# Kesten’s incipient infinite cluster for the three-dimensional, metric-graph
Gaussian free field, from critical level-set percolation, and for the Villain
model, from random cluster geometries and a Swendsen-Wang type algorithm

Pete Rigas

###### Abstract

We address one open problem in a recent work due to Ding and Wirth, the first
version of which was available in $2019$, relating to level-set percolation on
metric-graphs for the Gaussian free field in three dimensions, in which it was
shown that a percolation estimate that the authors employ for studying
connectivity properties of different heights of the metric graph Gaussian free
field is bounded above poly-logarithmically. In three dimensions, in order to
construct Kesten’s incipient infinite cluster which was first seminally
introduced for Bernoulli percolation in two dimensions, in $1986$, through the
equality of two probabilistic quantities, we make use of a streamlined version
of the $1986$ argument due to Basu and Sapozhnikov, which was first made
available in $2016$, that introduces properties of crossing probabilities for
demonstrating that the IIC exists for Bernoulli percolation on an infinite
connected, bounded degree graph, namely that crossing probabilities are quasi-
multiplicative. To make use of such arguments for demonstrating the existence
of the metric-graph GFF IIC in three dimensions, we also address another open
problem raised in a recent work, from October $2022$, due to Dubedat and
Falconet, which expresses an open problem pertaining to the construction of an
IIC-type limit for the Villain model.

##  1 Introduction

###  1.1 Overview

The Gaussian Free Field (GFF) is a random surface model that has attracted
significant attention within the mathematical community, with efforts
dedicated towards studying level-set connectivity properties through
construction of percolation interfaces [1,2,9,12,27,29,30], quantification of
extrema [6,8], connections to eigenvalues of the corners of random matrices
[7], the existence of a phase transition for percolation by constructing GFF
level-set dependent copies of edges on graphs whose randomness is integrated
out with an Edwards-Sokal coupled measure [14], asymptotics through a
renormalization scheme [15], and characterizations of the field through the
domain Markov property permitting for a harmonic decomposition over boundaries
of finite volumes [3,11,28]. In addition to such results, other well known
properties that are relevant for open questions raised in [11] pertain to the
behavior and organization of critical percolation clusters for $d=3$ on metric
graphs, which are analyzed with exploration martingales, include previous
works providing results on the chemical distance [10,20], sign cluster
arrangements [17,18], percolation critical exponents [16], and a 0-1 law [26].
Additional topics of interest are included in [19,22,23,31,32].

In order to construct Kesten’s incipient infinite cluster for the metric graph
GFF, originally seminally established for two-dimensional Bernoulli
percolation [24], for the case $d=3$ we study critical percolation clusters of
the metric graph GFF by examining continuous time trajectories of the
exploration martingale, whose time dependent filtration $\mathcal{I}$ varies
with respect to each step of the exploration process. From the initial
exploration step, estimates for the quadratic variation, which is expressed as
a superposition of exploration martingales, is dependent on the harmonic
measure along the boundary of each box explored on the metric graph associated
to $\textbf{Z}^{d}$, where at each vertex the transition probability is equal
to $\frac{1}{2d}$ from that of the simple random walk, and otherwise for the
edges behaves like a standard Brownian motion $\widetilde{B}_{t}$.

Particularly, for behaviors of metric graph GFF level-sets, over appropriate
scales, given a finite volume with side length $N$, the harmonic measure, or
alternatively, the crossing probability, is dependent upon a sequence of
heights $h(N)\longrightarrow 0$. To introduce other relevant properties of the
GFF, for a Gaussian process on $G=G(V,E)$ is characterized as having zero mean
and covariance between arbitrary points $u$ and $v$, with respective fields
$\phi_{u}$ and $\phi_{v}$, given by the Green’s function of the Laplacian,

$\displaystyle\textbf{E}[\phi_{u}\phi_{v}]=G(u,v)=\textbf{E}_{u}\big{[}\int_{0}^{\infty}\textbf{1}_{S_{t}=v}\big{]}\text{
, }$

for $d>3$, given a simple random walk $S_{t}$. For $d=3$, the stopping time of
the random walk, instead of being infinite, is given by
$\tau=\mathrm{inf}\\{t:S_{t}\in\partial V\\}$, which for $d=2$ is a finite
box, while for $d\geq 3$, $V=\textbf{Z}^{d}$. From the GFF on a graph,
$\widetilde{B}_{t}$ yields the GFF on metric graphs (where the metric graph of
$G$ is denoted with $\widetilde{G}$) respectively $\widetilde{\phi_{u}}$ and
$\widetilde{\phi_{v}}$, whose covariance depends on $G(u,v)$ from $G$, is
given by,

$\displaystyle
G(w_{1},w_{2})=(1-r_{1})(1-r_{2})G(u_{1},u_{2})+r_{1}r_{2}G(v_{1},v_{2})+(1-r_{1})r_{2}G(u_{1},v_{2})+\cdots$
$\displaystyle r_{1}(1-r_{2})G(v_{1},u_{2})+2d(r_{1}\wedge
r_{2}-r_{1}r_{2})\textbf{1}_{(u_{1},v_{1})=(u_{2},v_{2})}\text{ , }$

where the vertices of the Green’s function are given by adjacent pairs of
vertices $(u_{1},v_{1})$ and $(u_{2},v_{2})$, with $w_{1}\in e(u_{1},v_{1})$,
$w_{2}\in e(u_{2},v_{2})$, $r_{1}=|w_{1}-u_{1}|$, $r_{2}=|w_{2}-u_{2}|$, and
$d\geq 3$ [25]. Otherwise, for $d=2$ we enforce the stopping time
$\widetilde{\tau}=\mathrm{inf}\\{t:\widetilde{B_{t}}\in\partial V\\}$. From
the construction of the GFF whose covariance function is respectively defined
above for graphs and metric graphs, we introduce previous results on
percolation probabilities.

###  1.2 Statement of previous results on level-set percolation probabilities
for the metric graph extension of the GFF

We collect necessary results for a quantification of how critical percolation
clusters on level-sets from the metric graph GFF. To study connectivity
properties in the infinite volume limit, define the box
$V_{r}\equiv[-r,r]\cap\textbf{Z}^{d}$ centered about the origin for $d\geq 3$
and $r\geq 1$, and is otherwise a finite box for $d=2$. We define the GFF
level-sets as a random surface defined over the vertices of
$\widetilde{\textbf{Z}^{d}}$, with $\widetilde{E}^{\geq
h}_{N}=\\{v\in\widetilde{\textbf{Z}^{d}}:\widetilde{\phi}_{N,v}\geq h\\}$.
Finally, we examine the chemical distance between two arbitrary vertices on
$\widetilde{Z}^{d}$, $u$ and $v$, in which $u$ and $v$ are connected with
respect to the graph distance if
$\mathcal{D}_{N,h}=\mathrm{inf}\\{\mathcal{D}_{N,h}(u,v):u\in A,v\in
B,A,B\subset\widetilde{\textbf{Z}^{d}}\\}<\infty$, while the vertices are
disconnected if $\mathcal{D}_{N,h}=\infty$, indicating that the metric graph
GFF does not attain a crossing to the $\partial V$ with height $\geq h$. The
percolation probability associated to such a crossing from $u$ to $v$ is
introduced below.

Theorem 1 (percolation probability bounded away from $0$ uniformly in $N$,
[11, Theorem 1]) For any height $h\in\textbf{R}$, and
$0<\alpha<\beta<\gamma<1$, denote
$\textbf{P}^{\alpha,\gamma}_{N,h}\big{[}\cdot\big{]}$ as the metric-graph GFF
law of $\widetilde{\phi}_{N}$ conditioned on the crossing event that $u$ and
$v$ are connected, namely $\\{V_{\alpha N}\overset{\geq
h}{\longleftrightarrow}\partial V_{\gamma N}\\}$, with respect to the graph
distance $\mathcal{D}_{N,h}$. For any $\epsilon>0$, there exists some $C$ so
that in the infinite volume limit, crossings across $V_{N}$ satisfy,

$\displaystyle\underset{{N\longrightarrow+\infty}}{\mathrm{lim\text{
}sup}}\text{ }\textbf{P}^{\alpha,\gamma}_{N,h}[\mathcal{D}_{N,h}(V_{\alpha
N},\partial V_{\beta N})>CN(\mathrm{log}N)^{\frac{1}{4}}]\leq\epsilon\text{ .
}$

Theorem 2 (exponential decay of percolation probabilities; poly-logarithmic
upper bound for percolation probabilities above the $h=0$ level-set for three
dimensions, and greater than three dimensions, [11, Theorems 3 $\&$ 4]) For a
strictly positive sequence of heights $h_{N}$, satisfying,

$\displaystyle h_{N}\leq C\text{ }\text{ , }$

for sufficient $C>0$, the three dimensional percolation probability is bounded
above exponentially, in which there exists some $c$ so that,

$\displaystyle
p_{N,h}\leq\mathrm{exp}\big{(}-\frac{ch^{2}N}{\mathrm{log}N}\big{)}\text{ , }$

for $N>1$, while for all dimensions greater than $3$ the upper bound takes a
similar form,

$\displaystyle p_{N,h}\leq\mathrm{exp}\big{(}-ch^{2}N\big{)}\text{ . }$

Above the $h=0$ level-set, the percolation probabilities to a surface whose
boundary is at length $k$ of the exploration process satisfies,

$\displaystyle\frac{c}{\sqrt{N}}\leq p_{N,0}\leq
C\sqrt{\frac{\mathrm{logN}}{N}}\text{ , }$

for $d=3$, and otherwise,

$\displaystyle\frac{c}{\sqrt{n}}\leq p_{N,0}\leq\frac{C}{\sqrt{n}}\text{ , }$

for $d>3$.

Definition 1 (exploration martingale, [11]) For vertices $v\in V$ from
$\widetilde{G}$, the exploration martingale is the conditional expectation
$M_{A,t}=\textbf{E}[X_{A}|\mathcal{F}_{\mathcal{I}_{t}}]$, where the
observable $X_{A}$ takes the form,

$\displaystyle X_{A}=\frac{1}{|A|}\sum_{v\in A}\widetilde{\phi}_{v}\text{ , }$

where $A\subset V$. The filtration of the conditional expectation satisfies,

$\displaystyle\mathcal{F}_{\mathcal{I}_{t}}=\\{\mathcal{E}\in\mathcal{F}_{\widetilde{G}}:\mathcal{E}\cap\\{\mathcal{I}_{t}\subset
U\\}\in\mathcal{F}_{U}\text{ for all open }\mathcal{I}_{0}\subset U\\}\text{ ,
}$

where $\mathcal{I}_{0}$ denotes the initial starting point of the random walk
on $h$ before exploration has begun, while $\mathcal{I}_{t}$ denotes the step
of the exploration process of the random walk at $t$. From the observable
above, the exploration martingale takes the form,

$\displaystyle M_{A,t}=\textbf{E}[X_{A}|\mathcal{F}_{\mathcal{I}_{t}}]\text{ ,
}$

where in the final quantity in the equality above, $M_{A,t}$ can be decomposed
through contributions from a harmonic measure along the boundary of
exploration and values of the GFF at vertices of $\partial\mathcal{I}_{t}$.

Definition 2 (quadratic variation, [11,Corollary 10]) The quadratic variation
of the exploration martingale is defined as,

$\displaystyle\langle
M_{A}\rangle_{t}=\mathrm{Var}[X_{A}|\mathcal{F}_{\mathcal{I}_{0}}]-\mathrm{Var}[X_{A}|\mathcal{F}_{\mathcal{I}_{t}}]\text{
. }$

Besides the almost-sure continuity of the exploration martingale and quadratic
variation [11, Lemma 9], from the GFF covariance, subtracting the Green’s
functions associated with random walks at times $s$ and $t$, where $0\leq
s\leq t$, and $v,v^{\prime}\in A$, admits the following harmonic measure
decomposition,

$\displaystyle
G_{s}(v,v^{\prime})-G_{t}(v,v^{\prime})\equiv\sum_{w\in\mathcal{I}_{t}}\mathrm{Hm}_{t}(v,w)G(w,v^{\prime})-\sum_{w^{\prime}\in\mathcal{I}_{s}}\mathrm{Hm}_{s}(v,w^{\prime})G(w^{\prime},v^{\prime})\text{
, }$

from the accompanying sets of explorations on $\widetilde{G}$,
$\mathcal{I}_{s}$ and $\mathcal{I}_{t}$. Next, we gather results surrounding
difference quotients for the Harmonic measure $\mathrm{Hm}$, in addition to
the cumulative distribution and survivor functions for the standard normal
distribution, each of which we respectively denote with $\Phi$ and
$\bar{\Phi}$.

Definition 3 (harmonic measure quotients, level-set stopping time, [11, Lemma
14]) Define the two functions,

$\displaystyle f_{1}(N)=\text{ }\underset{{K\longrightarrow\text{ }+\text{
}\infty}}{\mathrm{lim\text{
}sup}}\frac{\mathrm{Hm}(x_{K},V_{N})-\mathrm{Hm}(x_{K},0)}{G(0,x_{K})}\text{ ,
}$

and,

$\displaystyle f_{2}(N)=\text{ }\underset{{K\longrightarrow\text{ }+\text{
}\infty}}{\mathrm{lim\text{
}inf}}\inf_{\mathcal{I}}\frac{\mathrm{Hm}(x_{K},\mathcal{I})-\mathrm{Hm}(x_{k},0)}{G(0,x_{K})}\text{
. }$

Also, for a standard Brownian motion $B$ previously introduced over $G$, the
associated stopping time with an arbitrary level-set height $h$ is,

$\displaystyle\tau_{h}=\mathrm{inf}\big{\\{}\tau\geq 0:B_{t}\leq
ht-\frac{\widetilde{\phi}_{0}-h}{\sigma^{2}_{d}}\big{\\}}\text{ , }$

where at each time step $t$ of evolution $B$ is independent of
$\widetilde{\phi}_{0}$.

Definition 4 (stopping time-normal distribution equality, [11, Proposition
15]) Fix $m\in\textbf{R}$, $b>0$, and stopping time
$\tau=\mathrm{inf}\\{t>0:B_{t}\leq mt-b\\}$. For some $T>0$, the following
equality holds,

$\displaystyle\textbf{P}[\tau\leq
T]=\Phi\big{[}\frac{b}{\sqrt{t}}-m\sqrt{T}\big{]}+\mathrm{exp}(2bm)\Phi\big{[}\frac{b}{\sqrt{T}}+m\sqrt{T}\big{]}\text{
. }$

Corollary 1 (stopping time- percolation probability estimates, [11, Corollary
16]) From the quotient of harmonic measures given in Definition 3, the
percolation probability to the boundary of a box of length $N$ and level-set
$h$ satisfies,

$\displaystyle\textbf{P}\big{[}\tau_{h}\geq f_{1}(N)\big{]}\leq
p_{N,h}\leq\textbf{P}\big{[}\tau_{h}\geq f_{2}(N)\big{]}\text{ , }$

from the level-set stopping time.

