# Spin imaging of Poiseuille flow of viscous electronic fluid

K.S. Denisov<EMAIL_ADDRESS>K.A Baryshnikov  P.S. Alekseev  Ioffe
Institute, 194021 St.Petersburg, Russia

###### Abstract

Recent progress in fabricating high-quality conductors with small densities of
defects has initiated the studies of the viscous electron fluid and has
motivated the search for the evidences of the hydrodynamic regime of electron
transport. In this work we come up with the spin imaging technique allowing us
to attest to the emergence of electron hydrodynamic flows. Based on numerical
calculations we demonstrate that the injected electron spin density is
inhomogeneous across the channel when the viscous electron fluid forms the
Poiseuille flow. We also argue that the Hanle curves at different positions
across the channel acquire relative phase shifts resulting from the variation
of the electron drift velocity in inhomogeneous hydrodynamic flows. The
studied effects can be employed to evidence and study the viscous electron
fluid non-invasively.

In high-quality conductors with small densities of defects, electrons can form
a viscous fluid at low temperatures due to frequent electron-electron
collisions and/or elasticity effects from the inter-particle interaction. The
charge transport in such fluid is carried out by inhomogeneous hydrodynamic
flows, controlled by particular shapes of samples, while its resistance
becomes proportional to the viscosity coefficient. These ideas were first
proposed and partially theoretically studied for bulk metals with strong
electron-phonon coupling [1]. Recently, this topic has become of interest as
the hydrodynamic regime of electron transport has been realized in high-
quality samples of graphene [2, 3, 4, 5, 6, 7, 8], quasi-two dimensional metal
PdCoO2 [9], Weyl semimetal WP2 [10], and high-mobility GaAs quantum wells [11,
12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. These experiments motivated
many theoretical works (see, for example [24, 25, 26, 27, 28, 29, 30, 31, 32,
33, 34, 35, 36, 37, 38, 39, 40]), which were aimed to formulation and search
for the evidences of the hydrodynamic regime as well as to studying of various
types and regimes of flows of the electron fluid.

Figure 1:  Scheme of the proposed spin imaging technique. Blue and bright blue
curves depict the electron flow velocity $V(y)$ under electric field
${\bm{E}}$ for Poiseuille and Ohmic regimes, respectively. The spin
polarization $S_{z}(y)$ (indicated by red arrows) keeps an inhomogeneous
structure at some distance from the injector due to hydrodynamic flow of
electrons and can be detected by a local spin probe. Applying an in-plane
magnetic field ${\bm{B}}$ one additionally detects the phase shifts in the
Hanle curves tracked at different positions across the sample due to the
variation of $V(y)$, here $V_{\ast}=V_{x}(W/2)$.

The evidences of formation of a viscous electron fluid are based, first, on an
inhomogeneity of space distributions of its flows, leading to the specific
properties of observed sample resistances. The simplest of these properties is
the cubic dependence of the conductance on the sample width. This dependence
was observed for the first time in [9] for stripes of PdCoO2. In samples of
peculiar geometry of edges and contacts, whirlpools can appear, similarly to
water flows in rivers. Herewith opposite direction of the current and the
voltage drop appear for some pairs of contacts. This effect of the “absolute
negative resistance” was proposed as an evidence of viscous flows of electrons
in [6] and was observed for graphene samples in [2]. Second, the dependencies
of the electron viscosity on magnetic field and flow frequency are very
specific and can be used to characterize the viscous electron fluid. The giant
negative magnetoresistance observed on high-mobility GaAs quantum wells [11,
12, 13, 14] was explained by this effect and thereby was employed to detect
the hydrodynamic transport [15]. For ac flows of the electron fluid, the
viscosity exhibits the resonance at the doubled electron cyclotron frequency
[38, 37]. Such resonance manifests itself in responses of its conductance on
incident radiation, that was apparently observed in [21, 22, 23]. In
particular in a strongly non-ideal electron fluid the ac flow is formed by
transverse shear stress waves, whose dispersion law reflects the resonance in
the viscosity coefficients. In recent works [7, 8] direct observations of the
profiles of the Hall electric field and the current density for a Poiseuille
flow of 2D electrons in graphene stripes by means of space resolved
measurements of electric and magnetic field were reported.

All these methods are quite difficult to use: they require either an analysis
of data on a number of specially designed samples with a given geometry or
applying sufficiently strong magnetic fields. Therefore, simpler, weaker-
invasive methods are wanted.

In this work we propose a spin-injection-based method to detect the
hydrodynamic regime of electric transport in an electron fluid (see Fig. 1).
We argue that the space distribution of the injected spins in a pure sample
can be employed to visualize a viscous flow (”spin imaging technique”).
Namely, the distributions of the electron spin density and its magnetic field
dependence for the Poiseuille flow of the viscous electron fluid, being
inhomogeneous by the section of a sample, strongly differs from the Ohmic
regime. We demonstrate this concept by performing the numerical calculations
with realistic parameters for high-mobility samples. The advantage of the
proposed method is that the injected spin distribution provides almost no
effect on the magnitude and profile of a given electron flow. In this way, the
proposed technique is ”light and non-invasive”, as compared with the already
existing methods of detection of the viscous electron fluid.

We consider a flow of 2D viscous electron fluid in high-mobility samples,
where the time of electron scattering on impurities can be longer than that
corresponding to electron-electron collisions. In this regime the electron
momentum relaxation takes place predominantly or mostly at the channel
boundaries with the subsequent formation of the Poiseuille flow. In case of
long and sufficiently wide samples (realized, for example, in experiments [7,
8, 16, 17, 18]) the velocity distribution can be found from the Navier-Stokes
equation. The result for low-frequency flows in samples with some density of
disorder and fully rough edges takes the form [1]:

$V_{x}(y)=\frac{e\,E_{x}}{m\,\tau_{tr}}\,\Big{\\{}1-\frac{\cosh[(y-W/2)/l_{G}]}{\cosh[(W/2)/l_{G}]}\Big{\\}}\>,$
(1)

where $m$ is an electron effective mass, $W$ is the channel width, $\tau_{tr}$
is the momentum relaxation time in the bulk due to scattering of electrons on
disorder or phonons, $l_{G}=\sqrt{\eta\tau_{tr}}$ is the Gurzhi length,
$\eta=v_{F}^{2}\tau_{ee}/4$ is the viscosity of the electron fluid determined
by the electron-electron collisions time, $\tau_{ee}$ is the time of
relaxation of the shear stress due to inter-particle collisions, and $v_{F}$
is the Fermi velocity. Equation (1) describes the homogeneous Ohmic flow
$V_{\rm Ohm}=e\,E_{x}/(m\,\tau_{tr})$ at $l_{G}\ll W$, while the Poiseuille
parabolic distribution appears in the opposite limit $l_{G}\gtrsim W$ (see
Fig. 1).

Figure 2:  (a): Calculated distribution of viscous electron fluid spin
polarization in the channel. (b): Profiles of the spin density at different
$x$-positions: $x=0.8W$ (1), $x=1.2W$ (2), $x=1.6W$ (3), and $x=2W$ (4). The
parameters $\tau_{s}=200$ ps, $D_{s}=60$ cm2/s ($L_{s}\approx 1.1~{}\mu$m),
$V_{\ast}=4\times 10^{6}$ cm/s ($L_{d}=8~{}\mu$m) and $W=8\mu$m.

In this work we address to systems with no pronounced effects of spin-orbital
coupling on electric current distribution and neglect the extra-boundary spin
accumulation due to the spin Hall effect [41, 42] or any rotational viscosity
effects [43, 44, 45, 46, 47]. The latter could lead to a vorticity of the
electron flow, which induces a torque acting on electron spin and results in
generation of the spin density [48]. By other words, our consideration is
valid for systems where the electric current profile is settled according to
Eq. (1), while the distribution of the spin density follows the local drift
velocity of the electron fluid and does not affect its orbital motion. In this
approximation the distribution of the spin density $\bm{S}$ can be determined
based on the drift-diffusion model [49] formulated in the following equation

$\dot{\bm{S}}+\nabla_{i}\bm{q}^{i}=\left[\bm{\omega}_{c}\times\bm{S}\right]-\bm{S}/\tau_{s}$
(2)

where $\bm{q}^{i}$ is the spin current (the flow of the value $\bm{S}$ along
the direction $x_{i}$), $\bm{\omega}_{c}$ is the Larmor precession frequency
due to the in-plane magnetic field, and $\tau_{s}$ is the spin relaxation
time, which is assumed to be isotropic. The spin current $\bm{q}^{i}$ contains
two contributions

$\bm{q}^{i}=-D_{s}\nabla_{i}\bm{S}+V_{i}\bm{S},$ (3)

where the first term describes the spin-diffusion with coefficient $D_{s}$ and
the second term stems from the drag of the spin density with the drift
velocity $\bm{V}=\bm{e}_{x}V_{x}(y)$ determined by Eq. (1). For further
consideration we also use the spin diffusion length
$L_{s}=\sqrt{D_{s}\tau_{s}}$ and the spin drift length
$L_{d}=V_{\ast}\tau_{s}$ in the centre of channel, $V_{\ast}=V_{x}(W/2)$.

In general, the spin diffusion length in high mobility samples can be
extremely large due to long momentum relaxation times $\tau_{tr}$. For
instance, in clean graphene-based lateral spin valves the spin can diffuse to
a distance of up to $1~{}\mu$m [50, 51, 52]. However, when the viscous
electron fluid is formed, $D_{s}$ is significantly reduced [53] as it is
mostly govern by electron-electron scattering, herewith the spin relaxation
time $\tau_{s}$ is controlled by D’yakonov-Perel-like mechanism giving
$1/\tau_{s}\sim\Omega_{so}^{2}\tau_{ee}$ [54, 55]. The decrease of $L_{s}$ due
to the inter-particle scattering is favorable for the spin imaging of the
Poiseuille flow, as the weakening of the spin diffusion prevents distortion of
an inhomogeneous spin pattern.

Figure 3: The dependence of spin polarization contrast across the channel on
its width $W$. The approximation $\Delta_{c}$ from Eq. (4) is shown by the
dotted curve. The inset shows the dependence $\Delta_{c}$ on $L_{s}/W$ for the
Poiseuille regime.

The distribution of spin density $S_{z}$ emerging due to the Poiseuille flow
of electrons in the hydrodynamical regime at $\omega_{c}=0$ is demonstrated in
Fig. 2(a). The parameters are relevant for high-mobility GaAs quantum wells
[53, 55] and are described in the caption to Fig 2, the channel width
$W=8\;\mu$m.

As boundary conditions we used the absence of spin current $q_{y}^{z}=0$ at
$y=0,W$, we also assumed that $q_{x}^{z}$ is proportional to the electric
current flowing through the contact at $x=0$ [49]. In our case we take
${q}_{x}^{z}\propto V_{x}(y)$ being consistent with the hydrodynamic
distribution of the electric current in the bulk of the channel. We recognize
that the electric current distribution right in the vicinity of a contact can
alter from the Poiseuille form. However, beyond some transition region (no
longer than $x\lesssim W$) the drift velocity will be settled according to Eq.
(1) and $S_{z}$ will keep strongly inhomogeneous shape due to an effective
dragging by $\bm{E}$ in the center of the channel.

