# New lower bounds for the (near) critical Ising and $\varphi^{4}$ models’
two-point functions

Hugo Duminil-Copin111Institut des Hautes Études Scientifiques<EMAIL_ADDRESS>22footnotemark: 2222Université de Genève<EMAIL_ADDRESS><EMAIL_ADDRESS>, Romain Panis22footnotemark: 2

###### Abstract

We study the nearest-neighbour Ising and $\varphi^{4}$ models on
$\mathbb{Z}^{d}$ with $d\geq 3$ and obtain new lower bounds on their two-point
functions at (and near) criticality. Together with the classical infrared
bound, these bounds turn into up-to constant estimates when $d\geq 5$. When
$d=4$, we obtain an “almost” sharp lower bound corrected by a logarithmic
factor. As a consequence of these results, we show that $\eta=0$ and $\nu=1/2$
when $d\geq 4$, where $\eta$ is the critical exponent associated with the
decay of the model’s two-point function at criticality and $\nu$ is the
critical exponent of the correlation length $\xi(\beta)$. When $d=3$, we
improve previous results and obtain that $\eta\leq 1/2$. As a byproduct of our
proofs, we also derive the blow-up at criticality of the so-called bubble
diagram when $d=3,4$.

##  1 Introduction

We are interested in two ferromagnetic and real-valued spin models on
$\mathbb{Z}^{d}$ that are amongst the most studied in statistical mechanics:
the Ising model and the (discrete) $\varphi^{4}$ model. These models are
formally defined as follows: given $\Lambda\subset\mathbb{Z}^{d}$ finite,
$\mathcal{S}\subset\mathbb{R}$, a probability measure $\rho$ on $\mathcal{S}$,
and an inverse temperature $\beta\geq 0$, we define a probability measure on
$\mathcal{S}^{\Lambda}$ according to the formula

$\langle
F(\tau)\rangle_{\Lambda,\rho,\beta}:=\frac{1}{\mathbf{Z}_{\Lambda,\rho,\beta}}\int_{\mathcal{S}^{\Lambda}}F(\tau)\exp\Big{(}\beta\sum_{\begin{subarray}{c}x,y\in\Lambda\\\
x\sim
y\end{subarray}}\tau_{x}\tau_{y}\Big{)}\prod_{x\in\Lambda}\mathrm{d}\rho(\tau_{x}),$
(1.1)

where $F:\mathcal{S}^{\Lambda}\rightarrow\mathbb{R}$,
$\mathbf{Z}_{\Lambda,\rho,\beta}$ is the _partition function_ of the model
ensuring that $\langle 1\rangle_{\Lambda,\rho,\beta}=1$, and $x\sim y$ means
that $|x-y|_{2}=1$ (where $|\cdot|_{2}$ is the $\ell^{2}$ norm on
$\mathbb{R}^{d}$). The Ising model corresponds to choosing
$\mathcal{S}=\\{-1,1\\}$ and
$\rho(\mathrm{d}x)=\tfrac{1}{2}(\delta_{-1}+\delta_{1})$ above; while the
$\varphi^{4}$ model corresponds to choosing $\mathcal{S}=\mathbb{R}$ and
$\rho=\rho_{g,a}$ where $g>0$, $a\in\mathbb{R}$, and

$\rho(\mathrm{d}t)=\rho_{g,a}(\mathrm{d}t)=\frac{e^{-gt^{4}-at^{2}}\mathrm{d}t}{\int_{\mathbb{R}}e^{-gs^{4}-as^{2}}\mathrm{d}s}.$
(1.2)

In the case of the Ising model (resp. the $\varphi^{4}$ model), we write
$\sigma$ (resp. $\varphi$) instead of $\tau$ for the spin (resp. field)
variable. We drop the subscript $\rho$ in the above notations.

It is a well-known fact (see [Gri67]) that the measures
$\langle\cdot\rangle_{\Lambda,\beta}$ admit a weak limit when
$\Lambda\nearrow\mathbb{Z}^{d}$. We denote it by
$\langle\cdot\rangle_{\beta}$. When $d\geq 2$, the model undergoes a (second-
order) phase transition (see [AF86, ADCS15, GPPS22]) at a critical parameter
$\beta_{c}=\beta_{c}(\rho)\in(0,\infty)$ that can be defined as follows (see
[ABF87])

$\beta_{c}:=\inf\Big{\\{}\beta\geq
0,\>\chi(\beta):=\sum_{x\in\mathbb{Z}^{d}}\langle\tau_{0}\tau_{x}\rangle_{\beta}=\infty\Big{\\}}.$
(1.3)

The connection between the Ising model and the $\varphi^{4}$ model is
anticipated to be highly intricate, with both models believed to fall within
the same universality class. Renormalisation group arguments, as detailed in
[Gri70, Kad93] and the recent book [BBS19], suggest that many of their
properties, including _critical exponents_ , precisely coincide at their
respective critical points. Griffiths and Simon [SG73] laid the groundwork for
these profound connections by demonstrating that the $\varphi^{4}$ model
emerges as a specific near-critical scaling limit of a collection of mean-
field Ising models. This allows for a rigorous transfer of numerous useful
properties from the Ising model, such as correlation inequalities, to the
$\varphi^{4}$ model. Conversely, the Ising model can be derived as a limit of
the $\varphi^{4}$ model using the following (weak) convergence:

$\frac{\delta_{-1}+\delta_{1}}{2}=\lim_{g\rightarrow\infty}\frac{e^{-g(\varphi^{2}-1)^{2}+g}\textup{d}\varphi}{\int_{\mathbb{R}}e^{-gs^{4}+2gs^{2}}\mathrm{d}s}.$
(1.4)

The upper critical dimension of these models should be equal to $4$ meaning
that they should exhibit _mean-field behaviour_ above dimension $d\geq 5$, and
a _marginal_ (i.e. with potential logarithmic corrections) mean-field
behaviour in dimension $d=4$. In such setups, the expected behaviour of the
models undergoes a notable simplification, where lattice geometry ceases to
exert a significant influence. A prominent approach to investigating the mean-
field regime involves the computation of the critical exponents of the model.
Despite substantial progress, determining these exponents systematically in
the mean-field regime has remained a challenging task. Noteworthy methods such
as the _lace expansion_ and the _renormalisation group method_ have emerged as
powerful alternatives, showcasing significant successes. In the realm of spin
models on $\mathbb{Z}^{d}$, their success covers the weakly-coupled [GK85,
BBS14, ST16, BBS19, MPS23], high-dimensional [Sak07, Sak15, Sak22], or
sufficiently spread-out [Sak07, CS15, CS19, Sak22] setups. Other milestones
toward the study of the mean-field regime of these models include the proofs
of triviality of the scaling limits at criticality in dimension $d\geq 5$
independently obtained by Aizenman [Aiz82] and Fröhlich [Frö82], together with
the recent work of Aizenman and Duminil-Copin [ADC21] which establishes the
corresponding result in dimension $d=4$ (see also [Pan23] for a treatment of
models of _effective_ dimension $d_{\text{eff}}\geq 4$).

The situation in dimension $d=3$ is more complicated and much less understood.
Obtaining a rigorous understanding of the critical $3d$ Ising model remains
one of the main challenges of statistical mechanics. In the physics
literature, recent progress has been made regarding this problem using the so-
called _conformal bootstrap_ , see [ESPP+12, ESPP+14, RSDZ17].

In this work, we study the critical exponent related to the decay of the
model’s two-point function at criticality when $d\geq 3$. It is expected that
at criticality the two-point function of the above models decays
algebraically, thus justifying the introduction of the critical exponent
$\eta$ defined as follows: for all $x\in\mathbb{Z}^{d}\setminus\\{0\\}$,

$\langle\tau_{0}\tau_{x}\rangle_{\beta_{c}}=\frac{1}{|x|^{d-2+\eta-o(1)}},$
(1.5)

where $|\cdot|$ denotes the infinite norm on $\mathbb{R}^{d}$ and $o(1)$ is a
quantity tending to zero as $|x|$ tends to infinity. Since these models are
reflection positive (see [FSS76, FILS78, Bis09]), the _infrared bound_ and the
Messager–Miracle-Solé (MMS) inequalities333To be more precise, it relies on
the following simple consequences of the MMS inequalities which hold for all
$\beta>0$ (see [MMS77] or e.g. [ADC21, Proposition 5.1]): $(i)$ the sequence
$(\langle\tau_{0}\tau_{k\mathbf{e}_{1}}\rangle_{\beta})_{k\geq 0}$ is
decreasing; $(ii)$ for any $x\in\mathbb{Z}^{d}$, one has
$\langle\tau_{0}\tau_{(|x|_{1},0_{\bot})}\rangle_{\beta}\leq\langle\tau_{0}\tau_{x}\rangle_{\beta}\leq\langle\tau_{0}\tau_{(|x|,0_{\bot})}\rangle_{\beta},$
where $|\cdot|_{1}$ denotes the $\ell^{1}$ norm on $\mathbb{R}^{d}$. [MMS77]
provide the existence of $C=C(d)>0$ such that for all $\beta\leq\beta_{c}$,
and all $x\in\mathbb{Z}^{d}\setminus\\{0\\}$,

$\langle\tau_{0}\tau_{x}\rangle_{\beta}\leq\frac{C}{|x|^{d-2}}.$ (1.6)

Moreover, as a consequence of the Simon–Lieb inequality [Sim80, Lie04], Simon
proved the existence of $c=c(d)>0$ such that for all
$x\in\mathbb{Z}^{d}\setminus\\{0\\}$,

$\langle\tau_{0}\tau_{x}\rangle_{\beta_{c}}\geq\frac{c}{|x|^{d-1}}.$ (1.7)

These two results show that (if it exists) the critical exponent $\eta$
satisfies $0\leq\eta\leq 1$. We now survey existing results regarding the
computation of $\eta$.

In dimension $d=2$, the transfer matrix formalism leads to exact solutions of
the Ising model [Ons44, KO49, Yan52]. This formalism was later used by Wu
[Wu66, MW73] to demonstrate that $\eta=1/4$. Much more information was
obtained on the two-point function through the proof of conformal invariance
of Smirnov [Smi10], see for instance [CHI15] for a proof of conformal
invariance of the spin correlations.

When $d=3$, the results are confined to the bounds $0\leq\eta\leq 1$. However,
the conformal bootstrap method provides precise predictions [KPSDV16] on the
value of $\eta$, with the most accurate numerical estimate being $\eta\approx
0.0362978(20)$.

When $d\geq 4$, the mean-field regime can be derived. Lace expansion methods
[Sak07, Sak15, Sak22] were applied in very large dimension to obtain, not only
that for both models $\eta=0$, but also exact asymptotics for the critical
two-point function, showing that it is equivalent (at large scales) to
$A/|x|_{2}^{d-2}$ (where $A>0$ is a model-dependent constant). For the case of
the weakly-coupled $\varphi^{4}$ model (i.e. with small coupling $g$), lace
expansion was successfully implemented for $d\geq 5$ in [BHH21]. The
renormalisation group method was applied up to dimension $d=4$ to obtain exact
asymptotics in this setup, see [ST16]. Away from these perturbative regimes,
the best bounds on $\eta$ are the ones given by (1.6) and (1.7).