###  1.3 Paper organization

We study the previously mentioned open problem raised in [11] pertaining to
the three-dimensional, metric graph GFF, through analysis of the quantities
defined in the previous subsection, namely the exploration martingale, and
harmonic measure which is introduced as an alternative for quantifying
crossing probabilities across level-sets of the metric graph GFF within the
critical height window. To do this, we discuss results from seminal arguments
due to [24] for constructing the IIC for Bernoulli percolation, in which two
probabilistic quantities are shown to be equal, one of which is the limit of
supercritically defined conditional crossing probabilities, which, through a
sequence of parameters $>\frac{1}{2}$ approaching criticality at
$p_{c}\equiv\frac{1}{2}$, is shown to equal another conditional crossing
probability taken exactly at $p_{c}$. For the metric graph GFF, to study
criticality about the height threshold about $h\equiv 0$, we make use of
arguments due to [11] which analyze the behavior of the percolation
probability from sequences of levels of the metric graph GFF strictly above,
and below, the aforementioned critical height threshold.

##  2 Properties of the IIC from Kesten’s 1986 work

###  2.1 Two dimensional Bernoulli percolation objects

A streamlined presentation of arguments provided in [24] is provided in [3],
the key quantities of which are summarized below. From classical assumptions,
fix some $p\in[0,1]$, and denote $\textbf{P}_{p}[\cdot]$ as the probability
measure for sampling a Bernoulli bond percolation configuration of parameter
$p$, with critical parameter,

$\displaystyle
p_{c}\equiv\mathrm{inf}\big{\\{}p:\textbf{P}_{p}\big{[}|C(v)|\equiv+\infty\big{]}>0\big{\\}}\text{
}\text{ , }$

which is the smallest parameter for which the cardinality of the open cluster,
$C(v)$, for some $v\in V(\textbf{Z}^{2})$, is infinite with positive
probability. Finally, from the induced graph metric $\rho$ for
$\textbf{Z}^{2}$, denote the set,

$\displaystyle S(v,n)\equiv\text{ }\big{\\{}x\in
V(\textbf{Z}^{2}):\rho(v,x)=n\big{\\}}\text{ , }$

for positive integers $m\leq n$.

Definition 5 (cylinder events, [24]). A cylinder event $E$ is an event that is
dependent upon a finite number of vertices from $\textbf{Z}^{2}$.

Definition 6 (conditional crossing probability for Bernoulli percolation at
criticality, [24]). Introduce, for $p\equiv p_{c}$,

$\displaystyle\mathcal{P}_{1}\equiv\textbf{P}_{p_{c}}\big{[}E|w\longleftrightarrow
S(w,n)\big{]}\text{ }\text{ . }$

Definition 7 (conditional crossing probability when approaching the critical
parameter from above, [24]). Introduce, as $p\searrow p_{c}$, for $w\in V$,

$\displaystyle\mathcal{P}^{\prime}_{2}\equiv\underset{p\searrow
p_{c}}{\mathrm{lim}}\mathcal{P}_{2}\equiv\underset{p\searrow
p_{c}}{\mathrm{lim}}\textbf{P}_{p}\big{[}E||C(w)|\equiv+\infty\big{]}\text{
}\text{ . }$

In addition to the Definitions provided above, to demonstrate, beyond the
existence of Kesten’s IIC about the critical height threshold $h\equiv 0$ of
the metric graph GFF, we will also make use of the following assumption.

Assumption One (Lower bounds for crossing probabilities of Bernoulli
percolation at the critical parameter for Kolmogorov extension of the
incipient infinite cluster measure to configurations of edges). At the
critical parameter $p_{c}\equiv\frac{1}{2}$, the following assumption from
[23], consisting of two inequalities below, asserts,

$\displaystyle\textbf{P}_{p_{c}}\bigg{[}[-\Lambda,0]\times[0,n]\underset{[-\Lambda,3n+\Lambda]\times[0,n]}{\longleftrightarrow}[3n,3n+\Lambda]\times[0,n]\bigg{]}\geq\delta\text{
}\text{ , }$
$\displaystyle\textbf{P}_{p_{c}}\bigg{[}[0,n]\times[-\Lambda,0]\underset{[0,n]\times[-\Lambda,3n+\Lambda]}{\longleftrightarrow}[0,n]\times[3n,3n+\Lambda]\bigg{]}\geq\delta\text{
}\text{ , }$

for some sufficient, strictly positive, lower bound $\delta$, and constant
$\Lambda\equiv\Lambda(G)$, for which,

$\displaystyle\forall e\in G:|e|\leq\Lambda\text{ }\text{ . }$

Below, we present a modification of Assumption One, with Assumption Two. In
the following, as will be introduced in Definition 9, denote the metric graph
GFF measure, precisely at the critical height threshold, with
$\textbf{P}_{h\equiv 0}[\cdot]\equiv\textbf{P}_{0}[\cdot]$, below.

Theorem 3 (conditional cylinder events under the Bernoulli bond percolation
are equal, under the previously stated Assumption, [24]). Given the assumption
provided above, the limits satisfy,

$\displaystyle\mathcal{P}_{1}\equiv\mathcal{P}^{\prime}_{2}\text{ }\text{ . }$

Denoting the common value shared by the two limits above with $\nu(E)$, by
Kolmogorov extension, there exists only one infinite cluster for which,

$\displaystyle\nu\big{[}\forall w\in V,\exists!\text{
}C(w):|C(w)|=+\infty\big{]}\equiv 1\text{ }\text{ , }$

where the unique infinite cluster contains $w$.

###  2.2 Three dimensional metric-graph GFF objects

From objects introduced in the previous section, we now introduce analogously
defined objects for the metric graph GFF. From [24] it is known that the
critical threshold for the metric graph GFF is $h\equiv 0$. Under this choice
of parameters, in three-dimensions critical level-set percolation for the
metric graph GFF can be further examined. First, state the following property
that the metric graph GFF satisfies, namely the Strong Markov Property.

Theorem 4 (GFF Strong Markov Property, Theorem 8, [11]). For any random
compacted connected subset $K\subsetneq\widetilde{G}$, conditionally upon $K$
and the filtration $\mathcal{F}_{K}$, the following equality in distribution,

$\displaystyle\big{\\{}\widetilde{\phi_{v}}:v\in\widetilde{G}\text{
}\backslash\text{
}K\big{\\}}\overset{\mathrm{d}}{\equiv}\big{\\{}\textbf{E}\big{[}\widetilde{\phi_{v}}\text{
}\big{|}\text{ }\mathcal{F}_{K}\big{]}+v\in\widetilde{G}\text{
}\backslash\text{ }K\big{\\}}\text{ }\text{ , }$

holds, concerning the metric graph GFF $\widetilde{\phi_{v}}$, for
$v\in\widetilde{G}$, over the complementary metric graph
$\widetilde{G}\backslash K$ under Dirichlet boundary conditions. From the
equality in distribution above, the conditional expectation on the LHS admits
the decomposition,

$\displaystyle\textbf{E}\big{[}\widetilde{\phi_{v}}|\mathcal{F}_{k}]\equiv\underset{u\in\partial
K}{\sum}\mathrm{Hm}(v,u;K)\widetilde{\phi_{u}}\text{ }\text{ , }$

over a summation of contributions from the harmonic measure
$\mathrm{Hm}(\cdot,\cdot;\cdot)$.

In addition to the property above, it is also of importance to introduce the
following quantities from Definitions 8-10. From the metric graph
$\widetilde{G}$, let $\rho$ be the corresponding graph metric, for which,
given some $v\in\widetilde{G}$, define,

$\displaystyle\mathcal{B}(v,n)\equiv\big{\\{}x\in V:\rho(v,x)\leq
n\big{\\}}\text{ }\text{ , }$

corresponding to the box that is centered about $v$, which is of some strictly
positive length $n$, in addition to,

$\displaystyle\partial\mathcal{B}(v,n)\equiv\big{\\{}x\in V:\rho(v,x)\equiv
n\big{\\}}\text{ }\text{ , }$

corresponding to the boundary of the box introduced in the previous item.
Finally, define, for $m\leq n$,

$\displaystyle\mathcal{A}(v,m,n)\equiv\mathcal{B}(v,n)\backslash\mathcal{B}(v,m-1)\equiv\big{\\{}x\in
V:m-1\leq\rho(v,x)\leq n\big{\\}}\text{ }\text{ , }$

corresponding to the finite volume enclosed by annulus contained within
$\mathcal{B}(v,n)$, and $\mathcal{B}(v,m)$, respectively.

Definition 8 (cylinder events that are a function of the height of the metric
graph GFF). Introduce, for some $h>0$,

$\displaystyle\mathscr{E}(h)\equiv\mathscr{E}\text{ }\text{ , }$

corresponding to the cylinder event of the metric graph GFF.

For Definition 9 below, denote another vertex of $\widetilde{G}$ with $w$.

Definition 9 (Kesten’s IIC for the metric graph GFF in three dimensions, from
conditional crossing probabilities for level-set percolation of the metric
graph GFF at the critical $h\equiv 0$ threshold; the metric graph GFF analogue
to Definition 6 originally provided for Bernolli percolation in two
dimensions). Introduce, for $h\equiv 0$,

$\displaystyle\textbf{P}^{\alpha,\beta}_{N,0}\big{[}\mathscr{E}|w\overset{\geq
0}{\longleftrightarrow}\mathcal{B}(w,n)\big{]}\equiv\textbf{P}_{N,0}\big{[}\mathscr{E}|w\overset{\geq
0}{\longleftrightarrow}\mathcal{B}(w,n)\big{]}\equiv\textbf{P}_{0}\big{[}\mathscr{E}|w\overset{\geq
0}{\longleftrightarrow}\mathcal{B}(w,n)\big{]}\text{ }\text{ , }$

corresponding to the level-set percolation crossing event across the critical
GFF height threshold, conditionally upon the existence of a configuration such
that $\big{\\{}w\longleftrightarrow\mathcal{B}(w,n)\big{\\}}$ occurs.

For Definition 10 below, denote the open cluster of the metric graph GFF,
centered about some $v$, with $\mathcal{C}(v)$.

Definition 10 (supercritical conditional crossing probability for level-set
percolation connectivity events of the metric graph GFF above the critical
$h\equiv 0$ threshold, the metric graph GFF analogue to Definition 7
originally provided for Bernoulli percolation). Introduce, for a sequence of
heights $\\{h_{N}\\}\longrightarrow 0$, where $N$ is the length of the side of
a finite volume,

$\displaystyle\underset{h_{N}\searrow
0}{\mathrm{lim}}\textbf{P}^{\alpha,\beta}_{N,h_{N}}\big{[}\mathscr{E}||\mathcal{C}(v)|=+\infty\big{]}\equiv\underset{h_{N}\searrow
0}{\mathrm{lim}}\textbf{P}_{N,h_{N}}\big{[}\mathscr{E}||\mathcal{C}(v)|=+\infty\big{]}\equiv\underset{h_{N}\searrow
0}{\mathrm{lim}}\textbf{P}_{h_{N}}\big{[}\mathscr{E}||\mathcal{C}(v)|=+\infty\big{]}\text{
}\text{ , }$

corresponding to the limit of the conditional probability that a cylinder
event $\mathscr{E}$ occurs, given the existence of an open cluster,
$\mathcal{C}$, centered about $v$, with infinite cardinality.

Defintion 11 (conditional crossing probability for level-set percolation in
order to decouple configurations from infinity). Given a sequence of heights
of the metric graph GFF, beginning with $h_{0}$ from
$\big{\\{}h_{N}\big{\\}}$, with a corresponding sequence of metric graphs
$\\{\widetilde{G}_{N}\\}$, introduce the following quantity, as $h_{N}\searrow
0$,

$\displaystyle\underset{h\in[0,h_{0}]}{\mathrm{sup}}\textbf{P}_{N}\big{[}\mathscr{E}_{2}|\mathscr{E}_{1}\big{]}\text{
}\text{ , }$

corresponding to the conditional probability of a cylinder event
$\mathscr{E}_{2}\big{(}h\big{)}\equiv\mathscr{E}_{2}$ occurring, conditionally
upon the occurrence of $\mathscr{E}_{1}\big{(}h\big{)}\equiv\mathscr{E}_{1}$,
which are two cylinder events, that are respectively given by,

$\displaystyle\mathscr{E}_{1}\equiv\big{\\{}\mathcal{B}(v,m)\overset{\geq
h}{\longleftrightarrow}\mathcal{B}(v,n)\big{\\}}\text{ }\text{ , }$

for $m\leq n$, corresponding to the existence of a crossing event between the
boxes $\mathcal{B}(v,m)$ and $\mathcal{B}(v,n)$, and by,

$\displaystyle\mathscr{E}_{2}\equiv\bigg{\\{}\forall v\in V,\exists w_{1}\neq
v,\cdots,w_{n}\neq w_{n-1}\neq\cdots\neq w_{1}\neq v\in
V:\mathcal{C}(v)\cap\bigg{(}\text{ }\underset{w_{1}\leq w\leq
w_{n}}{\bigcap}\text{ }\mathcal{C}(w)\bigg{)}\equiv\emptyset\bigg{\\}}\text{
}\text{ , }$

corresponding to the simultaneous existence of at least two disjoint open
clusters besides the existence of the open cluster $\mathcal{C}(v)$.

To proceed, in light of the metric graph GFF objects presented thus far, we
formulate assumptions that were originally provided for construction of the
IIC for Bernoulli percolation in two dimensions, from [24], in terms of the
properties that the metric graph GFF would be expected to fulfill, for the
construction of the IIC to remain applicable in three dimensions.

Assumptions on the metric graph GFF IIC in three dimensions. The metric graph
GFF IIC is based upon the following two assumptions imposed upon
$\widetilde{G}$:

  * $\bullet$

Assumption one (uniqueness of metric graph GFF open clusters): For any height
$h$ of the metric graph GFF within the critical window, there exists almost
surely at most $1$ infinite open cluster.

  * $\bullet$

Assumption two (quasi-multiplicativity of crossing probabilities for level-set
percolation in the metric graph GFF, which the authors of [3] state is
expected to hold for low dimensional integer-lattices of dimension $<6$): Fix
some $v\in V$ and $\delta>0$. There exists a strictly positive constant
$\mathscr{C}_{*}$, such that for a range of height parameters of the metric
graph free field defined above the critical threshold $h\equiv 0$, given by
$h\in[0,\delta]$, the probability of the level-set connectivity event,

$\displaystyle\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\mathscr{Y}\big{\\}}\text{ }\text{ , }$

occurring admits the following lower bound,

$\displaystyle\textbf{P}_{h}\big{[}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\mathscr{Y}\big{]}\geq\textbf{P}_{h}\big{[}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\partial\mathcal{B}(v,)\big{]}\mathscr{C}_{*}\textbf{P}_{h}\big{[}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\partial\mathcal{B}(v,)\big{]}\text{ }\text{ , }$

for a connected subset $\mathscr{Z}\subset V$, with
$\mathscr{Z}\supseteq\mathcal{A}(v,m,)$,
$\mathscr{X}\subset\mathscr{Z}\cap\mathcal{B}(v,m)$,
$\mathscr{Y}\subset\mathscr{Z}\backslash\mathcal{B}(v,)$.