This feature is clearly seen in Fig. 2(b), where we demonstrate the profiles
of the spin density across the channel at different distances from the left
boundary. The tail of the spin distribution presented in Fig. 2(b) can be
approximated by the expression

$S_{z}(x,y)=A\,e^{-x/L_{\ast}}\left[1-\Delta_{c}\cos{\left(\frac{2\pi
y}{W}\right)}\right],$ (4)

where $A$ is a constant being independent of $x$ and $y$, the factor
$e^{-x/L_{\ast}}$ describes the spin density decay with an averaged drift
length $L_{\ast}=\langle V_{x}\tau_{s}\rangle=2L_{d}/3$ and the parameter
$\Delta_{c}$ determines the contrast of the spin imaging. It follows from Eq.
4 that imaging of the Poiseuille flow by spin injection is possible provided
that $\Delta_{c}\sim 1$.

To determine the critical region beyond which $\Delta_{c}$ vanishes we get an
approximated analytical solution for Eq. (2). We use the biharmonic
approximation for $y$-dependence of $S_{z}(x,y)$ and $V_{x}(y)$ and take into
account only the drift component of $q_{x}^{z}$. For the slowest decaying
solution at the tail of the spin density we obtain
$\Delta_{c}=6/[\,2\pi^{4}\xi^{2}+\sqrt{18+72\pi^{2}\xi^{2}+4\pi^{8}\xi^{4}}\,]$,
where $\xi=L_{s}/W$. The expression Eq. (4) with $\Delta_{c}$ from above is
justified in range $1\lesssim\xi\lesssim 0.05$, where the right boundary is
determined by the failure of biharmonic approximation due to a weakened spin
diffusion. The spin distribution plotted in Fig. 2 have the parameters
$\xi=0.14$, $\Delta_{c}=0.52$ and the expression from Eq. (4) fits well with
the numerical solution starting from $x/W\gtrsim 1.2$.

The dependence of $\Delta_{c}$ on the ratio $L_{s}/W$ is shown on the inset of
Fig. 3. It is seen that the spin contrast decreases significantly
$\Delta_{c}\lesssim 0.2$ already at $\xi\gtrsim 0.3$, instead of $\xi\gtrsim
1$ as might be expected. Since $\Delta_{c}$ is determined by single ratio
$L_{s}/W$, the applicability of the spin imaging approach for sufficiently
large in-plane electric fields (when $L_{d}\gg L_{s}$) is govern by a simple
criteria $W\gtrsim 3L_{s}$, suggesting that a more plausible situation is
realized for sufficiently wide samples. The spin contrast across the
transition between Poiseuille and Ohmic transport regimes is illustrated in
Fig. 3, where we present the dependence of
$\Delta=[\,S_{z}(y_{c})-S_{z}(y_{e})\,]/[\,S_{z}(y_{c})+S_{z}(y_{e})\,]$ on
$W$; here $y_{c}=W/2$ is in the center of the channel and $y_{e}=W/10$ is
nearby its boundary. A nonmonotonic character of $\Delta$ stems from the fast
initial increase of $\Delta_{c}(L_{s}/W)$ relevant for the Poiseuille flow,
which is further suppressed at larger $W$ due to the transition to the Ohmic
regime.

Figure 4:  Distribution of the spin density in magnetic field directed along
the $y$ axis. (a,b): Space-resolved densities $S_{z}$ and $S_{x}$ at
$\omega_{c}\tau_{s}=3.5$. (c,d): Magnetic field dependences of $S_{z}$ and
$S_{x}$ calculated at $x=12\;\mu$m away from the injector. The Hanle curves on
the right panels are taken at the center ($y=0.5W$, blue color) and near the
edge ($y=0.1W$, orange color) of the channel. The positions of spin detection
in each case are depicted by big dots of corresponded colors on the left
panels.

Another spin-related evidence of the Poiseuille flow of the electron fluid
revealed by the spin imaging technique is the appearance of nonzero relative
phase shifts for the Hanle curves (the dependence of $S_{z,x}$ on
$\omega_{c}\tau_{s}$) probed at different positions across the channel. Below
we explain this idea in detail. We keep to the drift-dominated regime, at that
the difference between two Larmor frequencies $\Delta\omega_{c}$ at which the
Hanle curve exhibits the neighboring peaks or dips at a fixed point of spin
probe in space can be estimated as $\Delta\omega_{c}=(2\pi V_{x})/x_{0}$,
where $x_{0}$ is the distance from the injector. In case of the Poiseuille
electronic flow the drift velocity $V_{x}(y)$ from Eq. (1) changes
significantly across the channel leading to the variation of
$\Delta\omega_{c}$. An ultimate manifestation of this feature would be the
shift $\Delta\omega_{c}$ as a function of $y$-coordinate at fixed distance
$x_{0}$ resulting in the “desynchronization” of the Hanle curves.

We proceed with considering this scenario in more detail. We apply an in-plane
magnetic field $\bm{\omega}_{c}=\omega_{c}\bm{e}_{y}$ and keep only the
injection of $S_{z}$ (the boundary conditions are the same as Fig. 2; the spin
current $q_{x}^{x}=0$ is absent). In Fig. 4(a,b) we demonstrate inhomogeneous
oscillating spatial patterns of $S_{z},S_{x}$ inside the electronic channel at
fixed magnetic field $\omega_{c}\tau_{s}=3.5$ (red and blue colors stand for
the positive and negative signs, respectively), other parameters are the same
as in Fig. 2. The procession structure of the spin density visible in Fig. 4
is specific for the drift-dominated regime of the spin transport [49].
Importantly, $S_{z,x}$ are sign-altering and inhomogeneous at the same moment,
one can clearly observe that at fixed distance $x_{0}$ the spin density can
have different signs in the center and nearby the boundary of the channel.
This feature is explicitly connected with the discussed variation in drift
velocities emerging for the hydrodynamic regime of viscous electron fluid
transport.

The relative shifts of the oscillations period $\Delta\omega_{c}$ of the Hanle
curves are shown in Fig. 4(c,d). The Hanle curves are calculated at two
spatial positions across the channel, see Fig. 4(a,b). When
$\omega_{c}\tau_{s}\sim 4$, these shifts are up to $\pi$. We argue that the
presented desynchronization of the Hanle curves upon hydrodynamical response
can be used to confirm independently the formation of the viscous electron
fluid.

In conclusion, we have proposed an approach to visualize hydrodynamic
electronic flows by measuring the spin polarization distribution across the
transport channel. Our calculations show that measuring both the spin
polarization contrast and Hanle curves at the center and at the boundary of
the channel allow one to disclose the hydrodynamic regime. We believe that the
proposed method paves the way towards non-invasive studies of hydrodynamic
viscous electron fluids in samples of different geometry and microscopic
structure.

Acknowledgements. This work has been supported by the Russian Science
Foundation (Project 18-72-10111). P.S.A. thanks the Theoretical Physics and
Mathematics Advancement Foundation ”BASIS”.

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

# On the two-point function of the Ising model with infinite range-
interactions

Yacine Aoun  Section de Mathématiques, Université de Genève, Rue du Conseil-
Général 7-9, 1205 Genève, Switzerland<EMAIL_ADDRESS>and  Kamil
Khettabi  Section de Mathématiques, Université de Genève, Rue du Conseil-
Général 7-9, 1205 Genève, Switzerland<EMAIL_ADDRESS>
###### Abstract.

In this article, we prove some results concerning the truncated two-point
function of the infinite-range Ising model above and below the critical
temperature. More precisely, if the coupling constants are of the form
$J_{x}=\psi(x)\mathsf{e}^{-\rho(x)}$ with $\rho$ some norm and $\psi$ an
subexponential correction, we show under appropriate assumptions that given
$s\in\mathbb{S}^{d-1}$, the Laplace transform of the two-point function in the
direction $s$ is infinite for $\beta=\beta_{\rm sat}(s)$ (where $\beta_{\rm
sat}(s)$ is a the biggest value such that the inverse correlation length
$\nu_{\beta}(s)$ associated to the truncated two-point function is equal to
$\rho(s)$ on $[0,\beta_{\rm sat}(s)))$. Moreover, we prove that the two-point
function satisfies Ornstein-Zernike asymptotics for $\beta=\beta_{\rm sat}(s)$
on $\mathbb{Z}$. As far as we know, this constitutes the first result on the
behaviour of the two-point function at $\beta_{\rm sat}(s)$. Finally, we show
that there exists $\beta_{0}$ such that for every $\beta>\beta_{0}$,
$\nu_{\beta}(s)=\rho(s)$. All the results are new and their proofs are built
on different results and ideas developed in [11, 2].

##  1\. Introduction

In the present paper, we study the behaviour of the two-point function in
Ising models with infinite-range interactions. In [2] (see also [1]), the
first author and collaborators considered a general class of lattice spin
systems (including the Ising model) on $\mathbb{Z}^{d}$ with interactions of
the form $J_{x}=\psi(x)\mathsf{e}^{-\rho(x)}$, where $\psi(x)$ is a
subexponential correction and $\rho$ is a norm on $\mathbb{R}^{d}$. Let
$\langle\sigma_{0}\sigma_{x}\rangle_{\beta}$ be the usual Ising two-point
function with free boundary conditions at inverse temperature $\beta$ without
an external field, and $\nu_{\beta}(\hat{x})$ be the associated inverse
correlation length in the direction $\hat{x}=x/\|x\|$ where $\|\cdot\|$ is the
euclidian norm. It is easy to see that one always has
$\nu_{\beta}(\hat{x})\leq\rho(x)$. In [2], we developed an explicit necessary
and sufficient condition (see Theorem 2.2) to ensure the existence of a non-
trivial saturation transition, i.e. the strict positivity of $\beta_{\rm
sat}(\hat{x})=\sup\\{\beta\geq 0:\nu_{\beta}(\hat{x})=\rho(\hat{x})\\}$. For
instance, a sufficient condition for the latter to happen is to have
$\psi(x)=\mathsf{O}(\|x\|^{-(d+\varepsilon)})$ for some $\varepsilon>0$. By
definition one always has $\beta_{\rm sat}(\hat{x})\leq\beta_{\mathrm{c}}$
where $\beta_{\mathrm{c}}$ is the usual transition point of the Ising model.
Note that if $\beta_{\rm sat}(\hat{x})>0$, the function
$\beta\mapsto\nu_{\beta}(\hat{x})$ is non-analytic. Moreover, we proved in [2]
that if $\beta_{\rm sat}(\hat{x})>0$, then the Ornstein-Zernike asymptotics
(see (1)) do not hold at arbitrarily high temperature. In subsequent works [3,
4], we studied the behavior of the two point function in the saturated regime
$(0,\beta_{\rm sat}(\hat{x}))$ and in the non-saturated regime $(\beta_{\rm
sat}(\hat{x}),\beta_{\mathrm{c}})$. Under appropriate assumptions, for
$\beta\in(\beta_{\rm sat}(\hat{x}),\beta_{\mathrm{c}})$, we proved in [3] that
the two-point function has the Ornstein-Zernike asymptotics: there exists
$c:=c(\hat{x},\beta)>0$ such that

$\langle\sigma_{0}\sigma_{x}\rangle_{\beta}=c\|x\|^{-\frac{d-1}{2}}\mathsf{e}^{-\nu_{\beta}(x)}(1+\mathsf{o}_{\|x\|}(1)).$
(1)

The OZ asymptotics were predicted in the physics literature in [16], and were
expected to hold generally when the interactions decay exponentially fast in
the distance. In [4], we proved that this is not the case in the whole
saturated regime: under approriate assumptions, for $\beta\in(0,\beta_{\rm
sat}(\hat{x}))$, there exists $C(\beta,\hat{x})>0$ such that

$\langle\sigma_{0}\sigma_{x}\rangle_{\beta}=CJ_{x}(1+\mathsf{o}_{\|x\|}(1)).$
(2)

This leaves us with a natural question of determining the asymptotics of the
two-point function at $\beta_{\rm sat}(\hat{x})$. The techniques used for
proving (2) and (1) break down at $\beta_{\rm sat}(\hat{x})$. On the one hand,
in [3], we derived (1) under the mass-gap assumption
$\nu_{\beta}(\hat{x})<\rho(\hat{x})$, which is violated at $\beta_{\rm
sat}(\hat{x})$ since by continuity of the function
$\beta\mapsto\nu_{\beta}(\hat{x})$, one has
$\nu_{\beta}(\hat{x})=\rho(\hat{x})$. On the other hand, we used differential
inequalities (inspired by the ideas of [10, 14]) and the fact that for any
$\beta_{0}\in(0,\beta_{\rm sat}(\hat{x}))$ there exists an open interval
containing $\beta_{0}$ on which the function
$\beta\mapsto\nu_{\beta}(\hat{x})$ is constant to derive (2). In the present
article, we provide partial answers for the behavior of the two-point function
at $\beta_{\rm sat}(\hat{x})$: under suitable assumptions, we prove that the
Laplace transform associated to the two-point function is infinite. Moreover,
we prove that (1) holds up to multiplicative constants on $\mathbb{Z}$. This
is the first example where the OZ asymptotics are shown to hold in the absence
of a mass-gap. In particular, it shows that the mass-gap is not a necessary
condition for OZ asymptotics to hold.