In this paper, we obtain new (near) critical lower bounds on the two-point
functions of these models when $d\geq 3$. In particular, we establish a sharp
lower bound when $d\geq 5$, and an almost sharp lower bound (with a
logarithmic correction) when $d=4$, showing that $\eta=0$ in these dimensions.
We also obtain a new bound on $\eta$ when $d=3$: if it exists, $\eta\leq 1/2$.
When $d\geq 4$, these results allow us to compute the critical exponent $\nu$
associated with the blow-up near criticality of the correlation lengths
$L(\beta),\>\xi(\beta)$, and $\xi_{p}(\beta)$ for $p>0$ (see (1.9), (1.19),
and (1.20) below). To the best of our knowledge, this result is new in our
setup. Let us mention that the exponents $\eta$ and $\nu$ are usually more
complicated to derive than the exponents $\alpha,\beta,\delta,\gamma$ (for the
second derivative of the free energy, the magnetisation, and the
susceptibility, see [Aiz82, AG83, AF86, BBS14]) as they depend on the graph
metric rather than intrinsic distances. Although only stated in the case of
the Ising and the $\varphi^{4}$ models, we will show that these results extend
for measures $\rho$ that belong to the Griffiths–Simon class of measures, see
Section 3.1.

###  1.1 The main theorem: a new inequality for two-point functions

Before stating the main results of this work, we introduce the distance below
which, for $\beta<\beta_{c}$, the model recovers critical features. The
following distance, called the _sharp length_ , was introduced in [DCT16,
Pan23].

######  Definition 1.1 (Sharp length).

Let $\rho$ be corresponding to either the Ising or the $\varphi^{4}$ model.
Let $\beta>0$. Let $S$ be a finite subset of $\mathbb{Z}^{d}$ containing $0$
and set

$\varphi_{\rho,\beta}(S):=\beta\sum_{\begin{subarray}{c}x\in S\\\ y\notin
S,\>y\sim x\end{subarray}}\langle\tau_{0}\tau_{x}\rangle_{S,\rho,\beta}.$
(1.8)

Define the sharp length by

$L(\beta)=L(\rho,\beta):=\inf\left\\{k\geq 1,\>\exists
S\subset\mathbb{Z}^{d},\>0\in S,\>\textup{diam}(S)\leq
2k,\>\varphi_{\rho,\beta}(S)<1/2\right\\},$ (1.9)

where $\textup{diam}(S):=\max\\{|x-y|,\>x,y\in S\\}$.

With this definition at hand, the Simon–Lieb inequality together with the
infrared bound (1.6) yield the following result: there exist $c,C>0$ such
that, if $\beta\leq\beta_{c}$ and $x\in\mathbb{Z}^{d}\setminus\\{0\\}$,

$\langle\tau_{0}\tau_{x}\rangle_{\beta}\leq C\Big{(}\frac{1}{|x|\wedge
L(\beta)}\Big{)}^{d-2}\exp\Big{(}-c\frac{|x|}{L(\beta)}\Big{)}.$ (1.10)

Our main result is the following inequality. If $n\geq 1$, introduce the
hyperplane $\mathbb{H}_{n}:=\\{x\in\mathbb{Z}^{d},\>x_{1}=n\\}$ and denote by
$\mathcal{R}_{n}$ the orthogonal reflection with respect to $\mathbb{H}_{n}$.
Also introduce the box $\Lambda_{n}:=[-n,n]^{d}\cap\mathbb{Z}^{d}$.

######  Theorem 1.2.

Let $d\geq 3$. There exist $c_{0},N_{0}>0$ such that for all
$\beta\leq\beta_{c}$ and for all $N_{0}\leq n\leq L(\beta)$,

$\beta\sum_{\begin{subarray}{c}x,y\in\Lambda_{n}\\\ y\sim
x\end{subarray}}\left(\langle\tau_{0}\tau_{x}\rangle_{\beta}-\langle\tau_{0}\tau_{\mathcal{R}_{n}(x)}\rangle_{\beta}\right)\langle\tau_{y}\tau_{\mathcal{R}_{n}(y)}\rangle_{\beta}\geq
c_{0}.$ (1.11)

In fact, the bound will take a slightly different form for the $\varphi^{4}$
model, which does not affect its applications, see Proposition 3.3 for
details.

We briefly explain the strategy of proof of the above inequality. We prove the
result for the Ising model first and then extend it to the $\varphi^{4}$ model
(and in fact, to all models in the Griffiths–Simon class of measures) using
the Griffiths–Simon approximation (see Proposition 3.2). Focusing now on the
Ising model, the usual way of obtaining a lower bound on the model’s two-point
function at criticality goes by noticing that
$\varphi_{\beta_{c}}(\Lambda_{n})\geq 1$, and additionally using Griffiths’
inequality together with the MMS inequalities. Our method is a refinement of
this inequality which consists in using the above observation for a well-
chosen— random— set $S$ instead of $\Lambda_{n}$. The randomness of this set
is obtained through the use of the _random current representation_ of the
Ising model (see Section 2.1). The result will follow from an appropriate
application of the switching principle to a reflected current, see Section
2.2. Such reflected currents have been used in [ADCTW19] to derive a
Messager–Miracle-Solé inequality without using reflection positivity.

The regularity properties provided by reflection positivity (see [ADC21,
Section 5] or [Pan23, Section 3]) allow to turn the inequality of Theorem 1.2
into a pointwise lower bound on the two-point function that is valid for all
dimensions $d\geq 3$. Let $\mathbf{e}_{i}$ be the unit vector of $i$-th
coordinate equal to $1$.

######  Theorem 1.3.

Let $d\geq 3$. There exist $c_{1},N_{1}>0$ such that for all
$\beta\leq\beta_{c}$ and for all $N_{1}\leq n\leq L(\beta)$,

$\langle\tau_{0}\tau_{n\mathbf{e}_{1}}\rangle_{\beta}\geq\frac{c_{1}}{\displaystyle\chi_{4n}(\beta)+n^{d-2}\sum_{1\leq
k\leq 2n}k\langle\tau_{0}\tau_{k\mathbf{e}_{1}}\rangle_{\beta}}.$ (1.12)

###### Proof.

Let $c_{0},N_{0}>0$ be given by Theorem 1.2. Let $\beta\leq\beta_{c}$ and
$n\geq N_{0}$. The constants $C_{i}>0$ below only depend on the dimension.
Divide the sum on the left-hand side of (1.11) according to whether $-n\leq
x_{1}\leq n/2$ or $n/2<x_{1}\leq n$. By the MMS inequalities,

$\sum_{\begin{subarray}{c}x,y\in\Lambda_{n}\\\ x_{1}\leq n/2\\\ y\sim
x\end{subarray}}\left(\langle\tau_{0}\tau_{x}\rangle_{\beta}-\langle\tau_{0}\tau_{\mathcal{R}_{n}(x)}\rangle_{\beta}\right)\langle\tau_{y}\tau_{\mathcal{R}_{n}(y)}\rangle_{\beta}\leq\sum_{\begin{subarray}{c}x,y\in\Lambda_{n}\\\
\>x_{1}\leq n/2\\\ y\sim
x\end{subarray}}\langle\tau_{0}\tau_{x}\rangle_{\beta}\langle\tau_{y}\tau_{\mathcal{R}_{n}(y)}\rangle_{\beta}\leq\chi_{n}(\beta)\langle\tau_{0}\tau_{(n/4)\mathbf{e}_{1}}\rangle_{\beta}.$
(1.13)

Then, using the spectral representation of these models and the MMS
inequalities, we obtain the following gradient estimate444The gradient
estimate is obtained similarly as in [ADC21, Proposition 5.9]: using the
spectral representation of the Ising model, for all
$x_{\bot}\in\mathbb{Z}^{d-1}$, there exists a measure
$\mu=\mu(x_{\bot},\beta)$ on $[0,1]$ such that for all $k\geq 1$,
$\langle\tau_{0}\tau_{(k,x_{\bot})}\rangle_{\beta}=\int_{0}^{1}\lambda^{k}\text{d}\mu(\lambda).$
The above display implies that
$\langle\tau_{0}\tau_{(k,x_{\bot})}\rangle_{\beta}-\langle\tau_{0}\tau_{(k+1,x_{\bot})}\rangle_{\beta}=\dfrac{1}{k}\int_{0}^{1}k\lambda^{k}(1-\lambda)\text{d}\mu(\lambda).$
The estimate (1.14) follows by telescoping and replacing
$k\lambda^{k}(1-\lambda)$ by $C\lambda^{k/2}$ for some constant $C>0$.

$\langle\tau_{0}\tau_{x}\rangle_{\beta}-\langle\tau_{0}\tau_{\mathcal{R}_{n}(x)}\rangle_{\beta}\leq
C_{1}\frac{|x-\mathcal{R}_{n}(x)|}{n}\langle\tau_{0}\tau_{(n/4)\mathbf{e}_{1}}\rangle_{\beta}.$
(1.14)

Using (1.14), we get

$\displaystyle\sum_{\begin{subarray}{c}x,y\in\Lambda_{n}\\\ x_{1}>n/2\\\ y\sim
x\end{subarray}}\left(\langle\tau_{0}\tau_{x}\rangle_{\beta}-\langle\tau_{0}\tau_{\mathcal{R}_{n}(x)}\rangle_{\beta}\right)\langle\tau_{y}\tau_{\mathcal{R}_{n}(y)}\rangle_{\beta}$
$\displaystyle\leq
C_{2}\frac{\langle\tau_{0}\tau_{(n/4)\mathbf{e}_{1}}\rangle_{\beta}}{n}\sum_{\begin{subarray}{c}x\in\Lambda_{n}\\\
\>x_{1}\geq
n/2\end{subarray}}|x-\mathcal{R}_{n}(x)|\langle\tau_{x}\tau_{\mathcal{R}_{n}(x)}\rangle_{\beta}$
$\displaystyle\leq
C_{3}\langle\tau_{0}\tau_{(n/4)\mathbf{e}_{1}}\rangle_{\beta}n^{d-2}\sum_{1\leq
k\leq n/2}k\langle\tau_{0}\tau_{k\mathbf{e_{1}}}\rangle_{\beta}.$ (1.15)

Plugging (1.13) and (1.15) in (1.11) yields

$\frac{c_{1}}{\beta}\leq
C_{4}\langle\tau_{0}\tau_{(n/4)\mathbf{e}_{1}}\rangle_{\beta}\Big{(}\chi_{n}(\beta)+n^{d-2}\sum_{1\leq
k\leq n/2}k\langle\tau_{0}\tau_{k\mathbf{e}_{1}}\rangle_{\beta}\Big{)},$
(1.16)

from which the proof follows readily. ∎

###  1.2 Applications

We now list a number of applications and we include their (short) derivation.