From the two assumptions provided above, we make use of the FKG property of
discrete and continuous Gaussian free fields, as mentioned following the
statement of Theorem 1.1 in [21], and also in [28]. We denote this property of
the GFF as (FKG).

Also, introduce the following Theorem.

Theorem 5 (establishing equivalence between the random variables provided in
Definition 9, and in Definition 10, under appropriate assumptions on the
three-dimensional metric graph). Suppose that $\widetilde{G}$ satisfies
Assumption one and Assumption two provided above, with some choice of $v\in V$
and suitable $\delta>0$. Then, for any cylinder event $\mathscr{E}$, one
obtains the equivalence,

$\displaystyle\textbf{P}_{0}\big{[}\mathscr{E}|w\overset{\geq
0}{\longleftrightarrow}\mathcal{B}(w,n)\big{]}\equiv\underset{h_{N}\searrow
0}{\mathrm{lim}}\textbf{P}_{h_{N}}\big{[}\mathscr{E}||\mathcal{C}(v)|=+\infty\big{]}\text{
}\text{ , }$

between the conditional crossing event given on the LHS, precisely at $h\equiv
0$, and the conditional crossing event on the RHS, for a sequence of heights
$\longrightarrow 0$. Under the equality provided above, by Kolmogorov
extension as provided in the second condition of Theorem 3,

$\displaystyle\nu_{h}\big{[}\forall
w\in\widetilde{V},\exists!C(\widetilde{w}):|C(\widetilde{w})|=+\infty\big{]}\equiv
1\text{ }\text{ , }$

for the vertex set $\widetilde{V}$ of the metric graph $\widetilde{G}$.

Proof of Theorem 5. We make use of the arguments described initially for
Bernoulli percolation, provided in [24], in addition to the modifications to
that argument provided in [3] for demonstrating that the IIC for Bernoulli
percolation also exists on infinite, connected, bounded degree, graphs, with
the following. Namely, it suffices to demonstrate, by Kolmogorov extension,
that,

$\displaystyle\textbf{P}_{h}\big{[}\mathscr{E}|w\overset{\geq
h}{\longleftrightarrow}\mathcal{B}(w,n)\big{]}\longrightarrow\nu_{h}(\mathscr{E})\text{
}\text{ , }$ (1)

holds for $h\in[0,\delta]$, and $\delta>0$, as provided in Assumption two, in
which the above convergence is uniform. Towards accomplishing this, for some
increasing event, $i>0$, and parameters satisfying Assumption two, denote, for
$N_{i+1}\equiv
2^{i+1}\sqrt{\mathrm{log}n_{i+1}}>2^{i}\sqrt{\mathrm{log}n_{i}}\equiv N_{i}$,
$\mathcal{B}_{i}\equiv\mathcal{B}(v,N_{i})$ corresponding to the box centered
about $v$ of length $N_{i}$,
$\partial\mathcal{B}_{i}\equiv\partial\mathcal{B}(v,N_{i})$ corresponding to
the boundary of the $\mathcal{B}(v,N_{i})$ centered about $v$ of length
$N_{i}$, and
$\mathcal{A}_{i}\equiv\mathcal{A}(v,N_{i+1},N_{i})\equiv\mathcal{B}(v,N_{i+1})\backslash\mathcal{B}(v,N_{i})$
corresponding to the annulus centered about $v$ of length $N_{i+1}-N_{i}$.
Furthermore, introduce,

$\displaystyle\epsilon_{i}\equiv\underset{\mathscr{F}_{1},\mathscr{F}_{2}\in\mathcal{A}_{i}}{\underset{h\in[0,h_{0}]}{\mathrm{sup}}}\textbf{P}_{h}\big{[}\big{\\{}\mathscr{F}_{1}\overset{\geq
h}{\underset{\mathcal{A}_{i}}{\longleftrightarrow}}\mathscr{F}_{2}\big{\\}}^{c}|\partial\mathcal{B}_{i}\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{i+1}\big{]}\text{ }\text{ , }$ (2)

as the supremum of the conditional crossing probability, as provided above for
height parameters taken over the critical window $[0,h_{0}]$, with $h_{0}$
denoting the first height in the sequence $\\{h_{N}\\}$ which monotonically
decreases to the critical threshold $h\equiv 0$. We further manipulate this
quantity, observing,

$\displaystyle\textbf{P}_{h}\big{[}\big{\\{}w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}(w,n)\big{\\}}\cap\big{\\{}\mathscr{F}_{1}\overset{\geq
h}{\underset{\mathcal{A}_{1}}{\longleftrightarrow}}\mathscr{F}_{2}\big{\\}}\big{]}\leq\textbf{P}_{h}\big{[}w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{i}\big{]}\text{
}\textbf{P}_{h}\big{[}\big{\\{}\partial\mathcal{B}_{i}\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{i+1}\big{\\}}\cap\big{\\{}\mathscr{F}_{1}\overset{\geq
h}{\underset{\mathcal{A}_{1}}{\longleftrightarrow}}\mathscr{F}_{2}\big{\\}}\big{]}\text{
}\times\cdots$
$\displaystyle\textbf{P}_{h}\big{[}\partial\mathcal{B}_{i+1}\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}(w,n)\big{]}$
$\displaystyle\leq\epsilon_{i}\textbf{P}_{h}\big{[}w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{i}\big{]}\textbf{P}_{h}\big{[}\partial\mathcal{B}_{i}\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{i+1}\big{]}\textbf{P}_{h}\big{[}\partial\mathcal{B}_{i+1}\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}(w,n)\big{]}$
$\displaystyle\leq\epsilon_{i}\big{(}\mathscr{C}_{*}\big{)}^{2}\textbf{P}_{h}\big{[}w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}(w,n)\big{]}\text{ }\text{ , }$
($1^{*}$)

where the square of $\mathscr{C}_{*}$, in the uppermost bound, appears in the
statement of Assumption two for quasi-multiplicativity. Pursuant of the
decomposition steps provided for constructing the IICs for Bernoulli
percolation that are first provided in [24], and subsequently in [3],
introduce,

$\displaystyle\mathcal{C}_{i}\equiv\big{\\{}x\in\mathcal{B}_{i}:x\overset{\geq
h}{\underset{\mathcal{B}_{i+1}}{\longleftrightarrow}}\mathcal{B}_{i}\big{\\}}\text{
}\text{ , }$

corresponding to points within $\mathcal{B}_{i+1}$ that are connected to
$\mathcal{B}_{i}$, in addition to, from the denomination
$\partial\mathcal{B}_{i+1}\equiv\partial\mathcal{B}(v,N_{i}+1)$,

$\displaystyle\mathcal{D}_{i}\equiv\big{\\{}\forall
x\in\partial{B}_{i+1},\exists x\sim
y\in\mathcal{C}_{i}:\big{\\{}x\overset{\ngeq
h}{\longleftrightarrow}y\big{\\}}\big{\\}}\text{ }\text{ , }$

corresponding to the points in $\partial{B}_{i+1}$ for which there exists an
edge $(xy)$ which does not contain a crossing of at least height $h_{N}$. From
these two random objects, introduce, for $\mathscr{U}\subset\mathcal{B}_{i}$
and $\mathscr{R}\subset\partial\mathcal{B}_{i+1}$,

$\displaystyle\mathcal{F}_{i}\big{(}\mathscr{U},\mathscr{R}\big{)}\equiv\big{\\{}\mathcal{C}_{i}\equiv\mathscr{U}\big{\\}}\cup\big{\\{}\mathcal{D}_{i}\equiv\mathscr{R}\big{\\}}\text{
}\text{ , }$

corresponding to the event that $\mathcal{C}_{i}\equiv\mathscr{U}$, and that
$\mathcal{D}_{i}\equiv\mathscr{R}$ simultaneously. From all admissible pairs
$(\mathscr{U},\mathscr{R})$, given $\mathcal{C}_{i}$ and
$\mathscr{F}_{i}(\mathscr{U},\mathscr{R}\big{)}\neq\emptyset$ for which
$\big{\\{}\mathcal{C}_{i}\equiv\mathscr{U}\big{\\}}$, introduce the union,

$\displaystyle\mathcal{F}_{i}\big{(}\mathscr{U},\mathscr{V},\underset{i}{\prod}\big{)}\equiv\mathcal{F}_{i}\equiv\underset{(\mathscr{U},\mathscr{R})\in\prod_{i}}{\bigcup}\mathcal{F}_{i}\big{(}\mathscr{U},\mathscr{R}\big{)}\text{
}\text{ , }$

over the space $\prod_{i}$ of all admissible $(\mathscr{U},\mathscr{R})$. For
suitable $n$ with $n>N_{i+1}+N_{0}$,

$\displaystyle\textbf{P}_{h}\big{[}\mathscr{E}\cap\big{\\{}w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}(w,n)\big{\\}}\cap\big{\\{}\mathscr{F}_{1}\underset{\mathcal{A}_{1}}{\overset{\geq
h}{\longleftrightarrow}}\mathscr{F}_{2}\big{\\}}\big{]}\equiv\underset{(\mathscr{U},\mathscr{R})\in\prod_{i}}{\sum}\textbf{P}_{h}\big{[}\mathscr{E}\cap\big{\\{}w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}(w,n)\big{\\}}\cap\big{\\{}\big{\\{}\mathcal{C}_{i}\equiv\mathscr{U}\big{\\}}\cup\big{\\{}\mathcal{D}_{i}\equiv\mathscr{R}\big{\\}}\big{\\}}\big{]}\text{
}\text{ , }$

which, upon making the observation that,

$\displaystyle\textbf{P}_{h}\big{[}\mathscr{E}\cap\big{\\{}w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}(w,n)\big{\\}}\cap\big{\\{}\big{\\{}\mathcal{C}_{i}\equiv\mathscr{U}\big{\\}}\cup\big{\\{}\mathcal{D}_{i}\equiv\mathscr{R}\big{\\}}\big{\\}}\big{]}\equiv\textbf{P}_{h}\big{[}\mathscr{E}\cap\big{\\{}w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{i+1}\big{\\}}\cap\big{\\{}\big{\\{}\mathcal{C}_{i}\equiv\mathscr{U}\big{\\}}\cup\big{\\{}\mathcal{D}_{i}\equiv\mathscr{R}\big{\\}}\big{\\}}\big{]}\times\cdots$
$\displaystyle\textbf{P}_{h}\big{[}\mathscr{R}\overset{\geq
h}{\underset{\mathcal{B}(w,n)\backslash{\mathscr{U}}}{\longleftrightarrow}}\partial\mathcal{B}_{n}\big{]}\text{
}\text{ , }$

is equivalent to the following summation over $(\mathscr{U},\mathscr{V})$,

$\displaystyle\underset{(\mathscr{U},\mathscr{R})\in\prod_{i}}{\sum}\textbf{P}_{h}\big{[}\mathscr{E}\cap\big{\\{}w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{i+1}\big{\\}}\cap\big{\\{}\big{\\{}\mathcal{C}_{i}\equiv\mathscr{U}\big{\\}}\cup\big{\\{}\mathcal{D}_{i}\equiv\mathscr{R}\big{\\}}\big{\\}}\big{]}\textbf{P}_{h}\big{[}\mathscr{R}\overset{\geq
h}{\underset{\mathcal{B}(w,n)\backslash\mathscr{U}}{\longleftrightarrow}}\partial\mathcal{B}_{n}\big{]}\text{
}\text{ . }$

To quantify the absolute value between random variables that have been
previously manipulated, form the difference, between the probability of the
intersection of the increasing event $\mathscr{E}$ occurring, intersected
with,

$\displaystyle\mathscr{E}\cap\big{\\{}w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}(w,n)\big{\\}}\text{ }\text{ , }$

and each possible realization of,

$\displaystyle\mathscr{E}\cap\big{\\{}w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{i+1}\big{\\}}\cap\big{\\{}\big{\\{}\mathcal{C}_{i}\equiv\mathscr{U}\big{\\}}\cup\big{\\{}\mathcal{D}_{i}\equiv\mathscr{R}\big{\\}}\big{\\}}\text{
}\text{ , }$

for admissible $(\mathscr{U},\mathscr{V})\in\prod_{i}$ occurring, with,

$\displaystyle\big{|}\textbf{P}_{h}\big{[}\mathscr{E}\cap\big{\\{}w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}(w,n)\big{\\}}\big{]}-\underset{(\mathscr{U},\mathscr{V})\in\prod_{i}}{\sum}\textbf{P}_{h}\big{[}\mathscr{E}\cap\big{\\{}w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{i+1}\big{\\}}\cap\underset{\mathcal{F}_{i}}{\underbrace{\big{\\{}\big{\\{}\mathcal{C}_{i}\equiv\mathscr{U}\big{\\}}\cup\big{\\{}\mathcal{D}_{i}\equiv\mathscr{R}\big{\\}}\big{\\}}}}\big{]}\times\cdots$
$\displaystyle\textbf{P}_{h}\big{[}\mathscr{R}\overset{\geq
h}{\underset{\mathcal{B}(w,n)\backslash\mathscr{U}}{\longleftrightarrow}}\partial\mathcal{B}_{n}\big{]}\big{|}$
(DIFF) $\displaystyle\leq\text{
}\textbf{P}_{h}\big{[}\mathscr{E}\cap\mathcal{F}_{i}\cap\big{\\{}w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}(w,n)\big{\\}}\big{]}$
$\displaystyle\overset{(\mathrm{QM})}{\leq}\textbf{P}_{h}\big{[}\mathcal{F}^{c}_{i}\cap\big{\\{}w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}(w,n)\big{\\}}\big{]}\big{(}\mathscr{C}_{*}\big{)}^{-2}\textbf{P}_{h}\big{[}\mathscr{E}\big{]}\text{
}\text{ , }$ ($2^{*}$)

where the final expression, upon making the observation that the following
conditional probability,

$\displaystyle{\textbf{P}_{h}\big{[}\mathcal{F}^{c}_{i}|w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}(w,n)]}\equiv\frac{\textbf{P}_{h}\big{[}\mathcal{F}^{c}_{i}\cap\big{\\{}w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}(w,n)\big{\\}}\big{]}}{\textbf{P}_{h}\big{[}\mathcal{F}^{c}_{i}\big{]}}\text{
}\text{ , }$ ($3^{*}$)

is equivalent to the ratio of probabilities given on the RHS above, yields the
expression,