Note that in the discussion above, the saturation phenomenon is only shown to
happen at high temperatures. In the present work, we prove the existence of a
non-trivial saturation regime at arbitrarily low temperatures as well. Let
$\langle\sigma_{0};\sigma_{x}\rangle_{\beta}$ be the truncated two-point
function of the Ising model with $+$ boundary conditions and
$\nu_{\beta}(\hat{x})$ the associated inverse correlation length. We prove the
existence of $\beta_{\rm sat}^{*}:=\beta_{\rm sat}^{*}(\hat{x})<\infty$ such
that for every $\beta>\beta_{\rm sat}^{*}$, we have
$\nu_{\beta}(\hat{x})=\rho(\hat{x})$.

##  2\. Models and notations

###  2.1. Graphs

Most of our results naturally extend to a wider set-up but we restrict
attention to $\mathbb{Z}^{d}$. We will always see $\mathbb{Z}^{d}$ as
canonically embedded inside $\mathbb{R}^{d}$ and will denote $\|\cdot\|$ the
Euclidean norm on $\mathbb{R}^{d}$. $\rho$ will denote a norm on
$\mathbb{R}^{d}$ (and will be one of the parameters in our analysis).

We consider the graph $(\mathbb{Z}^{d},E_{d})$ with edge set
$E_{d}=\bigl{\\{}\\{i,j\\}\subset\mathbb{Z}^{d}\bigr{\\}}$, which we will
often write simply $\mathbb{Z}^{d}$. Let $\Lambda_{N}=\\{-N,\dots,N\\}^{d}$
and $\Lambda_{N}(x)=x+\Lambda_{N}$.

Given a subgraph $\Lambda$, let $\Lambda^{c}=\mathbb{Z}^{d}\backslash\Lambda$
and

$E_{\Lambda}=\bigl{\\{}\\{i,j\\}\in
E_{d}\,:\,\\{i,j\\}\subset\Lambda\bigr{\\}}.$

Given $x,y,z\in\mathbb{Z}^{d}$, a sequence
$\gamma=(\gamma_{0},\gamma_{1},\dots,\gamma_{n})\in(\mathbb{Z}^{d})^{n+1}$ is
called a path from $x$ to $y$ if $\gamma_{0}=x$ and $\gamma_{n}=y$. We say
that $n$ is the length of the path, and denote it by $\absolutevalue{\gamma}$.
We say that $\gamma$ is edge self-avoiding if
$\\{\gamma_{i},\gamma_{i+1}\\}=\\{\gamma_{j},\gamma_{j+1}\\}\Rightarrow i=j$.

###  2.2. Interaction

We consider a weight function (the interaction, or the set of coupling
constants) $J:E_{d}\to\mathbb{R}_{+}$ of the form
$J_{i,j}=\psi(i-j)\mathsf{e}^{-\rho(i-j)}$ where $\psi$ satisfies

$\lim\limits_{\|x\|\rightarrow\infty}\dfrac{\log(\psi(x))}{\|x\|}=0.$

Moreover, we will assume that the interaction satifies the following
properties:

  * •

No self-interaction: $J_{0}=0$,

  * •

Rotational invariance: $J$ is invariant by a rotation of $\pi/2$ around any
coordinate axis.

###  2.3. Percolation configurations

Given a subset $\Lambda$ of $\mathbb{Z}^{d}$, the percolation configuration
$\omega$ is defined as a function from $E_{\Lambda}$ to $\\{0,1\\}$. Given an
edge $\\{i,j\\}\in E_{\Lambda}$, we say that the edge $\\{i,j\\}$ is open in
$\omega$ if $\omega_{i,j}=1$ and closed otherwise. Given the subsets $A,B,C$
of $\Lambda$, we will denote by $\\{A\overset{C}{\leftrightarrow}B\\}$ the
subset of percolation configurations $\omega$ such that there exists a path
from $A$ to $B$ consisting of open edges of $C$. If $C=\Lambda$, we will
remove it from the notation. We will write
$\\{x\overset{C}{\leftrightarrow}y\\}$ instead of
$\\{\\{x\\}\overset{C}{\leftrightarrow}\\{y\\}\\}$. Finally, we will define
the connected component of $x$ by
$\mathcal{C}_{x}:=\\{y\in\mathbb{Z}^{d}:x\leftrightarrow y\\}$.

###  2.4. Constants

$c,C,c^{\prime},C^{\prime},\dots$ will denote constants whose value can change
from line to line. Unless explicitly stated otherwise, they depend only on
$d,\beta,h,J$.

###  2.5. Ising Model

The Ising model at inverse temperature $\beta\geq 0$ without a magnetic field
with free boundary condition on $\mathbb{Z}^{d}$ is the probability measure on
$\Omega:=\\{-1,+1\\}^{\mathbb{Z}^{d}}$ given by the weak limit of the finite-
volume measures (for $\sigma\in\\{-1,+1\\}^{\Lambda_{N}}$ and
$\Lambda_{N}=[-N,N]^{d}\cap\mathbb{Z}^{d}$)

$\mu^{{\scriptscriptstyle\rm
f}}_{\Lambda_{N};\beta}(\sigma)=\frac{1}{Z^{{\scriptscriptstyle\rm
f}}_{\Lambda_{N};\beta}}\mathsf{e}^{-\beta\mathscr{H}^{{\scriptscriptstyle\rm
f}}_{N}(\sigma)},$

with Hamiltonian

$\mathscr{H}^{{\scriptscriptstyle\rm
f}}_{N}(\sigma)=-\sum_{\\{i,j\\}\subset\Lambda_{N}}J_{ij}\sigma_{i}\sigma_{j}-\sum_{i\in\Lambda_{N}}\sigma_{i}$

and partition function $Z^{{\scriptscriptstyle\rm f}}_{\Lambda_{N};\beta,}$.
We also define the Ising measure at inverse temperature $\beta\geq 0$ with $+$
boundary condition and without a magnetic field by

$\mu^{+}_{\Lambda_{N};\beta}(\sigma)=\frac{1}{Z^{+}_{\Lambda_{N};\beta}}\mathsf{e}^{-\beta\mathscr{H}^{+}_{N}(\sigma)},$

with Hamiltonian

$\mathscr{H}_{N,h}^{+}(\sigma)=-\sum_{\\{i,j\\}\subset\Lambda_{N}}J_{ij}\sigma_{i}\sigma_{j}-\sum_{\begin{subarray}{c}i\in\Lambda_{N}\\\
j\in\Omega\setminus\Lambda_{N}\end{subarray}}J_{ij}\sigma_{i}.$

For $\eta\in\\{+,f\\}$, the limit
$\mu^{\eta}_{\beta}=\lim_{N\to\infty}\mu^{\eta}_{\Lambda_{N};\beta}$ is always
well defined and agrees with the unique infinite-volume measure whenever
$\beta<\beta_{\mathrm{c}}$, the critical point of the model; we refer to [12]
for more details. We will be interested in the behaviour of the truncated two-
point function of the model

$\langle\sigma_{0};\sigma_{x}\rangle_{\beta}:=\mathrm{Cov}(\sigma_{0},\sigma_{x}),$

where the covariance is taken with respect to $\mu^{+}_{\beta}$. We also
introduce the correlation length associated to the latter in the direction
$s\in\mathbb{S}^{d-1}$

$\nu_{\beta}(s):=-\lim_{n\to\infty}\frac{1}{n}\log\langle\sigma_{0};\sigma_{ns}\rangle_{\beta}.$

The existence of this limit follows from the subadditivity proved in [13]. The
subaddivity also provides the following bound

$\langle\sigma_{0};\sigma_{ns}\rangle_{\beta}\leq\mathsf{e}^{-n\nu_{\beta}(s)}.$
(3)

Let us also introduce

$\langle\sigma_{0}\sigma_{x}\rangle_{\beta}:=\mathbb{E}[\sigma_{0}\sigma_{x}].$

When $\beta<\beta_{\mathrm{c}}$, the truncated two-point function is just
equal to the usual two-point function,

$\langle\sigma_{0};\sigma_{x}\rangle_{\beta}=\langle\sigma_{0}\sigma_{x}\rangle_{\beta}.$

The following result was proved in [4].

######  Theorem 2.1.

Fix $\beta<\beta_{\mathrm{c}}$ and $s\in\mathbb{S}^{d-1}$. Then
$\nu_{\beta}(s)>0$.