#### Lower bounds for the spin-spin correlations

To begin, the infrared bound (1.6) allows to obtain a more explicit
formulation of Theorem 1.3. It can be interpreted as the fact that $\eta=0$
when $d\geq 4$.

######  Theorem 1.4 (Pointwise lower bound in dimension $d\geq 4$).

Let $d\geq 4$. There exists $c=c(d)>0$ such that for all $\beta\leq\beta_{c}$
and for all $x\in\mathbb{Z}^{d}$ with $2\leq|x|\leq L(\beta)$,

$\displaystyle\langle\tau_{0}\tau_{x}\rangle_{\beta}\geq\begin{cases}\displaystyle\frac{c}{|x|^{d-2}}&\text{
if }d\geq 5,\\\ \displaystyle\frac{c}{|x|^{2}\log|x|}&\text{ if
}d=4.\end{cases}$ (1.17)

###### Proof.

Let $c_{1},N_{1}>0$ be given by Theorem 1.3. Let $\beta\leq\beta_{c}$ and
$N_{1}\leq n\leq L(\beta)$. Using the infrared bound (1.6) gives

$\chi_{4n}(\beta)+n^{d-2}\sum_{1\leq k\leq
2n}k\langle\tau_{0}\tau_{k\mathbf{e}_{1}}\rangle_{\beta}\leq
C_{1}\Big{(}n^{2}+n^{d-2}\sum_{k=1}^{2n}\frac{1}{k^{d-3}}\Big{)}\leq\begin{cases}\displaystyle
C_{2}n^{d-2}&\text{ if }d\geq 5,\\\ \displaystyle C_{3}n^{2}\log n&\text{ if
}d=4.\end{cases}$ (1.18)

The proof follows readily by Theorem 1.3, the MMS inequalities, and by
choosing $c>0$ small enough (in particular to include the case $|x|\leq
N_{1}$). ∎

When $d=3$ we do not obtain a more explicit pointwise lower bound, but we
still improve on the existing bound on $\eta$.

######  Theorem 1.5.

Let $d=3$. If the critical exponent $\eta$ exists, it satisfies
$\eta\leq\frac{1}{2}$.

###### Proof.

The proof follows by plugging the estimate provided by the existence of $\eta$
in (1.12). ∎

#### Behaviour of the correlation lengths

Besides the sharp length $L(\beta)$, there are other natural typical lengths
that one may define. The following quantity, called the the _correlation
length_ , is well defined for $\beta<\beta_{c}$

$\xi(\beta):=-\lim_{n\rightarrow\infty}\left(\frac{1}{n}\log\langle\tau_{0}\tau_{n\mathbf{e}_{1}}\rangle_{\beta}\right)^{-1}.$
(1.19)

For $p>0$, one may also define

$\xi_{p}(\beta):=\Big{(}\frac{1}{\chi(\beta)}\sum_{x\in\mathbb{Z}^{d}}|x|^{p}\langle\tau_{0}\tau_{x}\rangle_{\beta}\Big{)}^{1/p}.$
(1.20)

It is expected that there exist $\nu_{1},\nu_{2}$, and $\nu_{3}>0$ such that
for $\beta<\beta_{c}$

$L(\beta)=(\beta_{c}-\beta)^{-\nu_{1}/2+o(1)},\qquad\xi(\beta)=(\beta_{c}-\beta)^{-\nu_{2}/2+o(1)},\qquad\xi_{p}(\beta)=(\beta_{c}-\beta)^{-\nu_{3}/2+o(1)},$
(1.21)

where $o(1)$ tends to zero as $\beta$ tends to $\beta_{c}$, and where
$\nu_{3}$ may depend on $p>0$. The relation between these quantities is not
clear a priori. Theorem 1.4 allows to compute these exponents when $d\geq 4$.

######  Theorem 1.6.

Let $d\geq 4$. Let $p>0$. Then, for $\beta<\beta_{c}$,

$L(\beta)=(\beta_{c}-\beta)^{-1/2+o(1)},$ (1.22)
$\xi(\beta)=(\beta_{c}-\beta)^{-1/2+o(1)},$ (1.23)
$\xi_{p}(\beta)=(\beta_{c}-\beta)^{-1/2+o(1)},$ (1.24)

where $o(1)$ tends to $0$ as $\beta$ tends to $\beta_{c}$.

###### Proof.

By [Aiz82, AG83] we know that $\chi(\beta)=(\beta_{c}-\beta)^{-1+o(1)}$.
Theorem 1.4 and (1.10) yield that $\chi(\beta)=L(\beta)^{2+o(1)}$. Combined
with the estimate on $\chi(\beta)$ this yields (1.22). The estimate (1.24)
follows by similar arguments. We obtain (1.23) by proving that
$L(\beta)=\xi(\beta)^{1+o(1)}$. Using (1.10) yields that $\xi(\beta)\leq
C_{1}L(\beta)$ for some $C_{1}=C_{1}(d)>0$. Moreover, by classical sub-
additivity arguments, one has that
$\langle\tau_{0}\tau_{n\mathbf{e}_{1}}\rangle_{\beta}\leq\exp(-n/\xi(\beta))$.
Together with the MMS inequalities, this yields

$\varphi_{\beta}(\Lambda_{n})\leq C_{2}n^{d-1}e^{-n/\xi(\beta)},$ (1.25)

which implies that $L(\beta)\leq C_{3}\xi(\beta)\log\xi(\beta)$. ∎

######  Remark 1.7.

When $d\geq 5$, there exists $c_{1},C_{1}>0$ such that for all
$\beta<\beta_{c}$, $c(\beta_{c}-\beta)^{-1}\leq\chi(\beta)\leq
C(\beta_{c}-\beta)^{-1}$, see [Aiz82]. This observations improves (1.22) and
yields the existence of $c_{2},C_{2}>0$ such that for all $\beta<\beta_{c}$,
$c_{2}(\beta_{c}-\beta)^{-1/2}\leq L(\beta)\leq
C_{2}(\beta_{c}-\beta)^{-1/2}$. Combined with (1.10) and Theorem 1.4, we
obtain the following near-critical estimate on the two-point function: there
exist $c,C>0$ such that for all $\beta<\beta_{c}$,

$\langle\tau_{0}\tau_{x}\rangle_{\beta}\leq\frac{C}{|x|^{d-2}}e^{-c(\beta_{c}-\beta)^{1/2}|x|},\qquad
x\in\mathbb{Z}^{d}\setminus\\{0\\},$ (1.26)
$\langle\tau_{0}\tau_{x}\rangle_{\beta}\geq\frac{c}{|x|^{d-2}},\qquad
1\leq|x|\leq L(\beta).$ (1.27)

Such estimates have been essential for the study of the _torus plateau_ in
various models of statistical mechanics [Sla23, HMS23, Liu23], and are used in
[LPS] to establish it in the case of the Ising model in dimension $d\geq 5$.

#### Divergence of the bubble diagram

As described by Aizenman in his seminal work [Aiz82], one may define the
_bubble diagram_

$B(\beta):=\sum_{x\in\mathbb{Z}^{d}}\langle\tau_{0}\tau_{x}\rangle_{\beta}^{2},$
(1.28)

whose finiteness at criticality allows to establish that some critical
exponents exist and take their mean-field values. The condition
$B(\beta_{c})<\infty$ also implies triviality of the critical scaling limits
of the model, see [Pan23, Theorem D.2]. The infrared bound yields that this
condition is satisfied for $d\geq 5$. The estimates for $d=2$ imply that the
bubble diagram diverges. In the following result, we complete the picture by
proving that the bubble diagram diverges for $d=3,4$.

######  Theorem 1.8 (Divergence of the bubble diagram).

Let $d=3,4$. Then, $B(\beta_{c})=\infty.$

###### Proof.

We proceed by contradiction and assume that $B(\beta_{c})$ is finite. Using
Cauchy–Schwarz inequality one has

$\sum_{\begin{subarray}{c}x,y\in\Lambda_{n}\\\ x_{1}\leq n/2\\\ y\sim
x\end{subarray}}\langle\tau_{0}\tau_{x}\rangle_{\beta_{c}}\langle\tau_{y}\tau_{\mathcal{R}_{n}(y)}\rangle_{\beta_{c}}\leq
C_{1}\sqrt{B(\beta_{c})}\sqrt{B_{\geq n/2}(\beta_{c})},$ (1.29)

where $B_{\geq k}(\beta)$ is the sum of the squares of spin-spin correlations
for $x\notin\Lambda_{k}$. Since $B(\beta_{c})<\infty$, the sum on the left-
hand side of (1.29) tends to $0$ as $n$ tends to infinity. Using Theorem 1.2
together with the MMS inequalities, there exists $c_{1}>0$ such that for $n$
large enough,

$\displaystyle c_{1}$ $\displaystyle\leq
C_{2}n^{d-2}\langle\tau_{0}\tau_{(n/4)\mathbf{e}_{1}}\rangle_{\beta_{c}}\sum_{1\leq
k\leq n}k\langle\tau_{0}\tau_{k\mathbf{e}_{1}}\rangle_{\beta_{c}}.$ (1.30)

Assume first that $d=4$. Using Cauchy–Schwarz inequality and the MMS
inequalities once more gives

$\sum_{1\leq k\leq
n}k\langle\tau_{0}\tau_{k\mathbf{e}_{1}}\rangle_{\beta_{c}}\leq
C_{3}\sqrt{\log n}\sqrt{\sum_{1\leq k\leq
n}k^{3}\langle\tau_{0}\tau_{k\mathbf{e}_{1}}\rangle_{\beta_{c}}^{2}}\leq
C_{4}\sqrt{\log n}\sqrt{B(\beta_{c})}.$ (1.31)

Putting the two previous displayed equations together gives

$\langle\tau_{0}\tau_{(n/4)\mathbf{e}_{1}}\rangle_{\beta_{c}}\geq\frac{c_{2}}{n^{d-2}\sqrt{\log
n}}.$ (1.32)

Using the MMS inequalities we obtain that a similar bound holds for all
$x\in\mathbb{Z}^{d}$. We find that $B(\beta_{c})=\infty$, which is a
contradiction.

