E_{\alpha}$ be the event

$E_{\alpha}=\\{\tilde{u}\in(\sqrt{\beta}\mathbb{B}^{N-2})^{\mathbb{Z}^{d}}:\|\tilde{u}_{x}\|\leq\sqrt{\beta}/2,\forall
x\in\mathrm{supp}\,\alpha^{N-1}\cup\mathrm{supp}\,\alpha^{N}\\},$

and using (37), which implies that $\mu(E_{\alpha}^{c})=O(e^{-c\beta})$ for
some $c>0$, we find

$\langle
a_{\alpha}(S)\rangle=\langle\mathds{1}_{E_{\alpha}}\,A_{\alpha}(\tilde{u},\phi)\rangle+O(e^{-c\beta}).$
(44)

By expanding each factor $\sqrt{1-T\|\tilde{u}_{x}\|^{2}}$,
$\sin(\sqrt{T}\phi_{x})$, $\cos(\sqrt{T}\phi_{x})$ appearing in the right hand
side of (43) in power series in $T$, as

$\begin{split}&\sqrt{1-T\|\tilde{u}_{x}\|^{2}}\stackrel{{\scriptstyle(\infty)}}{{=}}\sum_{n\geq
0}T^{n}(-1)^{n}{1/2\choose n}\|\tilde{u}_{x}\|^{2n},\\\
&\sin(\sqrt{T}\phi_{x})=\sqrt{T}\phi_{x}\sum_{n\geq
0}T^{n}\frac{(-1)^{n}}{(2n+1)!}\phi_{x}^{2n},\\\
&\cos(\sqrt{T}\phi_{x})=\sum_{n\geq
0}T^{n}\frac{(-1)^{n}}{(2n)!}\phi_{x}^{2n},\end{split}$ (45)

we obtain the formal power series expansion for $A_{\alpha}(\tilde{u},\phi)$,
in the form

$A_{\alpha}(\tilde{u},\phi)\stackrel{{\scriptstyle(\infty)}}{{=}}\sum_{s\geq
0}T^{s}\sum_{p,\tilde{p}}a_{\alpha,s}^{p,\tilde{p}}\phi^{p}\tilde{u}^{\tilde{p}},$
(46)

for suitable real coefficients $a^{p,\tilde{p}}_{\alpha,s}$, which are non
zero only if $\|p\|_{1}+\|\tilde{p}\|_{1}=2s$ (the sum over $p,\tilde{p}$ runs
over even tuples in $\mathfrak{P},\tilde{\mathfrak{P}}$). By the Taylor
remainder’s theorem, for any $n\geq 0$, there exists $C_{n,\alpha}>0$ such
that, for any $\tilde{u}$ in the support of $E_{\alpha}$,

$\Big{|}A_{\alpha}(\tilde{u},\phi)-\sum_{s=0}^{n}T^{s}\sum_{p,\tilde{p}}a_{\alpha,s}^{p,\tilde{p}}\phi^{p}\tilde{u}^{\tilde{p}}\Big{|}\leq
C_{n,\alpha}T^{n+1}\sum_{x\in\mathrm{supp}\alpha}(\|\tilde{u}_{x}\|^{2(n+1)}+\phi_{x}^{2(n+1)}).$

Therefore, using again (37) and the moments bounds in the first two items of
Theorem 4.3, we obtain (41)-(42), as desired. ∎

##  5\. Integration by part

From now on, we shall assume $T\leq T_{0}$, with $T_{0}$ the same as in the
statement of Theorem 4.3, without recalling it each time. The main results of
this section are the following two integration-by-part lemmas, which are the
key ingredients of the inductive computation of the coefficients of the low
temperature expansion for the correlation functions, described in Section 6.

######  Lemma 5.1 (Integration by part: non-gradient).

Let $p,p^{\prime}\geq 0$ be integers. Let $F$ be a monomial in $\phi$ of
degree $2p+1$, and $\mathcal{F}$ be a monomial in $(\tilde{u}_{x}^{k})_{x,k}$
with even degree at most $2p^{\prime}$, both with coefficient $1$. Then, for
any $x_{0}\in\mathbb{Z}^{d}$, integers $\gamma>n\geq 0$, and $\epsilon>0$,

$\displaystyle\langle\phi_{x_{0}}F\mathcal{F}\rangle$
$\displaystyle=\sum_{y\in\mathrm{supp}_{F}}G(x_{0},y)\langle\partial_{y}F\mathcal{F}\rangle+$
$\displaystyle\ +\mathds{1}_{n\geq
1}\sum_{k=1}^{n}T^{k}\sum_{x_{1},e}\nabla_{x_{1}}^{\mathrm{e}}G^{m_{\gamma}}(x_{0},\cdot)\sum_{r+p^{\prime}=k}c^{\prime}_{r}\langle
F\mathcal{F}(\nabla_{x_{1}}^{e}\phi)^{2r+1}\mathcal{G}^{p^{\prime}}_{x_{1},e}\rangle$
$\displaystyle\ +R_{n+1}(F\mathcal{F},x_{0})$

where $m_{\gamma}=T^{\gamma}$, $c^{\prime}_{r}=\frac{(-1)^{r+1}}{(2r+1)!}$,
$\mathcal{G}^{p^{\prime}}_{x,e}$ was defined after (17), and
$|R_{n+1}(F\mathcal{F},x_{0})|\leq CT^{n+1-\epsilon}$ with $C$ depending only
on $\epsilon,\gamma,p+p^{\prime}$.

######  Lemma 5.2 (Integration by part: gradient).

Let $p^{\prime}\geq 0$ be an integer. Let $\mathcal{F}$ be a monomial in
$(\tilde{u}_{x}^{k})_{x,k}$ of even degree at most $2p^{\prime}$ with
coefficient $1$. Let $p:\mathbb{Z}^{d}\to\mathbb{Z}_{+}$ be odd, and let
$q:\mathbb{E}_{d}\to\mathbb{Z}_{+}$ be even, both of finite support. Then, for
any $x_{0}\in\mathbb{Z}^{d},e^{\prime}\in\mathrm{B}_{+}$, $\alpha>n\geq 0$
integers, and $\epsilon>0$,

$\displaystyle\langle(\nabla_{x_{0}}^{e^{\prime}}\phi)\phi^{p}(\nabla\phi)^{q}\mathcal{F}\rangle=\sum_{y\in\mathrm{supp}_{p}\cup\mathrm{supp}_{q}}\nabla_{x_{0}}^{e^{\prime}}G(\cdot,y)\langle\partial_{y}(\phi^{p}(\nabla\phi)^{q})\mathcal{F}\rangle+$
$\displaystyle\qquad+\mathds{1}_{n\geq
1}\sum_{k=1}^{n}T^{k}\sum_{x_{1},e}\nabla_{x_{0},x_{1}}^{e^{\prime},e}G^{m}\sum_{r+p^{\prime}=k}c^{\prime}_{r}\langle\phi^{p}(\nabla\phi)^{q}(\nabla_{x_{1}}^{e}\phi)^{2r+1}\mathcal{G}^{p^{\prime}}_{x_{1},e}\mathcal{F}\rangle$
$\displaystyle\qquad+R_{n+1}^{\prime}(p,q,\mathcal{F},x_{0},e^{\prime})$

where $m=e^{-(\log T)^{2}}$, and $R_{n+1}^{\prime}$ satisfies

$|R_{n+1}^{\prime}(p,q,\mathcal{F},x_{0},e^{\prime})|\leq
CT^{\alpha}+CT^{n+1-\epsilon}\sum_{x\in\mathrm{supp}_{p}}\sum_{y\in\mathrm{supp}_{q}\cup\\{x_{0}\\}}\frac{\log^{2}(1+|x-y|)}{1+|x-y|}$

with $C$ depending only on
$\epsilon,\alpha,\left\|p\right\|_{\scriptscriptstyle
1}+\left\|q\right\|_{\scriptscriptstyle 1}+p^{\prime},n$.

In both proofs, we will use Gaussian integration by parts with respect to a
regularized version of the Gaussian measure described in Section 3.1, the
regularization consisting in a mass term, which we add and subtract to the
Gaussian weight. The first term in both lemmas is the standard Gaussian
integration by parts, the second comes form the perturbation $W_{L;h}$ (see
Section 3.1) and the remainder term contains the effect of the regularization
(mass) and of the perturbation $\mathds{1}_{|\phi|\leq\sqrt{\beta}\pi}$ (see
again Section 3.1). The proofs of the two lemmas will use the following a
priori bounds on the decay of correlations, which will be proven in Appendix
B.

######  Lemma 5.3.

Let $\epsilon>0,\alpha>0$, $p^{\prime}\geq 0$,
$p,q:\mathbb{Z}^{d}\to\mathbb{Z}_{+}$ odd of finite support, and
$q^{\prime}:\mathbb{E}^{d}\to\mathbb{Z}_{+}$ odd of finite support. There
exists $C=C(\epsilon,\alpha,p^{\prime}+\left\|p\right\|_{\scriptscriptstyle
1}+\left\|q\right\|_{\scriptscriptstyle 1})$ and
$C^{\prime}=C(\epsilon,\alpha,p^{\prime}+\left\|p\right\|_{\scriptscriptstyle
1}+\left\|q^{\prime}\right\|_{\scriptscriptstyle 1})$ such that, for any
$T<T_{0}$, and any $\mathcal{F}$ even monomial in $\tilde{u}$ of degree at
most $2p^{\prime}$,

$\displaystyle|\langle\mathcal{F}\phi^{p}\phi^{q}\rangle|\leq
CT^{\alpha}+C\beta^{\epsilon}\sum_{\begin{subarray}{c}x:\ p_{x}\textnormal{
odd},\\\ y:\ q_{y}\textnormal{
odd}\end{subarray}}\frac{\log(1+|x-y|)}{1+|x-y|},$ (47)
$\displaystyle|\langle\mathcal{F}\phi^{p}(\nabla\phi)^{q^{\prime}}\rangle|\leq
C^{\prime}T^{\alpha}+C^{\prime}\beta^{\epsilon}\sum_{\begin{subarray}{c}x:\
p_{x}\textnormal{ odd},\\\
y\in\mathrm{supp}_{q^{\prime}}\end{subarray}}\frac{\log(1+|x-y|)}{1+|x-y|}.$
(48)

###### Proof of Lemma 5.1.

Let us start with $L$ finite and $h>0$, and let us represent
$\langle\phi_{x_{0}}F\mathcal{F}\rangle_{L;h}$ as in (20). For any $0<m<1$, we
rewrite the integral over $\phi$ in the numerator of (20) (in the case of our
observable of interest) as follows:

$\int\limits_{[-\sqrt{\beta}\pi,\sqrt{\beta}\pi]^{\Lambda_{L}}}\\!\\!\\!\\!d\phi\,e^{-\frac{1}{2}(\phi,(G_{L}^{m})^{-1}\phi)+\frac{m^{2}}{2}\|\phi\|^{2}+W_{L;h}(\tilde{u},\phi)}\phi_{x_{0}}F(\phi).$

If we now integrate $\phi_{x_{0}}$ by parts with respect to the reference
Gaussian weight $e^{-\frac{1}{2}(\phi,(G_{L}^{m})^{-1}\phi)}$, we obtain
(denote $\partial_{y}\equiv\frac{\partial}{\partial\phi_{y}}$):

$\displaystyle\langle\phi_{x_{0}}F\mathcal{F}\rangle_{L;h}$
$\displaystyle=\sum_{y\in\mathrm{supp}_{F}}G^{m}_{L}(x_{0},y)\langle\partial_{y}F\mathcal{F}\rangle_{L;h}+\sum_{z\in\Lambda_{L}}G^{m}_{L}(x_{0},z)\langle
F\partial_{z}W_{L;h}\mathcal{F}\rangle_{L;h}$
$\displaystyle\quad+m^{2}\sum_{z\in\Lambda_{L}}G^{m}_{L}(x_{0},z)\langle\phi_{z}F\mathcal{F}\rangle_{L;h}$
$\displaystyle\quad+\sum_{z\in\Lambda_{L}}G^{m}_{L}(x_{0},z)\langle(\delta(\phi_{z}+\sqrt{\beta}\pi)-\delta(\phi_{z}-\sqrt{\beta}\pi))F\mathcal{F}\rangle_{L;h}.$

Now, taking $L\to\infty$ for $h\in D^{*}$, and then $h\to 0^{+}$ along a
sequence in $D^{*}$, all the terms in the two sides converge to their infinite
volume, zero field, limits. Bounding the last term via the third item of
Theorem 4.3 and the Gaussian estimate (77), we obtain:

$\displaystyle\langle\phi_{x_{0}}F\mathcal{F}\rangle$
$\displaystyle=\sum_{y\in\mathrm{supp}_{F}}G^{m}(x_{0},y)\langle\partial_{y}F\mathcal{F}\rangle+\sum_{z\in\mathbb{Z}^{d}}G^{m}(x_{0},z)\langle
F\partial_{z}W\mathcal{F}\rangle$
$\displaystyle\quad+m^{2}\sum_{z\in\mathbb{Z}^{d}}G^{m}(x_{0},z)\langle\phi_{z}F\mathcal{F}\rangle+R_{\infty}(m,F\mathcal{F}),$

where in the second term in the right side we denoted by $W$ the formal
$L\to\infty$, $h\to 0$ limit of (16), and $|R_{\infty}(m,F\mathcal{F})|\leq
m^{-2}(C\beta)^{1/2+p+p^{\prime}}e^{-c{\beta}}$ for some $c,C>0$ independent
of $\beta$ and $m$ (as long as $T<T_{0}$). We denote the three other terms in
the R.H.S. $I,II,III$. Using (78) and the moment bounds in the first two items
of Theorem 4.3, we find

$I=\sum_{y\in\mathrm{supp}_{F}}G(x_{0},y)\langle\partial_{y}F\mathcal{F}\rangle+R_{I}(m,F\mathcal{F},x_{0})$

with $|R_{I}(m,F\mathcal{F},x_{0})|\leq C_{p+p^{\prime}}m$ for some
$C_{p+p^{\prime}}$ depending only on $p+p^{\prime}$. We then treat $III$ when
$m=m_{\gamma}=T^{\gamma}$. Using Lemma 5.3, one obtains that for any
$\epsilon>0$,

$|\langle\phi_{z}F\mathcal{F}\rangle|\leq
CT^{\gamma}+C\beta^{\epsilon/2}\sum_{y\in\mathrm{supp}_{F}}\frac{\log(1+|z-y|)}{1+|z-y|},$

with $C$ depending only on $p+p^{\prime},\epsilon,\gamma$. In particular,
since $\log(1+r)/(1+r)\leq C_{\epsilon^{\prime}}(1+r)^{\epsilon^{\prime}-1}$
for any $\epsilon^{\prime}>0$, one has that, by choosing
$\epsilon^{\prime}=\epsilon/(2\gamma)$ and using (77) and (81) (and
remembering that we set $m=T^{\gamma}$),

$|III|\leq CT^{\gamma-\epsilon}$

with $C$ depending only on $p+p^{\prime},\epsilon,\gamma$. It remains to treat
$II$. We first compute