####  2.5.1. FK-Ising model

Intimately related to the Ising model is the FK-Ising model (i.e. the Random-
Cluster model with $q=2$). The latter is a measure on percolation
configurations on $\mathbb{Z}^{d}$ depending on a parameter
$\beta\in\mathbb{R}_{\geq 0}$ that will be denoted by $\Phi_{\beta}$ and is
obtained as the weak limit of the finite-volume measures

$\Phi_{\Lambda_{N};\beta}(\omega)=\frac{1}{Z^{\text{FK}}_{\Lambda_{N};\beta}}\prod_{\\{i,j\\}\in\omega}(\mathsf{e}^{\beta
J_{ij}}-1)2^{\kappa(\omega)},$ (4)

where $\kappa(\omega)$ is the number of connected components in the graph with
vertex set $\Lambda_{N}$ and edge set $\omega$ and
$Z^{\text{FK}}_{\Lambda_{N};\beta}$ is the partition function. One has the
following correspondance between the Ising model without a magnetic field and
the FK-Ising model

$\langle\sigma_{0}\sigma_{x}\rangle_{\beta}=\Phi_{\beta}(0\leftrightarrow x).$

It is a standard consequence that one in particular has

$\langle\sigma_{0};\sigma_{x}\rangle_{\beta}\geq\Phi_{\beta}(0\leftrightarrow
x,0\mathrel{\ooalign{$\leftrightarrow$\cr$/$}}\infty).$ (5)

During the proofs, we will need several well-known properties of the FK-Ising
model:

Finite energy property: Fix $\Lambda\subset\mathbb{Z}^{d}$ and $\beta>0$. For
any $e\in E_{\Lambda}$ and $\eta\in\\{0,1\\}^{E_{\Lambda}\backslash\\{e\\}}$,
one has

$\dfrac{\mathsf{e}^{\beta J_{e}}-1}{\mathsf{e}^{\beta
J_{e}}+1}\leq\Phi_{\Lambda;\beta}(\omega_{e}=1\hphantom{,}|\omega_{E_{\Lambda}\backslash\\{e\\}}=\eta)\leq
1-\mathsf{e}^{-\beta J_{e}}.$ (6)

FKG inequality: We say that a $\mathcal{F}_{\Lambda}$-measurable event $A$ is
increasing if $\mathds{1}_{A}$ in increasing with respect to the lexographical
order on $\\{0,1\\}^{E_{\Lambda}}$. Given two increasing events $A,B$, the FKG
inequality states that

$\Phi_{\Lambda;\beta}(A)\Phi_{\Lambda;\beta}(B)\leq\Phi_{\Lambda;\beta}(A\cap
B).$

Simon–Lieb inequality: Given a finite subset $S$ containing $0$, one has [11]

$\Phi_{\beta}(u\xleftrightarrow{\Lambda}v)\leq\\\ \sum_{x\in S}\sum_{y\notin
S}\Phi_{\beta}(u\xleftrightarrow{S}x)\beta
J_{x,y}\Phi(y\xleftrightarrow{\Lambda}v).$ (7)

These properties in particular imply the existence of $C:=C(\beta)>0$ such
that

$CJ_{x}\leq\langle\sigma_{0};\sigma_{x}\rangle_{\Lambda,\beta}.$ (8)

Indeed, one has

$\Phi_{\Lambda;\beta}(\omega_{\\{0,x\\}}=1,\absolutevalue{\mathcal{C}_{0}}=2)\leq\Phi_{\beta}(0\leftrightarrow
x,0\mathrel{\ooalign{$\leftrightarrow$\cr$/$}}\infty).$

The finite energy property then implies the existence of $C:=C(\beta)>0$ such
that

$CJ_{x}\leq\Phi_{\Lambda;\beta}(\omega_{\\{0,x\\}}=1,\absolutevalue{\mathcal{C}_{0}}=2).$

All these inequalities combined with (5) gives (8). Notice that (8) in
particular implies that $\nu_{\beta}(s)\leq\rho(s)$ for any $\beta>0$.

####  2.5.2. Random current

Let $\Lambda$ be a finite subgraph of $\mathbb{Z}^{d}$. We consider an
additional vertex $\mathfrak{g}$ in the graph $\Lambda$ and denote by
$\Lambda^{\mathfrak{g}}$ the graph obtained by adding an edge between each
$x\in\Lambda$ and $\mathfrak{g}$. A current
$\mathbf{n}=(\mathbf{n}_{xy})_{x,y\in E_{\Lambda^{\mathfrak{g}}}}$ on
$\Lambda^{\mathfrak{g}}$ is an element of
$\mathbb{N}^{E_{\Lambda^{\mathfrak{g}}}}$. For $x\in\Lambda^{\mathfrak{g}}$,
set $X(\mathbf{n},x):=\sum_{y\in\Lambda^{\mathfrak{g}}}\mathbf{n}_{xy}$. We
define

$\partial\mathbf{n}:=\\{x\in\Lambda^{\mathfrak{g}}:X(\mathbf{n},x)\text{ is
odd}\\}.$

In the case of the Ising model on a finite box $\Lambda\subset\mathbb{Z}^{d}$
with $+$ boundary condition, we set
$J_{x\mathfrak{g}}:=\sum_{y\in\Lambda^{c}}J_{xy}$. This will allow us to
reinterpret the $+$ boundary conditions as the presence of a new vertex,
namely $\mathfrak{g}$. We also define the weight of a current $\mathbf{n}$ on
$\Lambda^{\mathfrak{g}}$ to be the quantity

$w_{\Lambda^{\mathfrak{g}};\beta}(\mathbf{n}):=\prod_{xy\in
E_{\Lambda^{\mathfrak{g}}}}\frac{(\beta
J_{xy})^{\mathbf{n}_{xy}}}{\mathbf{n}_{xy}!}.$

Taylor-expanding $\mathsf{e}^{\beta J_{ij}\sigma_{i}\sigma_{j}}$ and
resumming, one gets

$\langle\sigma_{A}\rangle_{\Lambda,\beta}=\left\\{\begin{array}[]{ll}\frac{Z_{\Lambda}(A)}{Z_{\Lambda}(\varnothing)}&\mbox{if}\leavevmode\nobreak\
|A|\leavevmode\nobreak\ \mbox{is even}\\\
\frac{Z_{\Lambda}(A\cup\\{g\\})}{Z_{\Lambda}(\varnothing)}&\mbox{otherwise}\end{array}\right.$

where
$Z_{\Lambda}(F):=\sum_{\mathbf{n}:\partial\mathbf{n}=F}w_{\Lambda;\beta}(\mathbf{n})$
for any subset $F\subset\Lambda$. We will refer to this correspondance as the
random-current representation. Given a subset
$A\subset\Lambda^{\mathfrak{g}}$, one can define a probability law on currents
on $\Lambda^{\mathfrak{g}}$ with sources $A$ by

$\mathbb{P}^{A}_{\Lambda^{\mathfrak{g}};\beta}(\mathbf{n})=\frac{w_{\Lambda^{\mathfrak{g}};\beta}(\mathbf{n})\mathsf{1}_{\partial\mathbf{n}=A}}{Z_{\Lambda}(A)}.$

We will use the notation
$\mathbb{P}_{\Lambda^{\mathfrak{g}};\beta}^{\varnothing,\\{0,x\\}}$ for the
product measure
$\mathbb{P}_{\Lambda^{\mathfrak{g}},\beta}^{\varnothing}\times\mathbb{P}_{\Lambda^{\mathfrak{g}},\beta}^{\\{0,x\\}}$.
This is therefore a law on pairs of currents $(\mathbf{n}_{1},\mathbf{n}_{2})$
such that $\partial\mathbf{n}_{1}=\varnothing$ and
$\partial\mathbf{n}_{2}=\\{0,x\\}$. In particular, $0$ and $x$ are connected
in $\mathbf{n}_{2}$ since those are the only vertex with odd degree. Such a
pair $\mathbf{n}=(\mathbf{n}_{1},\mathbf{n}_{2})$ can be seen as the sum
$\mathbf{n}_{1}+\mathbf{n}_{2}$. It is well known (see for instance [8] that

$\langle\sigma_{0};\sigma_{x}\rangle_{\Lambda;\beta}=\langle\sigma_{0}\sigma_{x}\rangle_{\Lambda;\beta}\mathbb{P}^{\varnothing,\\{0,x\\}}_{\Lambda^{\mathfrak{g}};\beta}\left[0\mathrel{\ooalign{$\leftrightarrow$\cr$/$}}\mathfrak{g}\right].$
(9)

Note that every current $\mathbf{n}$ can be seen as a percolation realization
$(\omega_{e})_{e\in E_{\Lambda}}$, by declaring an edge $e$ is said to be open
if and only if $\mathbf{n}_{e}>0$.

#### Partial finite energy property

One can show that [17]

$\mathbb{P}^{A}_{\Lambda^{\mathfrak{g}},\beta}\left(\mathbf{n}_{e}>0\leavevmode\nobreak\
|\leavevmode\nobreak\ \mathbf{n}_{f}=m(f),\leavevmode\nobreak\ \forall f\neq
e\right)\geq\frac{\cosh(\beta J_{e})-1}{\cosh(\beta J_{e})},$ (10)

for any edge $e=\\{a,b\\}\in E_{\Lambda}$ and any function
$m:\Lambda\setminus\\{e\\}\rightarrow\mathbb{N}$ compatible with $A$. This in
particular implies that

$\mathbb{P}^{A}_{\Lambda^{\mathfrak{g}};\beta}\left(\mathbf{n}_{e}=0\leavevmode\nobreak\
|\leavevmode\nobreak\ \mathbf{n}_{f}=m(f),\leavevmode\nobreak\ \forall f\neq
e\right)\leq 2\mathsf{e}^{-\beta J_{e}}.$ (11)

Furthermore, recall that if $\beta>\beta_{c}$, then for any set $B$ with
$\absolutevalue{B}\in\\{0,2\\}$, there exists $C^{\prime}>0$ such that
$C^{\prime}\leq\langle\sigma_{B}\rangle_{\Lambda;\beta,h}\leq 1$ (we set
$\sigma_{\varnothing}=1$). There exists $C>0$ such that for any set
$\\{e_{1},...,e_{k}\\}$ of edges in $\Lambda$, one has

$\mathbb{P}^{B}_{\Lambda^{\mathfrak{g}};\beta,h}\left(\mathbf{n}_{e_{1}}\geq
1,...,\mathbf{n}_{e_{k}}\geq 1\right)\leq C\beta^{k}\prod_{i=1}^{k}J_{e_{i}}.$

Indeed, summing on all currents $\mathbf{n}$ with $\partial\mathbf{n}=B$
satisfying $\mathbf{n}_{e_{i}}\geq 1$ for $1\leq i\leq k$, one gets

$\displaystyle\sum_{\mathbf{n}}\frac{w(\mathbf{n})}{Z_{\Lambda}(B)}$
$\displaystyle\leq\left(\beta^{k}\prod_{i=1}^{k}J_{e_{i}}\right)\sum_{\tilde{\mathbf{n}}}\frac{w(\tilde{\mathbf{n}})}{Z_{\Lambda}(B)}\leq\frac{\langle\sigma_{S}\rangle_{\Lambda;\beta}}{\langle\sigma_{B}\rangle_{\Lambda;\beta}}\beta^{k}\prod_{i=1}^{k}J_{e_{i}}\leq
C\beta^{k}\prod_{i=1}^{k}J_{e_{i}},$

where the second sum is on the currents $\tilde{\mathbf{n}}$ having as sources
set the symmetric difference $S:=A\Delta e_{1}\Delta...\Delta e_{k}$. Putting
these two results together, one thus gets

$\mathbb{P}^{B}_{\Lambda^{\mathfrak{g}};\beta}\left(\mathbf{n}_{e_{1}}\geq
1,...,\mathbf{n}_{e_{k}}\geq
1,\mathbf{n}_{f_{1}}=...=\mathbf{n}_{f_{l}}=0\right)\leq
C2^{l}\beta^{k}\prod_{i=1}^{k}J_{e_{i}}\prod_{j=1}^{l}\mathsf{e}^{-\beta
J_{f_{j}}},$ (12)

for any family $\\{f_{1},...,f_{l}\\}$ of edges.

####  2.5.3. Convex geometry

It will be convenient to introduce a few quantities associated to the norm
$\rho$. First, two convex sets are important: the unit ball
$\mathscr{U}\subset\mathbb{R}^{d}$ associated to $\rho$ and the corresponding
_Wulff shape_

$\mathscr{W}=\\{t\in\mathbb{R}^{d}\,:\,\forall x\in\mathbb{R}^{d},\,t\cdot
x\leq\rho(x)\\}.$

Given a direction $s\in\mathbb{S}^{d-1}$, we say that the vector
$t\in\mathbb{R}^{d}$ is dual to $s$ if $t\in\partial\mathscr{W}$ and $t\cdot
s=\rho(s)$. A direction $s$ possesses a unique dual vector $t$ if and only if
$\mathscr{W}$ does not possess a facet with normal $s$. Equivalently, there is
a unique dual vector when the unit ball $\mathscr{U}$ has a unique supporting
hyperplane at $s/\rho(s)$. (See Fig. 1 for an illustration.) We refer to [15]
for the necessary backround on the convex geometry.