If $d=3$, using Cauchy–Schwarz inequality as in (1.31) gives

$\sum_{1\leq k\leq
n}k\langle\tau_{0}\tau_{k\mathbf{e}_{1}}\rangle_{\beta_{c}}\leq
C_{5}\sqrt{n}\sqrt{B(\beta_{c})},$ (1.33)

and hence, using (1.30) again,

$\langle\tau_{0}\tau_{(n/4)\mathbf{e}_{1}}\rangle_{\beta_{c}}\geq\frac{c_{3}}{n^{3/2}}.$
(1.34)

This contradicts, once again, the finiteness of the bubble diagram. ∎

######  Remark 1.9.

It is possible to use Theorem 1.3 to obtain a quantitative estimate on the
blow-up of the bubble diagram when $d=3,4$: more precisely, there exists
$c=c(d)>0$ such that, for all $\beta\leq\beta_{c}$, for all $1\leq n\leq
L(\beta)$,

$B_{n}(\beta)\geq\begin{cases}\displaystyle c\sqrt{\log n}&\text{ if }d=3,\\\
\displaystyle c\sqrt{\log\log n}&\text{ if }d=4.\end{cases}$ (1.35)

##  2 Proof of Theorem 1.2 for the Ising model

As mentioned above, the proof builds on the random current representation of
the Ising model. We briefly recall this classical expansion and refer to
[DC17] for a more detailed introduction.

###  2.1 The random current representation

Let $\Lambda$ be a finite subset of $\mathbb{Z}^{d}$.

######  Definition 2.1.

A current $\mathbf{n}$ on $\Lambda$ is a function defined on the set
$E(\Lambda):=\\{\\{x,y\\},\>x,y\in\Lambda\textup{ and }x\sim y\\}$ and taking
its values in $\mathbb{N}=\\{0,1,\ldots\\}$. We let $\Omega_{\Lambda}$ be the
set of currents on $\Lambda$. The set of _sources_ of $\mathbf{n}$, denoted by
$\partial\mathbf{n}$, is defined as

$\partial\mathbf{n}:=\Big{\\{}x\in\Lambda\>,\>\sum_{y\sim
x}\mathbf{n}_{x,y}\textup{ is odd}\Big{\\}}.$ (2.1)

We also set

$w_{\beta}(\mathbf{n}):=\prod_{\begin{subarray}{c}\\{x,y\\}\subset\Lambda\\\
x\sim y\end{subarray}}\dfrac{\beta^{\mathbf{n}_{x,y}}}{\mathbf{n}_{x,y}!}.$
(2.2)

If $A\subset\Lambda$, define a probability measure
$\mathbf{P}_{\Lambda,\beta}^{A}$ on $\Omega_{\Lambda}$ as follows: for a
current $\mathbf{n}$ on $\Lambda$,

$\mathbf{P}_{\Lambda,\beta}^{A}[\mathbf{n}]:=\mathds{1}_{\partial\mathbf{n}=A}\frac{w_{\beta}(\mathbf{n})}{Z_{\Lambda,\beta}^{A}},$
(2.3)

where
$Z_{\Lambda,\beta}^{A}:=\sum_{\partial\mathbf{n}=A}w_{\beta}(\mathbf{n})$ is a
normalisation constant. If $\mathcal{E}\subset\Omega_{\Lambda}$, we will also
write
$Z_{\Lambda,\beta}^{A}[\mathcal{E}]:=\sum_{\partial\mathbf{n}=A}w_{\beta}(\mathbf{n})\mathds{1}_{\mathbf{n}\in\mathcal{E}}$.
As proved in [ADCS15], if $A$ is a finite (even) subset of $\mathbb{Z}^{d}$,
the sequence of probability measures
$(\mathbf{P}_{\Lambda,\beta}^{A})_{\Lambda\subset\mathbb{Z}^{d}}$ admits a
weak limit as $\Lambda\nearrow\mathbb{Z}^{d}$ that we denote by
$\mathbf{P}_{\beta}^{A}$.

It is possible to expand the correlation functions of the Ising model to
relate them to currents: for $A\subset\Lambda$, if $\sigma_{A}:=\prod_{x\in
A}\sigma_{x}$ we get

$\left\langle\sigma_{A}\right\rangle_{\Lambda,\beta}=\dfrac{\sum_{\partial\mathbf{n}=A}w_{\beta}(\mathbf{n})}{\sum_{\partial\mathbf{n}=\emptyset}w_{\beta}(\mathbf{n})}.$
(2.4)

A current $\mathbf{n}$ with empty source set can be seen as the edge count of
a multigraph obtained as a union of loops. Adding sources to a current
configuration comes down to adding a collection of paths connecting pairwise
the sources. We will identify $\mathbf{n}$ with its multigraph $\mathcal{N}$
in which the vertex set is $\Lambda$ and where there are exactly
$\mathbf{n}_{x,y}$ edges between $x$ and $y$. We will write
$w_{\beta}(\mathcal{N}):=w_{\beta}(\mathbf{n})$ and
$\partial\mathcal{N}:=\partial\mathbf{n}$.

As it turns out, connectivity properties of the multigraph induced by a
current will play a crucial role in the analysis of the underlying Ising
model. For a multigraph $\mathcal{N}$ on $\Lambda$ and $x,y\in\Lambda$, we
write $x\overset{\mathcal{N}}{\longleftrightarrow}y$ if $x$ is connected to
$y$ in $\mathcal{N}$. More generally, if $A,B\subset\Lambda$, we write
$A\overset{\mathcal{N}}{\longleftrightarrow}B$ if there exists $x\in A$ and
$y\in B$ such that $x\overset{\mathcal{N}}{\longleftrightarrow}y$.

The main interest of the above expansion lies in the following result called
the switching principle, see [ADCTW19, Lemma 2.1]. This combinatorial result
first appeared in [GHS70] to prove the concavity of the magnetisation of an
Ising model with a positive external field, but the probabilistic picture
attached to it flourished in [Aiz82].

######  Lemma 2.2 (Switching principle).

For any multigraphs $\mathcal{M},\mathcal{K}$ with vertex set $\Lambda$, any
$A\subset\Lambda$, and any function $f$ of a current: if
$\mathcal{K}\subset\mathcal{M}$, then

$\sum_{\begin{subarray}{c}\mathcal{N}\subset\mathcal{M}\\\
\partial\mathcal{N}=A\end{subarray}}f(\mathcal{N})=\sum_{\begin{subarray}{c}\mathcal{N}\subset\mathcal{M}\\\
\partial\mathcal{N}=A\Delta\partial\mathcal{K}\end{subarray}}f(\mathcal{N}\Delta\mathcal{K}).$
(2.5)

###  2.2 Proof of the theorem

We now turn to the proof of Theorem 1.2 in the case of the Ising model. We
will follow [ADCTW19] and consider “folded” single currents on
$\mathbb{Z}^{d}$. We partition the edge-set $E(\mathbb{Z}^{d})$ of
$\mathbb{Z}^{d}$ into three disjoint sets:

$\displaystyle E_{-}(\mathbb{H}_{n})$ $\displaystyle:=\Big{\\{}\\{u,v\\}\in
E(\mathbb{Z}^{d})\textup{ s.t.~{}at least one endpoint is strictly on the left
of $\mathbb{H}_{n}$}\Big{\\}},$ $\displaystyle E_{+}(\mathbb{H}_{n})$
$\displaystyle:=\Big{\\{}\\{u,v\\}\in E(\mathbb{Z}^{d})\textup{ s.t.~{}at
least one endpoint is strictly on the right of $\mathbb{H}_{n}$}\Big{\\}},$
$\displaystyle E_{0}(\mathbb{H}_{n})$
$\displaystyle:=E(\mathbb{Z}^{d})\setminus\left(E_{-}(\mathbb{H}_{n})\cup
E_{+}(\mathbb{H}_{n})\right).$

Decompose $\mathbf{n}\in\Omega_{\mathbb{Z}^{d}}$ into its restrictions
$\mathbf{n}_{-}$, $\mathbf{n}_{+}$ and $\mathbf{n}_{0}$ to the above three
subsets of $E(\mathbb{Z}^{d})$. For the purpose of later applying the
switching principle, it is convenient to consider the multigraph
representation of these objects. Consider the multigraph $\mathcal{M}_{n}$
obtained by taking the union of the multigraph $\mathcal{N}_{-}$ associated
with $\mathbf{n}_{-}$ and the reflection $\mathcal{R}_{n}(\mathcal{N}_{+})$ of
the multigraph $\mathcal{N}_{+}$ associated with $\mathbf{n}_{+}$. In this
context, the switching principle yields the following result that we will use
several times below.

######  Lemma 2.3 (Switching principle for reflected currents).

Let $\beta>0$. Let $\Lambda\subset\mathbb{Z}^{d}$ be finite and symmetric
under $\mathcal{R}_{n}$. Assume that $A\subset\Lambda$ and $x\in\Lambda$ are
strictly on the left of $\mathbb{H}_{n}$. Then,

$Z_{\Lambda,\beta}^{A}[x\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}]=Z^{A\Delta\\{x,\mathcal{R}_{n}(x)\\}}_{\Lambda,\beta}[x\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}].$
(2.6)

###### Proof.

By definition,

$Z^{A}_{\Lambda,\beta}[x\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}]=\sum_{\mathbf{n}_{0}}w_{\beta}(\mathbf{n}_{0})\sum_{\begin{subarray}{c}\mathbf{n}_{-},\>\mathbf{n}_{+}\\\
\partial(\mathbf{n}_{-}+\mathbf{n}_{+})=\partial\mathbf{n}_{0}\Delta
A\end{subarray}}\mathds{1}_{x\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}}\,w_{\beta}(\mathbf{n}_{-})w_{\beta}(\mathbf{n}_{+}).$
(2.7)

When it exists, call $\Gamma=\Gamma(\mathcal{M}_{n},x)$ the earliest path
connecting $x$ to $\mathbb{H}_{n}$ in $\mathcal{M}_{n}$ (according to some
fixed ordering). In order to be in the setup of application of Lemma 2.2, we
condition on the sources of $\mathbf{n}_{-}$ and $\mathbf{n}_{+}$ as well as
on $\Gamma$ and adopt the multigraph notations for the currents. Together with
(2.7) we obtain,

$\displaystyle
Z^{A}_{\Lambda,\beta}[x\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}]$
$\displaystyle=\sum_{\mathbf{n}_{0}}w_{\beta}(\mathbf{n}_{0})\sum_{\begin{subarray}{c}D\subset\mathbb{H}_{n}\cap\Lambda\end{subarray}}\sum_{\begin{subarray}{c}\alpha\in\mathbb{H}_{n}\\\
\gamma:x\rightarrow\alpha\end{subarray}}\sum_{\begin{subarray}{c}\partial\mathbf{n}_{-}=D\Delta\partial\mathbf{n}_{0}\Delta
A\\\
\partial\mathbf{n}_{+}=D\end{subarray}}w_{\beta}(\mathbf{n}_{-})w_{\beta}(\mathbf{n}_{+})\mathds{1}_{\Gamma=\gamma}$
$\displaystyle=\sum_{\mathbf{n}_{0}}w_{\beta}(\mathbf{n}_{0})\sum_{\begin{subarray}{c}D\subset\mathbb{H}_{n}\cap\Lambda\end{subarray}}\sum_{\gamma:x\rightarrow\alpha}\sum_{\partial\mathcal{M}_{n}=\partial\mathbf{n}_{0}\Delta
A}w_{\beta}(\mathcal{M}_{n})\mathds{1}_{\Gamma=\gamma}\sum_{\begin{subarray}{c}\mathcal{N}\subset\mathcal{M}_{n}\\\
\partial\mathcal{N}=D\end{subarray}}1$
$\displaystyle=\sum_{\mathbf{n}_{0}}w_{\beta}(\mathbf{n}_{0})\sum_{\begin{subarray}{c}D\subset\mathbb{H}_{n}\cap\Lambda\end{subarray}}\sum_{\gamma:x\rightarrow\alpha}\sum_{\partial\mathcal{M}_{n}=\partial\mathbf{n}_{0}\Delta
A}w_{\beta}(\mathcal{M}_{n})\mathds{1}_{\Gamma=\gamma}\sum_{\begin{subarray}{c}\mathcal{N}\subset\mathcal{M}_{n}\\\
\partial\mathcal{N}=D\Delta\\{x,\alpha\\}\end{subarray}}1$

where we used the switching principle from Lemma 2.2 in the third line. As a
result, we obtained,