$\displaystyle\textbf{P}_{h}\big{[}\mathcal{F}^{c}_{i}\big{]}\textbf{P}_{h}\big{[}\mathcal{F}^{c}_{i}|w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}(w,n)\big{]}\text{
}\big{(}\mathscr{C}_{*}\big{)}^{-2}\text{
}\textbf{P}_{h}\big{[}\mathscr{E}\big{]}$
$\displaystyle\underset{\mathscr{F}_{1},\mathscr{F}_{2}\in\mathcal{A}_{i}}{\equiv}\textbf{P}_{h}\big{[}\mathcal{F}^{c}_{i}\big{]}\text{
}\textbf{P}_{h}\big{[}\big{\\{}\mathscr{F}_{1}\overset{\geq
h}{\underset{\mathcal{A}_{i}}{\longleftrightarrow}}\mathscr{F}_{2}\big{\\}}^{c}|w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}(w,n)\big{]}\big{(}\mathscr{C}_{*}\big{)}^{-2}\textbf{P}_{h}\big{[}\mathscr{E}\big{]}$
$\displaystyle\overset{(*)}{\leq}\epsilon^{\prime}_{i}\big{(}\mathscr{C}_{*}\big{)}^{-2}\textbf{P}_{h}\big{[}\mathscr{E}\big{]}$
$\displaystyle\equiv\epsilon^{\prime}_{i}\big{(}\mathscr{C}_{*}\big{)}^{-2}\frac{\textbf{P}_{h}\big{[}\mathscr{E}\big{]}\textbf{P}_{h}\big{[}w\longleftrightarrow\partial\mathcal{B}(w,n)\big{]}}{\textbf{P}_{h}\big{[}w\longleftrightarrow\partial\mathcal{B}(w,n)\big{]}}$
$\displaystyle\overset{(\mathrm{FKG})}{\leq}\epsilon^{\prime}_{i}\big{(}\mathscr{C}_{*}\big{)}^{-2}\frac{\textbf{P}_{h}\big{[}\mathscr{E}\cap\big{\\{}w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}(w,n)\big{\\}}\big{]}}{\textbf{P}_{h}\big{[}w\longleftrightarrow\partial\mathcal{B}(w,n)\big{]}}$
$\displaystyle\overset{(**)}{\leq}\epsilon^{\prime}\big{(}\mathscr{C}^{\prime}_{*}\big{)}^{-2}\textbf{P}_{h}\big{[}\mathscr{E}\cap\big{\\{}w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}(w,n)\big{\\}}\big{]}\text{ }\text{
, }$

where in $(*)$, the constant in the upper bound satisfies,

$\displaystyle\epsilon^{\prime}_{i}\geq\epsilon_{i}\text{
}\textbf{P}_{h}\big{[}\mathcal{F}^{c}_{i}\big{]}\text{ }\text{ , }$

while in $(**)$,

$\displaystyle\big{(}\mathscr{C}^{\prime}_{*}\big{)}^{-2}\geq\frac{\big{(}\mathscr{C}_{*}\big{)}^{-2}}{\textbf{P}_{h}\big{[}w\longleftrightarrow\partial\mathcal{B}(w,n)\big{]}}\text{
}\text{ , }$

in addition to the fact that the cylinder event $\mathscr{E}$, and,

$\displaystyle\big{\\{}w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}(w,n)\big{\\}}\text{ }\text{ , }$

being increasing events implicates the following lower bound,

$\displaystyle\textbf{P}_{h}\big{[}\mathscr{E}\cap\big{\\{}w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}(w,n)\big{\\}}\big{]}\geq\textbf{P}_{h}\big{[}\mathscr{E}\big{]}\textbf{P}_{h}\big{[}w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}(w,n)\big{]}\text{ }\text{ . }$

To proceed from final upper bound obtained from (**), introduce,

$\displaystyle u^{\prime}(\mathscr{U},\mathscr{R},h)\equiv
u_{h}^{\prime}(\mathscr{U},\mathscr{R})\equiv\textbf{P}_{h}\big{[}\mathscr{E}\cap\big{\\{}w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{i+1}\big{\\}}\cap\mathcal{F}_{i}\big{]}\text{
}\text{ , }$ $\displaystyle u^{\prime\prime}(\mathscr{U},\mathscr{R},h)\equiv
u_{h}^{\prime\prime}(\mathscr{U},\mathscr{R})\equiv\textbf{P}_{h}\big{[}\big{\\{}w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{i+1}\big{\\}}\cap\mathcal{F}_{i}\big{]}\text{
}\text{ , }$
$\displaystyle\gamma(\mathscr{U},\mathscr{R},n,h)\equiv\gamma_{h}(\mathscr{U},\mathscr{R},n)\equiv\textbf{P}_{h}\big{[}\mathscr{R}\underset{\mathcal{B}(w,n)\backslash\mathscr{U}}{\overset{\geq
h}{\longleftrightarrow}}\partial\mathcal{B}(w,n)\big{]}\text{ }\text{ , }$

which respectively corresponding to the probability of the intersection of the
three events provided occurring, the probability of the intersection of the
two events occurring, and finally, the probability of a connection between
$\mathscr{R}\longleftrightarrow\partial\mathcal{B}(w,n)$ occurring, within
$\mathcal{B}(w,n)\backslash\mathscr{U}$. Fix
$(\mathscr{U},\mathscr{R})\in\prod_{i}$, $j>i+2$, and $n>N_{j+1}+N_{0}$.
Observe that one can form a similar expression to $(\textit{DIFF})$ with,

$\displaystyle\big{|}\gamma(\mathscr{U},\mathscr{R},n,h)-\underset{(\mathscr{U}^{\prime},\mathscr{R}^{\prime})\in\prod_{j}}{\sum}\textbf{P}_{h}\big{[}\big{\\{}\mathscr{R}\underset{\mathcal{B}_{j+1}\backslash\mathscr{U}}{\overset{\geq
h}{\longleftrightarrow}}\partial\mathcal{B}_{j+1}\big{\\}}\cap\mathcal{F}_{j-1}\cap\mathcal{F}_{j}\big{(}\mathscr{U}^{\prime},\mathscr{R}^{\prime}\big{)}\big{]}\big{|}\text{
}\text{ , }$

which, upon similiar rearrangements as given previously, yields the upper
bound,

$\displaystyle\big{(}\mathscr{C}^{\prime}_{*}\big{)}^{-2}\bigg{(}\underset{\epsilon_{j-1}}{\underbrace{\underset{\mathscr{F}_{1},\mathscr{F}_{2}\in\mathcal{A}_{i}}{\underset{h\in[0,h_{0}]}{\mathrm{sup}}}\textbf{P}_{h}\big{[}\big{\\{}\mathscr{F}_{1}\overset{\geq
h}{\underset{\mathcal{A}_{i}}{\longleftrightarrow}}\mathscr{F}_{2}\big{\\}}^{c}|\partial\mathcal{B}_{j-1}\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{j}\big{]}}}+\underset{\epsilon_{j}}{\underbrace{\underset{\mathscr{F}_{1},\mathscr{F}_{2}\in\mathcal{A}_{i}}{\underset{h\in[0,h_{0}]}{\mathrm{sup}}}\textbf{P}_{h}\big{[}\big{\\{}\mathscr{F}_{1}\overset{\geq
h}{\underset{\mathcal{A}_{i}}{\longleftrightarrow}}\mathscr{F}_{2}\big{\\}}^{c}|\partial\mathcal{B}_{j+1}\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{j+2}\big{]}}}\bigg{)}\times\cdots$
$\displaystyle\textbf{P}_{h}\big{[}\mathscr{R}\underset{\mathcal{B}(w,n)\backslash\mathscr{U}}{\overset{\geq
h}{\longleftrightarrow}}\partial\mathcal{B}(w,n)\big{]}\text{ }\text{ . }$

Furthermore, executing this procedure repeatedly, fix some $k,N>0$, with
$k,N\in\textbf{N}$. For a sequence of parameters $j_{1},\cdots,j_{N}$, with
$j_{k+1}>j_{k}+2$, and as under a previous assumption, $n>N_{j_{k}+1}+N_{0}$,
that,

$\displaystyle\underset{(\mathscr{U}^{\prime},\mathscr{V}^{\prime})\in\prod_{j}}{\sum}\gamma(\mathscr{U},\mathscr{R},n,h)\text{
}\text{ , }$

for the conditional probability $\mathscr{M}$ appearing in the summation
above, which is defined with,

$\displaystyle\mathscr{M}\big{(}\mathscr{U},\mathscr{R};\mathscr{U}^{\prime},\mathscr{R}^{\prime}\big{)}\equiv\textbf{P}_{h}\big{[}\big{\\{}\mathscr{R}\underset{\mathcal{B}_{j+1}\backslash\mathscr{U}}{\overset{\geq
h}{\longleftrightarrow}}\partial\mathcal{B}_{i+1}\big{\\}}\cap\mathcal{F}_{j-1}\cap\mathcal{F}_{j}\big{(}\mathscr{U}^{\prime},\mathscr{R}^{\prime}\big{)}\big{]}\text{
}\text{ , }$ ($4^{*}$)

is bounded from below with,

$\displaystyle\gamma(\mathscr{U},\mathscr{R},n,h)-\big{(}\mathscr{C}^{\prime}_{*}\big{)}^{-2}\big{(}\epsilon_{j-1}+\epsilon_{j}\big{)}\gamma(\mathscr{U},\mathscr{R},n,h)\equiv\gamma(\mathscr{U},\mathscr{R},n,h)\big{(}1-\big{(}\mathscr{C}^{\prime}_{*}\big{)}^{-2}\big{(}\epsilon_{j-1}+\epsilon_{j}\big{)}\big{)}\text{
}\text{ , }$

and above with,

$\displaystyle\gamma(\mathscr{U},\mathscr{R},n,h)+\big{(}\mathscr{C}^{\prime}_{*}\big{)}^{-2}\big{(}\epsilon_{j-1}+\epsilon_{j}\big{)}\gamma(\mathscr{U},\mathscr{R},n,h)\equiv\gamma(\mathscr{U},\mathscr{R},n,h)\big{(}1+\big{(}\mathscr{C}^{\prime}_{*}\big{)}^{-2}\big{(}\epsilon_{j-1}+\epsilon_{j}\big{)}\big{)}\text{
}\text{ , }$

hence constituting a tight bound on,

$\displaystyle\underset{(\mathscr{U}^{\prime},\mathscr{V}^{\prime})\in\prod_{j}}{\sum}\gamma(\mathscr{U},\mathscr{R},n,h)\text{
}\text{ . }$

Next, observe, for $j_{k+1}$ introduced previously, the sequence of pairs,
from the collection,

$\displaystyle\big{\\{}\big{(}\mathscr{U}^{\prime}_{j_{1}},\mathscr{R}^{\prime}_{j_{1}}\big{)}\in\prod_{j_{1}},\cdots,\big{(}\mathscr{U}^{\prime}_{j_{k+1}},\mathscr{R}^{\prime}_{j_{k+1}}\big{)}\in\prod_{j_{k+1}}\big{\\}}\text{
}\text{ , }$

which implies, for $h\in[0,h_{0}]$,

$\displaystyle\frac{u_{h}^{\prime}(\mathscr{U},\mathscr{R},h)}{u_{h}^{\prime\prime}(\mathscr{U},\mathscr{R},h)}\underset{(\mathscr{U}^{\prime}_{j_{k+1}},\mathscr{R}^{\prime}_{j_{k+1}})\in\prod_{j_{k+1}}}{\underset{\vdots}{\underset{(\mathscr{U}^{\prime}_{j_{1}},\mathscr{R}^{\prime}_{j_{1}})\in\prod_{j_{1}}}{\underset{2\leq
k\leq
N}{\prod}}}}\frac{\mathscr{M}_{h}\big{(}\mathscr{U}_{1},\mathscr{R}_{1};\mathscr{U}^{\prime}_{2},\mathscr{R}^{\prime}_{2}\big{)}}{\gamma_{h}\big{(}\mathscr{U}_{k},\mathscr{R}_{k};\mathscr{U}^{\prime}_{k},\mathscr{R}^{\prime}_{k}\big{)}}\frac{\mathscr{M}_{h}\big{(}\big{(}\mathscr{U}_{1}\big{)}_{k},\big{(}\mathscr{R}_{1}\big{)}_{k};\big{(}\mathscr{U}^{\prime}_{2}\big{)}_{k},\big{(}\mathscr{R}^{\prime}_{2}\big{)}_{k}\big{)}}{\mathscr{M}_{h}\big{(}\big{(}\mathscr{U}_{1}\big{)}_{k-1},\big{(}\mathscr{R}_{1}\big{)}_{k-1};\big{(}\mathscr{U}^{\prime}_{2}\big{)}_{k-1},\big{(}\mathscr{R}^{\prime}_{2}\big{)}_{k-1}\big{)}}\times\cdots$
$\displaystyle\big{(}\frac{\mathscr{M}_{h}\big{(}\mathscr{U}_{1},\mathscr{R}_{1};\mathscr{U}^{\prime}_{2},\mathscr{R}^{\prime}_{2}\big{)}}{\gamma_{h}\big{(}\mathscr{U}_{k},\mathscr{R}_{k};\mathscr{U}^{\prime}_{k},\mathscr{R}^{\prime}_{k}\big{)}}\big{)}^{-1}\text{
}\text{ , }$

is tightly bound from below with,

$\displaystyle\mathrm{exp}\big{(}-\epsilon\big{)}\textbf{P}_{h}\big{[}\mathscr{E}|w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}(w,n)\big{]}\text{ }\text{ , }$

and is tightly bound from above with,

$\displaystyle\mathrm{exp}\big{(}\epsilon\big{)}\textbf{P}_{h}\big{[}\mathscr{E}|w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}(w,n)\big{]}\text{ }\text{ , }$

for suitable $\epsilon>0$. As demonstrated in [24], and subsequently adapted
for establishing the existence of the IIC for Bernoulli percolation in [3]
under different assumptions on the underlying graph structure, it suffices to
argue than an inequality of the form,

$\displaystyle\big{(}\mathscr{M}_{h}\big{(}\mathscr{U}_{1},\mathscr{R}_{1};\mathscr{U}^{\prime}_{1},\mathscr{R}^{\prime}_{1}\big{)}\text{
}\mathscr{M}_{h}\big{(}\mathscr{U}_{2},\mathscr{R}_{2};\mathscr{U}^{\prime}_{2},\mathscr{R}^{\prime}_{2}\big{)}\big{)}\text{
}\big{(}\mathscr{M}_{h}\big{(}\mathscr{U}_{1},\mathscr{R}_{1};\mathscr{U}^{\prime}_{2},\mathscr{R}^{\prime}_{2}\big{)}\mathscr{M}_{h}\big{(}\mathscr{U}_{2},\mathscr{R}_{2};\mathscr{U}^{\prime}_{1},\mathscr{R}^{\prime}_{1}\big{)}\big{)}^{-1}\leq\kappa^{2}\text{
}\text{ , }$

holds, in which the upper bound is independent of $\epsilon$ and $k$. To
demonstrate that the upper bound provided above holds, consider similar random
sets that were introduced previously, each of which are given by,