$\displaystyle\sum_{z\in\mathbb{Z}^{d}}G^{m}(x_{0},z)\partial_{z}W$
$\displaystyle=\sum_{z\in\mathbb{Z}^{d}}G^{m}(x_{0},z)\partial_{z}\sum_{x\in\mathbb{Z}^{d}}\sum_{e}\Big{[}\beta\rho_{x}\rho_{x+e}\cos(\sqrt{T}\nabla_{x}^{e}\phi)+\frac{1}{2}(\nabla_{x}^{e}\phi)^{2}\Big{]}$
$\displaystyle=\sum_{x\in\mathbb{Z}^{d}}\sum_{e}\nabla_{x}^{e}G^{m}(x_{0},\cdot)\Big{[}-\rho_{x}\rho_{x+e}\sqrt{\beta}\sin(\sqrt{T}\nabla_{x}^{e}\phi)+\nabla_{x}^{e}\phi\Big{]},$

where $\rho_{x}=\sqrt{1-T\|\tilde{u}_{x}\|^{2}}$. Now plugging in
$m=m_{\gamma}=T^{\gamma}$, and proceeding as in the proof of Lemma 4.4 (i.e.,
in brief: Taylor expanding to order $2\gamma-1$ both the $\sin$ and the square
root in the $\rho_{x}\rho_{x+e}$ term, and using the bound on moments in
Theorem 4.3 combined with Hölder’s inequality), one finds that by the Gaussian
estimate on the sum of gradients (79),

$\Big{|}\sum_{k=1}^{2\gamma-1}T^{k}\sum_{x_{1},\mathrm{e}}\nabla_{x_{1}}^{\mathrm{e}}G^{m_{\gamma}}(x_{0},\cdot)\sum_{r+p^{\prime}=k}c^{\prime}_{r}\langle
F\mathcal{F}(\nabla_{x_{1}}^{\mathrm{e}}\phi)^{2r+1}\mathcal{G}^{p^{\prime}}_{x_{1},\mathrm{e}}\rangle-
II\Big{|}\leq CT^{\gamma},$

for some $C$ depending only on $\gamma+p+p^{\prime}$, where
$c^{\prime}_{r}=\frac{(-1)^{r+1}}{(2r+1)!}$,
$\mathcal{G}^{p^{\prime}}_{x,\mathrm{e}}$ was defined after (17), and
$\sum_{x_{1}}$ denotes summatin over $\mathbb{Z}^{d}$. Now, one can use Lemma
5.3 combined with (79) and (81) in the same fashion as before on the $k$s with
$n<k\leq 2\gamma-1$, to obtain that for any $\epsilon>0$ there exists $C$
depending on $\epsilon$ and $p+p^{\prime}+\gamma$ only,

$\Big{|}\sum_{k=1}^{n}T^{k}\sum_{x_{1},\mathrm{e}}\nabla_{x_{1}}^{\mathrm{e}}G^{m_{\gamma}}(x_{0},\cdot)\sum_{r+p^{\prime}=k}c^{\prime}_{r}\langle
F\mathcal{F}(\nabla_{x_{1}}^{\mathrm{e}}\phi)^{2r+1}\mathcal{G}^{p^{\prime}}_{x_{1},\mathrm{e}}\rangle-
II\Big{|}\leq CT^{n+1-\epsilon}.$

Gathering all the above, and letting $m=m_{\gamma}=T^{\gamma}$ also in the
error terms $R_{\infty}$ and $R_{I}$, gives the result. ∎

###### Proof of Lemma 5.2.

Write $F_{p,q}=\phi^{p}(\nabla\phi)^{q}$. Starting as in the proof of Lemma
5.1, for $m=e^{-(\log T)^{2}}$,

$\displaystyle\langle(\nabla_{x_{0}}^{\mathrm{e}^{\prime}}\phi)F_{p,q}\mathcal{F}\rangle$
$\displaystyle=\sum_{y\in\mathrm{supp}_{F_{p,q}}}\nabla_{x_{0}}^{\mathrm{e}^{\prime}}G^{m}(\cdot,y)\langle\partial_{y}F_{p,q}\mathcal{F}\rangle+\sum_{z\in\mathbb{Z}^{d}}\nabla_{x_{0}}^{\mathrm{e}^{\prime}}G^{m}(\cdot,z)\langle
F_{p,q}\partial_{z}W\mathcal{F}\rangle$
$\displaystyle\quad+m^{2}\sum_{z\in\mathbb{Z}^{d}}\nabla_{x_{0}}^{\mathrm{e}^{\prime}}G^{m}(\cdot,z)\langle\phi_{z}F_{p,q}\mathcal{F}\rangle+R_{\infty}(m,p,q,\mathcal{F},x_{0})$
$\displaystyle=I+II+III+R_{\infty}(m,p,q,\mathcal{F},x_{0}),$

with $|R_{\infty}(m,p,q,\mathcal{F},x_{0})|\leq
m^{-1}(C\beta)^{p^{\prime}+\frac{1}{2}(\left\|p\right\|_{\scriptscriptstyle
1}+\left\|q\right\|_{\scriptscriptstyle 1})}e^{-c{\beta}}$, for some $C,c>0$
independent of $\beta$ and $m$. Then, using (79), one has

$|III|\leq C_{p^{\prime}+\left\|p\right\|_{\scriptscriptstyle
1}+\left\|q\right\|_{\scriptscriptstyle 1}}m.$

Moreover, using (78) and Theorem 4.3, one has

$\big{|}I-\sum_{y\in\mathrm{supp}_{F_{p,q}}}\nabla_{x_{0}}^{\mathrm{e}^{\prime}}G(\cdot,y)\langle\partial_{y}F_{p,q}\mathcal{F}\rangle\big{|}\leq
C_{p^{\prime}+\left\|p\right\|_{\scriptscriptstyle
1}+\left\|q\right\|_{\scriptscriptstyle 1}}m.$

To finish the proof of Lemma 5.2, consider $II$ and proceed as in the proof of
Lemma 5.1 to obtain

$\Big{|}\sum_{k=1}^{\alpha}T^{k}\sum_{x_{1},\mathrm{e}}\nabla_{x_{0},x_{1}}^{\mathrm{e}^{\prime},\mathrm{e}}G^{m}\sum_{r+p^{\prime}=k}c^{\prime}_{r}\langle
F_{p,q}\mathcal{F}(\nabla_{x_{1}}^{\mathrm{e}}\phi)^{2r+1}\mathcal{G}^{p^{\prime}}_{x_{1},\mathrm{e}}\rangle-
II\Big{|}\leq CT^{\alpha+1}(\log T)^{2},$

where we used (80) and $m=e^{-(\log T)^{2}}$, and $C$ depends only on
$p^{\prime}+\left\|p\right\|_{\scriptscriptstyle
1}+\left\|q\right\|_{\scriptscriptstyle 1}$ and $\alpha$. We then apply Lemma
5.3 and (82) to obtain that, for any $r,p^{\prime}$ with
$r+p^{\prime}\leq\alpha$ and any $\epsilon>0$,

$\Big{|}\sum_{x_{1},\mathrm{e}}\nabla_{x_{0},x_{1}}^{\mathrm{e}^{\prime},\mathrm{e}}G^{m}\langle
F_{p,q}\mathcal{F}(\nabla_{x_{1}}^{\mathrm{e}}\phi)^{2r+1}\mathcal{G}^{p^{\prime}}_{x_{1},\mathrm{e}}\rangle\Big{|}\leq
CT^{\alpha}+C\beta^{\epsilon}\sum_{x\in\mathrm{supp}_{p}}\sum_{y\in\mathrm{supp}_{q}\cup\\{x_{0}\\}}\frac{\log^{2}(1+|x-y|)}{1+|x-y|}$

with $C$ depending only on $p^{\prime}+\left\|p\right\|_{\scriptscriptstyle
1}+\left\|q\right\|_{\scriptscriptstyle 1}$, $\alpha$, and $\epsilon$, and
$\sum_{x_{1}}$ denotes summation over $\mathbb{Z}^{d}$. This allows us to get
rid of the terms with $n+1\leq k\leq\alpha$. Combining all the previous
estimates gives the lemma. ∎

##  6\. Inductive expansion

In this section we prove of our main results, Theorem 1.3 and Corollaries 1.4
and 1.5, on the basis of an iterative application of the integration-by-parts
lemmas stated and proved in Section 5 and of the moment bounds summarized in
Theorem 4.3. We start with the proof of Corollary 1.4, which illustrates the
general method in a simple, non-trivial, case. The general case is proved by
induction, and discussed step-by-step in the following subsections. We recall
that, whenever we write $\sum_{x_{1}}$, we always mean
$\sum_{x_{1}\in\mathbb{Z}^{d}}$, and similarly for $\sum_{x_{i}}$, etc. As
already mentioned above, we explicitly focus on the case $N>2$, the case $N=2$
being significantly simpler, in that the $u$ variables are not present; the
interpretation and adaptation of the following discussion to the case $N=2$ is
left to the reader.

###  6.1. Magnetization to second order

We first present the proof of Corollary 1.4. Shortcuts are taken as the full
procedure will be described in the remaining of the section. First, rewrite
$S^{N}_{0}$ in terms of the coordinates $(\tilde{u}_{0},\phi_{0})$, so that

$\langle
S_{0}^{N}\rangle=\langle\sqrt{1-T\|\tilde{u}_{0}\|^{2}}\cos(\sqrt{T}\phi_{0})\rangle.$

We expand the function under the average in the right hand side in Taylor
series in $T$ up to second order and, applying Lemma 4.4 (see also Remark
4.1), we obtain

$\langle
S_{0}^{N}\rangle=1-\frac{1}{2}T\langle\phi_{0}^{2}\rangle-\frac{1}{2}T\langle\|\tilde{u}_{0}\|^{2}\rangle-\frac{1}{8}T^{2}\langle\|\tilde{u}_{0}\|^{4}\rangle+\frac{1}{4}T^{2}\langle\|\tilde{u}_{0}\|^{2}\phi_{0}^{2}\rangle+\frac{1}{24}T^{2}\langle\phi_{0}^{4}\rangle+O(T^{3}).$
(49)

Treat each term separately. First, using Lemma 5.1 with $n=0$,

$\begin{split}&\langle\phi_{0}^{4}\rangle=3G(0,0)\langle\phi_{0}^{2}\rangle+O(T^{1-\epsilon})=3G(0,0)^{2}+O(T^{1-\epsilon}),\\\
&\langle\|\tilde{u}_{0}\|^{2}\phi_{0}^{2}\rangle=G(0,0)\langle\|\tilde{u}_{0}\|^{2}\rangle+O(T^{1-\epsilon}).\end{split}$
(50)

Then, using the leftover $O(N-1)$ symmetry of the model in the subspace
orthogonal to the magnetic field, eq.(18), the moment bounds of Theorem 4.3,
and applying Lemma 5.1 once more, we obtain

$\begin{split}\langle\|\tilde{u}_{0}\|^{2}\rangle&=(N-2)\big{(}\langle\phi_{0}^{2}\rangle-\frac{T}{3}\langle\phi_{0}^{4}\rangle-T\langle\|\tilde{u}_{0}\|^{2}\phi_{0}^{2}\rangle\big{)}+O(T^{2})\\\
&=(N-2)\big{(}\langle\phi_{0}^{2}\rangle-
TG(0,0)^{2}-TG(0,0)^{2}(N-2)\big{)}+O(T^{2-\epsilon}),\end{split}$ (51)

and

$\begin{split}\langle\|\tilde{u}_{0}\|^{4}\rangle&=(N-2)\langle\phi_{0}^{4}\rangle+(N-2)(N-3)\langle(\tilde{u}_{0}^{1})^{2}\phi_{0}^{2}\rangle+O(T)\\\
&=(N-2)3G(0,0)^{2}+(N-2)(N-3)G(0,0)^{2}+O(T^{1-\epsilon}).\end{split}$ (52)

So far we used: (1) the leftover $O(N-1)$ symmetry in the directions
orthogonal to the magnetic field, in order to generate $\phi$s when only
$\tilde{u}$s remain; (2) Lemma 5.1, but only in its “trivial” form ($n=0$). We
then need to generate the expansion of $\langle\phi_{0}^{2}\rangle$ to first
order. This is where the controlled inductive procedure starts to show up.
Using Lemma 5.1 with $n=1$ (and $\gamma=2$), we obtain

$\begin{split}&\langle\phi_{0}^{2}\rangle=G(0,0)+O(T^{2-\epsilon})\\\
&\quad+T\sum_{x_{1},e}\nabla_{x_{1}}^{\mathrm{e}}G^{T^{2}}(0,\cdot)\Big{(}\frac{1}{3!}\langle\phi_{0}(\nabla_{x_{1}}^{e}\phi)^{3}\rangle+\frac{1}{2}\langle\phi_{0}\nabla_{x_{1}}^{e}\phi(\|\tilde{u}_{x_{1}}\|^{2}+\|\tilde{u}_{x_{1}+e}\|^{2})\rangle\Big{)}.\end{split}$
(53)

One can then use Lemma 5.2 with $n=0$ on the terms inside the sum to obtain

$\begin{split}&\langle\phi_{0}(\nabla_{x_{1}}^{e}\phi)^{3}\rangle=\nabla_{x_{1}}^{e}G(\cdot,0)\langle(\nabla_{x_{1}}^{e}\phi)^{2}\rangle+2\nabla^{e,e}_{x_{1},x_{1}}G\langle\phi_{0}\nabla_{x_{1}}^{e}\phi\rangle+R_{1}(x_{1})\\\
&\langle\phi_{0}\nabla_{x_{1}}^{e}\phi(\|\tilde{u}_{x_{1}}\|^{2}+\|\tilde{u}_{x_{1}+e}\|^{2})\rangle=\nabla_{x_{1}}^{e}G(\cdot,0)\big{(}\langle\|\tilde{u}_{x_{1}}\|^{2}\rangle+\langle\|\tilde{u}_{x_{1}+e}\|^{2}\rangle\big{)}+R_{2}(x_{1})\end{split}$
(54)

with $R_{i}(x_{1})$ satisfying, for $i=1,2$,

$|R_{i}(x_{1})|\leq
CT^{3}+CT^{1-\epsilon}\frac{\log^{2}(1+|x_{1}|)}{1+|x_{1}|}.$ (55)

Using again Lemmas 5.1 and 5.2 with $n=0$ and the residual $O(N-1)$ symmetry
in the directions orthogonal to the magnetic field, in (54) we can rewrite

$\begin{split}&\langle(\nabla_{x_{1}}^{e}\phi)^{2}\rangle=\nabla^{e,e}_{x_{1},x_{1}}G+O(T^{1-\epsilon}),\\\
&\langle\phi_{0}\nabla_{x_{1}}^{e}\phi\rangle=\nabla^{e}_{x_{1}}G(0,\cdot)+R_{3}(x_{1}),\\\
&\langle\|\tilde{u}_{x_{1}}\|^{2}\rangle=\langle\|\tilde{u}_{x_{1}+e}\|^{2}\rangle=(N-2)G(0,0)+O(T^{1-\epsilon}),\end{split}$
(56)

with $R_{3}(x_{1})$ bounded as in (55). Plugging (56) in (54), and then the
resulting expressions in (LABEL:def), we find that, thanks to (79) and (81),
the contributions from the terms involving the error terms $R_{i}(x_{1})$ and
$O(T^{1-\epsilon})$ to the second line of (LABEL:def) can all bounded by
$CT^{2-\epsilon}$, for any $\epsilon>0$ and some $C>0$ depending on
$\epsilon$, but independent of $T$. Therefore, we have