$\displaystyle
Z^{A}_{\Lambda,\beta}[x\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}]$
$\displaystyle=\sum_{\mathbf{n}_{0}}w_{\beta}(\mathbf{n}_{0})\sum_{\begin{subarray}{c}D\subset\mathbb{H}_{n}\cap\Lambda\end{subarray}}\sum_{\begin{subarray}{c}\partial\mathbf{n}_{-}=D\Delta\partial\mathbf{n}_{0}\Delta
A\Delta\\{x,\alpha\\}\\\
\partial\mathbf{n}_{+}=D\Delta\\{\mathcal{R}_{n}(x),\alpha\\}\end{subarray}}w_{\beta}(\mathbf{n}_{-})w_{\beta}(\mathbf{n}_{+})\mathds{1}_{x\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}}$
$\displaystyle=\sum_{\mathbf{n}_{0}}w_{\beta}(\mathbf{n}_{0})\sum_{\begin{subarray}{c}\mathbf{n}_{-},\mathbf{n}_{+}\\\
\partial(\mathbf{n}_{-}+\mathbf{n}_{+})=\partial\mathbf{n}_{0}\Delta
A\Delta\\{x,\mathcal{R}_{n}(x)\\}\end{subarray}}w_{\beta}(\mathbf{n}_{-})w_{\beta}(\mathbf{n}_{+})\mathds{1}_{x\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}}$
$\displaystyle=Z^{A\Delta\\{x,\mathcal{R}_{n}(x)\\}}_{\Lambda,\beta}[x\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}],$

which concludes the proof. ∎

$\mathbb{H}_{n}$

Figure 1: The set $\mathcal{S}_{n}$ is represented in grey. On its boundary, a
point $x$ (in black) is next to a point $y$ (in red) which connects to
$\mathbb{H}_{n}$ in $\mathcal{M}_{n}$, i.e. which does not belong to
$\mathcal{S}_{n}(+\mathbf{e}_{1})$.

The above quantities were all defined with respect to the direction
$\mathbf{e}_{1}$. In order to generalise these objects, let us write
$\mathcal{M}_{n}(+\mathbf{e}_{1})=\mathcal{M}_{n}$,
$\mathcal{R}_{n}(+\mathbf{e}_{1})=\mathcal{R}_{n}$,
$\mathbb{H}_{n}(+\mathbf{e}_{1})=\mathbb{H}_{n}$, and define

$\mathcal{S}_{n}(+\mathbf{e}_{1}):=\Big{\\{}x\in\Lambda_{n},\>x\overset{\mathcal{M}_{n}(+\mathbf{e}_{1})}{\centernot\longleftrightarrow}\mathbb{H}_{n}(+\mathbf{e}_{1})\Big{\\}}.$
(2.8)

Similarly, define $\mathbb{H}_{n}(\pm\mathbf{e}_{i})$,
$\mathcal{R}_{n}(\pm\mathbf{e}_{i})$, $\mathcal{M}_{n}(\pm\mathbf{e}_{i})$,
and $\mathcal{S}_{n}(\pm\mathbf{e}_{i})$ for $1\leq i\leq d$ in the other
$2d-1$ directions. Let (see Figure 1)

$\mathcal{S}_{n}:=\bigcap_{1\leq i\leq
d}\left(\mathcal{S}_{n}(+\mathbf{e}_{i})\cap\mathcal{S}_{n}(-\mathbf{e}_{i})\right).$
(2.9)

Note that by definition, $\mathcal{S}_{n}\subset\Lambda_{n-1}$. Recall the
definitions of $\mathbf{P}^{\emptyset}_{\beta}$ and $\varphi_{\beta}(S)$ from
above.

By definition of $L(\beta)$, if $0\leq n\leq L(\beta)$ and
$S\subset\Lambda_{n}$, then $\varphi_{\beta}(S)\geq 1/2$. As a result, if
$1\leq n\leq L(\beta)$, then

$\tfrac{1}{2}\mathbf{P}_{\beta}^{\emptyset}[0\in\mathcal{S}_{n}]\leq\mathbf{E}_{\beta}^{\emptyset}[\mathds{1}_{0\in\mathcal{S}_{n}}\varphi_{\beta}(\mathcal{S}_{n})].$
(2.10)

The first step is to observe that the left-hand side in (2.10) is bounded away
from $0$.

######  Lemma 2.4.

There exist $c,N_{0}>0$ such that for $\beta\leq\beta_{c}$ and $n\geq N_{0}$,

$\mathbf{P}_{\beta}^{\emptyset}[0\in\mathcal{S}_{n}]\geq c.$ (2.11)

###### Proof.

Using a union bound and the symmetries of $\mathbf{P}_{\beta}^{\emptyset}$,

$\mathbf{P}_{\beta}^{\emptyset}[0\notin\mathcal{S}_{n}]\leq\sum_{i=1}^{d}(\mathbf{P}_{\beta}^{\emptyset}[0\notin\mathcal{S}_{n}(+\mathbf{e}_{i})]+\mathbf{P}_{\beta}^{\emptyset}[0\notin\mathcal{S}_{n}(-\mathbf{e}_{i})])=2d\mathbf{P}_{\beta}^{\emptyset}[0\notin\mathcal{S}_{n}(+\mathbf{e}_{1})].$
(2.12)

Fix for a moment $\Lambda\subset\mathbb{Z}^{d}$ finite, symmetric under
$\mathcal{R}_{n}$, and containing $\Lambda_{4n}$. The switching principle for
reflected currents (Lemma 2.3) implies

$Z^{\emptyset}_{\Lambda,\beta}[0\notin\mathcal{S}_{n}(+\mathbf{e}_{1})]=Z_{\Lambda,\beta}^{\emptyset}[0\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}]=\sum_{\mathbf{n}\in\Omega_{\Lambda},\>\partial\mathbf{n}=\emptyset}w_{\beta}(\mathbf{n})\mathds{1}_{0\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}}=Z_{\Lambda,\beta}^{\\{0,\mathcal{R}_{n}(0)\\}}.$
(2.13)

Dividing by $Z^{\emptyset}_{\Lambda,\beta}$, and using monotonicity of the
two-point function in $\Lambda$, we find that for $n\geq 1$,

$\mathbf{P}^{\emptyset}_{\Lambda,\beta}[0\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}]\leq\langle\sigma_{0}\sigma_{2n\mathbf{e}_{1}}\rangle_{\beta}.$
(2.14)

The proof follows readily from the infrared bound (1.6) and the limit
$\Lambda\nearrow\mathbb{Z}^{d}$. ∎

Combining (2.10) and Lemma 2.4, we obtain that for some $c,N_{0}>0$, if
$N_{0}\leq n\leq L(\beta)$,

$\frac{c}{2\beta}\leq\mathbf{E}^{\emptyset}_{\beta}\Big{[}\mathds{1}_{0\in\mathcal{S}_{n}}\sum_{\begin{subarray}{c}x\in\mathcal{S}_{n}\\\
y\sim
x,\>y\notin\mathcal{S}_{n}\end{subarray}}\langle\sigma_{0}\sigma_{x}\rangle_{\mathcal{S}_{n},\beta}\Big{]}.$
(2.15)

Notice that if $x\in\mathcal{S}_{n}$, $y\sim x$ and $y\notin\mathcal{S}_{n}$,
then there exist $1\leq i\leq d$ and $\varepsilon\in\\{\pm 1\\}$ such that
$y\notin\mathcal{S}_{n}(\varepsilon\mathbf{e}_{i})$. As a result,

$\displaystyle\mathds{1}_{0\in\mathcal{S}_{n}}\sum_{\begin{subarray}{c}x\in\mathcal{S}_{n}\\\
y\sim
x,\>y\notin\mathcal{S}_{n}\end{subarray}}\langle\sigma_{0}\sigma_{x}\rangle_{\mathcal{S}_{n},\beta}$
$\displaystyle\leq\mathds{1}_{0\in\mathcal{S}_{n}}\sum_{i=1}^{d}\sum_{\varepsilon\in\\{\pm
1\\}}\sum_{\begin{subarray}{c}x\in\mathcal{S}_{n}\\\ y\sim
x,\>y\notin\mathcal{S}_{n}(\varepsilon\mathbf{e}_{i})\end{subarray}}\langle\sigma_{0}\sigma_{x}\rangle_{\mathcal{S}_{n},\beta}$
$\displaystyle\leq\mathds{1}_{0\in\mathcal{S}_{n}}\sum_{i=1}^{d}\sum_{\varepsilon\in\\{\pm
1\\}}\sum_{\begin{subarray}{c}x\in\mathcal{S}_{n}(\varepsilon\mathbf{e}_{i})\cap\Lambda_{n-1}\\\
y\sim
x,\>y\notin\mathcal{S}_{n}(\varepsilon\mathbf{e}_{i})\end{subarray}}\langle\sigma_{0}\sigma_{x}\rangle_{\mathcal{S}_{n}(\varepsilon\mathbf{e}_{i}),\beta},$

where we used that
$\mathcal{S}_{n}\subset\mathcal{S}_{n}(\varepsilon\mathbf{e}_{i})\cap\Lambda_{n-1}$
together with Griffiths’ inequality on the second line. Using the symmetries
of $\mathbf{P}_{\beta}^{\emptyset}$ and (2.15), we get

$\frac{c}{4d\beta}\leq\mathbf{E}_{\beta}^{\emptyset}\Big{[}\mathds{1}_{0\in\mathcal{S}_{n}^{1}}\sum_{\begin{subarray}{c}x\in\mathcal{S}_{n}^{1}\\\
y\sim
x,\>y\in\Lambda_{n}\end{subarray}}\mathds{1}_{y\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}}\langle\sigma_{0}\sigma_{x}\rangle_{\mathcal{S}_{n}^{1},\beta}\Big{]},$
(2.16)

where
$\mathcal{S}_{n}^{1}:=\mathcal{S}_{n}(+\mathbf{e}_{1})\cap\Lambda_{n-1}$.
Theorem 1.2 follows from (2.16) and Lemma 2.5 below.