$\displaystyle\mathcal{X}_{j}=\big{\\{}x\in\mathcal{A}_{j-1}:x\underset{\mathcal{A}_{j-1}}{\overset{\geq
h}{{\longleftrightarrow}}}\partial\mathcal{B}_{j}\big{\\}}\text{ }\text{ , }$
$\displaystyle\mathcal{Y}_{j}=\big{\\{}\forall
y\in\partial\mathcal{B}(v,N_{j-1}-1)\text{ },\text{ }\exists x\sim
y:\big{\\{}x\overset{\ngeq h}{\longleftrightarrow}y\big{\\}}\big{\\}}\text{
}\text{ , }$

which can be used to formulate the event, as similarly provided in the union
of two events for defining $\mathcal{F}_{i}$, with,

$\displaystyle\mathcal{G}_{j}\big{(}X,Y\big{)}\equiv\big{\\{}\mathcal{X}_{j}\equiv
X\big{\\}}\cup\big{\\{}\mathcal{Y}_{j}\equiv Y\big{\\}}\text{ }\text{ , }$

for suitable $X$ and $Y$. From this choice of $X$ and $Y$, observe that the
occurrence of $\mathcal{G}_{j}$ provided above is dependent upon the edges
within $\mathcal{A}_{j-1}$, in which it is possible that,

$\displaystyle\big{\\{}\mathcal{X}_{j}\equiv
X\big{\\}}\cap\mathcal{F}_{j-1}\equiv\emptyset\text{ }\text{ , }$

or that,

$\displaystyle\big{\\{}\mathcal{X}_{j}\equiv
X\big{\\}}\subset\mathcal{F}_{j-1}\text{ }\text{ . }$

From the objects defined above, for each admissible pair $(X,Y)$,
$(\mathscr{U},\mathscr{R})\in\prod_{i}$,
$(\mathscr{U}^{\prime},\mathscr{R}^{\prime})\in\prod_{j}$, $\mathcal{F}_{j-1}$
is expressible with,

$\displaystyle\mathcal{F}_{j-1}\equiv\underset{(X,Y)\in\Gamma_{j}}{\bigcup}\mathcal{G}_{j}\big{(}X,Y\big{)}\text{
}\text{ , }$

for $\Gamma_{j}$, which is expressible with another union,

$\displaystyle\Gamma_{j}\equiv\underset{\mathcal{G}_{j}(X,Y)\neq\emptyset}{\underset{\\{\mathcal{X}_{j}\equiv
X\\}\subset\mathcal{F}_{j-1}}{\bigcup}}\big{\\{}X,Y\big{\\}}\text{ }\text{ ,
}$

which can then be used to define the following summation which is a product
over the probabilities,

$\displaystyle\mathscr{M}\big{(}\mathscr{U},\mathscr{R};\mathscr{U}^{\prime},\mathscr{R}^{\prime},h\big{)}\equiv\mathscr{M}_{h}\big{(}\mathscr{U},\mathscr{R};\mathscr{U}^{\prime},\mathscr{R}^{\prime}\big{)}\equiv\underset{(X,Y)\in\Gamma_{j}}{\sum}\textbf{P}_{h}\big{[}\mathscr{R}\underset{\mathcal{B}_{j}\backslash(X\cup\mathscr{U})}{\overset{\geq
h}{\longleftrightarrow}}Y\big{]}\times\cdots$
$\displaystyle\textbf{P}_{h}\big{[}\mathcal{G}_{j}(X,Y)\cap\mathcal{F}_{j}\big{(}\mathscr{U}^{\prime},\mathscr{R}^{\prime}\big{)}\cap\big{\\{}Y\overset{\geq
h}{\longleftrightarrow}\mathscr{R}^{\prime}\big{\\}}\big{]}$
$\displaystyle\overset{(\mathrm{FKG})}{\geq}\underset{(X,Y)\in\Gamma_{j}}{\sum}\textbf{P}_{h}\big{[}\mathscr{R}\underset{\mathcal{B}_{j}\backslash(X\cup\mathscr{U})}{\overset{\geq
h}{\longleftrightarrow}}Y\big{]}\textbf{P}_{h}\big{[}\mathcal{G}_{j}(X,Y)\big{]}\textbf{P}_{h}\big{[}\mathcal{F}_{j}\big{(}\mathscr{U}^{\prime},\mathscr{R}^{\prime}\big{)}\big{]}\times\cdots$
$\displaystyle\textbf{P}_{h}\big{[}Y\overset{\geq
h}{\longleftrightarrow}\mathscr{R}^{\prime}\big{]}\text{ }\text{ . }$

From the summation of probabilities above, the first term,

$\displaystyle\textbf{P}_{h}\big{[}\mathscr{R}\underset{\mathcal{B}_{j}\backslash(X\cup\mathscr{U})}{\overset{\geq
h}{\longleftrightarrow}}Y\big{]}\text{ }\text{ , }$ ($5^{*}$)

can be bound from below, by quasi-multiplicativity of crossing probabilities,

$\displaystyle\textbf{P}_{h}\big{[}\mathscr{R}\underset{\mathcal{B}_{j}\backslash(X\cup\mathscr{U})}{\overset{\geq
h}{\longleftrightarrow}}Y\big{]}\geq\textbf{P}_{h}\big{[}\mathscr{R}\underset{\mathcal{B}(v,2N_{i+1})\backslash\mathscr{U}}{\overset{\geq
h}{\longleftrightarrow}}\partial\mathcal{B}(v,2N_{i+1})\big{]}\text{
}\mathscr{C}_{*}\text{
}\textbf{P}_{h}\big{[}\mathcal{G}_{j}(X,Y)\cap\mathcal{F}_{j}(\mathscr{U}^{\prime},\mathscr{R}^{\prime})\cap\big{\\{}Y{\overset{\geq
h}{\longleftrightarrow}}\mathscr{R}^{\prime}\big{\\}}\big{]}\text{ }\text{ ,
}$ ($6^{*}$)

in which the final probability obtained from the quasi-multiplicativity lower
bound can be further bound from below with,

$\displaystyle\mathscr{C}_{*}\textbf{P}_{h}\big{[}\mathscr{R}\underset{\mathcal{B}_{j}\backslash(X\cup\mathscr{U})}{\overset{\geq
h}{\longleftrightarrow}}Y\big{]}\text{ }\text{ . }$

To obtain the desired lower and upper bounds for concluding the argument,
observe, for $\xi\leq 1$,

$\displaystyle\big{|}\xi\big{|}\big{|}\frac{u^{\prime}}{u^{\prime\prime}}\big{|}\big{|}\xi^{-1}\underset{2\leq
j\leq
k}{\prod}\frac{\mathscr{M}_{j}\big{(}\mathscr{U}_{j}\big{)}}{\mathscr{M}_{j}\big{(}\mathscr{U}_{j}\big{)}}-\frac{u^{\prime\prime}}{u^{\prime}}\big{|}\equiv\big{|}\frac{u^{\prime}}{u^{\prime\prime}}\underset{2\leq
j\leq
k}{\prod}\frac{\mathscr{M}_{j}\big{(}\mathscr{U}_{j}\big{)}}{\mathscr{M}_{j}\big{(}\mathscr{U}_{j}\big{)}}-\xi\big{|}\leq\big{(}\frac{\kappa-1}{\kappa+1}\big{)}^{k-1}\text{
}\text{ , }$

for the row vector,

$\displaystyle
u^{\prime}\equiv(u^{\prime}(1),\cdots,u^{\prime}(\lambda))\text{ }\text{ , }$

for the row vector,

$\displaystyle
u^{\prime\prime}\equiv(u^{\prime\prime}(1),\cdots,u^{\prime\prime}(\lambda))\text{
}\text{ , }$

and for finite volumes $\mathscr{U}_{j}\subset\mathcal{A}(j_{k})$, $\forall
j_{k}$, with $\lambda>0$. This implies that the lower bound takes the form,

$\displaystyle\mathrm{exp}\big{(}-\epsilon)\xi-\mathrm{exp}\big{(}-\epsilon)\big{(}\frac{\kappa-1}{\kappa+1}\big{)}^{k-1}\equiv\mathrm{exp}\big{(}-\epsilon)\big{(}\xi-\big{(}\frac{\kappa-1}{\kappa+1}\big{)}^{k-1}\big{)}\text{
}\text{ , }$

while the upper bound takes the form,

$\displaystyle\mathrm{exp}\big{(}\epsilon)\xi+\mathrm{exp}\big{(}\epsilon)\big{(}\frac{\kappa-1}{\kappa+1}\big{)}^{k-1}\equiv\mathrm{exp}\big{(}\epsilon)\big{(}\xi+\big{(}\frac{\kappa-1}{\kappa+1}\big{)}^{k-1}\big{)}\text{
}\text{ , }$

implying that a bound for,

$\displaystyle\textbf{P}_{h}\big{[}\mathscr{E}|w\longleftrightarrow\partial\mathcal{B}(w,n)\big{]}\text{
}\text{ , }$

holds, as,

$\displaystyle\mathrm{exp}\big{(}-\epsilon)\big{(}\xi-\big{(}\frac{\kappa-1}{\kappa+1}\big{)}^{k-1}\big{)}\leq\textbf{P}_{h}\big{[}\mathscr{E}|w\longleftrightarrow\partial\mathcal{B}(w,n)\big{]}\leq\mathrm{exp}\big{(}\epsilon)\big{(}\xi+\big{(}\frac{\kappa-1}{\kappa+1}\big{)}^{k-1}\big{)}\text{
}\text{ . }$

Hence, for $m,n>N_{i+1}+N_{0}$, the absolute value difference,

$\displaystyle\big{|}\frac{\textbf{P}_{h}\big{[}\mathscr{E}|w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}(w,m)\big{]}}{\textbf{P}_{h}\big{[}\mathscr{E}|w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}(w,n)\big{]}}-1\big{|}\equiv\big{|}\textbf{P}_{h}\big{[}\mathscr{E}|w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}(w,m)\big{]}-\textbf{P}_{h}\big{[}\mathscr{E}|w\overset{\geq
h}{\longleftrightarrow}\partial\mathcal{B}(w,n)\big{]}\big{|}\text{ }\text{ ,
}$

can be upper bounded with,

$\displaystyle\big{(}\mathrm{exp}\big{(}\epsilon\big{)}-\mathrm{exp}\big{(}-\epsilon\big{)}\big{)}+\big{(}\mathrm{exp}\big{(}\epsilon\big{)}+\mathrm{exp}\big{(}-\epsilon\big{)}\big{)}\big{(}\frac{\kappa-1}{\kappa+1}\big{)}^{k-1}\text{
}\text{ , }$ ($7^{*}$)

which implies the desired item in the statement of Theorem 5, as,

$\displaystyle\textbf{P}_{0}\big{[}\mathscr{E}|w\overset{\geq
0}{\longleftrightarrow}\mathcal{B}(w,n)\big{]}\equiv\underset{h_{N}\searrow
0}{\mathrm{lim}}\textbf{P}_{h_{N}}\big{[}\mathscr{E}||\mathcal{C}(v)|=+\infty\big{]}\Rightarrow\nu_{h}\big{[}\forall
w\in\widetilde{V},\exists!C(\widetilde{w}):|C(\widetilde{w})|=+\infty\big{]}\equiv
1\text{ }\text{ , }$

from which we conclude the proof.

##  3 Quasi-mulitiplicativity of crossing probabilities for level-set, metric
graph GFF percolation

In Section 3, with existence of the IIC established in Setion 2, we apply
arguments developed in [3] for proving that quasi-multiplicativity of crossing
probabilities for level-set percolation of the metric graph GFF, as for two
dimensional Bernoulli percolation, also hold. First, we begin by introducing
the setting over which arguments for establishing quasi-multiplicativity take
place.

Definition 12 (the three-dimensional environment). Denote
$\mathcal{S}\equiv\big{(}\mathcal{S}\big{(}\mathcal{V}\big{)},\mathcal{S}\big{(}\mathcal{E}\big{)}\big{)}\equiv\big{(}\mathcal{V},\mathcal{E}\big{)}\subsetneq\textbf{Z}^{3}$,
with,

$\displaystyle\mathcal{S}\equiv\textbf{Z}\times\big{\\{}0,\cdots,k\big{\\}}^{d-2}\text{
}\text{ , }$

for $k\geq 0$, and $d\geq 2$.

Besides three-dimensional objects introduced above, we also introduce similar
notions as introduced previously for the statement of quasi-multiplicativity,
namely a box over $\mathcal{S}$, the boundary of a box over $\mathcal{S}$, and
the annulus over $\mathcal{S}$.

Definition 13 (three-dimensional boxes, boundaries of three-dimensional boxes,
and three-dimensional annuli, for further analysis of crossing events). Denote
the box, with strictly positive side length $2n$, over $\textbf{Z}^{3}$, with,

$\displaystyle\mathcal{B}_{\mathcal{S}}\big{(}v,n\big{)}\equiv\big{\\{}x\in\mathcal{V}\big{(}[-n,n]^{2}\big{)}:\rho_{\mathcal{S}}\big{(}v,x)\leq
n\big{\\}}\text{ }\text{ , }$

for the restriction of the graph $\rho_{\mathcal{S}}$, with $\rho$ on
$\mathcal{V}\subsetneq\mathcal{V}$. As a result, denote the boundary with,

$\displaystyle\partial\mathcal{B}_{\mathcal{S}}\big{(}v,n\big{)}\equiv\partial\mathcal{B}_{\mathcal{S}}\big{(}n\big{)}\equiv\big{\\{}x\in\mathcal{V}\big{(}[-n,n]^{2}\big{)}:\rho_{\mathcal{S}}(v,x)=n\big{\\}}\text{
}\text{ , }$

and the three-dimensional annulus with, for $m>n>0$,

$\displaystyle\mathcal{A}_{\mathcal{S}}(v,m,n)\equiv\mathcal{A}_{\mathcal{S}}(m,n)\equiv\mathcal{B}_{\mathcal{S}}(v,n)\backslash\mathcal{B}_{\mathcal{S}}(v,m-1)\equiv\big{\\{}x\in\mathcal{V}\big{(}[-n,n]^{2}\big{)}:m-1\leq\rho_{\mathcal{S}}\big{(}v,x\big{)}\leq
n\big{\\}}\text{ }\text{ . }$

With the geometric objects defined above over $\mathcal{S}$, also introduce
the analogue of quasi-multiplicativity of crossing probabilities over slabs.