Figure 1. Left: The unit ball for the norm
$\rho(\cdot)=\left\|\cdot\right\|_{\scriptscriptstyle 1}$. Middle: the
corresponding Wulff shape $\mathscr{W}$ with two vectors $t_{1}$ and $t_{2}$
dual to $s=(1,0)$. Right: the set $\mathscr{W}$ with the unique vector $t$
dual to $s=\frac{1}{\sqrt{5}}(2,1)$.

####  2.5.4. Saturation transition

Recall that (8) implies that $\nu_{\beta}(s)\leq\rho(s)$ for every $\beta>0$.
As explained in the introduction, we consider the saturation point above the
critical temperature in the direction $s\in\mathbb{S}^{d-1}$ defined by

$\beta_{\rm
sat}(s)=\sup\\{\beta\in[0,\beta_{\mathrm{c}}]:\nu_{\beta}(s)=\rho(s)\\}.$

For $t\in\mathscr{W}$, we define

$\mathbb{G}_{\beta}(t)=\sum_{x\in\mathbb{Z}^{d}}\mathsf{e}^{t\cdot
x}\Phi_{\beta}(0\leftrightarrow x)\qquad\text{ and
}\qquad\mathbb{J}(t)=\sum_{x\in\mathbb{Z}^{d}}\mathsf{e}^{t\cdot x}J_{0,x},$

and an associated transition points

$\hat{\beta}_{\rm sat}(t)=\sup\\{\beta\geq
0\,:\,\mathbb{G}_{\beta}(t)<\infty\\},$

and

$\hat{\beta}_{\rm
sat}(s)=\smashoperator[]{\sup_{\begin{subarray}{c}t\in\mathscr{W}\\\ \text{t
dual to s}\end{subarray}}^{}}\hat{\beta}_{\rm sat}(t).$

It was proved in [4] that if $\psi(x)=\rho(x)^{-\alpha}$ with $\alpha>2d$ or
$\psi(x)=\mathsf{e}^{-c\rho(x)^{\eta}}$ with $\eta\in(0,1)$ and $c>0$, then
$\hat{\beta}_{\rm sat}(s)=\beta_{\rm sat}(s)$. We can now state the criterion
ensuring the existence of a non-trivial saturation point:

######  Theorem 2.2.

Let $J$ be exponentially decaying. Fix $s\in\mathbb{S}^{d-1}$.Then $\beta_{\rm
sat}(s)>0$ if and only if there exists a dual vector $t$ to $s$ such that
$\mathbb{J}(t)<\infty$.

Note that $\mathbb{J}(t)<\infty$ whenever
$\psi(x)=\mathsf{O}(\rho(x)^{-d-\varepsilon})$ for some $\varepsilon>0$. An
even more explicit (although a little bit less general) criterion ensuring the
finitude of $\mathbb{J}(t)$ was derived in [2]. It was proved in [11] that
$\nu_{\beta_{\mathrm{c}}}(s)=0$ for every $s\in\mathbb{S}^{d-1}$, and
therefore one always has $\beta_{\rm sat}(s)<\beta_{\mathrm{c}}$ by the
continuity of the function $\beta\mapsto\nu_{\beta}(s)$.

We also introduce a saturation point below the critical temperature in the
direciton $s$ defined by

$\beta_{\rm
sat}^{*}(s)=\sup\\{\beta\in[\beta_{\mathrm{c}},\infty):\nu_{\beta^{\prime}}(s)=\rho(s)\hphantom{,}\forall\beta^{\prime}>\beta\\}.$

##  3\. Main results and conjectures

######  Theorem 3.1.

For any $t\in\mathscr{W}$ such that $\hat{\beta}_{\rm sat}(t)>0$, there exists
$C:=C(\hat{\beta}_{\rm sat}(t))$ and a strictly increasing sequence
$(n_{k})_{k=1}^{\infty}$ such that

$\sum_{x\in\Lambda_{n_{k}}}\mathsf{e}^{t\cdot x}\Phi_{\hat{\beta}_{\rm
sat}(t)}(0\xleftrightarrow{\Lambda_{n_{k}}}x)\geq Ck.$

In particular,

$\mathbb{G}_{\hat{\beta}_{\rm sat}(t)}(t)=\infty.$

Moreover, if $\psi(x)=\mathsf{O}(\rho(x)^{-d-1-\varepsilon})$ for some
$\varepsilon>0$, one can choose $n_{k}=k$.

Theorem 3.1 has the following immediate Corollary.

######  Corollary 3.2.

Suppose that $\psi$ has one of the following forms:

  * •

$\psi(x)=\rho(x)^{-\alpha}$ with $\alpha>2d$

  * •

$\psi(x)=\mathsf{e}^{-\tilde{c}\rho(x)^{\eta}}$ with $\tilde{c}>0$ and
$\eta\in(0,1)$.

Then there exists $C>0$ such that for any $t$ dual to $s$

$\sum_{x\in\Lambda_{n}}\mathsf{e}^{t\cdot x}\Phi_{\beta_{\rm
sat}(s)}(0\xleftrightarrow{\Lambda_{n}}x)\geq Cn.$

In particular, $\mathbb{G}_{\beta_{\rm sat}(s)}(t)=\infty$ for any $t$ dual to
$s$.

###### Proof.

It was proved in [4] that $\beta_{\rm sat}(s)=\hat{\beta}_{\rm sat}(s)$ under
the assumptions of Corollary 3.2. Therefore, the conclusion follows from
Theorem 3.1 since $\hat{\beta}_{\rm sat}(t)\leq\hat{\beta}_{\rm sat}(s)$.

∎

The next result will give a description of the saturation phenomenon as a
function of the direction $s$: if $\mathscr{W}$ is regular locally in a
strictly saturated direction $s$ (in the sense that $\beta<\beta_{\rm
sat}(s)$), then there exists a neighborhood of $s$ for which all the
directions are strictly saturated.

######  Lemma 3.3.

Fix $t\in\mathscr{W}$ and assume that $\mathscr{W}$ is locally strictly convex
and $C^{1}$. Fix the unique direction $s\in\mathbb{S}^{d-1}$ dual to $t$.
Assume that $\beta_{\rm sat}(s)=\hat{\beta}_{\rm sat}(s)$ locally and that
there exists $\delta>0$ such that $\mathbb{J}(h)<\infty$ for all
$h\in\partial\mathscr{W}\cap B_{\delta}(t)$. Then, for every $\beta<\beta_{\rm
sat}(s)$, there exists $\varepsilon>0$ such that for any
$s^{\prime}\in\mathbb{S}^{d-1}\cap B_{\varepsilon}(s)$, $\beta<\beta_{\rm
sat}(s^{\prime})$.

The next result gives the asymptotics of the two-point function at $\beta_{\rm
sat}(1)$ on $\mathbb{Z}$.

######  Theorem 3.4.

Fix $d=1$. Suppose that

  * •

$\psi(x)=\absolutevalue{x}^{-\alpha}$ with $\alpha>2$

  * •

$\psi(x)=\mathsf{e}^{-\tilde{c}\absolutevalue{x}^{\eta}}$ with $\tilde{c}>0$
and $\eta\in(0,1)$.

Then, there exists $C_{-}>0$ such that for any $x\in\mathbb{Z}$, one has

$C_{-}\leq\mathsf{e}^{\rho(x)}\Phi_{\beta_{\rm sat}(1)}(0\leftrightarrow
x)\leq 1.$

Our next result shows that a non-trivial saturation regime can exist even at
arbitrarily low temperatures for the truncated two-point function.

######  Theorem 3.5.

Let $d\geq 2$ and $s\in\mathbb{S}^{d-1}$. Suppose that $J_{e}>0$ for any edge
of length 1. If there exists $t\in\partial\mathcal{W}$ dual to $s$ such that
$\mathbb{J}(t)<\infty$, then there exists $\beta_{0}$ such that $\beta_{\rm
sat}^{*}(s)<\beta_{0}$. Moreover, for any $\beta>\beta_{0}$ there exists
$C_{-},C_{+}>0$ such that

$C_{-}J_{ns}\leq\langle\sigma_{0};\sigma_{ns}\rangle_{\beta}\leq C_{+}J_{ns}.$

######  Remark 3.1.

Notice that in Theorem 3.5, we take $d\geq 2$. This assumption is necessary,
since by definition one has $\beta_{\rm sat}^{*}(s)\geq\beta_{\mathrm{c}}$,
and $\beta_{\mathrm{c}}=\infty$ on $\mathbb{Z}$. This is in contrast with what
happens at high temperatures, in which case Theorem 2.2 holds.

Theorem 3.5 is in contrast with what happens in the finite-range Ising model,
in which case it was proved in [6] that the truncated two-point function
satisfies OZ asymptotics on $\mathbb{Z}^{d}$ with $d\geq 3$ (see also [7]).

Our work suggests a number of conjectures and open problems that we summarize
now.

####  3.0.1. Behaviour at $\beta_{\rm sat}(s)$

Theorem 3.4 suggests that the OZ asymptotics should hold at $\beta_{\rm
sat}(s)$ whenever $\psi$ decays fast enough.

######  Conjecture 3.6.

For any $\psi$ decaying fast enough, the conclusion of Theorem 3.4 holds on
$\mathbb{Z}^{d}$.

However, this is easily seen not to be true in general. To see that, fix
$\rho(\cdot)=\|\cdot\|_{1}$. Using the results of [2], it can easily be seen
that for $\psi(x)=\rho(x)^{-\alpha}$, $\beta_{\rm sat}(e_{1})>0$ whenever
$\alpha>1$. However, one always has the lower bound

$CJ_{x}\leq\langle\sigma_{0}\sigma_{x}\rangle_{\beta}.$

This shows that OZ asymptotics cannot hold in this case whenever $d>3$.

######  Open problem 3.7.

Caracterise all possible behaviours of the two-point function at $\beta_{\rm
sat}(s)$ in function of the dimension and $\rho$.

We expect that the OZ asymptotics could fail at $\beta_{\rm sat}$ for two
different reasons:

  1. (1)

The dominant contribution to the FK-Ising two-point function comes from
configurations with $\absolutevalue{\mathcal{C}_{0}}=\mathsf{o}(n)$.

  2. (2)

The dominant contribution to the FK-Ising two-point function comes from
configurations with $\absolutevalue{\mathcal{C}_{0}}=\mathsf{O}(n)$ (as is the
case in the OZ regime), but the steps of the associated effective random walk
don’t have two moments, and so the usual local limit theorem does not hold
(however, there has been results on the non-OZ asymptotic behaviour of the
Green function in this case, see [5] and references therein).

We plan to come back to this issue in a simpler context of the killed random
walk (see section 5 for the definition of this model).

####  3.0.2. Behaviour for $\beta>\beta_{\mathrm{c}}$

In the case of exponentially decaying coupling constants, Theorem 3.5 implies
the exponential decay of the two-point function for $\beta$ large enough
whenever $\psi$ decays fast enough. We expect this to hold more generally
below the critical temperature.