######  Lemma 2.5.

For $\beta\leq\beta_{c}$,

$\mathbf{E}^{\emptyset}_{\beta}\Big{[}\mathds{1}_{0\in\mathcal{S}_{n}^{1}}\sum_{\begin{subarray}{c}x\in\mathcal{S}_{n}^{1}\\\
y\sim
x,\>y\in\Lambda_{n}\end{subarray}}\mathds{1}_{y\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}}\langle\sigma_{0}\sigma_{x}\rangle_{\mathcal{S}_{n}^{1},\beta}\Big{]}\leq\sum_{\begin{subarray}{c}x,y\in\Lambda_{n}\\\
y\sim
x\end{subarray}}(\langle\sigma_{0}\sigma_{x}\rangle_{\beta}-\langle\sigma_{0}\sigma_{\mathcal{R}_{n}(x)}\rangle_{\beta})\langle\sigma_{y}\sigma_{\mathcal{R}_{n}(y)}\rangle_{\beta}.$
(2.17)

###### Proof.

Fix $\Lambda\subset\mathbb{Z}^{d}$ finite, symmetric under $\mathcal{R}_{n}$,
which contains $\Lambda_{4n}$. Observe that

$\mathbf{E}^{\emptyset}_{\Lambda,\beta}\Big{[}\mathds{1}_{0\in\mathcal{S}_{n}^{1}}\sum_{\begin{subarray}{c}x\in\mathcal{S}_{n}^{1}\\\
y\sim
x,\>y\in\Lambda_{n}\end{subarray}}\mathds{1}_{y\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}}\langle\sigma_{0}\sigma_{x}\rangle_{\mathcal{S}_{n}^{1},\beta}\Big{]}=\sum_{\begin{subarray}{c}x,y\in\Lambda_{n}\\\
y\sim
x\end{subarray}}\mathbf{E}^{\emptyset}_{\Lambda,\beta}\Big{[}\mathds{1}_{0,x\in\mathcal{S}_{n}^{1},\>y\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}}\langle\sigma_{0}\sigma_{x}\rangle_{\mathcal{S}_{n}^{1},\beta}\Big{]}.$
(2.18)

Set $\mathcal{G}_{n}:=\big{\\{}x\in\Lambda,\>x\textup{ or
}\mathcal{R}_{n}(x)\overset{\mathcal{M}_{n}}{\centernot\longleftrightarrow}\mathbb{H}_{n}\big{\\}}$.
For $x\in\Lambda_{n}$ and $y\sim x$ with $y\in\Lambda_{n}$, write

$\displaystyle\mathbf{E}^{\emptyset}_{\Lambda,\beta}\Big{[}\mathds{1}_{0,x\in\mathcal{S}_{n}^{1},\>y\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}}\langle\sigma_{0}\sigma_{x}\rangle_{\mathcal{S}_{n}^{1},\beta}\Big{]}$
$\displaystyle\leq\mathbf{E}^{\emptyset}_{\Lambda,\beta}\Big{[}\mathds{1}_{0,x\in\mathcal{G}_{n},\>y\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}}\langle\sigma_{0}\sigma_{x}\rangle_{\mathcal{G}_{n},\beta}\Big{]}$
$\displaystyle=\sum_{\begin{subarray}{c}S\subset\Lambda\\\ S\ni 0,x\\\
S\not\ni
y\end{subarray}}\mathbf{E}^{\emptyset}_{\Lambda,\beta}\left[\mathds{1}_{\mathcal{G}_{n}=S}\frac{Z^{\\{0,x\\}}_{S,\beta}}{Z^{\emptyset}_{S,\beta}}\right].$
(2.19)

Notice that

$Z^{\emptyset}_{\Lambda,\beta}[\mathcal{G}_{n}=S]=Z^{\emptyset}_{S,\beta}Z^{\emptyset}_{\Lambda\setminus
S,\beta}[\mathcal{G}_{n}=S]\quad\text{and}\quad
Z^{\\{0,x\\}}_{S,\beta}Z^{\emptyset}_{\Lambda\setminus
S,\beta}[\mathcal{G}_{n}=S]=Z^{\\{0,x\\}}_{\Lambda,\beta}[\mathcal{G}_{n}=S].$
(2.20)

As a result,

$\mathbf{E}^{\emptyset}_{\Lambda,\beta}\Big{[}\mathds{1}_{\mathcal{G}_{n}=S}\frac{Z^{\\{0,x\\}}_{S,\beta}}{Z^{\emptyset}_{S,\beta}}\Big{]}=\frac{Z^{\emptyset}_{\Lambda,\beta}[\mathcal{G}_{n}=S]}{Z^{\emptyset}_{\Lambda,\beta}}\frac{Z^{\\{0,x\\}}_{S,\beta}}{Z^{\emptyset}_{S,\beta}}=\frac{Z^{\\{0,x\\}}_{\Lambda,\beta}[\mathcal{G}_{n}=S]}{Z^{\emptyset}_{\Lambda,\beta}}.$
(2.21)

The right-hand side of (2.19) rewrites

$\sum_{\begin{subarray}{c}S\subset\Lambda\\\ S\ni 0,x\\\ S\not\ni
y\end{subarray}}\frac{Z^{\\{0,x\\}}_{\Lambda,\beta}[\mathcal{G}_{n}=S]}{Z^{\emptyset}_{\Lambda,\beta}}=\frac{Z^{\\{0,x\\}}_{\Lambda,\beta}[0,x\overset{\mathcal{M}_{n}}{\centernot\longleftrightarrow}\mathbb{H}_{n},\>y\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}]}{Z^{\emptyset}_{\Lambda,\beta}}.$
(2.22)

Now, we analyse
$Z^{\\{0,x\\}}_{\Lambda,\beta}[0,x\overset{\mathcal{M}_{n}}{\centernot\longleftrightarrow}\mathbb{H}_{n},\>y\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}]$
by conditioning on the set $\mathcal{C}_{n}(0)$ defined as the union of the
cluster of $0$ in $\mathcal{M}_{n}$ and its reflection with respect to
$\mathbb{H}_{n}$. Write,

$Z^{\\{0,x\\}}_{\Lambda,\beta}[0,x\overset{\mathcal{M}_{n}}{\centernot\longleftrightarrow}\mathbb{H}_{n},\>y\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}]=\sum_{\begin{subarray}{c}S\cap\mathbb{H}_{n}=\emptyset\\\
S\ni 0,x\\\ S\not\ni
y\end{subarray}}Z^{\\{0,x\\}}_{\Lambda,\beta}[\mathcal{C}_{n}(0)=S,\>y\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}].$
(2.23)

Finally,

$Z^{\\{0,x\\}}_{\Lambda,\beta}[\mathcal{C}_{n}(0)=S,\>y\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}]=Z^{\\{0,x\\}}_{\Lambda,\beta}[\mathcal{C}_{n}(0)=S]\frac{Z_{\Lambda\setminus
S,\beta}^{\emptyset}[y\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}]}{Z^{\emptyset}_{\Lambda\setminus
S,\beta}}.$ (2.24)

Applying the switching principle for reflected currents (Lemma 2.3),

$Z_{\Lambda\setminus
S,\beta}^{\emptyset}[y\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}]=Z^{\\{y,\mathcal{R}_{n}(y)\\}}_{\Lambda\setminus
S,\beta}.$ (2.25)

Moreover, using one last time Lemma 2.3,

$\sum_{\begin{subarray}{c}S\cap\mathbb{H}_{n}=\emptyset\\\ \>S\ni
0,x\end{subarray}}Z^{\\{0,x\\}}_{\Lambda,\beta}[\mathcal{C}_{n}(0)=S]=Z^{\\{0,x\\}}_{\Lambda,\beta}[0,x\overset{\mathcal{M}_{n}}{\centernot\longleftrightarrow}\mathbb{H}_{n}]=Z^{\\{0,x\\}}_{\Lambda,\beta}-Z^{\\{0,\mathcal{R}_{n}(x)\\}}_{\Lambda,\beta}.$
(2.26)

Putting (2.23)-(2.26) in (2.22) and using Griffiths’ inequality to replace
$\Lambda\setminus S$ by $\Lambda$, we get

$\mathbf{E}^{\emptyset}_{\Lambda,\beta}\Big{[}\mathds{1}_{0,x\in\mathcal{S}_{n}^{1},\>y\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}}\langle\sigma_{0}\sigma_{x}\rangle_{\mathcal{S}_{n}^{1},\beta}\Big{]}\leq\Big{(}\langle\sigma_{0}\sigma_{x}\rangle_{\Lambda,\beta}-\langle\sigma_{0}\sigma_{\mathcal{R}_{n}(x)}\rangle_{\Lambda,\beta}\Big{)}\langle\sigma_{y}\sigma_{\mathcal{R}_{n}(y)}\rangle_{\Lambda,\beta}.$
(2.27)

We conclude by taking $\Lambda\nearrow\mathbb{Z}^{d}$. ∎

##  3 Proof of Theorem 1.2 for the $\varphi^{4}$ model

As explained in the introduction, the discrete $\varphi^{4}$ model can be
recovered from taking the limit of well-chosen Ising models. In fact, with
this procedure we may recover more models which we now describe.

###  3.1 The Griffiths–Simon class of measures

######  Definition 3.1 (The Griffiths–Simon class of measures).

A Borel measure $\rho$ on $\mathbb{R}$ falls into the _Griffiths–Simon (GS)
class_ of measures if it satisfies one of the following conditions:

  1. $(i)$

there exists an integer $N\geq 1$, a renormalisation constant $Z>0$,
$(J_{i,j})_{1\leq i,j\leq N}\in(\mathbb{R}^{+})^{N^{2}}$, and $(Q_{n})_{1\leq
n\leq N}\in\mathbb{R}^{N}$ such that for every
$F:\mathbb{R}\rightarrow\mathbb{R}^{+}$ bounded and measurable,

$\int_{\mathbb{R}}F(\tau)\textup{d}\rho(\tau)=\frac{1}{Z}\sum_{\sigma\in\\{\pm
1\\}^{N}}F\left(\sum_{n=1}^{N}Q_{n}\sigma_{n}\right)\exp\left(\sum_{i,j=1}^{N}J_{i,j}\sigma_{i}\sigma_{j}\right),$
(3.1)

  2. $(ii)$

the measure $\rho$ can be obtained as a weak limit of probability measures of
the above type, and it is of sub-Gaussian growth: for some $\alpha>2$,

$\int_{\mathbb{R}}e^{|\tau|^{\alpha}}\textup{d}\rho(\tau)<\infty.$ (3.2)

Measures that fulfill condition $(i)$ are described as being of the “Ising
type”.

The following result was proved in [SG73] (see also [KPP23]).