Theorem 6 (establishing equivalence between the metric-graph GFF limits from
Theorem 5 in the three-dimensional environment). Fix $\mathcal{S}$, $d\geq 2$,
$k\geq 0$, and $\delta<1$. Then, from the two limits introduced in Theorem 5,
namely,

$\displaystyle\textbf{P}_{0}\big{[}\mathscr{E}|w\overset{\geq
0}{\longleftrightarrow}\mathcal{B}(w,n)\big{]}\text{ }\text{ , }$

corresponding to the metric-graph GFF probability measure at the critical $h$
parameter, and,

$\displaystyle\underset{h_{N}\searrow
0}{\mathrm{lim}}\textbf{P}_{h_{N}}\big{[}\mathscr{E}||\mathcal{C}(v)|=+\infty\big{]}\text{
}\text{ , }$

corresponding to the limit of a sequence subcritically dependent metric-graph
GFF probability measures from a sequence of heights, $\\{h_{N}\\}$, each exist
and are equal.

To demonstrate that the result above holds for the metric-graph IIC, introduce
the following.

Lemma 1 (three-dimensional quasi-multiplicativity). From the choice of
$\mathcal{S}$, $d$, $k$, and $\delta$ introduced in Theorem 6, additionally
introduce heights $h\in[0,h_{0}]$, $m\in\textbf{Z}_{+}$, and a strictly
positive $\mathcal{C}_{*}$, from which,

$\displaystyle\textbf{P}_{h}\big{[}\mathscr{X}{\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}}\mathscr{Y}\big{]}\geq\textbf{P}_{h}\big{[}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{\mathcal{S}}\big{(}2\sqrt{\mathrm{log}m}\big{)}\big{]}\text{
}\mathcal{C}_{*}\text{
}\textbf{P}_{h}\big{[}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{\mathcal{S}}\big{(}2\sqrt{\mathrm{log}m}\big{)}\big{]}\text{
}\text{ , }$

holds, given $\mathscr{Z}\subset\mathcal{S}$,
$\mathscr{Z}\supset\mathcal{A}(m,2\sqrt{\mathrm{log}m}+1)$,
$\mathscr{X}\subsetneq\mathscr{Z}\cap\partial\mathcal{B}_{\mathcal{S}}\big{(}m\big{)}$,
and
$\mathscr{Y}\subset\mathscr{Z}\backslash\mathcal{B}_{\mathcal{S}}\big{(}2\sqrt{\mathrm{log}m}+1\big{)}$.

Proof of Lemma 1. To establish quasi-multiplicativity in three-dimensions,
instead of directly establishing the inequality between crossing probabilities
above, observe, that it suffices to demonstrate,

$\displaystyle\textbf{P}_{h}\big{[}\mathscr{X}{\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}}\mathscr{Y}\big{]}\geq\textbf{P}_{h}\big{[}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{\mathcal{S}}\big{(}2\sqrt{\mathrm{log}m}+1\big{)}\big{]}\text{
}\mathcal{C}_{*}\text{
}\textbf{P}_{h}\big{[}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{\mathcal{S}}\big{(}2\sqrt{\mathrm{log}m}\big{)}\big{]}\text{
}\text{ , }$

in which the first probability due to the application of quasi-
multiplicativitiy above,

$\displaystyle\textbf{P}_{h}\big{[}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{\mathcal{S}}\big{(}2\sqrt{\mathrm{log}m}+1\big{)}\big{]}\text{
}\text{ , }$

can itself be lower bound by another application of quasi-multiplicativity,
as,

$\displaystyle\textbf{P}_{h}\big{[}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{\mathcal{S}}\big{(}2\sqrt{\mathrm{log}m}+1\big{)}\big{]}\geq\textbf{P}_{h}\big{[}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{\mathcal{S}}\big{(}2\sqrt{\mathrm{log}m}\big{]}\text{
}\mathscr{C}_{*}\text{
}\textbf{P}_{h}\big{[}\partial\mathcal{B}_{\mathcal{S}}\big{(}\frac{4}{3}\sqrt{\mathrm{log}m}\big{)}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{\mathcal{S}}\big{(}2\sqrt{\mathrm{log}m}+1\big{)}\big{]}\text{
}\text{ . }$

From the second application of quasi-multiplicativity above, adapting results
from [4], yields, for $\textbf{P}_{h}\big{[}\cdot\big{]}$,

$\displaystyle\textbf{P}_{h}\big{[}\partial\mathcal{B}_{\mathcal{S}}\big{(}\frac{4}{3}\sqrt{\mathrm{log}m}\big{)}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{\mathcal{S}}\big{(}2\sqrt{\mathrm{log}m}\big{)}\big{]}\geq\textbf{P}_{0}\big{[}\partial\mathcal{B}_{\mathcal{S}}\big{(}\frac{4}{3}\sqrt{\mathrm{log}m}\big{)}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{\mathcal{S}}\big{(}2\sqrt{\mathrm{log}m}\big{)}\big{]}\text{
}\text{ , }$

for $h\in[0,h_{0}]$. Hence, from adaptations of arguments provided in [4] for
the three-dimensional environment, one has,

$\displaystyle\textbf{P}_{0}\big{[}\partial\mathcal{B}_{\mathcal{S}}\big{(}\frac{4}{3}\sqrt{\mathrm{log}m}\big{)}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{\mathcal{S}}\big{(}2\sqrt{\mathrm{log}m}\big{)}\big{]}>0\text{
}\text{ . }$

Next, consider,

$\displaystyle\textbf{P}_{h}\big{[}\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{\mathcal{S}}\big{(}2\sqrt{\mathrm{log}m}+1\big{)}\big{\\}}\cap\big{\\{}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{\mathcal{S}}\big{(}2\sqrt{\mathrm{log}m})\big{\\}}\cap\mathcal{E}_{1}\big{]}\text{
}\text{ , }$ ($\mathcal{E}_{1}$ probability)

where the event $\mathcal{E}_{1}$, is given by,

$\displaystyle\mathcal{E}_{1}\equiv\bigg{\\{}\mathcal{V}_{1}\underset{\mathcal{A}_{\mathcal{S}}(2\sqrt{\mathrm{log}m},2\sqrt{\mathrm{log}m}+1)}{\overset{<h}{\longleftrightarrow}}\mathcal{V}_{2}\bigg{\\}}\text{
}\text{ , }$

where the vertices in the connectivity event defined above, within the annulus
$\mathcal{A}_{\mathcal{S}}(2\sqrt{\mathrm{log}m},2\sqrt{\mathrm{log}m}+1)$,
are drawn from the vertex set,

$\displaystyle\mathcal{V}_{1},\mathcal{V}_{2}\in\mathcal{V}\big{(}\mathcal{S}\big{)}\text{
}\text{ , }$

the collection of all vertices from the three-dimensional environment. This
intersection of crossing events, admits, by (FKG), the lower bound,

$\displaystyle\textbf{P}_{h}\big{[}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{\mathcal{S}}\big{(}2\sqrt{\mathrm{log}m}+1\big{)}\big{]}\textbf{P}_{h}\big{[}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{\mathcal{S}}\big{(}2\sqrt{\mathrm{log}m})\big{]}\textbf{P}_{h}\big{[}\mathcal{E}_{1}\big{]}\text{
}\text{ . }$

From the lower bound above, furthermore, observe, that the intersection of
events given with the probability below,

$\displaystyle\textbf{P}_{h}\big{[}\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\bar{\Gamma}\big{\\}}\cap\big{\\{}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\bar{\Gamma}\big{\\}}\cap\mathcal{E}_{1}\big{]}\text{
}\text{ , }$ ($\bar{\Gamma}$ probability)

admits a bound from below, of the form, by ($\mathrm{FKG}$),

$\displaystyle\textbf{P}_{h}\big{[}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\bar{\Gamma}\big{]}\textbf{P}_{h}\big{[}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\bar{\Gamma}\big{]}\textbf{P}_{h}\big{[}\mathcal{E}_{1}\big{]}\text{
}\text{ , }$

for a path $\bar{\Gamma}\neq\Gamma$, satisfying $\bar{\Gamma}>\Gamma$, where
$\Gamma$ denotes the minimal open circuit, given by the path infimum,

$\displaystyle\Gamma\equiv\underset{v_{0},\cdots,v_{n}\in\mathcal{V}}{\underset{\gamma\equiv\\{v_{0}\overset{<h}{\longleftrightarrow}v_{n}\\}}{\underset{\mathrm{circuits}\text{
}\gamma}{\mathrm{inf}}}}\big{\\{}\rho_{\mathcal{S}}\big{(}\gamma,\partial\mathcal{B}_{\mathcal{S}}\big{(}2\sqrt{\mathrm{log}m}\big{)}\big{)}\big{\\}}\text{
}\text{ , }$

which, as an infimum over circuits $\gamma$ restricted within the annulus
$\mathcal{A}_{\mathcal{S}}(2\sqrt{\mathrm{log}m},2\sqrt{\mathrm{log}m}+1)$,
for which,

$\displaystyle\gamma\cap\mathcal{A}_{\mathcal{S}}(2\sqrt{\mathrm{log}m},2\sqrt{\mathrm{log}m}+1)\neq\emptyset\text{
}\text{ , }$

are nonempty subsets over $\mathcal{S}$, as of admissible $\gamma$ satisfy,

$\displaystyle\gamma\equiv\underset{i:0\leq i\leq
n}{\bigcup}\gamma_{i}\equiv\underset{i:0\leq i\leq
n}{\bigcup}\big{\\{}\big{\\{}v_{i}\overset{<h}{\longleftrightarrow}v_{i+1}\big{\\}}\big{\\}}\text{
}\text{ , }$

$\forall i$. The deterministic ordering of circuits is obtained from the
relation,

$\displaystyle\Gamma_{1}\geq\Gamma_{2}\Longleftrightarrow\big{\\{}\forall
v_{1},v_{2}\in\mathcal{V}\big{(}\mathcal{S}\big{)},\exists
v_{1}\in\Gamma_{1},v_{2}\in\Gamma_{2}:\rho_{\mathcal{S}}\big{(}v_{1},\partial\mathcal{B}_{\mathcal{S}}\big{(}2\sqrt{\mathrm{log}m}\big{)}\big{)}\geq\rho_{\mathcal{S}}\big{(}v_{2},\partial\mathcal{B}_{\mathcal{S}}\big{(}2\sqrt{\mathrm{log}m}\big{)}\big{)}\big{\\}}\text{
}\text{ , }$

for two circuits $\Gamma_{1}$ and $\Gamma_{2}$, in which for two paths
contained around $\mathcal{B}$. For a subset $W\neq\emptyset$, furthermore,
denote,

$\displaystyle\mathcal{S}\supset
W\supsetneq\bar{W}\equiv\big{\\{}z\equiv\big{(}z_{1},\cdots,z_{d}\big{)}\in\mathcal{S}:\big{(}z_{1},z_{2},x_{3},\cdots,x_{d}\big{)}\in
W\text{ for some }x_{3},\cdots,x_{d}\big{\\}}\text{ }\text{ , }$

corresponding to a nonempty subset of the three-dimensional environment for
which there exists points $x_{3},\cdots,x_{d}\in W$. From quantities
introduced above relating to the ordering of paths in $\mathcal{S}$, express a
lower bound for ($\bar{\Gamma}$ probability), in which, from previous
manipulation,

$\displaystyle(\bar{\Gamma}\text{
}\textit{probability})\geq\textbf{P}_{h}\big{[}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{\mathcal{S}}\big{(}2\sqrt{\mathrm{log}m}+1\big{)}\big{]}\textbf{P}_{h}\big{[}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\partial\mathcal{B}_{\mathcal{S}}\big{(}2\sqrt{\mathrm{log}m})\big{]}\textbf{P}_{h}\big{[}\mathcal{E}_{1}\big{]}\overset{(\mathrm{FKG})}{\leq}(\mathcal{E}_{1}\text{
}\textit{probability})\text{ }\text{ . }$

As a result, it suffices to demonstrate, for strictly positive
$\mathcal{C}_{1}$,

$\displaystyle\textbf{P}_{h}\big{[}\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\bar{\Gamma}\big{\\}}\cap\big{\\{}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\bar{\Gamma}\big{\\}}\cap\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\not\longleftrightarrow}\mathscr{Y}\big{\\}}\cap\mathcal{E}_{1}\big{]}\geq\mathcal{C}_{1}\textbf{P}_{h}\big{[}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\mathscr{Y}\big{]}\text{ }\text{ , }$ ($\bar{\Gamma}$
Probability I)

in which the lower bound probability consists of solely the pushforward, under
$\textbf{P}_{h}\big{[}\cdot\big{]}$, of,

$\displaystyle\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\mathscr{Y}\big{\\}}\text{ }\text{ , }$

in comparison to the upper bound, which consists of the intersection, pushed
forwards under $\textbf{P}_{h}\big{[}\cdot\big{]}$, of,

$\displaystyle\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\bar{\Gamma}\big{\\}}\cap\big{\\{}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\bar{\Gamma}\big{\\}}\cap\mathcal{E}_{1}\text{ }\text{
, }$

with the intersection of the three events provided above. To rewrite
($\bar{\Gamma}$ Probability I), observe,

$\displaystyle\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\not\longleftrightarrow}\mathscr{Y}\big{\\}}\cap\big{\\{}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\not\longleftrightarrow}\Gamma\big{\\}}\subsetneq\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\bar{\Gamma}\big{\\}}\text{ }\text{ , }$
$\displaystyle\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\not\longleftrightarrow}\Gamma\big{\\}}\cap\big{\\{}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\Gamma\big{\\}}\subsetneq\big{\\{}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\bar{\Gamma}\big{\\}}\text{ }\text{ , }$
$\displaystyle\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\Gamma\big{\\}}\cap\big{\\{}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\not\longleftrightarrow}\Gamma\big{\\}}\subsetneq\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\not\longleftrightarrow}\mathscr{Y}\big{\\}}\cap\mathcal{E}_{1}\text{
}\text{ , }$

in which, from the three consecutively defined events provided above, passing
to the pushforward under the measure $\textbf{P}_{h}\big{[}\cdot\big{]}$
yields,