######  Conjecture 3.8.

If there exists $c>0$ such that $J_{x}\leq\mathsf{e}^{-c\|x\|}$, then
$\nu_{\beta}(s)>0$ for $\beta>\beta_{\mathrm{c}}$ and $s\in\mathbb{S}^{d-1}$.

Understanding the behaviour of the truncated two-point function without an
external field non pertubatively below critical temperature is challenging.
The exponential decay of the two-point function in the finite-range Ising
models was established only recently in [9]. For $\beta<\beta_{c}$, the
conclusion of Conjecture 3.8 was established in [4] using the random cluster
representation of the Ising model and the OSSS inequality for monotonic
measures (see [10]). Since the double random-current is not known to be
monotonic, one cannot use the same reasoning to prove Conjecture 3.8.

We also expect the same dichotomy of behaviour of the two-point function
between $\beta<\beta_{\rm sat}(s)$ and $\beta\in(\beta_{\rm
sat}(s),\beta_{\mathrm{c}})$ (see (2) and (1)) to happen below the critical
temperature.

######  Conjecture 3.9.

  1. (1)

For $\beta>\beta_{\rm sat}^{*}(s)$, there exists $C>0$ such that

$\langle\sigma_{0};\sigma_{ns}\rangle_{\beta}=CJ_{ns}(1+\mathsf{o}_{n}(1)).$

  2. (2)

For $\beta\in(\beta_{\mathrm{c}},\beta_{\rm sat}^{*}(s))$, there exists $C>0$
such that

$\langle\sigma_{0};\sigma_{ns}\rangle_{\beta}=Cn^{-\frac{d-1}{2}}\mathsf{e}^{-\nu_{\beta}(x)}(1+\mathsf{o}_{n}(1)).$

###  3.1. Organisation of the paper

In Section 4, we will prove Theorem 3.1, Theorem 3.4 and Lemma 3.3 using the
so-called $\varphi(S)$ argument. In Section 5, we will prove Theorem 3.5 by
comparing directly the random-current representation of the truncated two-
point function to the Green function associated to a well-chosen killed random
walk. Note that different parts are essentially independent.

##  4\. $\varphi(S)$ argument

In this section, we are going to prove Theorem 3.1, Lemma 3.3 and Theorem 3.4.
Generalizing what has been done in [11], given a finite subset $S$ containing
$0$, $t\in\partial\mathscr{W}$ and $\beta>0$, let us define

$\varphi_{\beta}(S,t)=\beta\sum_{x\in S}\sum_{y\notin S}\mathsf{e}^{t\cdot
x}\Phi_{\beta}(0\xleftrightarrow{S}x)J_{x,y}\mathsf{e}^{t\cdot(y-x)}.$

Moreover, we define

$\tilde{\beta}_{\mathrm{sat}}(t)=\sup\\{\beta\geq 0\,:\,\text{there exists a
finite $S$ containing $0$ such that $\varphi_{\beta}(S,t)<1$}\\}.$

We will need the following lemma:

######  Lemma 4.1.

Fix $t\in\partial\mathscr{W}$ and $\beta>0$. Assume that $\hat{\beta}_{\rm
sat}(t)>0$.

  1. (1)

If there exists a finite subset $S\ni 0$ such that $\varphi_{\beta}(S,t)<1$,
then there exists $C=C(S)>0$ such that

$\mathbb{G}(t)\leq\dfrac{C}{1-\varphi_{\beta}(S,t)}<\infty.$

  2. (2)

There exist $c>0$ and a strictly increasing sequence $(n_{k})$ such that

$c\sum_{k=1}^{l}\varphi_{\beta}(\Lambda_{n_{l}},t)\leq\sum_{x\in\Lambda_{n_{l}}}\mathsf{e}^{t\cdot
x}\Phi_{\beta}(0\xleftrightarrow{\Lambda_{n_{l}}}x).$

If $\psi(x)=\mathsf{O}(\rho(x)^{-d-1-\varepsilon})$ for some $\varepsilon>0$,
one can take $n_{k}=k$.

In particular, $\tilde{\beta}_{\mathrm{sat}}(t)=\hat{\beta}_{\rm sat}(t)$.

Before proving Lemma 4.1, let us see how it implies Theorem 3.1 and Lemma 3.3.

###### Proof of Theorem 3.1.

Since $\varphi_{\beta}(S,t)$ is a continuous function in $\beta$ and $[0,1)$
is open in $[0,\infty)$, it follows that at $\tilde{\beta}_{\mathrm{sat}}(t)$,
for every $\Lambda\ni 0$, we have
$\varphi_{\tilde{\beta}_{\mathrm{sat}}(t)}(\Lambda,t)\geq 1$. This in turn
implies, by the second part of Lemma 4.1, the conclusion of Theorem 3.1. ∎

###### Proof of Theorem 3.4.

We will only show the result for $x>0$, since the result follows for $x$
negative by symmetry. The right inequality follows directly from (3) and
$\nu_{\beta_{\rm sat}(1)}(1)=\rho(1)$. For the left inequality, remark that
since we assumed that $\alpha>2$, it follows from Corollary 3.2 that for every
$x\geq 1$:

$\sum_{k=1}^{x}\mathsf{e}^{k}\Phi_{\beta_{\rm sat}(1)}(0\leftrightarrow k)\geq
Cx-\sum_{k=0}^{x}\mathsf{e}^{-k}\Phi_{\beta_{\rm
sat}(1)}(0\leftrightarrow-k).$

Since $\mathsf{e}^{k}\Phi_{\beta_{\rm sat}(1)}(0\leftrightarrow k)\in[0,1]$ by
(3), this implies that there exists $R>0$ and $c>0$ such that for any
$m\in\mathbb{N}$, there exists $k\in\\{m,\dots,m+R\\}$ such that one has

$\mathsf{e}^{k}\Phi_{\beta_{\rm sat}(1)}(0\leftrightarrow k)\geq c.$

The result then follows by the finite energy (6) and FKG for every
$x\in\mathbb{N}$. ∎

###### Proof of Lemma 3.3.

In order to prove Lemma 3.3, note that, by assumption, $\beta<\beta_{\rm
sat}(s)=\hat{\beta}_{\mathrm{sat}}(t)=\tilde{\beta}_{\mathrm{sat}}(t)$, where
the last equality is given by Lemma 4.1. It follows that there exists a finite
$S$ containing $0$ such that $\varphi_{\beta}(S,t)<1$. Since $\mathbb{J}$ is
locally finite (around $t$) and $S$ is finite, it follows by continuity that
$\varphi_{\beta}(S,h)<1$ for $h\in
B_{\varepsilon^{\prime}}(t)\cap\mathcal{W}_{\rho}$. This implies that
$\beta<\beta_{\rm sat}(s^{\prime})$ for $s^{\prime}$ in some small
neighborhood around $s$ since $\mathscr{W}$ is locally strictly convex and
$\beta_{\rm sat}=\tilde{\beta}_{\mathrm{sat}}$ locally, which is the desired
result. ∎

###### Proof of Lemma 4.1.

We follow here ideas developed in [11]. First, suppose that there exists $S$
containing $0$ such that $\varphi_{\beta}(S,t)<1$. Let
$\Lambda\subset\mathbb{Z}^{d}$ and let

$\tilde{\chi}(\Lambda,t,\beta)=\max\Bigl{\\{}\sum_{v\in\Lambda}\mathsf{e}^{t\cdot(v-u)}\Phi_{\beta}(u\xleftrightarrow{\Lambda}v)\,:\,u\in\Lambda\Bigr{\\}}.$

Let us fix $u\in\Lambda$ and denote by $S_{u}$ the translation of $S$ by $u$.
Fix $v\in\Lambda\setminus S_{u}$. If $u$ is connected to $v$, then there
exists $x\in S_{u}$ and $y\notin S_{u}$ such that $u$ is connected to $x$ in
$S$, $\\{x,y\\}$ is open and $y$ is connected to $v$. Using the union bound
and the Simon–Lieb inequality (7), we get

$\mathsf{e}^{t\cdot(v-u)}\Phi_{\beta}(u\xleftrightarrow{\Lambda}v)\leq\\\
\sum_{x\in S_{u}}\sum_{y\notin
S_{u}}\mathsf{e}^{t\cdot(x-u)}\Phi_{\beta}(u\xleftrightarrow{S_{u}}x)\mathsf{e}^{t\cdot(y-x)}\beta
J_{x,y}\mathsf{e}^{t\cdot(v-y)}\Phi(y\xleftrightarrow{\Lambda}v).$

Summing over $v\in\Lambda\setminus S_{u}$, we get

$\sum_{v\in\Lambda\setminus
S_{u}}\mathsf{e}^{t\cdot(v-u)}\Phi_{\beta}(u\xleftrightarrow{\Lambda}v)\leq\varphi_{\beta}(S,t)\tilde{\chi}(\Lambda,t,\beta),$

where we used the invariance under translations. Since $S$ is finite, there
exists $C:=C(S)>0$ such that

$\sum_{v\in\Lambda}\mathsf{e}^{t\cdot(v-u)}\Phi_{\beta}(u\xleftrightarrow{\Lambda}v)\leq
C+\varphi_{\beta}(S,t)\tilde{\chi}(\Lambda,t,\beta).$

Now, we can optimize over $u$ to get

$\tilde{\chi}(\Lambda,t,\beta)\leq
C+\varphi_{\beta}(S,t)\tilde{\chi}(\Lambda,t,\beta),$

which can be rewritten as

$\tilde{\chi}(\Lambda,t,\beta)\leq\frac{C}{1-\varphi_{\beta}(S,t)}.$

Taking the limit $\Lambda\uparrow\mathbb{Z}^{d}$, we obtain

$\mathbb{G}_{\beta}(t)\leq\frac{C}{1-\varphi_{\beta}(S,t)}<\infty,$

where the last inequality follows from the assumption
$\varphi_{\beta}(S,t)<1$.