######  Proposition 3.2.

Let $g>0$ and $a\in\mathbb{R}$. The probability measure $\rho_{g,a}$ on
$\mathbb{R}$ given by

$\textup{d}\rho_{g,a}(\varphi)=\frac{1}{z_{g,a}}e^{-g\varphi^{4}-a\varphi^{2}}\textup{d}\varphi,$
(3.3)

where $z_{g,a}$ is a renormalisation constant, belongs to the GS class.

###  3.2 Random current representation of Ising-type models in the GS class

Fix $\rho$ in the GS class of the Ising-type, and $\beta>0$. The measure
$\langle\cdot\rangle_{\Lambda,\rho,\beta}$ can be represented as an Ising
measure555In this measure, if $x\sim y$, the spins $(x,i)$ and $(y,j)$
interact through the coupling constant $\beta Q_{i}Q_{j}$. on
$\Lambda_{N}:=\Lambda\times K_{N}$ that we denote by
$\langle\cdot\rangle_{\Lambda_{N},\rho,\beta}$. In that case, we identify
$\tau_{x}$ with averages of the form

$\sum_{i=1}^{N}Q_{i}\sigma_{(x,i)},$ (3.4)

where $Q_{i}\geq 0$ for $1\leq i\leq N$. For $x\in\mathbb{Z}^{d}$, we will
denote $B_{x}:=\\{(x,i),\>1\leq i\leq N\\}$. This point of view allows to use
the random current representation. We introduce a measure
$\mathsf{P}^{xy}_{\Lambda,\rho,\beta}$ on $\Omega_{\Lambda\times K_{N}}$ which
we define in the following two steps procedure:

  1. -

first, sample two integers $1\leq i,j\leq N$ with probability given by

$\frac{Q_{i}Q_{j}\langle\sigma_{(x,i)}\sigma_{(y,j)}\rangle_{\Lambda_{N},\rho,\beta}}{\langle\tau_{x}\tau_{y}\rangle_{\Lambda,\rho,\beta}},$
(3.5)

  2. -

then, sample a current according to the “standard” current measure
$\mathbf{P}^{\\{(x,i),(y,j)\\}}_{\Lambda_{N},\rho,\beta}$ introduced above.

As before, it is possible to define the infinite volume version of the above
measure. We will denote it $\mathsf{P}^{xy}_{\rho,\beta}$. Samples of
$\mathsf{P}^{xy}_{\rho,\beta}$ are random currents with _random sources_ in
$B_{x}$ and $B_{y}$. The interest of this measure lies in the fact that it
allows to derive bounds on connection probabilities that are formulated in
terms of the correlation functions of the field variable $\tau$.

###  3.3 Proof of the theorem for Ising-type models in the GS class

We restate the result, as it is slightly different than for the Ising model
(notice the introduction of the variable $y^{\prime}$).

######  Proposition 3.3.

Let $d\geq 3$ and let $\rho$ be a measure of the Ising type in the GS class.
There exist $c,N_{0}>0$ such that, for all $\beta\leq\beta_{c}(\rho)$ and for
all $N_{0}\leq n\leq L(\beta,\rho)$,

$c\leq\beta\sum_{\begin{subarray}{c}x\in\Lambda_{n}\\\ y\sim x\\\
y^{\prime}\sim
y\end{subarray}}\left(\langle\tau_{0}\tau_{x}\rangle_{\rho,\beta}-\langle\tau_{0}\tau_{\mathcal{R}_{n}(x)}\rangle_{\rho,\beta}\right)\langle\tau_{y}\tau_{\mathcal{R}_{n}(y^{\prime})}\rangle_{\rho,\beta}.$
(3.6)

We begin with some notations: for a subset $\mathcal{E}$ of
$\Omega_{\Lambda\times K_{N}}$, set

$\mathsf{Z}^{\emptyset}_{\Lambda,\rho,\beta}[\mathcal{E}]:=Z^{\emptyset}_{\Lambda_{N},\rho,\beta}[\mathcal{E}]\text{
and
}\mathsf{Z}^{\\{u,v\\}}_{\Lambda,\rho,\beta}[\mathcal{E}]=\sum_{i,j=1}^{N}Q_{i}Q_{j}Z^{\\{(u,i),(v,j)\\}}_{\Lambda_{N},\rho,\beta}[\mathcal{E}].$
(3.7)

Note that

$\mathsf{P}_{\Lambda,\rho,\beta}^{\\{u,v\\}}[\mathcal{E}]=\sum_{i,j=1}^{N}\frac{Q_{i}Q_{j}\langle\sigma_{(u,i)}\sigma_{(v,j)}\rangle_{\Lambda_{N},\rho,\beta}}{\langle\tau_{u}\tau_{v}\rangle_{\Lambda,\rho,\beta}}\mathbf{P}^{\\{(u,i),(v,j)\\}}_{\Lambda_{N},\rho,\beta}[\mathcal{E}]=\frac{\mathsf{Z}^{\\{u,v\\}}_{\Lambda,\rho,\beta}[\mathcal{E}]}{\mathsf{Z}_{\Lambda,\rho,\beta}^{uv}},$
(3.8)

and

$\frac{\mathsf{Z}^{\\{u,v\\}}_{\Lambda,\rho,\beta}}{\mathsf{Z}^{\emptyset}_{\Lambda,\rho,\beta}}=\langle\tau_{u}\tau_{v}\rangle_{\Lambda,\rho,\beta}.$
(3.9)

Finally, we write $\mathbb{H}_{n}(N):=\mathbb{H}_{n}\times K_{N}$.

We start by the switching principle in this context.

######  Lemma 3.4 (Switching principle for reflected currents of Ising-type
models in the GS class).

Let $\Lambda\subset\mathbb{Z}^{d}$ finite and symmetric with respect to
$\mathcal{R}_{n}$. Let $x\in\Lambda^{-}$. Then,

  * $(i)$

$\mathsf{Z}^{\emptyset}_{\Lambda,\rho,\beta}[B_{x}\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}(N)]\leq\beta\sum_{x^{\prime}\sim
x}\mathsf{Z}^{\\{x,\mathcal{R}_{n}(x^{\prime})\\}}_{\Lambda,\rho,\beta}$,

  * $(ii)$

$\mathsf{Z}^{\\{0,x\\}}_{\Lambda,\rho,\beta}[\partial\mathbf{n}\cap
B_{x}\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}(N)]=\mathsf{Z}^{\\{0,\mathcal{R}_{n}(x)\\}}_{\Lambda,\rho,\beta}$

###### Proof.

For $(i)$, we follow the strategy employed in [ADC21, Lemma A.7]. Write,

$\displaystyle\mathsf{Z}^{\emptyset}_{\Lambda,\rho,\beta}[B_{x}\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}(N)]$
$\displaystyle\leq\sum_{i=1}^{N}Z^{\emptyset}_{\Lambda_{N},\rho,\beta}[(x,i)\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}(N)]$
$\displaystyle\leq\sum_{x^{\prime}\sim x}\sum_{i,j=1}^{N}(\beta
Q_{i}Q_{j})Z^{\\{(x,i),(x^{\prime},j)\\}}_{\Lambda_{N},\rho,\beta}[(x^{\prime},j)\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}(N)]$
$\displaystyle\leq\sum_{x^{\prime}\sim x}\sum_{i,j=1}^{N}(\beta
Q_{i}Q_{j})Z^{\\{(x,i),(\mathcal{R}_{n}(x^{\prime}),j)\\}}_{\Lambda_{N},\rho,\beta},$

where on the second line we used the fact that any path going out of $B_{x}$
has to visit $B_{x^{\prime}}$ for some $x^{\prime}\sim x$, and on the third
line we used666For full disclosure, we actually used a straightforward
generalisation of Lemma 2.3 to graphs of the form $\Lambda\times K_{N}$ that
are symmetric with respect to $\mathcal{R}_{n}$, in the sense that for all
$x\in\Lambda$ and all $1\leq i\leq N$, $(x,i)\in\Lambda_{N}$ if and only if
$\mathcal{R}_{n}((x,i)):=(\mathcal{R}_{n}(x),i)\in\Lambda_{N}$. Lemma 2.3.

For $(ii)$, we essentially reproduce the argument of Lemma 2.3 and omit the
proof. ∎

With this result, we obtain Proposition 3.3 by copying the argument used
above. We will modify the notations above to make them adapted to our setup.
We now define,

$\mathcal{S}_{n}(+\mathbf{e}_{1}):=\Big{\\{}x\in\Lambda_{n},\>B_{x}\overset{\mathcal{M}_{n}(+\mathbf{e}_{1})}{\centernot\longleftrightarrow}\mathbb{H}_{n}(+\mathbf{e}_{1})\times
K_{N}\Big{\\}},$ (3.10)

and modify accordingly the definitions of $\mathcal{S}_{n}$ and
$\mathcal{S}_{n}^{1}$.

###### Proof of Proposition 3.3.

With the help of the previous lemma, the proof is basically the same as for
the Ising case. The only modification lies in the statement and in the proof
of the adaptation of Lemma 2.5, which reads as follows. ∎

######  Lemma 3.5.

For $\beta\leq\beta_{c}(\rho)$,

$\mathsf{E}^{\emptyset}_{\rho,\beta}\Big{[}\mathds{1}_{0\in\mathcal{S}_{n}^{1}}\sum_{\begin{subarray}{c}x\in\mathcal{S}_{n}^{1}\\\
y\sim
x,\>y\in\Lambda_{n}\end{subarray}}\mathds{1}_{B_{y}\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}(N)}\langle\tau_{0}\tau_{x}\rangle_{\mathcal{S}_{n}^{1},\rho,\beta}\Big{]}\\\
\leq\beta\sum_{\begin{subarray}{c}x,y\in\Lambda_{n}\\\ y\sim x\\\
y^{\prime}\sim
y\end{subarray}}\left(\langle\tau_{0}\tau_{x}\rangle_{\rho,\beta}-\langle\tau_{0}\tau_{\mathcal{R}_{n}(x)}\rangle_{\rho,\beta}\right)\langle\tau_{y}\tau_{\mathcal{R}_{n}(y^{\prime})}\rangle_{\rho,\beta}.$
(3.11)

###### Proof.

Fix $\Lambda\subset\mathbb{Z}^{d}$ finite, symmetric under $\mathcal{R}_{n}$,
which contains $\Lambda_{4n}$. Observe that

$\mathsf{E}^{\emptyset}_{\Lambda,\rho,\beta}\Big{[}\mathds{1}_{0\in\mathcal{S}_{n}^{1}}\sum_{\begin{subarray}{c}x\in\mathcal{S}_{n}^{1}\\\
y\sim
x,\>y\in\Lambda_{n}\end{subarray}}\mathds{1}_{B_{y}\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}(N)}\langle\tau_{0}\tau_{x}\rangle_{\mathcal{S}_{n}^{1},\rho,\beta}\Big{]}=\sum_{\begin{subarray}{c}x,y\in\Lambda_{n}\\\
y\sim
x\end{subarray}}\mathsf{E}^{\emptyset}_{\Lambda,\beta}\Big{[}\mathds{1}_{0,x\in\mathcal{S}_{n}^{1},\>B_{y}\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}(N)}\langle\tau_{0}\tau_{x}\rangle_{\mathcal{S}_{n}^{1},\beta}\Big{]}.$
(3.12)