$\displaystyle\textbf{P}_{h}\big{[}\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\not\longleftrightarrow}\mathscr{Y}\big{\\}}\cap\big{\\{}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\not\longleftrightarrow}\Gamma\big{\\}}\big{]}\leq\textbf{P}_{h}\big{[}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\bar{\Gamma}\big{]}\text{ }\text{ , }$
$\displaystyle\textbf{P}_{h}\big{[}\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\not\longleftrightarrow}\Gamma\big{\\}}\cap\big{\\{}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\Gamma\big{\\}}\big{]}\leq\textbf{P}_{h}\big{[}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\bar{\Gamma}\big{]}\text{ }\text{ , }$
$\displaystyle\textbf{P}_{h}\big{[}\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\Gamma\big{\\}}\cap\big{\\{}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\not\longleftrightarrow}\Gamma\big{\\}}\big{]}\leq\textbf{P}_{h}\big{[}\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\not\longleftrightarrow}\mathscr{Y}\big{\\}}\cap\mathcal{E}_{1}\big{]}\text{
}\text{ , }$

which, upon further inspection, implies,

$\displaystyle\textbf{P}_{h}\big{[}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\not\longleftrightarrow}\mathscr{Y}\big{]}\text{
}\textbf{P}_{h}\big{[}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\Gamma\big{]}\overset{(\mathrm{FKG})}{\leq}\textbf{P}_{h}\big{[}\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\not\longleftrightarrow}\mathscr{Y}\big{\\}}\cap\big{\\{}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\not\longleftrightarrow}\Gamma\big{\\}}\big{]}\text{ }\text{ , }$
$\displaystyle\textbf{P}_{h}\big{[}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\not\longleftrightarrow}\Gamma\big{]}\textbf{P}_{h}\big{[}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\Gamma\big{]}\overset{(\mathrm{FKG})}{\leq}\textbf{P}_{h}\big{[}\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\not\longleftrightarrow}\Gamma\big{\\}}\cap\big{\\{}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\Gamma\big{\\}}\big{]}\text{ }\text{ , }$
$\displaystyle\textbf{P}_{h}\big{[}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\Gamma\big{]}\textbf{P}_{h}\big{[}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\Gamma\big{]}\overset{(\mathrm{FKG})}{\leq}\textbf{P}_{h}\big{[}\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\Gamma\big{\\}}\cap\big{\\{}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\Gamma\big{\\}}\big{]}\text{ }\text{ , }$

in which the LHS lower bound of each inequality can be further bound from
below by applying $(\mathrm{FKG})$ three times. As a result of these facts,
introduce another closely related to ($\bar{\Gamma}$ Probability I), for
strictly positive, suitable $\mathcal{C}_{1}$, as previously provided, in
which,

$\displaystyle\textbf{P}_{h}\big{[}\mathcal{E}_{1}\cap\mathscr{E}_{1}\cap\mathscr{E}_{2}\cap\mathscr{E}_{3}\cap\mathscr{E}_{4}\cap\mathscr{E}_{5}\big{]}\leq\mathcal{C}_{1}\textbf{P}_{h}\big{[}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\mathscr{Y}\big{]}\text{ }\text{ , }$ ($\bar{\Gamma}$
Probability II)

for the following five events,

$\displaystyle\mathscr{E}_{1}\equiv\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\bar{\Gamma}\big{\\}}\text{ }\text{ , }$
$\displaystyle\mathscr{E}_{2}\equiv\big{\\{}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\bar{\Gamma}\big{\\}}\text{ }\text{ , }$
$\displaystyle\mathscr{E}_{3}\equiv\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\not\longleftrightarrow}\mathscr{Y}\big{\\}}$
$\displaystyle\mathscr{E}_{4}\equiv\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\not\longleftrightarrow}\Gamma\big{\\}}\text{ }\text{ , }$
$\displaystyle\mathscr{E}_{5}\equiv\big{\\{}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\not\longleftrightarrow}\Gamma\big{\\}}\text{ }\text{ , }$

in which, in comparison to the intersection of three crossing events,

$\displaystyle\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\bar{\Gamma}\big{\\}}\cap\big{\\{}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\bar{\Gamma}\big{\\}}\cap\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\not\longleftrightarrow}\mathscr{Y}\big{\\}}\text{ }\text{ , }$

that is intersected with $\mathcal{E}_{1}$ in ($\bar{\Gamma}$ Probability I),
satisfies,

$\displaystyle\textbf{P}_{h}\big{[}\mathcal{E}_{1}\big{]}\textbf{P}_{h}\big{[}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\mathscr{Y}\big{]}\textbf{P}_{h}\big{[}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\bar{\Gamma}\big{]}\textbf{P}_{h}\big{[}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\not\longleftrightarrow}\mathscr{Y}\big{]}\textbf{P}_{h}\big{[}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\not\longleftrightarrow}\Gamma\big{]}\textbf{P}_{h}\big{[}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\not\longleftrightarrow}\Gamma\big{]}$
$\displaystyle\overset{(\mathrm{FKG})}{\leq}\textbf{P}_{h}\big{[}\mathcal{E}_{1}\big{]}\textbf{P}_{h}\big{[}\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\mathscr{Y}\big{\\}}\cap\big{\\{}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\bar{\Gamma}\big{\\}}\cap\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\not\longleftrightarrow}\mathscr{Y}\big{\\}}\cap\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\not\longleftrightarrow}\Gamma\big{\\}}\cap\big{\\{}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\not\longleftrightarrow}\Gamma\big{\\}}\big{]}$
$\displaystyle\equiv\textbf{P}_{h}\big{[}\mathcal{E}_{1}\big{]}\textbf{P}_{h}\big{[}\mathscr{E}_{1}\cap\mathscr{E}_{2}\cap\mathscr{E}_{3}\cap\mathscr{E}_{4}\cap\mathscr{E}_{5}\big{]}$
$\displaystyle\overset{(\mathrm{FKG})}{\leq}(\bar{\Gamma}\textit{Probability
II})\equiv\textbf{P}_{h}\big{[}\mathcal{E}_{1}\cap\big{\\{}\mathscr{E}_{1}\cap\mathscr{E}_{2}\cap\mathscr{E}_{3}\cap\mathscr{E}_{4}\cap\mathscr{E}_{5}\big{\\}}\big{]}$
$\displaystyle\leq\textbf{P}_{h}\big{[}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\bar{\Gamma}\big{]}\textbf{P}_{h}\big{[}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\bar{\Gamma}\big{]}\textbf{P}_{h}\big{[}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\not\longleftrightarrow}\mathscr{Y}\big{]}\textbf{P}_{h}\big{[}\mathcal{E}_{1}\big{]}$
$\displaystyle\overset{(\mathrm{FKG})}{\leq}\textbf{P}_{h}\big{[}\big{\\{}\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\bar{\Gamma}\big{\\}}\cap\big{\\{}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\bar{\Gamma}\big{\\}}\cap\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\not\longleftrightarrow}\mathscr{Y}\big{\\}}\big{\\}}\cap\mathcal{E}_{1}\big{]}$
$\displaystyle\equiv(\bar{\Gamma}\textit{Probability I})\text{ }\text{ , }$

because of the fact that,

$\displaystyle 0<\textbf{P}_{h}\big{[}\mathscr{E}_{4}\big{]}<1\text{ }\text{ ,
}$ $\displaystyle 0<\textbf{P}_{h}\big{[}\mathscr{E}_{5}\big{]}<1\text{
}\text{ . }$

Therefore, it suffices to construct a map,

$\displaystyle
f_{h}:\big{\\{}\mathcal{E}_{1}\cap\mathscr{E}_{1}\cap\mathscr{E}_{2}\cap\mathscr{E}_{3}\cap\mathscr{E}_{4}\cap\mathscr{E}_{5}\big{\\}}\longrightarrow\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\mathscr{Y}\big{\\}}\text{ }\text{ , }$

in which the pullback under $f^{-1}_{h}$ denotes the events contained within
the intersection
$\big{\\{}\mathcal{E}_{1}\cap\mathscr{E}_{1}\cap\mathscr{E}_{2}\cap\mathscr{E}_{3}\cap\mathscr{E}_{4}\cap\mathscr{E}_{5}\big{\\}}$.
For some $\omega$ in the preimage space, and the pushforward of $\omega$,
$f_{h}(\omega)$, under $f_{h}(\cdot)$, to demonstrate that such an inequality
holds, $f_{h}$ should satisfy,

  * $\bullet$

Property 1: The intersection between $\omega$, and the pushforward of $\omega$
under $f_{h}$, satisfies the following constraint,

$\displaystyle\big{|}\omega\cap f_{h}(\omega)\big{|}<\mathscr{D}\text{ }\text{
, }$

for some finite $\mathscr{D}$, in which the intersection between the
configuration $\omega$ and its pushforward, $f_{h}(\omega)$, can differ by at
most $\mathscr{D}$ edges.

  * $\bullet$

Property 2: Building off Property 1, a configuration $\omega$ from the
preimage space, and the pushforward under $f_{h}(\cdot)$, $f_{h}(\omega)$,
satisfies the following constraint,

$\displaystyle\big{\\{}\omega,\omega^{\prime}\in\\{\mathcal{E}_{1}\cap\mathscr{E}_{1}\cap\mathscr{E}_{2}\cap\mathscr{E}_{3}\cap\mathscr{E}_{4}\cap\mathscr{E}_{5}\big{\\}}:\big{|}f(\omega)\equiv
f(\omega^{\prime})\big{|}<\mathscr{D}\big{\\}}\text{ }\text{ , }$

for two configurations $\omega$ and $\omega^{\prime}$ from the preimage space
of $f_{h}$, in which it is stipulated that at most $\mathscr{D}$ edges between
$f(\omega)$ and $f(\omega^{\prime})$ are equal, in which equality amongst
edges of the pushforward configurations $f_{h}(\omega)$ and
$f_{h}(\omega^{\prime})$ is given by,

$\displaystyle\big{|}f_{h}(\omega)\equiv
f_{h}(\omega^{\prime})\big{|}=\mathscr{D}\Longleftrightarrow\forall
e_{1}(\omega)\in f_{h}(\omega),\cdots,e_{n}(\omega)\in
f_{h}(\omega),e_{1}(\omega^{\prime})\in f_{h}(\omega^{\prime}),$
$\displaystyle\cdots,e_{n}(\omega^{\prime})\in
f_{h}(\omega^{\prime}),\exists\mathscr{D}<n:e_{1}(\omega)\equiv
e_{1}(\omega^{\prime}),\cdots,e_{\mathscr{D}}(\omega)\equiv
e_{\mathscr{D}}(\omega^{\prime})\text{ }\text{ , }$

in which there is equality amongst the first $\mathscr{D}<n$ edges associated
with the pushfoward configurations $f_{h}(\omega)$ and
$f_{h}(\omega^{\prime})$.

With regards to Property 1 and Property 2, of each condition holds, observe
that ($\bar{\Gamma}$ Probability I) would hold with the constant in the upper
bound taking the form,

$\displaystyle\mathcal{C}_{1}\equiv\mathscr{D}\big{(}\mathrm{inf}\big{\\{}0,1-\delta\big{\\}}\big{)}^{-\mathscr{D}}\text{
}\text{ . }$

for suitable $\delta>0$. To demonstrate that Property 1 holds, first,
introduce, for some path of strictly positive length,

$\displaystyle\pi_{u,n}\in\pi_{\mathscr{U},\mathscr{N}}\equiv\underset{n\in\mathscr{N}\subset\textbf{N}}{\underset{u\in\mathscr{U}}{\bigcup}}\pi_{u,n}\equiv\underset{n\in\mathscr{N}\subset\textbf{N}}{\underset{u\in\mathscr{U}}{\bigcup}}\bigg{\\{}\forall
u\in\mathscr{U},n\in\mathscr{N},\exists
u_{1},\cdots,u_{\mathscr{D}},\cdots,u_{n}\in\mathcal{V}:\big{\\{}u_{1}\equiv
u_{n}\big{\\}}\cup\cdots$ $\displaystyle\big{\\{}\underset{0\leq i\leq
n}{\bigcup}\big{\\{}\pi_{u_{i}}\overset{\mathscr{Z},<h}{\longleftrightarrow}\pi_{u_{i+1}}\big{\\}}\big{\\}}\bigg{\\}}\text{
}\text{ , }$

where $u$ are drawn from the set, for
$\mathscr{X}^{\prime}\supsetneq\mathscr{X}$,

$\displaystyle\mathscr{U}\equiv\big{\\{}u\in\bar{\Gamma}:\textbf{P}_{h}\big{[}u\overset{\mathscr{Z},<h}{\longleftrightarrow}\mathscr{X}\big{]}>0,\textbf{P}_{h}\big{[}u\overset{\mathscr{Z},\mathscr{X},<h}{\longleftrightarrow}\mathscr{X}^{\prime}\big{]}>0,\textbf{P}_{h}\big{[}\big{\\{}u\overset{\mathscr{Z},<h}{\longleftrightarrow}\mathscr{X}\big{\\}}\cap\big{\\{}u\overset{\mathscr{Z},<h}{\longleftrightarrow}\mathscr{X}^{\prime}\big{\\}}\cap\big{\\{}\mathscr{X}^{\prime}\overset{\mathscr{Z},<h}{\longleftrightarrow}u\big{\\}}\big{]}=0\big{\\}}\text{
}\text{ . }$

Similarly, for another point $v$ instead of $u$, the self-avoiding path from
the level-set connectivity event of the metric-graph GFF would instead be
dependent upon the conditions,

$\displaystyle\pi_{v,n^{\prime}}\propto\big{\\{}v_{1}\equiv
v_{n^{\prime}}\big{\\}}\cup\bigg{\\{}\underset{0\leq i\leq
n^{\prime}}{\bigcup}\big{\\{}\pi_{v_{i}}\overset{\mathscr{Z},<h}{\longleftrightarrow}\pi_{v_{i+1}}\big{\\}}\bigg{\\}}\text{
}\text{ , }$

being fulfilled.

Case A

From $\pi_{u,n}$ and $\pi_{v,n^{\prime}}$, as two self-avoiding paths
associated with the existence of a level-set percolation crossing in the
metric-graph GFF, to demonstrate that Property 1 and Property 2 of
$f_{h}(\cdot)$ hold, fix some $u\in\mathscr{U}$, $v\in\mathscr{V}$, where, as
defined with $\mathscr{U}$ above, introduce, for
$\mathscr{V}^{\prime}\supsetneq\mathscr{V}$,

$\displaystyle\mathscr{V}\equiv\big{\\{}v\in\bar{\Gamma}:\textbf{P}_{h}\big{[}v\overset{\mathscr{Z},<h}{\longleftrightarrow}\mathscr{V}\big{]}>0,\textbf{P}_{h}\big{[}v\overset{\mathscr{Z},\mathscr{V},<h}{\longleftrightarrow}\mathscr{V}^{\prime}\big{]}>0,\textbf{P}_{h}\big{[}\big{\\{}v\overset{\mathscr{Z},<h}{\longleftrightarrow}\mathscr{V}\big{\\}}\cap\big{\\{}v\overset{\mathscr{Z},<h}{\longleftrightarrow}\mathscr{X}^{\prime}\big{\\}}\cap\big{\\{}\mathscr{V}^{\prime}\overset{\mathscr{Z},<h}{\longleftrightarrow}v\big{\\}}\big{]}=0\big{\\}}\text{
}\text{ , }$

corresponding to $v\in\bar{\Gamma}$ satisfying the three conditions provided
above. Pertaining to the pushforward of $\omega$ in the preimage space under
$f_{h}(\cdot)$ fix $u,v$ so that $\overline{\\{u\\}}\equiv\overline{\\{v\\}}$.
Under this assumption, $f_{h}(\omega)$ is modified from $\omega$ in the
following ways, which we denote below with Case A.