$\begin{split}&\langle\phi_{0}^{2}\rangle=G(0,0)+O(T^{2-\epsilon})\\\
&\quad+T\sum_{x_{1},e}\nabla_{x_{1}}^{e}G^{T^{2}}(0,\cdot)\Big{(}\frac{1}{2}\nabla_{x_{1}}^{e}G(\cdot,0)\nabla^{e,e}_{x_{1},x_{1}}G+(N-2)\nabla_{x_{1}}^{e}G(\cdot,0)G(0,0)\Big{)}.\end{split}$
(57)

Finally, noting that, for any $\epsilon>0$,
$\big{|}\nabla_{x_{1}}^{e}G^{T^{2}}(0,\cdot)-\nabla_{x_{1}}^{e}G(0,\cdot)\big{|}\leq
CT^{2-2\frac{1+\epsilon}{d-1}}\frac{1}{(1+|x_{1}|)^{1+\epsilon}}$ (which
follows from the second bound in Theorem C.1 and from the use of Lemma C.3
with $m=T^{2}$, $\gamma=d-1$ and $\gamma^{\prime}=1+\epsilon$), we can remove
the mass $T^{2}$ from the massive propagator up to an additional error of
order $O(T^{2+\frac{d-3}{d-1}-\frac{2\epsilon}{d-1}})$. In conclusion, for any
$\epsilon>0$,

$\begin{split}\langle
S_{0}^{N}\rangle&=1-\frac{T}{2}(N-1)G(0,0)+\frac{T^{2}}{8}(N-1)(3N-5)G(0,0)^{2}\\\
&-\frac{T^{2}}{2}(N-1)\sum_{x_{1},e}\big{(}\nabla_{x_{1}}^{e}G(0,\cdot)\big{)}^{2}\Big{(}\frac{1}{2}\nabla_{x_{1},x_{1}}^{e,e}G+(N-2)G(0,0)\Big{)}+O(T^{3-\epsilon}),\end{split}$

which concludes the proof of Corollary 1.4.

In the incoming subsections we will apply the same ideas used here in order to
compute at a generic order in the low temperature expansion the correlation
functions of the model. More precisely, we will inductively prove that, for
any $p\in\mathfrak{P}$, $\tilde{p}\in\tilde{\mathfrak{P}}$, $n\geq 0$,
$\epsilon>0$,

$\langle\phi^{p}\tilde{u}^{\tilde{p}}\rangle=\langle\phi^{p}\tilde{u}^{\tilde{p}}\rangle^{(n)}+O(T^{n+1-\epsilon}),$
(58)

where $\langle\ \rangle^{(n)}$ was defined in Definition 3.1. In view of Lemma
4.4, (58) implies our main result, Theorem 1.3, with an explicit bound on the
remainder and an explicit procedure for computing the coefficients $a_{i}$.

###  6.2. Inductive hypotheses

######  Definition 6.1 (Inductive hypotheses $\mathcal{H}_{M,K}$).

We say that $\mathcal{H}_{M,K}$ holds if for any $0<\epsilon<1$, and any
integer $\alpha>M$, the two following hold:

  * $\mathcal{H}_{M,K}^{1}$:

There exists $C=C(\epsilon,K,M)$ such that: for any integer $0\leq n\leq M$,
$p:\mathbb{Z}^{d}\to\mathbb{Z}_{+}$ even,
$\tilde{p}:\mathbb{Z}^{d}\times[N-2]$ even with
$\frac{\left\|\tilde{p}\right\|_{\scriptscriptstyle
1}+\left\|p\right\|_{\scriptscriptstyle 1}}{2}\leq K$,

$\big{|}\langle\phi^{p}\tilde{u}^{\tilde{p}}\rangle-\langle\phi^{p}\tilde{u}^{\tilde{p}}\rangle^{(n)}\big{|}\leq
CT^{n+1-\epsilon}.$ (59)

  * $\mathcal{H}_{M,K}^{2}$:

There exists $C^{\prime}=C^{\prime}(\epsilon,K,M,\alpha)$ and $c_{K}$ such
that: for any integer $0\leq n\leq M$, even
$\tilde{p}:\mathbb{Z}^{d}\times[N-2]\to\mathbb{Z}_{+}$, odd
$p:\mathbb{Z}^{d}\to\mathbb{Z}_{+}$, and odd
$q:\mathbb{E}_{d}\to\mathbb{Z}_{+}$ with
$\frac{\left\|\tilde{p}\right\|_{\scriptscriptstyle
1}+\left\|p\right\|_{\scriptscriptstyle
1}+\left\|q\right\|_{\scriptscriptstyle 1}}{2}\leq K$,

$\big{|}\langle\phi^{p}\tilde{u}^{\tilde{p}}(\nabla\phi)^{q}\rangle-\langle\phi^{p}\tilde{u}^{\tilde{p}}(\nabla\phi)^{q}\rangle^{(n)}\big{|}\leq\\\
\leq
C^{\prime}T^{\alpha}+C^{\prime}T^{n+1-\epsilon}\sum_{x\in\mathrm{supp}_{p}}\sum_{y\in\mathrm{supp}_{q}}\frac{\log^{2+n}(1+|x-y|)}{1+|x-y|}.$
(60)

We say that $\mathcal{H}_{M,\infty}$ holds if $\mathcal{H}_{M,K}$ holds for
every $K\geq 0$, and that $\mathcal{H}_{\infty,K}$ holds if
$\mathcal{H}_{M,K}$ holds for every $M\geq 0$.

######  Remark 6.1.

By linearity, $\mathcal{H}_{M,K}$ implies a control over polynomials in
$\tilde{u},\phi,\nabla\phi$. The constants in the error terms depend then on
the coefficients of the polynomial.

######  Remark 6.2.

We automatically have the validity of $\mathcal{H}_{\infty,0}$.

To lighten notation, we will not recall the domain and image of
$p,q,\tilde{p}$. The main result of this section is

######  Theorem 6.1.

$\mathcal{H}_{M,K}$ holds for any $M,K\geq 0$.

The proof of Theorem 6.1 will occupy the remainder of this section. Note that
what we are in fact only interested in is the validity of
$\mathcal{H}_{M,K}^{1}$. $\mathcal{H}_{M,K}^{2}$ is only needed for technical
reasons (which will be highlighted when proving the induction step for
$\mathcal{H}_{M,K}^{1}$).

###  6.3. Initiating the procedure

######  Lemma 6.2.

Suppose $\mathcal{H}_{0,K}$ holds. Then, $\mathcal{H}_{0,K+1}$ holds.

###### Proof.

Suppose $\mathcal{H}_{0,K}$ holds. Let $0<\epsilon<1,\alpha>0$. We prove
$\mathcal{H}_{0,K+1}^{2}$ and then $\mathcal{H}_{0,K+1}^{1}$.

####  $\mathcal{H}_{0,K+1}^{2}$ part.

Let $p$ be odd, $q=\mathds{1}_{(x_{0},e_{0})}+\tilde{q}$ with $\tilde{q}$
even, and $\tilde{p}$ even with
$\frac{\left\|\tilde{p}\right\|_{\scriptscriptstyle
1}+\left\|p\right\|_{\scriptscriptstyle
1}+\left\|q\right\|_{\scriptscriptstyle 1}}{2}\leq K+1$. Using Lemma 5.2 with
$n=0$, one obtains

$\big{|}\langle\phi^{p}(\nabla\phi)^{q}\tilde{u}^{\tilde{p}}\rangle-\sum_{y\in\mathrm{supp}_{p}\cup\mathrm{supp}_{\tilde{q}}}\nabla_{x_{0}}^{e_{0}}G(\cdot,y)\langle\partial_{y}(\phi^{p}(\nabla\phi)^{\tilde{q}})\tilde{u}^{\tilde{p}}\rangle\big{|}\leq\\\
\leq
CT^{\alpha}+CT^{1-\epsilon}\sum_{x\in\mathrm{supp}_{p}}\sum_{y\in\mathrm{supp}_{q}}\frac{\log^{2}(1+|x-y|)}{1+|x-y|}$

for some $C=C(\epsilon,\alpha,K+1)$. We can then use $\mathcal{H}_{0,K}^{1}$
to obtain

$|\langle(\nabla\phi)^{\tilde{q}}\partial_{y}\phi^{p}\tilde{u}^{\tilde{p}}\rangle-\langle(\nabla\phi)^{\tilde{q}}\partial_{y}\phi^{p}\tilde{u}^{\tilde{p}}\rangle^{(0)}|\leq
C^{\prime}T^{1-\epsilon}$

for some $C^{\prime}=C^{\prime}(\epsilon,K)$. So,

$\big{|}\sum_{y\in\mathrm{supp}_{p}}\nabla_{x_{0}}^{e_{0}}G(\cdot,y)\big{(}\langle(\nabla\phi)^{\tilde{q}}\partial_{y}\phi^{p}\tilde{u}^{\tilde{p}}\rangle-\langle(\nabla\phi)^{\tilde{q}}\partial_{y}\phi^{p}\tilde{u}^{\tilde{p}}\rangle^{(0)}\big{)}\big{|}\leq\\\
\leq
C^{\prime}T^{1-\epsilon}\sum_{y\in\mathrm{supp}_{p}}\frac{c}{(1+|x_{0}-y|)^{d-1}},$

where we used the decay of $\nabla_{x_{0}}^{e}G(\cdot,y)$, see the second
inequality in Theorem C.1. Now,

$\sum_{y}\nabla_{x_{0}}^{e_{0}}G(\cdot,y)\langle\phi^{p}\partial_{y}(\nabla\phi)^{\tilde{q}}\tilde{u}^{\tilde{p}}\rangle=\sum_{z,e}\tilde{q}_{z,e}\nabla_{x_{0},z}^{e_{0},e}G\langle\phi^{p}(\nabla\phi)^{\tilde{q}-\mathds{1}_{(z,e)}}\tilde{u}^{\tilde{p}}\rangle.$

We can then use $\mathcal{H}_{0,K}^{2}$, the uniform boundedness of
$|\nabla_{x_{0},z}^{e_{0},e}G|$, and the previous estimates to obtain

$|\langle\phi^{p}(\nabla\phi)^{q}\tilde{u}^{\tilde{p}}\rangle-\sum_{y\in\mathrm{supp}_{p}\cup\mathrm{supp}_{\tilde{q}}}\nabla_{x_{0}}^{e_{0}}G(\cdot,y)\langle\partial_{y}(\phi^{p}(\nabla\phi)^{\tilde{q}})\tilde{u}^{\tilde{p}}\rangle^{(0)}|\leq\\\
\leq
C^{\prime\prime}T^{\alpha}+C^{\prime\prime}T^{1-\epsilon}\sum_{x\in\mathrm{supp}_{p}}\sum_{y\in\mathrm{supp}_{q}}\frac{\log^{2}(1+|x-y|)}{1+|x-y|},$

with $C^{\prime\prime}=C^{\prime\prime}(\epsilon,\alpha,K+1)$. Finally, using
the extraction property (32),

$\displaystyle\langle\phi^{p}(\nabla\phi)^{q}\tilde{u}^{\tilde{p}}\rangle^{(0)}$
$\displaystyle=\Phi(\phi^{p}(\nabla\phi)^{q})\langle\tilde{u}^{\tilde{p}}\rangle^{(0)}$
$\displaystyle=\sum_{y}\nabla_{x_{0}}^{e_{0}}G(\cdot,y)\Phi\big{(}\partial_{y}(\phi^{p}(\nabla\phi)^{\tilde{q}})\big{)}\langle\tilde{u}^{\tilde{p}}\rangle^{(0)}$
$\displaystyle=\sum_{y}\nabla_{x_{0}}^{e_{0}}G(\cdot,y)\langle\partial_{y}(\phi^{p}(\nabla\phi)^{\tilde{q}})\tilde{u}^{\tilde{p}}\rangle^{(0)}$

where we used Gaussian integration by part in the second line. This concludes
the proof of $\mathcal{H}_{0,K+1}^{2}$.

####  $\mathcal{H}_{0,K+1}^{1}$ part.

Start with the case $p\neq 0$. Then, $p=\mathds{1}_{x_{0}}+p^{\prime}$ with
$p^{\prime}$ odd. Using Lemma 5.1 with $n=0$,

$\big{|}\langle\phi^{p}\tilde{u}^{\tilde{p}}\rangle-\sum_{y}G(x_{0},y)\langle\partial_{y}\phi^{p^{\prime}}\tilde{u}^{\tilde{p}}\rangle\big{|}\leq
CT^{1-\epsilon}$

with $C=C(\epsilon,K+1)$. Using then $\mathcal{H}_{0,K}^{1}$ and the
extraction property (32) as before, one obtains the wanted claim. It remains
to treat the $p=0$ case. In that case, there exists $k\in\\{1,\cdots,N-2\\}$
such that $\tilde{p}^{k}$ is non-zero (and even). Denote
$\bar{p}=(\tilde{p}^{1},\ldots,\tilde{p}^{k-1},0,\tilde{p}^{k+1},\ldots,\tilde{p}^{N-2})$.
Using the residual $O(N-1)$ symmetry in the directions $\\{1,\ldots,N-1\\}$,
eq.(30), we get

$\langle\tilde{u}^{\tilde{p}}\rangle=\beta^{\|{\tilde{p}^{k}}\|/2}\langle\prod_{x}\Big{(}\sqrt{1-T\|\tilde{u}_{x}\|^{2}}\sin\big(\sqrt{T}\phi_{x}\big{missing})\Big{)}^{\tilde{p}^{k}_{x}}\tilde{u}^{\bar{p}}\rangle.$
(61)

Now, we apply Lemma 4.4 with $n=\|{\tilde{p}^{k}}\|/2$ to the right hand side,
thus getting

$\langle\tilde{u}^{\tilde{p}}\rangle=\langle\phi^{\tilde{p}^{k}}\tilde{u}^{\bar{p}}\rangle+R_{1}(\tilde{p}),$

with $|R_{1}(\tilde{p})|\leq CT$, $C=C(K+1)$. The rest is the same as in the
$p\neq 0$ case. This conclude the proof of $\mathcal{H}_{0,K+1}^{1}$. ∎

######  Remark 6.3.

From Lemma 6.2 and the validity of $\mathcal{H}_{0,0}$, one obtains the
validity of $\mathcal{H}_{0,\infty}$.