Let us now turn to the second point. For any strictly increasing sequence
$(n_{k})$, one has

$\displaystyle\sum_{k=1}^{l}\varphi_{\beta}(\Lambda_{n_{k}},t)$
$\displaystyle=\sum_{k=1}^{l}\sum_{x\in\Lambda_{n_{k}}}\sum_{y\notin\Lambda_{n_{k}}}\mathsf{e}^{t\cdot
x}\Phi_{\beta}(0\xleftrightarrow{\Lambda_{n_{k}}}x)\mathsf{e}^{t\cdot(y-x)}\beta
J_{x,y}$ $\displaystyle\leq\sum_{x\in\Lambda_{n_{l}}}\mathsf{e}^{t\cdot
x}\Phi_{\beta}(0\xleftrightarrow{\Lambda_{n_{l}}}x)\sum_{k=1}^{l}\sum_{y\notin\Lambda_{n_{k}}}\mathsf{e}^{t\cdot(y-x)}\beta
J_{x,y}\mathds{1}_{x\in\Lambda_{n_{k}}}.$

Given $x\in\mathbb{Z}^{d}$, let us prove that the double sum over $k$ and $y$
is finite. The sum over $y$ is bounded by $\mathbb{J}(t)$ which is finite.
Indeed, for $\beta=\hat{\beta}_{\rm sat}(t)/2>0$ by hypothesis, one has by
finite energy

$\mathbb{G}_{\beta}(t)\geq C_{\beta}\mathbb{J}(t).$

This implies that

$\lim_{L\rightarrow\infty}\sum_{y\notin\Lambda_{L}}\mathsf{e}^{t\cdot(y-x)}J_{xy}=0.$

We can thus choose $n_{k}$ such that

$\sum_{k\geq
1}\sum_{y\notin\Lambda_{n_{k}}}\mathsf{e}^{t\cdot(y-x)}J_{x,y}=C<\infty.$

Moreover, if $\psi(x)=\mathsf{O}(\rho(x)^{-d-1-\varepsilon})$ for some
$\varepsilon>0$, this last sum is finite if one chooses $n_{k}=k$ since
$t\cdot y-\rho(y)\leq 0$ for any $y\in\mathbb{Z}^{d}$ by definition of the
dual vector $t$. Therefore, we get

$\sum_{k=1}^{l}\varphi_{\beta}(\Lambda_{n_{k}},t)\leq
C\sum_{x\in\Lambda_{n_{l}}}\mathsf{e}^{t\cdot
x}\Phi_{\beta}(0\xleftrightarrow{\Lambda_{n_{l}}}x),$

which proves the desired identity. ∎

##  5\. The existence of a saturation transition at low temperatures

In this section, we are going to prove Theorem 3.5. Through this section, we
are going to assume that $J_{e}>0$ for any edge of length 1. By rotational
invariance, we can assume without loss of generality that $J_{e}=1$ for every
edge of length $1$. We start by making a brief summary of the result proved in
[2] that we rely on. Given $\lambda>0$, we define the Green function of the
killed random walk model by

$G_{\lambda}^{\mathrm{KRW}}(x,y)=\sum_{\gamma:x\rightarrow
y}\prod_{i=1}^{\absolutevalue{\gamma}}\lambda J_{\gamma_{i-1},\gamma_{i}},$

where the sum is over edge self-avoiding paths from $x$ and $y$. We will need
the following result proved in [2].

######  Theorem 5.1.

Fix $s\in\mathbb{S}^{d-1}$. If there exists a dual vector $t$ to $s$ such that
$\mathbb{J}(t)<\infty$, then there exists $\lambda_{0}$, such that for every
$\lambda<\lambda_{0}$, there exists $C:=C(\lambda)>0$ such that

$G^{\mathrm{KRW}}_{\lambda}(0,x)\leq CJ_{x}.$

The next result bounds the truncated two-point function of the Ising model by
the Green functions introduced above

######  Lemma 5.2.

There exists $\beta_{0}$ such that for any $\beta>\beta_{0}$, there exists
$C_{+}>0$ such that

$\langle\sigma_{0};\sigma_{x}\rangle_{\beta}\leq
C_{+}G_{\lambda(\beta)}^{\mathrm{KRW}}(0,x),$

where $\lim_{\beta\rightarrow\infty}\lambda(\beta)=0$.

Before proving Lemma 5.2, let us show how it implies Theorem 3.5.

###### Proof of Theorem 3.5.

On the one hand, the lower bound in Theorem 3.5 follows directly from (8) for
any $\beta>0$. Fix $\beta_{1}$ such that $\lambda(\beta)<\lambda_{0}$ for any
$\beta>\beta_{1}$. Let $\beta_{2}=\max\\{\beta_{0},\beta_{1}\\}$. Then, for
any $\beta>\beta_{2}$

$\langle\sigma_{0};\sigma_{x}\rangle_{\beta}\leq
C_{+}G_{\lambda(\beta)}^{\mathrm{KRW}}(0,x)\leq cJ_{x},$

where we used Lemma 5.2 in the first inequality and Theorem 5.1 in the second
inequality. This gives the desired result. ∎

Heuristic proof of Lemma 5.2, Thanks to (9), we need to compare
$\mathbb{P}^{\varnothing,\\{0,x\\}}_{\Lambda^{\mathfrak{g}},\beta}[0\mathrel{\ooalign{$\leftrightarrow$\cr$/$}}\mathfrak{g}]$
to a Green function of a killed random walk. Recall that one has
$J_{x\mathfrak{g}}=\sum_{y\in\Lambda^{c}}J_{xy}$. Therefore, thanks to (11),
most of the points $y$ close to $\partial\Lambda$ will not be connected to
$\partial\Lambda$, which allows us to replace the event
$\\{0\mathrel{\ooalign{$\leftrightarrow$\cr$/$}}\mathfrak{g}\\}$ with the
event $\\{0\mathrel{\ooalign{$\leftrightarrow$\cr$/$}}\partial\Lambda\\}$.
This term can be estimated using Peierls-like argument: we will decompose
$\mathcal{C}_{0,x}$ into $C_{1},\dots,C_{k}$ where $C_{i}$’s are disjoint
nearest neighbor connected components of $\mathcal{C}_{0,x}$. We will extract
a path $\gamma$ from $0$ to $x$ in such a way that all points of $\gamma$ are
in $\cup_{i=1}^{k}C_{i}$ and that
$\absolutevalue{\gamma^{(i)}}=K\absolutevalue{\partial C_{i}}$ for some $K>0$
where $\gamma^{(i)}$ is the part of $\gamma$ in $C_{i}$. In this way, using
(12) and stantard perturbative estimates, we will extract for $\gamma^{(i)}$ a
cost of order

$e^{-c\beta\absolutevalue{\partial
C_{i}}}\beta^{\absolutevalue{\gamma^{(i)}}}\prod_{e\in\gamma^{(i)}}J_{e},$

which can compared easily to a Green function of a killed random walk with
parameter $\lambda(\beta)$ satisfying
$\lim_{\beta\rightarrow\infty}\lambda(\beta)=0$.

###### Proof of Lemma 5.2.

We are going to prove Lemma 5.2 only for $d=2$ where the use of planar duality
simplifies the notations. One can generalize the argument that follows for any
$d\geq 2$ in a standard way by introducing $d-1$ dimensional plaquettes (i.e.,
the $d-1$ dimensional faces of a $d$ dimensional hypercubes). We define the
dual graph of $\mathbb{Z}^{2}$ by

$(\mathbb{Z}^{2})^{\ast}=\mathbb{Z}^{2}+(1/2,1/2).$

The edges of $(\mathbb{Z}^{2})^{\ast}$ are called dual edges, and any dual
edge $e^{\ast}$ is perpendicular to an unique edge $e$ of $\mathbb{Z}^{2}$.
Therefore, there is a one-to one correspondance between percolation
configurations on $\mathbb{Z}^{2}$ and those on $(\mathbb{Z}^{2})^{\ast}$,
where a dual edge $e^{\ast}$ is open if and only if $e$ is closed.

We are going to use the random-current representation of the truncated two-
point function (9) (see section 2.5.2). We will only work with a single
current since one has

$\mathbb{P}^{\varnothing,\\{0,x\\}}_{\Lambda^{\mathfrak{g}}_{N},\beta}[0\mathrel{\ooalign{$\leftrightarrow$\cr$/$}}\mathfrak{g}]\leq\mathbb{P}^{\\{0,x\\}}_{\Lambda^{\mathfrak{g}}_{N},\beta}[0\mathrel{\ooalign{$\leftrightarrow$\cr$/$}}\mathfrak{g}].$

To any percolation configuration $\omega$ induced by a current $\mathbf{n}$ on
$\Lambda_{N}^{\mathfrak{g}}$ with sources $\\{0,x\\}$, we can associate a new
percolation configuration $(\hat{\omega}_{e})_{e\in E_{\Lambda_{N+1}}}$ as
follows:

$\hat{\omega}_{e}=\left\\{\begin{array}[]{ll}\omega_{e}&\mbox{if}\leavevmode\nobreak\
e\in E_{\Lambda_{N}}\\\ \omega_{x\mathfrak{g}}&\mbox{if}\leavevmode\nobreak\
e=\\{x,y\\},\leavevmode\nobreak\ x\in\partial\Lambda_{N},\leavevmode\nobreak\
y\in\partial\Lambda_{N+1},\leavevmode\nobreak\ |x-y|_{1}=1\\\
0&\mbox{otherwise. }\end{array}\right.$

We therefore have a surjective mapping $F:\omega\mapsto\hat{\omega}$ from the
set of currents on $\Lambda_{N}^{\mathfrak{g}}$ having sources $\\{0,x\\}$ to
the set of percolation configurations on $E_{\Lambda_{N+1}}$. The law of
$\hat{\omega}$ previously defined is therefore the push-forward measure of
$\mathbb{P}_{\Lambda_{N}^{\mathfrak{g}},\beta}^{\\{0,x\\}}$ by $F$. Said
differently, $\hat{\omega}\sim\mathbf{P}_{\Lambda_{N+1}}$, where
$\mathbf{P}_{\Lambda_{N+1}}$ is the probability measure defined by

$\mathbf{P}_{\Lambda_{N+1}}(A)=\mathbb{P}_{\Lambda_{N}^{\mathfrak{g}},\beta}^{\\{0,x\\}}(F^{-1}(A))$

for any $A\in{\\{0,1\\}}^{E_{\Lambda_{N+1}}}$. Remark that
$\mathbf{P}_{\Lambda_{N+1}}$ inherits the finite energy lower bound (12) from
$\mathbb{P}_{\Lambda_{N}}^{\\{0,x\\}}$. This allows us to reinterpret
$0\mathrel{\ooalign{$\leftrightarrow$\cr$/$}}\mathfrak{g}$ as the event that
$0$ is disconnected from $\partial\Lambda_{N+1}$. Indeed, observe that in
order to have a connection from $0$ to $\partial\Lambda_{N+1}$ in
$\hat{\omega}$, there must be a connection from $0$ to $\mathfrak{g}$ in
$\omega$. This implies in particular that

$\mathbb{P}^{\\{0,x\\}}_{\Lambda^{\mathfrak{g}}_{N}}[0\mathrel{\ooalign{$\leftrightarrow$\cr$/$}}\mathfrak{g}]\leq\mathbf{P}_{\Lambda_{N+1}}[0\mathrel{\ooalign{$\leftrightarrow$\cr$/$}}\partial\Lambda_{N+1},0\leftrightarrow
x].$

Figure 2. A realization of $\mathcal{C}_{0,x}$. The open dual edges are dashed
and the $\alpha_{i}$’s are in red.

Such an event can easily be described using dual blocking surfaces in a
Peierls-like argument. We will call a path basic if it only uses edges of
length $1$. Consider $\mathcal{C}_{0,x}$ the joint cluster of $0$ and $x$. For
any $y\in\mathcal{C}_{0,x}$, denote by $[y]$ the (random) set of points
$z\in\mathcal{C}_{0,x}$ such that there exists an open basic path joining $y$
to $z$. Choose an arbitrary order on $\mathbb{Z}^{d}$. Choose
$\gamma=(\gamma_{0},...,\gamma_{n})$ joining $0$ to $x$ to be an open self-
avoiding path minimal according to this order. We extract a new path from
$\gamma$ using the following procedure. Let $r_{0}=0$ and

$r_{1}:=\max\\{i\leavevmode\nobreak\ :\leavevmode\nobreak\
\gamma_{i}\in[0],\leavevmode\nobreak\ 0\leq i\leq n\\}.$

For $k\geq 1$, define recursively

$r_{k+1}:=\max\\{i\leavevmode\nobreak\ :\leavevmode\nobreak\
\gamma_{i}\in[\gamma_{r_{k}+1}],\leavevmode\nobreak\ r_{k}<i\leq n\\}.$

This procedure stops as soon as $r_{k}=n$. Let $m=m(\gamma)$ be such that
$r_{m}=n$. By construction, for any $1\leq k<m$ we have the inclusion
$\\{\gamma_{r_{k}+1},...,\gamma_{r_{k+1}}\\}\subset[\gamma_{r_{k}}]$, and the
sets $\left([\gamma_{r_{k}}]\right)_{k\geq 1}$ are all disjoint sets. For any
$k\geq 1$, there is a minimal self-avoiding basic path of open edges joining
$\gamma_{r_{k}+1}$ to $\gamma_{r_{k+1}}$, using only points in
$[\gamma_{r_{k}}]$, that is minimal with respect to the order we previously
chose. Denote by $\alpha_{k}$ such a path, and by $\lambda_{k}$ its length.
Denote by $\alpha_{0}$ (respectively $\lambda_{0}$) the self-avoiding basic
path joining $0$ to $\gamma_{r_{1}}$ (respectively its length).