For $x\in\Lambda_{n}$ and $y\sim x$ with $y\in\Lambda_{n}$, write

$\displaystyle\mathsf{E}^{\emptyset}_{\Lambda,\beta}\Big{[}\mathds{1}_{0,x\in\mathcal{S}_{n}^{1},\>B_{y}\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}}\langle\tau_{0}\tau_{x}\rangle_{\mathcal{S}_{n}^{1},\rho,\beta}\Big{]}$
$\displaystyle\leq\mathsf{E}^{\emptyset}_{\Lambda,\beta}\Big{[}\mathds{1}_{B_{0},B_{x}\subset\mathcal{G}_{n},\>B_{y}\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}(N)}\langle\tau_{0}\tau_{x}\rangle_{\mathcal{G}_{n},\beta}\Big{]}$
$\displaystyle=\sum_{\begin{subarray}{c}S\subset\Lambda\times K_{N}\\\
S\supset B_{0},B_{x}\\\ S\not\supset
B_{y}\end{subarray}}\mathsf{E}^{\emptyset}_{\Lambda,\rho,\beta}\left[\mathds{1}_{\mathcal{G}_{n}=S}\frac{\mathsf{Z}^{\\{0,x\\}}_{S,\rho,\beta}}{\mathsf{Z}^{\emptyset}_{S,\rho,\beta}}\right],$
(3.13)

where777There is a small abuse of notation here since the correlations of
$\tau$ where technically only defined on subsets of $\mathbb{Z}^{d}$ not
subsets of $\mathbb{Z}^{d}\times K_{N}$. We solve this issue by replacing
$\tau_{u}$ with $\sum_{i=1}^{N}\sigma_{(u,i)}$, and by writing, for
$G_{N}\subset\Lambda_{N}$ containing $B_{u}$ and $B_{v}$,
$\mathsf{Z}^{\\{u,v\\}}_{G_{N},\rho,\beta}:=\sum_{i,j=1}^{N}Q_{i}Q_{j}Z^{\\{(u,i),(v,j)\\}}_{G_{N},\rho,\beta}$.
$\mathcal{G}_{n}:=\big{\\{}(x,i)\in\Lambda_{N},\>(x,i)\textup{ or
}(\mathcal{R}_{n}(x),i)\overset{\mathcal{M}_{n}}{\centernot\longleftrightarrow}\mathbb{H}_{n}(N)\big{\\}}$.
Notice that,

$\mathsf{Z}^{\emptyset}_{\Lambda,\rho,\beta}[\mathcal{G}_{n}=S]=Z^{\emptyset}_{S,\rho,\beta}Z^{\emptyset}_{\Lambda_{N}\setminus
S,\rho,\beta}[\mathcal{G}_{n}=S],$ (3.14)

and
$\mathsf{Z}^{\\{0,x\\}}_{S,\rho,\beta}\mathsf{Z}^{\emptyset}_{\Lambda_{N}\setminus
S,\rho,\beta}[\mathcal{G}_{n}=S]=\mathsf{Z}^{\\{0,x\\}}_{\Lambda_{N},\rho,\beta}[\mathcal{G}_{n}=S]$.
As a result,

$\mathsf{E}^{\emptyset}_{\Lambda,\rho,\beta}\left[\mathds{1}_{\mathcal{G}_{n}=S}\frac{\mathsf{Z}^{\\{0,x\\}}_{S,\rho,\beta}}{\mathsf{Z}^{\emptyset}_{S,\rho,\beta}}\right]=\frac{\mathsf{Z}^{\\{0,x\\}}_{\Lambda,\rho,\beta}[\mathcal{G}_{n}=S]}{\mathsf{Z}^{\emptyset}_{\Lambda,\rho,\beta}}.$
(3.15)

The right-hand side of (3.13) rewrites

$\sum_{\begin{subarray}{c}S\subset\Lambda\times K_{N}\\\ S\supset
B_{0},B_{x}\\\ S\not\supset
B_{y}\end{subarray}}\frac{\mathsf{Z}^{\\{0,x\\}}_{\Lambda,\rho,\beta}[\mathcal{G}_{n}=S]}{\mathsf{Z}^{\emptyset}_{\Lambda,\rho,\beta}}=\frac{\mathsf{Z}^{\\{0,x\\}}_{\Lambda,\rho,\beta}[B_{0},B_{x}\overset{\mathcal{M}_{n}}{\centernot\longleftrightarrow}\mathbb{H}_{n}(N),\>B_{y}\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}]}{\mathsf{Z}^{\emptyset}_{\Lambda,\rho,\beta}}.$
(3.16)

Now, we analyse
$\mathsf{Z}^{\\{0,x\\}}_{\Lambda,\rho,\beta}[B_{0},B_{x}\overset{\mathcal{M}_{n}}{\centernot\longleftrightarrow}\mathbb{H}_{n}(N),\>B_{y}\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}(N)]$
by conditioning on the cluster of $B_{0}\cup B_{x}$ in $\mathcal{M}_{n}$ that
we denote by $\mathcal{C}_{n}(0,x)$. Write,

$\mathsf{Z}^{\\{0,x\\}}_{\Lambda,\rho,\beta}[B_{0},B_{x}\mathrel{\mathop{\centernot\longleftrightarrow}^{\mathcal{M}_{n}}}\mathbb{H}_{n}(N),\>B_{y}\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}(N)]\\\
=\sum_{S\cap\mathbb{H}_{n}(N)=\emptyset,\>S\supset
B_{0},B_{x}}\mathsf{Z}^{\\{0,x\\}}_{\Lambda,\rho,\beta}[\mathcal{C}_{n}(0,x)=S,\>B_{y}\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}(N)].$
(3.17)

Finally,

$\mathsf{Z}^{\\{0,x\\}}_{\Lambda,\rho,\beta}[\mathcal{C}_{n}(0,x)=S,\>B_{y}\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}(N)]=\mathsf{Z}^{\\{0,x\\}}_{\Lambda,\rho,\beta}[\mathcal{C}_{n}(0,x)=S]\frac{\mathsf{Z}_{\Lambda\setminus
S,\rho,\beta}^{\emptyset}[B_{y}\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}(N)]}{\mathsf{Z}^{\emptyset}_{\Lambda\setminus
S,\rho,\beta}}.$ (3.18)

Applying Lemma 3.4 gives

$\mathsf{Z}_{\Lambda\setminus
S,\rho,\beta}^{\emptyset}[B_{y}\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}(N)]\leq\mathcal{\beta}\sum_{y^{\prime}\sim
y}\mathsf{Z}^{\\{y,\mathcal{R}_{n}(y^{\prime})\\}}_{\Lambda\setminus
S,\beta}.$ (3.19)

Moreover, using one more time Lemma 3.4 on the last line,

$\displaystyle\sum_{S\cap\mathbb{H}_{n}=\emptyset,\>S\supset
B_{0},B_{x}}\mathsf{Z}^{\\{0,x\\}}_{\Lambda,\rho,\beta}[\mathcal{C}_{n}(0)=S]$
$\displaystyle=\mathsf{Z}^{\\{0,x\\}}_{\Lambda,\rho,\beta}[B_{0},B_{x}\overset{\mathcal{M}_{n}}{\centernot\longleftrightarrow}\mathbb{H}_{n}(N)]$
$\displaystyle\leq\mathsf{Z}^{\\{0,x\\}}_{\Lambda,\rho,\beta}-\mathsf{Z}^{\\{0,x\\}}_{\Lambda,\rho,\beta}[\partial\mathbf{n}\cap
B_{x}\xleftrightarrow{\>\mathcal{M}_{n}\>}\mathbb{H}_{n}(N)]$
$\displaystyle=\mathsf{Z}^{\\{0,x\\}}_{\Lambda,\rho,\beta}-\mathsf{Z}^{\\{0,\mathcal{R}_{n}(x)\\}}_{\Lambda,\rho,\beta}.$

Collecting the above work and taking the limit as
$\Lambda\nearrow\mathbb{Z}^{d}$, we get the result. ∎

###  3.4 Proof of the theorem for general models in the GS class

We now turn to the extension of Proposition 3.3 to all models in the GS class
(and in particular to the $\varphi^{4}$ model). It suffices to show that the
inequality is stable under weak limits $\rho_{k}\rightarrow\rho$ for
$(\rho_{k})_{k\geq 1}$ a sequence of Ising-type measures in the GS class. The
same kind of difficulty appeared in [ADC21, Pan23] and we will essentially
import the tools used there. The extension of the result follows readily from
this proposition.

######  Proposition 3.6 ( [Pan23, Proposition 8.7]).

Let $d\geq 3$. Let $\rho$ be a measure in the GS class. Let $(\rho_{k})_{k\geq
1}$ be a sequence of measures of the Ising type in the GS class that converges
weakly to $\rho$. Then,

  1. $(i)$

$\liminf\beta_{c}(\rho_{k})\geq\beta_{c}(\rho)$,

  2. $(ii)$

for every $\beta>0$, $\liminf L^{\prime}(\rho_{k},\beta)\geq L(\rho,\beta)$
(where $L^{\prime}$ is defined similarly as $L$ with $1/2$ replaced by $1/4$),

  3. $(iii)$

for every $\beta<\beta_{c}(\rho)$, for every $x,y\in\mathbb{Z}^{d}$,

$\lim_{k\rightarrow\infty}\langle\tau_{x}\tau_{y}\rangle_{\rho_{k},\beta}=\langle\tau_{x}\tau_{y}\rangle_{\rho,\beta}.$
(3.20)

The above statement was initially proved for $d\geq 4$ and the argument for
$(iii)$ used in [Pan23] does not extend to $d=3$. However, it is possible to
solve this issue by noticing that for $\beta<\beta_{c}(\rho)$, for $k$ large
enough, if $u\in\mathbb{Z}^{d}$,

$\langle\tau_{0}\tau_{u}\rangle_{\rho_{k},\beta}\leq
C\exp\left(-c\frac{|u|}{2L(\rho,\beta)}\right).$ (3.21)

Indeed, this follows directly from (1.10) and the observation that $\limsup
L(\rho_{k},\beta)\leq L(\rho,\beta)$.

######  Acknowledgements.

We thank Alexis Prévost for stimulating discussions leading to Remark 1.9. We
thank Trishen S. Gunaratnam, Tiancheng He, and Gordon Slade for useful
comments. This project has received funding from the Swiss National Science
Foundation, the NCCR SwissMAP, and the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (grant
agreement No. 757296). HDC acknowledges the support from the Simons
collaboration on localization of waves.

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