###  6.4. Induction step

To keep notation readable, we introduce the shorthands

$f_{i}\equiv(\nabla_{x_{i}}^{e_{i}}\phi)^{2r_{i}+2},\quad
g_{i}\equiv\mathcal{G}_{x_{i},e_{i}}^{p_{i}^{\prime}},\quad f_{A}=\prod_{i\in
A}f_{i},\quad g_{A}=\prod_{i\in A}g_{i}.$

We will use a simple identity on connected correlation of Gaussian:

######  Claim 1.

For any $F$ monomial in $\phi$ of odd degree,

$\Phi(\phi_{z}F;f_{1};\cdots;f_{k})=\sum_{y}G(z,y)\Phi(\partial_{y}F;f_{1};\cdots;f_{k})+\\\
+\sum_{i=1}^{k}(2r_{i}+2)\nabla_{x_{i}}^{e_{i}}G(z,\cdot)\Phi\big{(}F(\nabla_{x_{i}}^{e_{i}}\phi)^{2r_{i}+1};f_{1};\cdots;f_{i-1};f_{i+1};\cdots;f_{k}\big{)}.$
(62)

###### Proof.

It is a simple consequence of the sum-over-partitions formula (25) for
cumulants and of Gaussian integration by part. Letting $A_{0}$ be the element
of $\pi\in\mathcal{P}(\\{0,1,\ldots,k\\})$ containing $0$, we have

$\displaystyle\Phi(\phi_{z}F;f_{1};\cdots;f_{k})=\sum_{\pi\in\mathcal{P}(\\{0,\cdots,k\\})}(|\pi|-1)!(-1)^{|\pi|-1}\Big{(}\prod_{B\in\pi,0\notin
B}\Phi(f_{B})\Big{)}\Phi(\phi_{z}Ff_{A_{0}\setminus 0})$
$\displaystyle\quad=\sum_{\pi\in\mathcal{P}_{k}}(|\pi|-1)!(-1)^{|\pi|-1}\sum_{A\in\pi}\Big{(}\prod_{B\in\pi,B\neq
A}\Phi(f_{B})\Big{)}\big{[}\Phi(\phi_{z}Ff_{A})-\Phi(\phi_{z}F)\Phi(f_{A})\big{]}.$

Integrating by parts $\phi_{z}$ in the two terms in brackets gives

$\displaystyle\Phi(\phi_{z}F;f_{1};\cdots;f_{k})=\sum_{\pi\in\mathcal{P}_{k}}(|\pi|-1)!(-1)^{|\pi|-1}\sum_{A\in\pi}\Big{(}\prod_{B\in\pi,B\neq
A}\Phi(f_{B})\Big{)}\cdot$ $\displaystyle\qquad\cdot\Big{[}\sum_{i\in
A}(2r_{i}+2)\Phi(\phi_{z}\nabla_{x_{i}}^{e_{i}}\phi)\Phi\big{(}F(\nabla_{x_{i}}^{e_{i}}\phi)^{2r_{i}+1}f_{A\setminus
i}\big{)}+\sum_{y}G(z,y)\Phi(\partial_{y}F;f_{A})\Big{]}$
$\displaystyle\quad=\sum_{i=1}^{k}(2r_{i}+2)\Phi(\phi_{z}\nabla_{x_{i}}^{e_{i}}\phi)\Phi\big{(}F(\nabla_{x_{i}}^{e_{i}}\phi)^{2r_{i}+1};f_{1};\cdots;f_{i-1};f_{i+1};\cdots;f_{k}\big{)}+$
$\displaystyle\qquad+\sum_{y}G(z,y)\Phi(\partial_{y}F;f_{1};\cdots;f_{k}),$

as desired. ∎

######  Lemma 6.3.

Suppose $\mathcal{H}_{M+1,K}$ and $\mathcal{H}_{M,\infty}$ hold. Then,
$\mathcal{H}_{M+1,K+1}$ holds.

###### Proof.

Let $\epsilon>0,\alpha>M+1$. We only have to consider $n=M+1$. We again treat
first $\mathcal{H}_{M+1,K+1}^{2}$ and then prove $\mathcal{H}_{M+1,K+1}^{1}$.

####  $\mathcal{H}_{M+1,K+1}^{2}$ part.

As $q$ is odd, there exist $x_{0}\in\mathbb{Z}^{d}$, $e_{0}\in\mathrm{B}_{+}$
such that $q=\mathds{1}_{(x_{0},e_{0})}+\tilde{q}$ with $\tilde{q}$ even.
Then, by Lemma 5.2,

$\Big{|}\langle\nabla_{x_{0}}^{\mathrm{e}_{0}}\phi\phi^{p}(\nabla\phi)^{\tilde{q}}\tilde{u}^{\tilde{p}}\rangle-\sum_{y}\nabla_{x_{0}}^{e_{0}}G(\cdot,y)\langle\partial_{y}(\phi^{p}(\nabla\phi)^{\tilde{q}})\tilde{u}^{\tilde{p}}\rangle\\\
-\sum_{k=1}^{M+1}T^{k}\sum_{x_{1},e_{1}}\nabla_{x_{0},x_{1}}^{e_{0},e_{1}}G^{m}\sum_{r_{1}+p^{\prime}_{1}=k}c^{\prime}_{r_{1}}\langle(\nabla_{x_{1}}^{\mathrm{e}_{1}}\phi)^{2r_{1}+1}\phi^{p}(\nabla\phi)^{\tilde{q}}\tilde{u}^{\tilde{p}}g_{1}\rangle\Big{|}\leq\\\
\leq
CT^{\alpha}+CT^{M+2-\epsilon}\sum_{x\in\mathrm{supp}_{p}}\sum_{y\in\mathrm{supp}_{q}}\frac{\log^{2}(1+|x-y|)}{1+|x-y|},$
(63)

with $C=C(\epsilon,\alpha,K+1,M+1)$, and $m=e^{-(\log T)^{2}}$. We can then
use $\mathcal{H}_{M+1,K}^{1}$ to obtain

$\big{|}\langle(\nabla\phi)^{\tilde{q}}\partial_{y}\phi^{p}\tilde{u}^{\tilde{p}}\rangle-\langle(\nabla\phi)^{\tilde{q}}\partial_{y}\phi^{p}\tilde{u}^{\tilde{p}}\rangle^{(M+1)}\big{|}\leq
C^{\prime}T^{M+2-\epsilon},$

with $C^{\prime}=C^{\prime}(\epsilon,K,M+1)$. So, using the decay of
$|\nabla_{x_{0}}^{e_{0}}G(\cdot,y)|$ (see the second bound of Theorem C.1),

$\Big{|}\sum_{y}\nabla_{x_{0}}^{e_{0}}G(\cdot,y)\big{[}\langle(\nabla\phi)^{\tilde{q}}\partial_{y}\phi^{p}\tilde{u}^{\tilde{p}}\rangle-\langle(\nabla\phi)^{\tilde{q}}\partial_{y}\phi^{p}\tilde{u}^{\tilde{p}}\rangle^{(M+1)}\big{]}\Big{|}\leq
C^{\prime}T^{M+2-\epsilon}\sum_{y\in\mathrm{supp}_{p}}\frac{c_{d}}{(1+|x_{0}-y|)^{d-1}}$

with $c_{d}$ depending only on $d$. Next,

$\sum_{y}\nabla_{x_{0}}^{e_{0}}G(\cdot,y)\langle\phi^{p}\partial_{y}(\nabla\phi)^{\tilde{q}}\tilde{u}^{\tilde{p}}\rangle=\sum_{z,e}\tilde{q}_{z,e}\nabla_{x_{0},z}^{e_{0},e}G\langle\phi^{p}(\nabla\phi)^{\tilde{q}-\mathds{1}_{(z,\mathrm{e})}}\tilde{u}^{\tilde{p}}\rangle.$

So, by the uniform boundedness of $|\nabla_{x_{0},z}^{e_{0},e}G|$, and
$\mathcal{H}_{M+1,K}^{2}$,

$\Big{|}\sum_{y}\nabla_{x_{0}}^{e_{0}}G(\cdot,y)\big{[}\langle\phi^{p}\partial_{y}(\nabla\phi)^{\tilde{q}}\tilde{u}^{\tilde{p}}\rangle-\langle\phi^{p}\partial_{y}(\nabla\phi)^{\tilde{q}}\tilde{u}^{\tilde{p}}\rangle^{(M+1)}\big{]}\Big{|}\leq\\\
\leq
C^{\prime\prime}T^{\alpha}+C^{\prime\prime}T^{M+2-\epsilon}\sum_{y\in\mathrm{supp}_{\tilde{q}}}\sum_{x\in\mathrm{supp}_{p}}\frac{\log^{M+3}(1+|x-y|)}{1+|x-y|}$

with $C^{\prime\prime}=C^{\prime\prime}(\epsilon,\alpha,K+1,M+1)$. Finally,
using $\mathcal{H}_{M,\infty}^{2}$, there exists
$C^{\prime\prime\prime}=C^{\prime\prime\prime}(\epsilon,\alpha,K+1,M+1)$ such
that for $1\leq k\leq M$,

$\big{|}\langle(\nabla_{x_{1}}^{e_{1}}\phi)^{2r_{1}+1}\phi^{p}(\nabla\phi)^{\tilde{q}}\tilde{u}^{\tilde{p}}g_{1}\rangle-\langle(\nabla_{x_{1}}^{e_{1}}\phi)^{2r_{1}+1}\phi^{p}(\nabla\phi)^{\tilde{q}}\tilde{u}^{\tilde{p}}g_{1}\rangle^{(M+1-k)}\big{|}\leq\\\
\leq
C^{\prime\prime\prime}T^{\alpha}+C^{\prime\prime\prime}T^{M+2-k-\epsilon}\sum_{x\in\mathrm{supp}_{p}}\sum_{y\in\mathrm{supp}_{\tilde{q}}\cup\\{x_{1}\\}}\frac{\log^{M+3-k}(1+|x-y|)}{1+|x-y|}.$

Plugging all the estimates in (63), using triangular inequality, and using
Theorem C.2, more precisely (80) and (82), we obtain

$\begin{split}&\Big{|}\langle\nabla_{x_{0}}^{e_{0}}\phi\phi^{p}(\nabla\phi)^{\tilde{q}}\tilde{u}^{\tilde{p}}\rangle-\sum_{y}\nabla_{x_{0}}^{e_{0}}G(\cdot,y)\langle\partial_{y}(\phi^{p}(\nabla\phi)^{\tilde{q}})\tilde{u}^{\tilde{p}}\rangle^{(M+1)}\\\
&-\sum_{k=1}^{M+1}T^{k}\sum_{x_{1},e_{1}}\nabla_{x_{0},x_{1}}^{e_{0},e_{1}}G^{m}\sum_{r_{1}+p^{\prime}_{1}=k}c^{\prime}_{r_{1}}\langle(\nabla_{x_{1}}^{e_{1}}\phi)^{2r_{1}+1}\phi^{p}(\nabla\phi)^{\tilde{q}}\tilde{u}^{\tilde{p}}g_{1}\rangle^{(M+1-k)}\Big{|}\leq\\\
&\leq\tilde{C}T^{\alpha}+\tilde{C}T^{M+2-\epsilon}(\log
T)^{2}\sum_{x\in\mathrm{supp}_{p}}\sum_{y\in\mathrm{supp}_{q}}\frac{\log^{M+3}(1+|x-y|)}{1+|x-y|},\end{split}$
(64)

with $\tilde{C}=\tilde{C}(\epsilon,\alpha,K+1,M+1)$. If we now use the bound

$|\nabla_{x_{0},x_{1}}^{e_{0},e_{1}}G-\nabla_{x_{0},x_{1}}^{e_{0},e_{1}}G^{m}|\leq
Cm^{1/(2d)}(1+|x_{1}-x_{0}|)^{-d+1/2}$

following from Theorems C.2 and C.1 and Lemma C.3, the bound

$|\langle(\nabla_{x_{1}}^{e_{1}}\phi)^{2r_{1}+1}\phi^{p}(\nabla\phi)^{\tilde{q}}\tilde{u}^{\tilde{p}}g_{1}\rangle^{(M+1-k)}|\leq\sum_{y\in\mathrm{supp}_{p}\,\cup\,\mathrm{supp}_{\tilde{q}}}(1+|x_{1}-y|)^{-d+1+\epsilon}$

following from Lemma 3.2, and the summability result of Lemma C.4, we see that
in the second line of (64) we can replace the massive propagator
$\nabla_{x_{0},x_{1}}^{e_{0},e_{1}}G^{m}$ by
$\nabla_{x_{0},x_{1}}^{e_{0},e_{1}}G$ up to an error term $\leq CTm^{1/(2d)}$,
which can be re-absorbed in $\tilde{C}T^{\alpha}$ up to a redefinition of
$\tilde{C}$. It remains to prove that

$\sum_{y}\nabla_{x_{0}}^{e_{0}}G(\cdot,y)\langle\partial_{y}(\phi^{p}(\nabla\phi)^{\tilde{q}})\tilde{u}^{\tilde{p}}\rangle^{(M+1)}+\sum_{k=1}^{M+1}T^{k}\sum_{x_{1},e_{1}}\nabla_{x_{0},x_{1}}^{e_{0},e_{1}}G\
\cdot\\\
\cdot\sum_{r_{1}+p^{\prime}_{1}=k}c^{\prime}_{r_{1}}\langle(\nabla_{x_{1}}^{e_{1}}\phi)^{2r_{1}+1}\phi^{p}(\nabla\phi)^{\tilde{q}}\tilde{u}^{\tilde{p}}g_{1}\rangle^{(M+1-k)}\equiv\langle\nabla_{x_{0}}^{e_{0}}\phi\,\phi^{p}(\nabla\phi)^{\tilde{q}}\tilde{u}^{\tilde{p}}\rangle^{(M+1)}.$
(65)

Setting $F=\phi^{p}(\nabla\phi)^{\tilde{q}}$, we can now use the extraction
formula (32) to obtain:

$\langle\partial_{y}F\tilde{u}^{\tilde{p}}\rangle^{(M+1)}=\Phi(\partial_{y}F)\langle\tilde{u}^{\tilde{p}}\rangle^{(M+1)}+\\\
+\sum_{k=1}^{M+1}\frac{1}{k!}\sum_{s=k}^{M+1}T^{s}\sum_{\begin{subarray}{c}s_{1},\cdots,s_{k}\geq
1\\\ \sum s_{l}=s\end{subarray}}\sum_{\begin{subarray}{c}x_{1},\cdots,x_{k}\\\
e_{1},\cdots,e_{k}\end{subarray}}\
\sum_{\begin{subarray}{c}r_{1},p^{\prime}_{1},\cdots,r_{k},p^{\prime}_{k}\geq
0\\\
p^{\prime}_{l}+r_{l}=s_{l}\end{subarray}}\big{(}\prod_{l=1}^{k}c_{r_{l}}\big{)}\Phi\big{(}\partial_{y}F;f_{1};\cdots;f_{k}\big{)}\langle\tilde{u}^{\tilde{p}}\prod_{l=1}^{k}g_{l}\rangle^{(M+1-s)},$

where $c_{r}=\frac{(-1)^{r+1}}{(2r+2)!}=\frac{1}{2r+2}c_{r}^{\prime}$.
Similarly, renaming $k$ as $s_{1}$ in the second line of (65), for
$s_{1}=1,\cdots M+1$,