We now have a new self-avoiding path joining $0$ to $x$ defined by taking the
union of the paths $(\alpha_{i})_{i\geq 0}$. From now on, we will denote by
$\gamma$ this new path in order to lighten the notations. To any cluster
realization of the cluster $\mathcal{C}_{0,x}$ one can thus associate an open
path $\Gamma(\mathcal{C}_{0,x})$ joining $0$ to $x$ using this procedure.
Moreover, each $\alpha_{k}$ is contained in the interior of a dual basic path
of open edges. Denote by $\partial^{\ast}[\gamma_{r_{k}}]$ the shortest such
path and by $\text{Int}(\partial^{\ast}[\gamma_{r_{k}}])$ its interior. We
call $\partial^{\ast}[\gamma_{r_{k}}]$ the dual boundary of
$[\gamma_{r_{k}}]$. Note that the $[\gamma_{r_{k}}]$’s are disjoint and each
edge belonging to one of their dual boundaries can belong at most to two
different boundaries. Since all the $[\gamma_{r_{i}}]$’s are connected
subgraphs of a lattice and the $\alpha_{i}$’s are of minimal length, there
exists a family $\alpha_{1}^{\ast},...,\alpha_{m}^{\ast}$ of dual basic paths
with $|\alpha_{i}^{\ast}|\geq|\alpha_{i}|$ and
$\absolutevalue{\alpha^{\ast}_{i}}\neq 0$ for all $0\leq i\leq n$, such that,
for every $i\in\\{1,\dots,n\\}$, one has

$\alpha_{i}^{\ast}\subset\left(\partial^{\ast}[\gamma_{r_{i}}]\cup\text{Int}(\partial^{\ast}[\gamma_{r_{i}}])\right)\setminus\bigcup_{j\neq
i}\text{Int}(\partial^{\ast}[\gamma_{r_{j}}])$

and such that there exists a deterministic constant $K>0$ satisfying

$\frac{\absolutevalue{\\{e^{\ast}\in\alpha_{i}^{\ast}:\omega_{e^{\ast}}=1\\}}}{|\alpha_{i}^{\ast}|}\geq
K.$ (13)

Notice that it is possible that
$\alpha_{i}^{*}=\partial^{\ast}[\gamma_{r_{i}}]$. We are going to prove that
there exists $C,c>0$ such that

$\mathbf{P}_{\Lambda_{N+1}}\left[0\mathrel{\ooalign{$\leftrightarrow$\cr$/$}}\partial\Lambda_{N+1},0\leftrightarrow
x,\Gamma(\mathcal{C}_{0,x})=\gamma\right]\leq\prod_{i=1}^{|\gamma|}C\mathsf{e}^{-c\beta}\beta
J_{\gamma_{i-1}\gamma_{i}}.$

In order to prove this inequality, we are going to use the fact that all edges
in $\gamma$ are open (which will give the contribution in $\beta
J_{\gamma_{i-1},\gamma_{i}})$, that all (dual) edges in
$\partial^{\ast}[\gamma_{r_{i}}]$ are open and that there exists a strictly
positive proportion of (dual) edges in $\alpha_{i}^{*}$ that are open. Fix now
some $\alpha_{k}$. We are going to separate between two cases. Firstly, assume
$\absolutevalue{\partial^{\ast}[\gamma_{r_{k}}]}\geq\absolutevalue{\alpha_{k}}$.
In this case, using (12), one has

$\mathbf{P}_{\Lambda_{N+1}}(\omega_{f^{\ast}}=1\hphantom{,}\forall
f^{\ast}\in\partial^{\ast}[\gamma_{r_{k}}])\leq
C2^{\absolutevalue{\partial^{\ast}[\gamma_{r_{k}}]}}\mathsf{e}^{-\frac{\beta}{2}\absolutevalue{\partial^{\ast}[\gamma_{r_{k}}]}}\leq
C\mathsf{e}^{-c\beta\absolutevalue{\partial^{\ast}[\gamma_{r_{k}}]}},$ (14)

In the first inequality, the $\frac{1}{2}$ factor ensures that edges belonging
to two different boundaries are not counted twice in the upcoming bounds.
Therefore, the existence of an open dual basic path
$\partial^{\ast}[\gamma_{r_{k}}]$ of length at least
$\absolutevalue{\alpha_{k}^{\ast}}$ surrounding $\alpha_{k}$ is an event of
probability $C\mathsf{e}^{-c\beta\absolutevalue{\alpha_{k}}}$.

Secondly, assume that
$\absolutevalue{\partial^{\ast}[\gamma_{r_{k}}]}<\absolutevalue{\alpha_{k}}$.
In this case, using (13) and(12), the existence of $\alpha_{k}^{\ast}$ is an
event with probability bounded by

$|\partial^{\ast}[\gamma_{r_{k}}]\cup\text{Int}(\partial^{\ast}[\gamma_{r_{k}}])|C\mathsf{e}^{-c\beta
K|\alpha_{k}^{\ast}|}\leq C\mathsf{e}^{-c^{\prime}\beta
K\absolutevalue{\alpha_{k}^{\ast}}},$

where we used that the number of ways of choosing open edges in
$\alpha_{k}^{\ast}$ is given by $\sum_{r\geq
K|\alpha_{i}^{\ast}|}\binom{|\alpha_{i}^{\ast}|}{r}$.

Putting all of this together, denoting by $A_{N}$ the event
$\\{0\leftrightarrow
x\\}\cap\\{0\mathrel{\ooalign{$\leftrightarrow$\cr$/$}}\Lambda_{N+1}\\}$, the
union bound gives

$\displaystyle\mathbf{P}_{\Lambda_{N+1}}\left(A_{N},\leavevmode\nobreak\
\Gamma(\mathcal{C}_{0,x})=\gamma\right)$
$\displaystyle\leq\sum_{\alpha^{\ast}_{1},...,\alpha^{\ast}_{m(\gamma)}}\prod_{k=1}^{m(\gamma)}C\mathsf{e}^{-c\beta
K\left|\alpha^{\ast}_{k}\right|}\prod_{i=1}^{|\gamma|}\beta
J_{\gamma_{i-1}\gamma_{i}}$ (15)
$\displaystyle\leq\left(\prod_{k=1}^{m(\gamma)}\sum_{\alpha^{\ast}_{k}}C\mathsf{e}^{-c\beta
K\left|\alpha^{\ast}_{k}\right|}\right)\prod_{i=1}^{|\gamma|}\beta
J_{\gamma_{i-1}\gamma_{i}}$ (16)
$\displaystyle=\prod_{k=1}^{m(\gamma)}C\mathsf{e}^{-c^{\prime}\beta|\lambda_{k}|}\prod_{i=1}^{|\gamma|}\beta
J_{\gamma_{i-1}\gamma_{i}},$ (17)

where, in the last line, we used that the number of paths $\alpha_{k}^{\ast}$
of length $l$ is bounded by $(2d)^{l}$ and took $\beta$ large enough. Since
$\sum_{k}\lambda_{k}\geq\frac{\absolutevalue{\gamma}}{2}$, there exist two
positive constants $C$ and $c$ such that

$\mathbf{P}_{\Lambda_{N+1}}\left[A_{n},\Gamma(\mathcal{C}_{0,x})=\gamma\right]\leq\prod_{i=1}^{|\gamma|}C\mathsf{e}^{-c\beta}\beta
J_{\gamma_{i-1}\gamma_{i}}.$ (18)

Therefore, for any $\beta$ big enough, one has

$\mathbf{P}_{\Lambda_{N+1}}\left[A_{n}\right]\leq\sum_{n\geq
0}\sum_{\gamma\in\text{SAW}_{n}(0,x)}\prod_{i=1}^{|\gamma|}C\mathsf{e}^{-c\beta}\beta
J_{\gamma_{i-1}\gamma_{i}}\leq G_{\lambda(\beta)}^{\text{KRW}}(0,x),$

where $\lim_{\beta\rightarrow\infty}\lambda(\beta)=0$. Since we have

$\langle\sigma_{0};\sigma_{x}\rangle_{\Lambda_{N};\beta}=\langle\sigma_{0}\sigma_{x}\rangle_{\Lambda_{N};\beta}\mathbb{P}^{\varnothing,\\{0,x\\}}_{\Lambda^{\mathfrak{g}}_{N},\beta}\left[0\mathrel{\ooalign{$\leftrightarrow$\cr$/$}}\mathfrak{g}\right]\leq\langle\sigma_{0}\sigma_{x}\rangle_{\Lambda_{N};\beta}\mathbf{P}_{\Lambda_{N+1}}\left[A_{n}\right],$

there exists a constant $c_{\beta}>0$ such that
$\langle\sigma_{0};\sigma_{x}\rangle_{\Lambda_{N};\beta}\leq
c_{\beta}G_{\lambda(\beta)}^{\text{KRW}}(0,x)$ for $\beta$ big enough. Taking
the limit as $N\rightarrow\infty$, one finally gets

$\langle\sigma_{0};\sigma_{x}\rangle_{\beta}\leq
c_{\beta}G_{\lambda(\beta)}^{\text{KRW}}(0,x),$

which is the desired result. ∎

######  Remark 5.1.

In the case of the Ising model with striclty positive magnetic field $h$, one
could prove that there exists a non-trivial saturated regime in a
straightforward way. Indeed, one can derive a random-current representation of
the truncated two-point function in such a way that $J_{x,\mathfrak{g}}=h>0$
for any $x\in\mathbb{Z}^{d}$ and that

$\langle\sigma_{0};\sigma_{x}\rangle_{\Lambda;\beta,h}=\langle\sigma_{0},\sigma_{x}\rangle_{\Lambda;\beta,h}\mathbb{P}^{\varnothing,\\{0,x\\}}_{\Lambda^{\mathfrak{g}};\beta,h}\left[0\mathrel{\ooalign{$\leftrightarrow$\cr$/$}}\mathfrak{g}\right].$

In particular, in this case, for any connection $\gamma$ from $0$ to $x$ in
the right-hand side, $\gamma$ has to be disconnected from $\mathfrak{g}$ which
is an event of probability of order $\mathsf{e}^{-ch\beta}$. Therefore, Lemma
5.2 holds in this case as well, from which the desired conclusion follows.

## Acknowledgments

YA is supported by the Swiss NSF grant 200021_200422 is a member of the NCCR
SwissMAP. KK thanks the Excellence Fellowship program at the University of
Geneva for supporting him during his studies. Both authors very kindly thank
Yvan Velenik and Sébastien Ott for useful discussions. We also thank Yvan
Velenik for reading the first version of the present article and several
helpful comments.

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