$\langle(\nabla_{x_{1}}^{e_{1}}\phi)^{2r_{1}+1}F\tilde{u}^{\tilde{p}}g_{1}\rangle^{(M+1-s_{1})}=\Phi\big{(}(\nabla_{x_{1}}^{e_{1}}\phi)^{2r_{1}+1}F\big{)}\langle\tilde{u}^{\tilde{p}}g_{1}\rangle^{(M+1-s_{1})}\\\
+\mathds{1}_{M+1>s_{1}}\sum_{k=1}^{M+1-s_{1}}\frac{1}{k!}\sum_{s=k}^{M+1-s_{1}}T^{s}\sum_{\begin{subarray}{c}s_{2},\cdots,s_{k+1}\geq
1\\\ \sum
s_{l}=s\end{subarray}}\sum_{\begin{subarray}{c}x_{2},\cdots,x_{k+1}\\\
e_{2},\cdots,e_{k+1}\end{subarray}}\
\sum_{\begin{subarray}{c}r_{2},p^{\prime}_{2},\cdots,r_{k+1},p^{\prime}_{k+1}\geq
0\\\ p^{\prime}_{l}+r_{l}=s_{l}\end{subarray}}\cdot\\\
\cdot\big{(}\prod_{l=2}^{k+1}c_{r_{l}}\big{)}\Phi\big{(}(\nabla_{x_{1}}^{e_{1}}\phi)^{2r_{1}+1}F;f_{2};\cdots;f_{k+1}\big{)}\langle\tilde{u}^{\tilde{p}}\prod_{l=1}^{k+1}g_{l}\rangle^{(M+1-s_{1}-s)}.$

Plugging these in the L.H.S. of (65), and using the integration by parts
formula for the Gaussian measure $\Phi$, one obtains

$\displaystyle\Phi(\nabla_{x_{0}}^{e_{0}}\phi
F)\langle\tilde{u}^{\tilde{p}}\rangle^{(M+1)}+\sum_{k=1}^{M+1}\frac{1}{k!}\sum_{s=k}^{M+1}T^{s}\sum_{\begin{subarray}{c}s_{1},\cdots,s_{k}\geq
1\\\ \sum s_{l}=s\end{subarray}}\sum_{\begin{subarray}{c}x_{1},\cdots,x_{k}\\\
e_{1},\cdots,e_{k}\end{subarray}}\
\sum_{\begin{subarray}{c}r_{1},p^{\prime}_{1},\cdots,r_{k},p^{\prime}_{k}\geq
0\\\ p^{\prime}_{l}+r_{l}=s_{l}\end{subarray}}\
\big{(}\prod_{l=1}^{k}c_{r_{l}}\big{)}\times$ $\displaystyle\hskip
99.58464pt\times\sum_{y}\nabla_{x_{0}}^{e_{0}}G(\cdot,y)\Phi\big{(}\partial_{y}F;f_{1};\cdots;f_{k}\big{)}\langle\tilde{u}^{\tilde{p}}\prod_{l=1}^{k}g_{l}\rangle^{(M+1-s)}$
$\displaystyle+\sum_{s_{1}=1}^{M+1}T^{s_{1}}\sum_{x_{1},e_{1}}\sum_{r_{1}+p^{\prime}_{1}=s_{1}}c_{r_{1}}(2r_{1}+2)\nabla_{x_{0},x_{1}}^{e_{0},e_{1}}G\,\Phi\big{(}(\nabla_{x_{1}}^{\mathrm{e}_{1}}\phi)^{2r_{1}+1}F\big{)}\langle\tilde{u}^{\tilde{p}}g_{1}\rangle^{(M+1-s_{1})}$
$\displaystyle+\sum_{k=2}^{M+1}\frac{1}{k!}\sum_{s=k}^{M+1}T^{s}\sum_{\begin{subarray}{c}s_{1},\cdots,s_{k}\geq
1\\\ \sum s_{l}=s\end{subarray}}\sum_{\begin{subarray}{c}x_{1},\cdots,x_{k}\\\
e_{1},\cdots,e_{k}\end{subarray}}\
\sum_{\begin{subarray}{c}r_{1},p^{\prime}_{1},\cdots,r_{k},p^{\prime}_{k}\geq
0\\\ p^{\prime}_{l}+r_{l}=s_{l}\end{subarray}}\sum_{i=1}^{k}(2r_{i}+2)\times$
$\displaystyle\quad\times\big{(}\prod_{l=1}^{k}c_{r_{l}}\big{)}\nabla_{x_{0},x_{i}}^{e_{0},e_{i}}G\,\Phi\big{(}(\nabla_{x_{i}}^{e_{i}}\phi)^{2r_{i}+1}F;f_{1};\cdots;f_{i-1};f_{i+1};\cdots;f_{k}\big{)}\langle\tilde{u}^{\tilde{p}}\prod_{l=1}^{k}g_{l}\rangle^{(M+1-s)},$

where we symmetrized the role of index $1$ in the last line. Re-grouping the
terms with the same values of $k,s$, we can rewrite this as

$\Phi(\nabla_{x_{0}}^{e_{0}}\phi
F)\langle\tilde{u}^{\tilde{p}}\rangle^{(M+1)}\ +\
\sum_{k=1}^{M+1}\frac{1}{k!}\sum_{s=k}^{M+1}T^{s}\sum_{\begin{subarray}{c}s_{1},\cdots,s_{k}\geq
1\\\ \sum s_{l}=s\end{subarray}}\sum_{\begin{subarray}{c}x_{1},\cdots,x_{k}\\\
e_{1},\cdots,e_{k}\end{subarray}}\,\sum_{\begin{subarray}{c}r_{1},p^{\prime}_{1},\cdots,r_{k},p^{\prime}_{k}\geq
0\\\
p^{\prime}_{l}+r_{l}=s_{l}\end{subarray}}\langle\tilde{u}^{\tilde{p}}\prod_{l=1}^{k}g_{l}\rangle^{(M+1-s)}\times\\\
\times\big{(}\prod_{l=1}^{k}c_{r_{l}}\big{)}\Big{[}\sum_{i=1}^{k}(2r_{i}+2)\nabla_{x_{0},x_{i}}^{e_{0},e_{i}}G\,\Phi\big{(}(\nabla_{x_{i}}^{e_{i}}\phi)^{2r_{i}+1}F;f_{1};\cdots;f_{i-1};f_{i+1};\cdots;f_{k}\big{)}+\\\
+\sum_{y}\nabla_{x_{0}}^{e_{0}}G(\cdot,y)\Phi\big{(}\partial_{y}F;f_{1};\cdots;f_{k}\big{)}\Big{]}.$

But now, it follows from (62) that the term in brackets is simply
$\Phi\big{(}\nabla_{x_{0}}^{e_{0}}\phi F;f_{1};\cdots;f_{k}\big{)}$, which
implies (65) (by another look at the extraction formula (32)).

####  $\mathcal{H}_{M+1,K+1}^{1}$ part.

We first consider $p\neq 0$. We can then write $p=\mathds{1}_{x_{0}}+\bar{p}$
with $\bar{p}$ odd, $x_{0}\in\mathbb{Z}^{d}$. Using Lemma 5.1 with
$\gamma=2(d-1)(M+1)$, we have

$\Big{|}\langle\phi^{p}\tilde{u}^{\tilde{p}}\rangle-\sum_{y}G(x_{0},y)\langle\partial_{y}\phi^{\bar{p}}\tilde{u}^{\tilde{p}}\rangle\\\
-\sum_{k=1}^{M+1}T^{k}\sum_{x_{1},e_{1}}\nabla_{x_{1}}^{e_{1}}G^{m_{\gamma}}(x_{0},\cdot)\sum_{r_{1}+p_{1}^{\prime}=k}c^{\prime}_{r_{1}}\langle(\nabla_{x_{1}}^{e_{1}}\phi)^{2r_{1}+1}\phi^{\bar{p}}\tilde{u}^{\tilde{p}}g_{1}\rangle\Big{|}\leq
CT^{M+2-\epsilon}$

with $C=C(K+1,M+1,\epsilon)$, and $m_{\gamma}=T^{\gamma}=T^{2(d-1)(M+1)}$. We
can then use $\mathcal{H}_{M,\infty}^{2}$ to obtain that for $k=1,\cdots,n$,

$\big{|}\langle(\nabla_{x_{1}}^{e_{1}}\phi)^{2r_{1}+1}\phi^{\bar{p}}\tilde{u}^{\tilde{p}}g_{1}\rangle-\langle(\nabla_{x_{1}}^{e_{1}}\phi)^{2r_{1}+1}\phi^{\bar{p}}\tilde{u}^{\tilde{p}}g_{1}\rangle^{(M+1-k)}\big{|}\leq\\\
\leq
C^{\prime}T^{\gamma+M+1}+T^{M+2-k-\epsilon/2}C^{\prime}\sum_{y\in\mathrm{supp}_{\bar{p}}}\frac{\log^{M+3-k}(1+|y-x_{1}|)}{1+|y-x_{1}|},$

with $C^{\prime}=C^{\prime}(K+1,M+1,\epsilon)$, and use
$\mathcal{H}_{M+1,K}^{1}$ to obtain

$\big{|}\langle\partial_{y}\phi^{\bar{p}}\tilde{u}^{\tilde{p}}\rangle-\langle\partial_{y}\phi^{\bar{p}}\tilde{u}^{\tilde{p}}\rangle^{(M+1)}\big{|}\leq
C^{\prime\prime}T^{M+2-\epsilon},$

with $C^{\prime\prime}=C^{\prime\prime}(K+1,M+1,\epsilon)$. Using these and
(79), (81), we get

$\Big{|}\langle\phi^{p}\tilde{u}^{\tilde{p}}\rangle-\sum_{y}G(x_{0},y)\langle\partial_{y}\phi^{\bar{p}}\tilde{u}^{\tilde{p}}\rangle^{(M+1)}\\\
-\sum_{k=1}^{M+1}T^{k}\sum_{x_{1},\mathrm{e}_{1}}\nabla_{x_{1}}^{e_{1}}G^{m_{\gamma}}(x_{0},\cdot)\sum_{r_{1}+p_{1}^{\prime}=k}c^{\prime}_{r_{1}}\langle(\nabla_{x_{1}}^{e_{1}}\phi)^{2r_{1}+1}\phi^{\bar{p}}\tilde{u}^{\tilde{p}}g_{1}\rangle^{(M+1-k)}\Big{|}\leq
C^{\prime\prime\prime}T^{M+2-\epsilon},$

with $C^{\prime\prime\prime}=C^{\prime\prime\prime}(K+1,M+1,\epsilon)$. We can
then use the decay

$|\langle(\nabla_{x_{1}}^{e_{1}}\phi)^{2r_{1}+1}\phi^{\bar{p}}\tilde{u}^{\tilde{p}}g_{1}\rangle^{(M+1-k)}|\leq\sum_{y\in\mathrm{supp}_{\bar{p}}}\frac{c}{(1+|x_{1}-y|)^{d-5/4}}$

from Lemma 3.2, combined with

$|\nabla_{x_{1}}^{e_{1}}G^{m_{\gamma}}(x_{0},\cdot)-\nabla_{x_{1}}^{e_{1}}G(x_{0},\cdot)|\leq
cm_{\gamma}^{1/(2d-2)}(1+|x_{0}-x_{1}|)^{-d+3/2}$

(which follows from (78), the decay bound of Theorem C.1, and Lemma C.3), and
with Lemma C.4, to replace the massive propagator
$\nabla_{x_{1}}^{e_{1}}G^{m_{\gamma}}$ by its non-massive version,
$\nabla_{x_{1}}^{e_{1}}G$, up to an error smaller than
$c^{\prime}Tm_{\gamma}^{1/(2d-2)}=c^{\prime}T^{M+2}$. The equality

$\sum_{y}G(x_{0},y)\langle\partial_{y}\phi^{\bar{p}}\tilde{u}^{\tilde{p}}\rangle^{(M+1)}\\\
+\sum_{k=1}^{M+1}T^{k}\sum_{x_{1},\mathrm{e}_{1}}\nabla_{x_{1}}^{e_{1}}G(x_{0},\cdot)\sum_{r_{1}+p_{1}^{\prime}=k}c_{r_{1}}^{\prime}\langle(\nabla_{x_{1}}^{e_{1}}\phi)^{2r_{1}+1}\phi^{\bar{p}}\tilde{u}^{\tilde{p}}g_{1}\rangle^{(M+1-k)}\equiv\langle\phi^{p}\tilde{u}^{\tilde{p}}\rangle^{(M+1)}$

follows exactly as in the proof of $\mathcal{H}_{M+1,K+1}^{2}$, and concludes
the proof of $\mathcal{H}_{K+1,M+1}^{1}$ when $p\neq 0$. We now turn to the
$p=0$ case. Without loss of generality, we can assume that $\tilde{p}^{1}$ is
non-zero and even. Let $\tilde{q}_{x}^{k}=\tilde{p}_{x}^{k}$ for
$k=2,\cdots,N-2$ and $\tilde{q}_{x}^{1}=0$. We then use the remaining $O(N-1)$
symmetry through (18) to obtain

$\langle\tilde{u}^{\tilde{p}}\rangle=\langle(\tilde{u}^{1})^{\tilde{p}^{1}}\tilde{u}^{\tilde{q}}\rangle=\beta^{\|{\tilde{p}^{1}}\|/2}\langle\prod_{x}\big{(}\sqrt{1-T\|\tilde{u}_{x}\|^{2}}\sin\big(\sqrt{T}\phi_{x}\big{missing})\big{)}^{\tilde{p}_{x}^{1}}\tilde{u}^{\tilde{q}}\rangle.$

We now apply Lemma 4.4: letting $\phi^{\tilde{p}^{1}}(1+\sum_{s\geq
1}T^{s}\sum_{n,\tilde{n}}a^{n,\tilde{n}}_{\tilde{p}^{1},s}\phi^{n}\tilde{u}^{\tilde{n}})$
be the formal low temperature expansion of
$\beta^{\|{\tilde{p}^{1}}\|/2}\prod_{x}\Big{(}\sqrt{1-T\|\tilde{u}_{x}\|^{2}}\sin\big(\sqrt{T}\phi_{x}\big{missing})\Big{)}^{\tilde{p}^{1}_{x}}$
obtained via the first two of (45) (cf. with (46)), where the sum over
$n,\tilde{n}$ runs over even tuples in $\mathfrak{P},\tilde{\mathfrak{P}}$,
and the coefficients $a^{n,\tilde{n}}_{\tilde{p}^{1},s}$ are non zero only if
$\|n\|_{1}+\|\tilde{n}\|_{1}=2s$, we obtain

$\Big{|}\langle\tilde{u}^{\tilde{p}}\rangle-\langle\phi^{\tilde{p}^{1}}\tilde{u}^{\tilde{q}}\rangle-\sum_{s=1}^{M+1}T^{s}\sum_{n,\tilde{n}}a^{n,\tilde{n}}_{\tilde{p}^{1},s}\langle\phi^{n+\tilde{p}^{1}}\tilde{u}^{\tilde{n}+\tilde{q}}\rangle\Big{|}\leq
CT^{M+2},$

with $C=C(K,M)$. The claim then reduces to the $p\neq 0$ case and to
$\mathcal{H}^{1}_{M,\infty}$. This concludes the proof of the validity of
$\mathcal{H}_{M+1,K+1}^{1}$ and therefore of the Lemma. ∎

From Lemma 6.3, the validity of $\mathcal{H}_{1,0}$ (see Remark 6.2) and of
$\mathcal{H}_{0,\infty}$ (see Remark 6.3), the conclusion of Theorem 6.1
follows. In particular, this implies that (58) holds, for any $n\geq 0$ and
$\epsilon>0$, and, as remarked after (58), this also implies our main result,
Theorem 1.3.

###  6.5. Proof of Corollary 1.5

###### Proof.

It is sufficient to check convergence for finite degree polynomials in
$\sqrt{\beta}(S^{1},S^{2},$ $\ldots,S^{N-1})$, as they are dense in the
continuous local functions; and, by linearity, we can further reduce to finite
degree monomials. Therefore, using the change of variables from $S$ to
$(\phi,\tilde{u})$, one needs to control

$\langle\prod_{x}\Big{(}\big{(}\sqrt{\beta}\sqrt{1-T\|\tilde{u}_{x}\|^{2}}\sin\big(\sqrt{T}\phi_{x}\big{missing})\big{)}^{\alpha^{N-1}_{x}}\prod_{k=1}^{N-2}\tilde{u}_{x}^{\alpha^{k}_{x}}\Big{)}\rangle,$

for some $\alpha^{1},\ldots,\alpha^{N-1}:\mathbb{Z}^{d}\to\mathbb{Z}_{+}$ of
finite support. By Lemma 4.4, this is equal to
$\langle\prod_{x}\Big{(}\phi_{x}^{\alpha^{N-1}_{x}}\prod_{k=1}^{N-2}\tilde{u}_{x}^{\alpha^{k}_{x}}\Big{)}\rangle+O(T)$.
By Theorem 6.1 this is, in turn, equal to

$\langle\prod_{x}\Big{(}\phi_{x}^{\alpha^{N-1}_{x}}\prod_{k=1}^{N-2}\tilde{u}_{x}^{\alpha^{k}_{x}}\Big{)}\rangle^{(0)}+O(T^{1-\epsilon})=\Phi(\phi^{\alpha^{N-1}})\langle\prod_{x}\prod_{k=1}^{N-2}\tilde{u}_{x}^{\alpha^{k}_{x}}\rangle^{(0)}+O(T^{1-\epsilon}),$

for any $\epsilon>0$. Using the residual $O(N-1)$ symmetry (18) and iterating
$(N-2)$ times gives the result. ∎

## Acknowledgements

We gratefully acknowledges financial support of the European Research Council
through the ERC CoG UniCoSM, grant agreement n. 724939. A.G. also acknowledges
support from MIUR, through the PRIN 2017 project MaQuMA, PRIN201719VMAST01 and
thanks GNFM-INdAM Gruppo Nazionale per la Fisica Matematica. S.O. is supported
by the Swiss NSF grant 200021_182237 and is a member of the NCCR SwissMAP.
Most of this work was completed while S.O. was supported by an Swiss NSF early
PostDoc.Mobility Grant. He thanks the university Roma Tre for its hospitality.

##  Appendix A Proof of Proposition 1.2

Recall that

$\psi(v)\equiv\psi_{\beta}(v)=\lim_{L\to\infty}\frac{1}{|\Lambda_{L}|}\log
Z_{L;\beta}\big{(}e^{\sum_{x\in\Lambda_{L}}v\cdot S_{x}}\big{)}.$

By $O(N)$ invariance, $\psi(Rv)=\psi(v)$ for any matrix $R\in O(N)$; moreover,
if $\psi$ is differentiable at $hs$, for some $h>0$ and
$s\in\mathbb{S}^{N-1}$, then $\psi$ is differentiable at $hs^{\prime}$, for
any $s^{\prime}\in\mathbb{S}^{N-1}$. Recall also that $v\mapsto\psi(v)$ is a
convex function from $\mathbb{R}^{N}$ to $\mathbb{R}$. Therefore, $\psi$ is
differentiable almost everywhere. Define

$D^{*}=\\{h\in[0,\infty):\ \psi\textnormal{ is differentiable at }hs,\ \forall
s\in\mathbb{S}^{N-1}\\},$

which is (at least) dense in $[0,\infty)$. For $h\in D^{*}$, let
$\mathrm{J}_{h\hat{\mathrm{e}}_{N}}:\mathbb{R}^{N}\to\mathbb{R}$ denote the
derivative (gradient) of $\psi$ at $h\hat{\mathrm{e}}_{N}$. Since $\psi$ is
the limit of a sequence of convex differentiable functions, one has that for
every $s\in\mathbb{S}^{N-1}$,

$\mathrm{J}_{h\hat{\mathrm{e}}_{N}}(s)=\lim_{L\to\infty}\frac{1}{|\Lambda_{L}|}\frac{Z_{L;\beta}\big{(}\sum_{x\in\Lambda_{L}}S_{x}\cdot
se^{h\sum_{x\in\Lambda_{L}}S_{x}^{N}}\big{)}}{Z_{L;\beta}\big{(}e^{h\sum_{x\in\Lambda_{L}}S_{x}^{N}}\big{)}}=\lim_{L\to\infty}\langle
S_{0}\cdot s\rangle_{L;\beta,h}.$

In particular, by the residual $O(N-1)$ symmetry in the directions orthogonal
to $\hat{\mathrm{e}}_{N}$,

$\mathrm{J}_{h\hat{\mathrm{e}}_{N}}(\hat{\mathrm{e}}_{k})=0,\quad 1\leq k<N.$

A simple consequence of this fact is that, if we define
$m^{*}:=\mathrm{J}_{h\hat{\mathrm{e}}_{N}}(\hat{\mathrm{e}}_{N})$, then, for
any translation invariant Gibbs measure $\nu$ of the spin $O(N)$ model at
inverse temperature $\beta$ and magnetic field $h\mathrm{e}_{N}$,

$\nu(S_{0}\cdot s)=s^{N}m^{*}.$ (66)

The proof is a classical fact, which follows from differentiability of $\psi$,
see e.g. [5, Proof of Theorem 2.5] for the proof of a similar fact. A slightly
more subtle consequence is that

$\nu(|M_{n}-\nu(S_{0})|)\xrightarrow{n\to\infty}0,$ (67)

where $M_{n}=\frac{1}{(2n+1)^{d}}\sum_{x\in\\{-n,\dots,n\\}^{d}}S_{x}$. To
prove this, we use the ergodic decomposition of $\nu$: there exists a
probability measure $P_{\nu}$ on the space of ergodic Gibbs measures such that
$\nu(f)=\int dP_{\nu}(\eta)\eta(f)$ for every $\nu$-integrable $f$ (see [13,
Theorem 14.10]). Moreover, by the ergodic theorem, for any $\eta$ ergodic,
$\lim_{n\to\infty}\eta(|M_{n}-\eta(S_{0})|)=0$ (see [13, Appendix 14.A]). Now,
any $\eta$ ergodic is in particular translation invariant, therefore, by (66),
$\eta(S_{0})=(0,\dots,0,m^{*})=\nu(S_{0})$. So, one has the pointwise
convergence $\lim_{n\to\infty}\eta(|M_{n}-\nu(S_{0})|)=0$. To conclude, by
Vitali’s dominated convergence theorem one has

$\lim_{n\to\infty}\nu(|M_{n}-\nu(S_{0})|)=\lim_{n\to\infty}\int
dP_{\nu}(\eta)\eta(|M_{n}-\nu(S_{0})|)=0.$

Given (67), the proof of eqs.(5)-(6) follows [11]. Let $h\in D^{*}$, and let
$f:\mathbb{Z}^{d}\to\mathbb{R}^{N}$ be a finitely supported function. Then,
set $C_{f}=\sum_{x}f(x)\in\mathbb{R}^{N}$, and
$f_{n}(x)=f(x)-\frac{1}{(2n+1)^{d}}C_{f}\mathds{1}_{x\in\\{-n,\dots,n\\}^{d}}$.
Since $\sum_{x}f_{n}(x)=0$, one can apply (3) and take $L\to\infty$ to obtain

$\langle e^{\sum_{x}S_{x}\cdot f_{n}(x)}\rangle_{h}\leq
e^{\frac{1}{2\beta}(f_{n},Gf_{n})}.$

Now, as in [11, Corollary 2.5], $(f_{n},Gf_{n})\to(f,Gf)$ as $n\to\infty$
(using the decay of the Green function in $d\geq 3$). To conclude, observe
that $\sum_{x}S_{x}\cdot f_{n}(x)=\sum_{x}(S_{x}-M_{n})\cdot f(x)$. Then, for
every integer $p\geq 1$,

$\big{|}\langle\big{(}\sum_{x}S_{x}\cdot
f_{n}(x)\big{)}^{p}\rangle_{h}-\langle\big{(}\sum_{x}(S_{x}-\langle
S_{0}\rangle_{h})\cdot f(x)\big{)}^{p}\rangle_{h}\big{|}=\\\
=\big{|}\sum_{k=1}^{p}\binom{p}{k}\langle\big{(}\sum_{x}(S_{x}-\langle
S_{0}\rangle_{h})\cdot f(x)\big{)}^{p-k}\big{(}(\langle
S_{0}\rangle_{h}-M_{n})\cdot C_{f}\big{)}^{k}\rangle_{h}\big{|}\\\
\leq\sum_{k=1}^{p}\binom{p}{k}(2c_{f})^{p-k}(2|C_{f}|)^{k-1}|C_{f}|\langle|\langle
S_{0}\rangle_{h}-M_{n}|\rangle_{h}$

where $c_{f}=\sum_{x}|f(x)|$. Now, (67) implies that the last line converges
to $0$ as $n\to\infty$; therefore, the convergence $\langle
e^{\sum_{x}S_{x}\cdot f_{n}(x)}\rangle_{h}\to\langle e^{\sum_{x}(S_{x}-\langle
S_{0}\rangle_{h})\cdot f(x)}\rangle_{h}$ as $n\to\infty$ follows. The bound
(6) follows from (5) as in [11] (choose
$f(x)=\epsilon\delta_{x,0}\hat{\mathrm{e}}_{i})$, expand in $\epsilon$ at
second order, and sum over $i=1,\ldots,N$, recalling that
$\sum_{i=1}^{N}(S^{i}_{0})^{2}=1$).

##  Appendix B Upper bounds on correlation functions

This appendix contains the proof of Lemma 5.3. We first derive a bound on the
two point function. The log can be removed when $d\geq 4$ as is clear from the
proof.

######  Lemma B.1.

For any $d\geq 3$, there exists $c\geq 0$ such that for any distinct
$x,y\in\mathbb{Z}^{d}$, and any $k=1,\cdots,N$,

$\langle S_{x}^{k};S_{y}^{k}\rangle\leq\frac{c}{\beta}\frac{\log(n)}{n}$ (68)

where $n=\left\|x-y\right\|_{\scriptscriptstyle\infty}$.

###### Proof.

By translation invariance, we can reduce to the case $y=0$. Then, by symmetry,
one can assume $\left\|x\right\|_{\scriptscriptstyle\infty}=x_{1}$.
Furthermore, using reflection positivity, one has

$|\langle f_{0}f_{x}\rangle|=|\langle f_{0}\Theta
f_{x^{\perp}}\rangle|\leq\langle f_{0}f_{x^{\parallel}}\rangle^{1/2}\langle
f_{x^{\perp}}f_{x}\rangle^{1/2},$

where $\Theta$ is the reflection through the hyperplane $\\{y:\
y_{1}=x_{1}/2\\}$, $f_{x}=S_{x}^{k}$, $1\leq k<N$ or $f_{x}=S_{x}^{N}-\langle
S_{0}^{N}\rangle$, and $x=x^{\parallel}+x^{\perp}$,
$x^{\parallel}=(x\cdot\mathrm{e}_{1})\,\mathrm{e}_{1}$. Since, by translation
invariance, $\langle f_{x^{\perp}}f_{x}\rangle=\langle
f_{0}f_{x^{\parallel}}\rangle$, the general claim is implied by its
restriction to $x=n\mathrm{e}_{1}$.

First treat $k\neq N$. By symmetry, it is sufficient to prove the result for
$k=1$. Set
$g(x)=z\mathrm{e}_{1}\mathds{1}_{x\in\\{0,\ldots,n\mathrm{e}_{1}\\}}$. Use
then (5) for this $g$. Expand both sides of the inequality, simplify the
constant term, divide by $z^{2}$ and take $z\searrow 0$ to obtain

$\sum_{k,l=0}^{n}\langle
S_{k\mathrm{e}_{1}}^{1}S_{l\mathrm{e}_{1}}^{1}\rangle\leq\frac{1}{\beta}\sum_{k,l=0}^{n}G(k\mathrm{e}_{1},l\mathrm{e}_{1}),$

where $G(k\mathrm{e}_{1},l\mathrm{e}_{1})\leq c/(1+|k-l|)^{d-2}$. We can then
use the standard fact (following from reflection positivity, see [17] or [15])
that $\langle f_{0}f_{n\mathrm{e}_{1}}\rangle$ in monotonic in $n$ (for the
same $f$s as before) to obtain

$n^{2}\langle
S_{0}^{1}S_{n\mathrm{e}_{1}}^{1}\rangle\leq\frac{c}{\beta}n\sum_{k=0}^{n}(1+k)^{2-d}\leq\frac{c}{\beta}n\log(n).$

The proof for $k=N$ follows exactly the same path but the truncature becomes
non-trivial. ∎

###### Proof of Lemma 5.3.

To simplify notations, we conduct the proof with infinite volume notation.
What we are really doing is to work in finite volume with $h>0$ and take
limits afterwards. We will use several times that if one conditions on
everything but the sign field of $\phi$, the latter is distributed according
to a ferromagnetic Ising model, denoted $\sigma$ in what follows, and is
therefore coupled to a Random Cluster (RC) model [8]. We denote
$\langle\cdot\rangle^{\prime}$ the conditional measure of the sign field
$\sigma$, and $\Phi^{\prime}$ the associated Random Cluster measure. We first
prove the intermediate bound: there exist $C<\infty,c>0$ such that for any
$x,y\in\mathbb{Z}^{d}$ and $T>0$,

$\langle\phi_{x}\phi_{y}\rangle\leq\frac{C\log(1+|x-y|)}{1+|x-y|}+Ce^{-c{\beta}}.$
(69)

Indeed, introducing the event
$B=\\{|\theta_{x}|\leq\frac{\pi}{2}\\}\cap\\{|\theta_{y}|\leq\frac{\pi}{2}\\}\cap\\{\|u_{x}\|\leq\frac{\sqrt{3}}{2}\\}\cap\\{\|u_{y}\|\leq\frac{\sqrt{3}}{2}\\}$,
one has (using the fact that, on $B$, $\frac{1}{\rho_{x}\rho_{y}}\leq 4$, and
$\frac{|\theta_{x}\theta_{y}|}{|\sin\theta_{x}\sin\theta_{y}|}\leq\pi^{2}/4$)

$\displaystyle 0\leq\langle\phi_{x}\phi_{y}\rangle$
$\displaystyle=\beta\langle\mathds{1}_{B}|\theta_{x}\theta_{y}|\langle\sigma_{x}\sigma_{y}\rangle^{\prime}\rangle+\beta\langle\mathds{1}_{B^{c}}|\theta_{x}\theta_{y}|\langle\sigma_{x}\sigma_{y}\rangle^{\prime}\rangle$
$\displaystyle\leq\pi^{2}\beta\langle\mathds{1}_{B}\rho_{x}\rho_{y}|\sin\theta_{x}\sin\theta_{y}|\langle\sigma_{x}\sigma_{y}\rangle^{\prime}\rangle+\pi^{2}\beta\mu(B^{c})$
$\displaystyle\leq\pi^{2}\beta\Big{(}\langle
S_{x}^{N-1}S_{y}^{N-1}\rangle+\beta\mu(B^{c})\Big{)}$

To conclude, apply Lemma B.1 and observe that

$\mu(B^{c})\leq 2\mu(|\theta_{0}|>\pi/2)+2\mu(\|u_{0}\|>\sqrt{3}/2)\leq
Ce^{-c\beta}$

by a union bound and an application of eqs.(37)-(38). Before we turn to the
proof of our main claim, let us also observe the following: define

$\tilde{C}_{K}=\sup_{0<T<T_{0}}\sup_{\begin{subarray}{c}p,p^{\prime},\mathcal{F}\\\
\left\|p\right\|_{\scriptscriptstyle 1}+p^{\prime}\leq
K\end{subarray}}\langle|\phi^{p}\mathcal{F}|\rangle$ (70)

where the second sup is over $p:\mathbb{Z}^{d}\to\mathbb{Z}_{+}$,
$p^{\prime}\in\mathbb{Z}_{+}$ and $\mathcal{F}$ monomial in
$\tilde{u}_{x}^{k}$ with degree at most $2p^{\prime}$. By Theorem 4.3 and
Hölder’s inequality, $\tilde{C}_{K}<\infty$.

Let us now turn to the proof of (47). Let $A=\\{x:\ p_{x}\text{ is odd}\\}$
and $B=\\{x:\ q_{x}\text{ is odd}\\}$. Note that, since $\sum_{x}p_{x}$ and
$\sum_{x}q_{x}$ are odd, $|A|$ and $|B|$ are odd as well. Using the RC
representation of the Ising model associated with the $\sigma$ variables
discussed above, one has

$|\langle\mathcal{F}\phi^{p}\phi^{q}\rangle|\leq\langle|\mathcal{F}\phi^{p}\phi^{q}|\langle\sigma_{A}\sigma_{B}\rangle^{\prime}\rangle$
(71)

Let now $\mathcal{E}_{D}$ be the percolation event “each cluster contains an
even (possibly $0$) number of sites of $D$”. Since $|A|$ and $|B|$ are odd,
$\mathcal{E}_{A\cup B}$ implies the existence of $x\in A,y\in B$ such that $x$
and $y$ are in the same connected component, denoted $x\leftrightarrow y$. One
has

$\langle\sigma_{A}\sigma_{B}\rangle^{\prime}=\Phi^{\prime}(\mathcal{E}_{A\cup
B})\\\ \leq\sum_{x\in A}\sum_{y\in B}\Phi^{\prime}(x\leftrightarrow
y)=\sum_{x\in A}\sum_{y\in B}\langle\sigma_{x}\sigma_{y}\rangle^{\prime}.$

Plugging this bound in (71), one obtains

$|\langle\mathcal{F}\phi^{p}\phi^{q}\rangle|\leq\sum_{x\in A}\sum_{y\in
B}\langle|\mathcal{F}\phi^{p-\delta_{x}}\phi^{q-\delta_{y}}|\phi_{x}\phi_{y}\rangle.$
(72)

We then partition on whether
$|\mathcal{F}\phi^{p-\delta_{x}}\phi^{q-\delta_{y}}|\leq\beta^{\epsilon}$ or
not, and use Cauchy-Schwartz to obtain

$\displaystyle\langle|\mathcal{F}\phi^{p-\delta_{x}}\phi^{q-\delta_{y}}|\phi_{x}\phi_{y}\rangle$
$\displaystyle\leq\beta^{\epsilon}\langle\phi_{x}\phi_{y}\rangle+\langle|\mathcal{F}\phi^{p}\phi^{q}|^{2}\rangle^{1/2}\mu(|\mathcal{F}\phi^{p-\delta_{x}}\phi^{q-\delta_{y}}|>\beta^{\epsilon})^{1/2}$
(73)
$\displaystyle\leq\beta^{\epsilon}\langle\phi_{x}\phi_{y}\rangle+(\tilde{C}_{2K_{0}})^{1/2}\Big{(}\frac{\tilde{C}_{K_{0}\lceil
2\alpha/\epsilon\rceil}}{\beta^{2\alpha}}\Big{)}^{1/2}$ (74)

where we used the definition (70), we let
$K_{0}=\left\|p\right\|_{\scriptscriptstyle
1}+\left\|q\right\|_{\scriptscriptstyle 1}+p^{\prime}$, and in the last line
we used Markov’s inequality. Applying the upper bound on correlation (69) in
this last estimate and plugging the resulting bound in (72) and in (75), we
obtain (47). For (48), letting
$C^{\prime}=\bigcup_{(x,\mathrm{e}):q_{x,\mathrm{e}}^{\prime}\neq
0}\\{x,x+\mathrm{e}\\}$, we expand

$(\nabla\phi)^{q^{\prime}}=\sum_{C\subset
C^{\prime}}\sum_{\begin{subarray}{c}q:C\to\mathbb{Z}_{+}\\\
\left\|q\right\|_{\scriptscriptstyle
1}\leq\left\|q^{\prime}\right\|_{\scriptscriptstyle
1}\end{subarray}}c_{q^{\prime}}(q)\phi^{q}$

for appropriate coefficients $c_{q^{\prime}}(q)$. We can then write

$\langle\mathcal{F}\phi^{p}(\nabla\phi)^{q^{\prime}}\rangle=\sum_{C\subset
C^{\prime}}\sum_{\begin{subarray}{c}q:C\to\mathbb{Z}_{+}\\\
\left\|q\right\|_{\scriptscriptstyle
1}\leq\left\|q^{\prime}\right\|_{\scriptscriptstyle
1}\end{subarray}}c_{q^{\prime}}(q)\langle\mathcal{F}\phi^{p}\phi^{q}\rangle.$
(75)

Now, by symmetry, $\langle\mathcal{F}\phi^{p}\phi^{q}\rangle$ is $0$ if
$\sum_{x\in C}q_{x}$ is even. In particular, defining $C_{q}=\\{x\in C:\
q_{x}\text{ odd}\\}$, $|C_{q}|$ is odd. Then we bound
$\langle\mathcal{F}\phi^{p}\phi^{q}\rangle$ as in eqs.(71) to (73) and, using
(69), we obtain (48), as desired. ∎

##  Appendix C Gaussian estimates

###  C.1. Correlations and massive correlations

Let us first state and prove the following standard bounds on the massive
lattice Green’s function.

######  Theorem C.1.

For $d\geq 3$, there exists $c=c(d)$ such that for any $m\geq 0$,
$x\in\mathbb{Z}^{d}$, $e,e^{\prime}\in\mathrm{B}$,

$0\leq
G^{m}_{0x}\leq\frac{c}{(1+|x|)^{d-2}},\quad|\nabla_{x}^{\mathrm{e}}G^{m}(0,\cdot)|\leq\frac{c}{(1+|x|)^{d-1}},\quad|\nabla_{0,x}^{\mathrm{e}^{\prime},\mathrm{e}}G^{m}|\leq\frac{c}{(1+|x|)^{d}}.$

###### Proof.

We use the following integral representation of $G^{m}$ (see, for example,
[16]):

$G^{m}(0,x)=\int_{0}^{\infty}dte^{-(m^{2}+2d)t}\prod_{k=1}^{d}I_{x_{k}}(2t),$
(76)

where $I_{\nu}(t)$ is the modified Bessel function of the first kind. We shall
use that $I_{\nu}(t)\geq 0$, $I_{-\nu}(t)=I_{\nu}(t)$, and $I_{\nu}(t)\geq
I_{\nu+1}(t)$ for $\nu\in\mathbb{Z}_{+},t\in\mathbb{R}_{+}$ [18]. The
asymptotic $I_{\nu}(t)\sim O((2\pi t)^{-1/2}e^{t})$ valid for large real
argument makes the integral well defined at $m=0$ for any $d\geq 3$.

The claim at $m=0$ is classical and proven, for example, in [14]. The integral
representation (76) together with the monotonicity of $I_{\nu}(t)$ in $\nu$
implies $G^{m}_{0x}$, $|\nabla_{x}^{\mathrm{e}}G^{m}(0,\cdot)|$, and
$|\nabla_{0,x}^{\mathrm{e}^{\prime},\mathrm{e}}G^{m}|$ when
$\mathrm{e}\cdot\mathrm{e}^{\prime}=0$ are monotone decreasing in $m$;
therefore, in these cases, the bound for $m=0$ implies the one for $m\neq 0$.
By translation invariance, we are then left with bounding
$|\nabla_{0,x}^{\mathrm{e}^{\prime},\mathrm{e}}G^{m}|$ for
$\mathrm{e}=\mathrm{e}^{\prime}$; we can further restrict to
$\mathrm{e}=\mathrm{e}_{1}$ by symmetry. Consider first $x_{1}=0$. Then, using
(76),

$|\nabla_{0,x}^{\mathrm{e}_{1},\mathrm{e}_{1}}G^{m}|=2\int_{0}^{\infty}dte^{-(m^{2}+d)t}\big{(}I_{0}(t)-I_{1}(t)\big{)}\prod_{k=2}^{d}I_{x_{k}}(t),$

which is again decreasing in $m$. We then consider $x_{1}\geq 1$ (which will
cover all leftover cases by symmetry). W.l.o.g., we can suppose
$x_{2},\cdots,x_{d}\geq 0$ and $x_{2}=\max(x_{2},\cdots,x_{d})$. We first
treat $x_{1}\geq x_{2}$. Using the reflection positivity of the Gaussian
measure $\Phi_{m}$, and denoting by $\Theta$ the reflection through the
hyperplane $\\{y:\ y_{2}=\lfloor x_{1}/2\rfloor-y_{1}\\}$ (which passes
through sites of $\Lambda$ and is tilted by an angle $\pi/4$ w.r.t. the $i$-th
coordinate axis, $i>2$)), we find:

$|\nabla_{0,x}^{\mathrm{e}_{1},\mathrm{e}_{1}}G^{m}|=|\Phi_{m}(\nabla_{0}^{\mathrm{e}_{1}}\phi\nabla_{x}^{\mathrm{e}_{1}}\phi)|=|\Phi_{m}(\nabla_{0}^{\mathrm{e}_{1}}\phi\Theta\nabla_{\bar{x}}^{-\mathrm{e}_{2}}\phi)|\leq\\\
\leq\Phi_{m}(\nabla_{0}^{\mathrm{e}_{1}}\phi\Theta\nabla_{0}^{\mathrm{e}_{1}}\phi_{0})^{1/2}\Phi_{m}(\nabla_{\bar{x}}^{-\mathrm{e}_{2}}\phi\Theta\nabla_{\bar{x}}^{-\mathrm{e}_{2}}\phi)^{1/2}=\Phi_{m}(\nabla_{0}^{\mathrm{e}_{1}}\phi\nabla_{\tilde{x}}^{-\mathrm{e}_{2}}\phi)^{1/2}\Phi_{m}(\nabla_{\bar{x}}^{-\mathrm{e}_{2}}\phi\nabla_{x}^{\mathrm{e}_{1}}\phi)^{1/2}$

where $\bar{x}=(\lfloor x_{1}/2\rfloor-x_{2},\lfloor x_{1}/2\rfloor-
x_{1},x_{3},\cdots,x_{d})$ and $\tilde{x}=(\lfloor x_{1}/2\rfloor,\lfloor
x_{1}/2\rfloor,0,\cdots,0)$. Now, as $e\cdot\mathrm{e}_{2}=0$,
$\|\tilde{x}\|\geq cx_{1}\geq cd^{-1/2}\|x\|$, and $\|\bar{x}-x\|\geq
c^{\prime}x_{1}\geq c^{\prime}d^{-1/2}\|x\|$, we can apply the previously
obtained bounds to get the result. The case $x_{2}>x_{1}$ is treated similarly
using a reflection through $\\{y:\ y_{2}=\lfloor x_{2}/2\rfloor-y_{1}\\}$. ∎

We next gather the following estimates on the sum of $G^{m}$ and of their
derivatives, as well as on the difference between $G^{m}$ and $G^{0}$, which
are used systematically in the proof of our main result.

######  Theorem C.2.

There exists $c<\infty$ such that for any $0<m\leq 1$,

  1. (1)

$\sum_{x\in\mathbb{Z}^{d}}G^{m}(0,x)=\frac{1}{m^{2}}$ (77)

  2. (2)

For any $x$

$|G^{m}(0,x)-G(0,x)|\leq cm.$ (78)

  3. (3)

for $\mathrm{e}\in\mathrm{B}$,

$\sum_{x\in\mathbb{Z}^{d}}|\Phi_{m}\big{(}\phi_{0}(\phi_{x}-\phi_{x+\mathrm{e}})\big{)}|\leq
cm^{-1}.$ (79)

  4. (4)

for $\mathrm{e},\mathrm{e}^{\prime}\in\mathrm{B}$,

$\sum_{x\in\mathbb{Z}^{d}}|\Phi_{m}\big{(}(\phi_{0}-\phi_{\mathrm{e}})(\phi_{x}-\phi_{x+\mathrm{e}^{\prime}})\big{)}|\leq
c|\log(m)|.$ (80)

  5. (5)

For $0<\alpha<1$, there exists $C<\infty$ such that for any
$y\in\mathbb{Z}^{d}$,

$\displaystyle\begin{split}\sum_{x\in\mathbb{Z}^{d}}G^{m}(0,x)(1+|x-y|)^{-\alpha}\leq
Cm^{-2+\alpha},\\\
\sum_{x\in\mathbb{Z}^{d}}|\Phi_{m}\big{(}\phi_{0}(\phi_{x}-\phi_{x+\mathrm{e}})\big{)}|(1+|x-y|)^{-\alpha}\leq
Cm^{-1+\alpha}.\end{split}$ (81)

  6. (6)

For every $k\geq 0$ there exists $C<\infty$ such that for any
$y\in\mathbb{Z}^{d}$,

$\sum_{x\in\mathbb{Z}^{d}}|\Phi_{m}\big{(}(\phi_{0}-\phi_{\mathrm{e}^{\prime}})(\phi_{x}-\phi_{x+\mathrm{e}})\big{)}|\frac{\log^{k}(1+|x-y|)}{1+|x-y|}\leq
C\frac{\log^{k+1}(1+|y|)}{(1+|y|)}.$ (82)

###### Proof.

All estimates are proved in [7], except for the last one. More precisely, for
(77), see [7, Prop.A1(a)]; for (78), see [7, Prop.A5]; for (79), see [7,
Prop.A1(b)]; for (80), see [7, Prop.A1(d)]; for (81) see [7, Prop.A1(e)]. We
are left with proving (82): using Theorem C.1, we have the upper bound

$\sum_{x\in\mathbb{Z}^{d}}\frac{c}{(1+|x|)^{d}}\frac{\log^{k}(1+|x-y|)}{1+|x-y|}=\sum_{x:|x|\leq|y|/2}\cdot+\sum_{x:|x-y|<|y|/2}\cdot+\sum_{x:|x|>|y|/2,|x-y|\geq|y|/2}\cdot.$

In the first sum, one has $|y|/2\leq|x-y|\leq 3|y|/2$, which allows us to
bound from above the first term by
$C^{\prime}\frac{\log^{k}(1+|y|)}{1+|y|}\sum_{|x|\leq|y|/2}(1+|x|)^{-d}\leq
C\frac{\log^{k}(1+|y|)}{1+|y|}\log(1+|y|)$. In the same fashion, in the second
sum, $|x|>|y|/2$ so the second term is bounded from above by
$C^{\prime}\frac{\log^{k}(1+|y|)}{(1+|y|)^{d}}\sum_{|x-y|<|y|/2}(1+|x-y|)^{-1}\leq
C\frac{\log^{k}(1+|y|)}{1+|y|}$. In the last case, using Hölder’s inequality,
we see that the third sum is bounded from above by

$C^{\prime}\Big{(}\sum_{|x|>|y|/2}\frac{1}{(1+|x|)^{d+1}}\Big{)}^{\frac{d}{d+1}}\Big{(}\sum_{|x-y|\geq|y|/2}\frac{\log^{k(d+1)}(1+|x-y|)}{(1+|x-y|)^{d+1}}\Big{)}^{\frac{1}{d+1}}\leq
C\frac{\log^{k}(1+|y|)}{1+|y|}.$

This proves the claim ($C$ depends on $k,d$). ∎

In order to compare massive and massless propagators, we use Theorem C.1 in
combination with eq.(78) and the following Lemma.

######  Lemma C.3.

Suppose $f_{m}(x)$ is such that

  * •

$\sup_{m\geq 0}|f_{m}(x)|\leq C(1+|x|)^{-\gamma}$,

  * •

$|f_{m}(x)-f_{0}(x)|\leq Cm$,

for some $C,\gamma>0$. Then, for any $0<\gamma^{\prime}<\gamma$, there exists
$C^{\prime}>0$ such that

$|f_{m}(x)-f_{0}(x)|\leq
C^{\prime}m^{1-\gamma^{\prime}/\gamma}(1+|x|)^{-\gamma^{\prime}}.$ (83)

###### Proof.

One has
$|f_{m}(x)-f_{0}(x)|\leq|f_{m}(x)-f_{0}(x)|^{1-\gamma^{\prime}/\gamma}(2\sup_{m\geq
0}|f_{m}(x)|)^{\gamma^{\prime}/\gamma}$. From this, the two assumptions
readily imply (83). ∎

We conclude this subsection with a simple lemma that we used in many
instances.

######  Lemma C.4.

Let $a,b>0$ be such that $a+b>d$. Then, there exists $C=C(a,b)$ such that for
any $u\in\mathbb{Z}^{d}$,

$\sum_{x\in\mathbb{Z}^{d}}(1+|x|)^{-a}(1+|x-u|)^{-b}\leq
C\frac{\log(1+|u|)}{(1+|u|)^{a+b-{d}}}.$

###### Proof.

Separate the sum over $x$ into
$\sum_{|x|\leq|u|/2}+\sum_{|x-u|<|u|/2}+\sum_{\begin{subarray}{c}|x|>|u|/2\\\
|x-u|\geq|u|/2\end{subarray}}$. In the first sum, one has $|x-u|\geq|u|/2$,
which allows us to bound from above the first term by
$C\frac{\log(1+|u|)}{(1+|u|)^{a+b-d}}$ (the log only appears if $a=d$). In the
same fashion, in the second sum, $|x|>|u|/2$ so the second term is bounded
from above by $C\frac{\log(1+|u|)}{(1+|u|)^{a-d+b}}$ (the log only appears if
$b=d$). In the last case, one has $|x|\leq|x-u|+|u|\leq 3|x-u|$. So, the last
sum is bounded from above by $C\sum_{|x|>|u|/2}(1+|x|)^{-a-b}\leq
C(1+|u|)^{d-a-b}$. ∎

###  C.2. Gaussian connected correlations

Let $a_{1},\cdots,a_{n}\in\mathbb{Z}^{d}$ and denote by
$A=\\{a_{1},\cdots,a_{n}\\}$ the corresponding multi-set, i.e., the collection
of the elements $a_{1},\ldots,a_{n}$ counted with their multiplicities, in the
case that a given site of $\mathbb{Z}^{d}$ appears more than once in the list.
One has the Wick formula

$\Phi(\phi_{A})=\sum_{\pi\in\mathrm{Pairings}(A)}\prod_{\\{a_{1},a_{2}\\}\in\pi}\Phi(\phi_{a_{1}}\phi_{a_{2}}),$
(84)

where $\phi_{A}=\prod_{a\in A}\phi_{a}$, and $\mathrm{Pairings}(A)$ is the set
of partitions of $A$ into two-elements classes. Moreover, from (25) one also
has the corresponding formula for connected correlations: let
$A_{i}=\\{a_{1}^{i},\cdots,a_{k_{i}}^{i}\\}$, $i=1,\cdots,n$, be multi-sets of
sites. Then,

$\Phi(\phi_{A_{1}};\cdots;\phi_{A_{n}})=\sum_{\pi\in\mathrm{Pairings}^{c}(A_{1},\cdots,A_{n})}\prod_{\\{a_{1},a_{2}\\}\in\pi}\Phi(\phi_{a_{1}}\phi_{a_{2}}),$
(85)

where $\mathrm{Pairings}^{c}(A_{1},\cdots,A_{n})$ is the subset of
$\mathrm{Pairings}(A_{1}\cup\cdots\cup A_{n})$ consisting of the pairings
$\pi$ such that the graph with vertex set $\\{A_{1},\cdots,A_{n}\\}$ and edge
set $F_{\pi}=\\{\\{A_{i},A_{j}\\}:\exists\ a\in A_{i},b\in A_{j}\ \text{such
that}\ \\{a,b\\}\in\pi\\}$ is connected. The same formulas extend (by
linearity) to the case $A_{i}=A_{i}^{v}\sqcup A_{i}^{e}$ with $A_{i}^{v}$ a
multi-set of sites and $A_{i}^{e}$ a multi-set of edges (elements of
$\mathbb{E}_{d}$) and $\phi_{A_{i}}$ is replaced by
$\phi_{A_{i}^{v}}(\nabla\phi)_{A_{i}^{e}}$.

######  Lemma C.5.

Let $K,n\geq 0$ be integers. Let $\epsilon>0$. Then, there exists
$C=C(n,K,\epsilon)$ such that for any
$p,p^{\prime}:\mathbb{Z}^{d}\to\mathbb{Z}_{+}$ odd, any
$q:\mathbb{E}_{d}\to\mathbb{Z}_{+}$ odd, and $r_{1},\cdots,r_{n}>0$ integers
with $\left\|p\right\|_{\scriptscriptstyle
1}+\left\|p^{\prime}\right\|_{\scriptscriptstyle
1}+\left\|q\right\|_{\scriptscriptstyle 1}+\sum_{i=1}^{n}2r_{i}\leq K$,

$\sum_{\begin{subarray}{c}x_{1},\cdots,x_{n}\in\mathbb{Z}^{d}\\\
e_{1},\cdots,e_{n}\in\mathrm{B}_{+}\end{subarray}}\big{|}\Phi\big{(}\phi^{p}\phi^{p^{\prime}};(\nabla_{x_{1}}^{e_{1}}\phi)^{2r_{1}};\cdots;(\nabla_{x_{n}}^{e_{n}}\phi)^{2r_{n}}\big{)}\big{|}\leq
C\sum_{x\in\mathrm{supp}_{p}}\sum_{y\in\mathrm{supp}_{p^{\prime}}}\frac{1}{(1+|x-y|)^{d-2-\epsilon}},$
(86)

and

$\sum_{\begin{subarray}{c}x_{1},\cdots,x_{n}\in\mathbb{Z}^{d}\\\
e_{1},\cdots,e_{n}\in\mathrm{B}_{+}\end{subarray}}\big{|}\Phi\big{(}\phi^{p}(\nabla\phi)^{q};(\nabla_{x_{1}}^{e_{1}}\phi)^{2r_{1}};\cdots;(\nabla_{x_{1}}^{e_{n}}\phi)^{2r_{n}}\big{)}\big{|}\leq
C\sum_{x\in\mathrm{supp}_{p}}\sum_{y\in\mathrm{supp}_{q}}\frac{1}{(1+|x-y|)^{d-1-\epsilon}}.$
(87)

###### Proof.

For $n=0$ the summands in the left sides should be interpreted as
$|\Phi(\phi^{p}\phi^{p^{\prime}})|$ and $|\Phi(\phi^{p}(\nabla\phi)^{q})|$,
respectively. In this case, the claim readily follows from C.1 and (84). We
therefore suppose $n\geq 1$. As the two claims are proved in exactly the same
fashion, we prove only the first one and leave the adaptations to get the
second to the reader. We will use the identity (85) together with Lemma C.4.

It will useful to use a graphical language: introduce a set of sites
$\\{0,0^{\prime},1,\cdots,n\\}$ corresponding to
$\phi^{p},\phi^{p^{\prime}},(\nabla_{x_{1}}^{e_{1}}\phi)^{2r_{1}},\cdots,(\nabla_{x_{n}}^{e_{n}}\phi)^{2r_{n}}$.
Edges will be given by the pairings in (85). See Figure 1 for an illustration
with $n=4$.

Figure 1. $n=4$, $r_{1}=r_{2}=r_{3}=1$, $r_{4}=2$. Left: the “vertices” with
the outgoing half-edges. Right: a possible pairing of the elements with
$0,0^{\prime}$ not merged.

We consider two cases: first, if there is a pairing $\\{x,y\\}$ in $\pi$ such
that $x\in\mathrm{supp}_{p},y\in\mathrm{supp}_{p^{\prime}}$, we merge the two
vertices $0,0^{\prime}$ into one, to be denoted $00^{\prime}$ (see Figure 2).
In this case, since all the terms in the connected correlation have even
degree, one can extract (for a given pairing $\pi$) two sites
$u,z\in\mathrm{supp}_{p}\cup\mathrm{supp}_{p^{\prime}}$, and a path
$\gamma:00^{\prime}\to 00^{\prime}$ passing through every site in
$\\{1,\cdots,n\\}$, of length $|\gamma|\leq K+1$ (see Figure 2)

Figure 2. $n=4$, $r_{1}=r_{2}=r_{3}=1$, $r_{4}=2$. Left: a possible pairing of
the elements with $0,0^{\prime}$ merged. Right: a path extraction.

such that the contribution to the total weight of $\pi$ coming from the pair
$x,y$ is $\Phi(\phi_{x}\phi_{y})$ while the contribution coming from $\gamma$
is

$\Phi\big{(}\phi_{u}\nabla_{x_{\gamma_{1}}}^{e_{\gamma_{1}}}\phi\big{)}\Big{(}\prod_{k=1}^{M-2}\Phi\big{(}\nabla_{x_{\gamma_{k}}}^{e_{\gamma_{k}}}\phi\nabla_{x_{\gamma_{k+1}}}^{e_{\gamma_{k+1}}}\phi\big{)}\Big{)}\Phi\big{(}\phi_{z}\nabla_{x_{\gamma_{M-1}}}^{e_{\gamma_{M-1}}}\phi\big{)}$

(here $\gamma$ is a tuple $\gamma=(\gamma_{0},\gamma_{1},\ldots,\gamma_{M})$,
with $\gamma_{0}=\gamma_{M}=00^{\prime}$, $\gamma_{i}\in\\{1,\ldots,n\\}$ for
$i=1,\ldots,M-1$, such that the multi-set
$\\{\gamma_{1},\ldots,\gamma_{M-1}\\}$ contains $\\{1,\ldots,n\\}$). Moreover,
the contribution of the other pairs is bounded from above by $c^{K}$ for some
universal $c<\infty$. Therefore, in the case $0,0^{\prime}$ are merged, we
have the following upper bound on our target:

$c^{K}\sum_{\begin{subarray}{c}x\in\mathrm{supp}_{p}\\\
y\in\mathrm{supp}_{p^{\prime}}\end{subarray}}\Phi(\phi_{x}\phi_{y})\sum_{u,z\in\mathrm{supp}_{p+p^{\prime}}}\sum_{\gamma:u\to
z}\sum_{\begin{subarray}{c}x_{1},\cdots,x_{n}\\\
e_{1},\cdots,e_{n}\end{subarray}}\Big{|}\Phi(\phi_{u}\nabla_{x_{\gamma_{1}}}^{e_{\gamma_{1}}}\phi)\Phi(\phi_{z}\nabla_{x_{\gamma_{M-1}}}^{e_{\gamma_{M-1}}}\phi)\prod_{k=1}^{M-2}\Phi(\nabla_{x_{\gamma_{k}}}^{e_{\gamma_{k}}}\phi\nabla_{x_{\gamma_{k+1}}}^{e_{\gamma_{k+1}}}\phi)\Big{|},$

where the sum is over $\gamma$ as described previously. The contribution of a
fixed $\gamma$ is then bounded from above by

$\sum_{\begin{subarray}{c}x_{1},\cdots,x_{M-1}\\\
e_{1},\cdots,e_{M-1}\end{subarray}}\Big{|}\Phi(\phi_{u}\nabla_{x_{1}}^{e_{1}}\phi)\Phi(\phi_{z}\nabla_{x_{M-1}}^{e_{M-1}}\phi)\prod_{k=1}^{M-2}\Phi(\nabla_{x_{k}}^{e_{k}}\phi\nabla_{x_{k+1}}^{e_{k+1}}\phi)\Big{|}\leq\\\
\leq
C^{M}\sum_{x_{1},\cdots,x_{M-1}}(|u-x_{1}|+1)^{1-d}(|z-x_{M-1}|+1)^{1-d}\prod_{k=1}^{M-2}(|x_{k}-x_{k+1}|+1)^{-d}.$

Repeated use of Lemma C.4 gives the summability of the infinite sum, implying
the claim (in the case that $0,0^{\prime}$ are merged), since the
combinatorial factor coming from the sum over $\gamma$ depends only on $n,K$.

We next turn to the case where there is no pairing in $\pi$ linking $0$ to