  * $\bullet$

Modification one (metric-graph GFF analog for closing edges in configurations
as previously argued for Bernoulli percolation): For edges in $f(\omega)$ for
which the event,

$\displaystyle\textbf{P}_{h}\big{[}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\mathscr{Y}\big{]}\text{ }\text{ , }$

occurs with positive probability, close all edges belonging to the following
set, which satisfy the requirements,

$\displaystyle\mathscr{V}^{\prime}\equiv\big{\\{}\forall
v\in\mathcal{V},\exists v\in
f_{h}(\omega):v\cap\overline{\\{u\\}}\neq\emptyset\big{\\}}\text{ }\text{ , }$

corresponding to the set of edges belonging to $f_{h}(\omega)$ which have
nonempty intersection with $\overline{\\{u\\}}$, as well as the set,

$\displaystyle\mathscr{V}^{\prime}_{\pi_{u}}\equiv\mathscr{V}^{\prime}_{u}\equiv\big{\\{}\forall
u\in\mathcal{V},\exists
u\in\pi_{u}:u\cap\overline{\\{u\\}}\neq\emptyset\big{\\}}\text{ }\text{ , }$

corresponding to the set of edges belonging to $\pi_{u}$ for which the
intersection of $\pi_{u}$ with $\overline{\\{u\\}}$ is nonempty. From
$\mathscr{V}^{\prime}$ and $\mathscr{V}^{\prime}_{u}$, admissible $u$ and $v$
satisfying conditions from the intersection,

$\displaystyle\mathscr{V}^{\prime}\cap\mathscr{V}^{\prime}_{u}\text{ }\text{ ,
}$

correspond to vertices from the vertex set $\mathcal{V}$ of the three-
dimensional environment that, besides each having nonempty intersection with
$\overline{\\{u\\}}$, also lie within the intersection of vertices of
$f_{h}(\omega)$ and $\pi_{u}$, for which,

$\displaystyle\textbf{P}_{h}\big{[}v\overset{\mathscr{Z},<h}{\longleftrightarrow}\overline{\\{u\\}}\big{]}>0\text{
}\text{ , }$

for $v\in\mathscr{V}^{\prime}$, and also for which,

$\displaystyle\textbf{P}_{h}\big{[}u\overset{\mathscr{Z},<h}{\longleftrightarrow}\overline{\\{u\\}}\big{]}>0\text{
}\text{ , }$

for $u\in\mathscr{V}^{\prime}_{u}$. From the modifications introduced for
Modification one, the two probabilities above would instead have vertices, and
hence edges, of the pushforward configuration $f_{h}(\omega)$ satisfy,

$\displaystyle\textbf{P}_{h}\big{[}v\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\overline{\\{u\\}}\big{]}>0\text{ }\text{ , }$

and also,

$\displaystyle\textbf{P}_{h}\big{[}u\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\overline{\\{u\\}}\big{]}>0\text{ }\text{ , }$

for edges satisfying,

$\displaystyle\underset{e(v_{i},v_{i+1})\overset{i\equiv
n-1}{=}e(v_{n-1},\overline{\\{u\\}})}{\underset{e(v,v_{i})\overset{i\equiv
1}{=}e(v,v_{1})}{\underset{\forall 1\leq i\leq
n}{\underbrace{\textbf{1}_{e(v,v_{i})\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}e(v_{i},v_{i+1})}\equiv
1}}}}\propto\underset{v_{j}\overset{j\equiv\frac{n}{2}}{=}\text{
}\overline{\\{u\\}}}{\underset{v_{j}\overset{j\equiv 1}{=}v}{\underset{\forall
1\leq j\leq\frac{n}{2}}{\underbrace{\textbf{1}_{v_{j}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}v_{j+1}}\equiv 1}}}}\text{ }\text{ , }\text{ }\text{
}\text{ . }$

corresponding to the event $\\{v\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\overline{\\{u\\}}\\}$, while similarly, the indicator
for each index is satisfied for,

$\displaystyle\textbf{1}_{e(u,u_{j})\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}e(u_{j},u_{j+1})}\equiv 1\text{ }\text{ , }$

$\forall 1\leq j\leq n^{\prime}$,corresponding to the event
$\\{u\overset{\mathscr{Z},\geq h}{\longleftrightarrow}\overline{\\{u\\}}\\}$,
under a similar convention of indices as provided for
$\\{v\overset{\mathscr{Z},\geq h}{\longleftrightarrow}\overline{\\{u\\}}\\}$.

  * $\bullet$

Modification two (metric-graph GFF analog for opening line segments in three
dimensions as previously argued for Bernoulli percolation). In opposition to
Modification 1 for closing edges for metric-graph GFF level-set percolation
events, for Modification two, we consider the following procedure for opening
edges depending upon the height of the metric-graph GFF, in which, from the
edge set,

$\displaystyle\mathcal{E}\supsetneq\mathscr{E}^{\prime\prime}\equiv\big{\\{}\forall
e\in\mathcal{E},\exists
e\in\mathcal{E}\big{(}\overline{\\{u\\}}\big{)}:e\cap\gamma\neq\emptyset\big{\\}}\text{
}\text{ , }$

where,

$\displaystyle\big{|}\gamma\big{(}u,\overline{\big{\\{}u\big{\\}}}\big{)}\big{|}\equiv\big{|}\gamma\big{|}\equiv\underset{\Gamma\cap
u\cap\overline{\\{u\\}}\neq\emptyset}{\underset{{\Gamma}\subset\mathcal{S}}{\mathrm{inf}}}\big{\\{}\rho_{\Gamma}\big{(}u,\overline{\big{\\{}u\big{\\}}}\big{)}\big{\\}}\text{
}\text{ , }$

with edge set from $\mathscr{E}^{\prime\prime}$ given by,

$\displaystyle\mathcal{E}\big{(}\overline{\\{u\\}}\big{)}\equiv\big{\\{}e\in\mathcal{E}:e\cap\overline{\\{u\\}}\neq\emptyset\big{\\}}\text{
}\text{ , }$

for the restriction of the metric $\rho$ to $\Gamma\subset\mathcal{S}$, which
is defined by,

$\displaystyle\rho_{\Gamma}\big{(}u,\overline{\big{\\{}u\big{\\}}}\big{)}\equiv\big{\\{}\forall
v_{1},v_{2}\in\mathcal{V},\exists v_{1}\equiv
u,v_{2}\equiv\overline{\big{\\{}u\big{\\}}}:\rho_{\Gamma}\big{(}u,\overline{\big{\\{}u\big{\\}}}\big{)}\equiv\rho_{\mathcal{S}|_{\Gamma}}\big{(}u,\overline{\big{\\{}u\big{\\}}}\big{)}\big{\\}}\text{
}\text{ , }$

and,

$\displaystyle\mathcal{E}\supsetneq\mathcal{E}\big{(}\overline{\\{u\\}}\big{)}\equiv\big{\\{}e\in\mathcal{E}:e\cap\overline{\big{\\{}u\big{\\}}}\neq\emptyset\big{\\}}\text{
}\text{ , }$

as all edges belonging to $\mathscr{E}^{\prime\prime}$ that belong to the path
indicated above are opened.

  * $\bullet$

Modification three (opening edges from the shortest path emanating from $v$,
in comparison to opening edges from the shortest path emanating from $u$ as
provided in Modification two). For the final modification, to build off
arguments introduced for Modification two, introduce the following edge set,

$\displaystyle\mathscr{E}^{\prime\prime\prime}\big{(}v,e_{2}\big{)}\equiv\mathscr{E}^{\prime\prime\prime}\equiv\big{\\{}\forall
e_{1}\in\mathcal{E}\big{(}v\big{)},\exists
e_{2}\in\mathcal{E}\big{(}\overline{\big{\\{}u\big{\\}}}\big{)}:\Gamma^{\prime}\big{(}e_{1},e_{2}\big{)}\cap\rho_{\Gamma}\neq\emptyset\big{\\}}\text{
}\text{ , }$

where,

$\displaystyle\mathcal{E}\big{(}v\big{)}\equiv\big{\\{}e\in\mathcal{E}:e\cap
v\neq\emptyset\big{\\}}\text{ }\text{ , }$

and,

$\displaystyle\Gamma\big{(}e_{1},e_{2}\big{)}\equiv\big{\\{}\overline{\big{\\{}u\big{\\}}}\in\mathcal{E}:\Gamma^{\prime}\big{(}e_{1},e_{2}\big{)}\cap\overline{\big{\\{}u\big{\\}}}\neq\emptyset\big{\\}}\text{
}\text{ , }$

with the length of the path given by the infimum,

$\displaystyle\big{|}\Gamma^{\prime}(e_{1},e_{2})\big{|}\equiv\big{|}\Gamma^{\prime}\big{|}\equiv\underset{\Gamma\subset\mathcal{S}}{\underset{e_{1},e_{2}\in\mathcal{E}}{\mathrm{inf}}}\big{\\{}\rho_{\Gamma}\big{(}e_{1},e_{2}\big{)}\big{\\}}\text{
}\text{ , }$

in which we incorporate conditions from $\gamma$ introduced in Modification
two, namely that this shortest path, with respect to the infimum of the metric
$\rho$, intersect $v$ and $\overline{\big{\\{}u\big{\\}}}$. Concluding,
opening the edges in $\overline{\big{\\{}u\big{\\}}}$ with this property
entails that,

$\displaystyle\textbf{P}_{h}\big{[}e_{1}\overset{\mathscr{Z},<h}{\longleftrightarrow}\big{(}\Gamma^{\prime}\big{(}e_{1},e_{2}\big{)}\cup\gamma\big{)}\big{]}\text{
}\text{ , }$

occurs with positive probability, in addition to,

$\displaystyle\underset{\Gamma^{\prime}\subset\mathcal{S}}{\underset{\gamma\subset\mathcal{S}}{\mathrm{inf}}}\big{\\{}\textbf{P}_{h}\big{[}e_{1}\overset{\mathscr{Z},<h}{\longleftrightarrow}\big{(}\Gamma^{\prime}\big{(}e_{1},e_{2}\big{)}\cup\gamma\big{)}\big{]}\big{\\}}\text{
}\text{ , }$

occurring with positive probability, for an infimum taken over
$\gamma,\Gamma^{\prime}\subset\mathcal{S}$.

From Modification one, Modification two, and Modification three described
above, observe,

$\displaystyle\big{|}\omega\cap
f_{h}\big{(}\omega\big{)}\big{|}<\mathscr{D}\text{ }\text{ , }$

as the intersection between the number of common edges between $\omega$ and
$f_{h}\big{(}\omega\big{)}$ can be upper bounded by a suitable $\mathscr{D}$
as provided before describing the three modifications of $f_{h}(\omega)$. As a
result, upon performing the respective modifications of closing, opening, and
opening, certain edges depending upon whether the level-set percolation event
across height $\geq h$ or $<h$ occurs, $\overline{\big{\\{}u\big{\\}}}$ can be
uniquely reconstructed in $f_{h}(\omega)$, in which,

$\displaystyle\textbf{P}_{h}\big{[}\big{\\{}\overline{\big{\\{}z\big{\\}}}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\mathscr{X}\big{\\}}\cup\big{\\{}\overline{\big{\\{}z\big{\\}}}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\mathscr{Y}\big{\\}}\big{]}>0\text{ }\text{ , }$

as the two events displayed above, namely the union of
$\overline{\big{\\{}z\big{\\}}}$ being connected to $\mathscr{X}$, in addition
to $\overline{\big{\\{}z\big{\\}}}$ being connected to $\mathscr{Y}$, occurs
with positive probability.

Case B

On the other hand, if
$\overline{\mathscr{U}}\cap\overline{\mathscr{V}}\equiv\emptyset$, then a
similar series of modifications, as given previously, can be applied to the
pushfoward $f_{h}(\omega)$. For the following items, fix $u\in\mathscr{U}$,
and $v\in\mathscr{V}$. From such a choice of $u$ and $v$, one has,

$\displaystyle\textbf{P}_{h}\big{[}\overline{\big{\\{}u\big{\\}}}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\mathscr{Y}\big{]}\text{ }\text{ , }$

occurring with zero probability, in addition to,

$\displaystyle\textbf{P}_{h}\big{[}\overline{\big{\\{}v\big{\\}}}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\mathscr{X}\big{]}\text{ }\text{ , }$

occurring with zero probability.

  * $\bullet$

Modification one (closing edges, as an alternative modification to
$f_{h}\big{(}\omega\big{)}$ presented previously in Modification one).
Introduce the set of edges,

$\displaystyle\mathscr{E}^{\prime\prime\prime\prime}\equiv\big{\\{}e\in\mathcal{E}:e\cap\big{(}\overline{\big{\\{}u\big{\\}}}\cup\overline{\big{\\{}v\big{\\}}}\big{)}\neq\emptyset\big{\\}}\text{
}\text{ , }$

in which one examines the edges, of the three-dimensional environment that
have nonempty intersection with
$\overline{\big{\\{}u\big{\\}}}\cup\overline{\big{\\{}v\big{\\}}}$.
Furthermore, related objects for this modification to
$f_{h}\big{(}\omega\big{)}$ also include,

$\displaystyle\mathscr{E}^{\prime\prime\prime\prime\prime}_{u,<h}\equiv\mathscr{E}_{u,<h}\equiv\bigg{\\{}e\in\mathcal{E}:\big{(}e\cap\pi_{u}\neq\emptyset\big{)}\cup\bigg{(}\textbf{P}_{h}\big{[}\big{(}\overline{\big{\\{}u\big{\\}}}\cup\overline{\big{\\{}v\big{\\}}}\big{)}\overset{\mathscr{Z},<h}{\longleftrightarrow}\pi_{u}\big{]}\bigg{)}\bigg{\\}}\text{
}\text{ , }$

corresponding to the edges of the three-dimensional environment which have
nonempty intersection with $\pi_{u}$, and,

$\displaystyle\mathscr{E}^{\prime\prime\prime\prime\prime\prime}_{v,<h}\equiv\mathscr{E}_{v,<h}\equiv\bigg{\\{}e\in\mathcal{E}:\big{(}e\cap\pi_{v}\neq\emptyset\big{)}\cup\bigg{(}\textbf{P}_{h}\big{[}\big{(}\overline{\big{\\{}u\big{\\}}}\cup\overline{\big{\\{}v\big{\\}}}\big{)}\overset{\mathscr{Z},<h}{\longleftrightarrow}\pi_{v}\big{]}\bigg{)}\bigg{\\}}\text{
}\text{ , }$

corresponding to the edges of the three-dimensional environment which have
nonempty intersection with $\pi_{v}$, and,

$\displaystyle\mathscr{E}^{\prime\prime\prime\prime\prime\prime\prime}_{\Gamma,<h}\equiv\mathscr{E}_{\Gamma,<h}\equiv\bigg{\\{}e\in\mathcal{E}:\big{(}e\cap\Gamma\neq\emptyset\big{)}\cup\bigg{(}\textbf{P}_{h}\big{[}\big{(}\overline{\big{\\{}u\big{\\}}}\cup\overline{\big{\\{}v\big{\\}}}\big{)}\overset{\mathscr{Z},<h}{\longleftrightarrow}\Gamma\big{]}\bigg{)}\bigg{\\}}\text{
}\text{ , }$

corresponding to the edges of the three-dimensional environment which From
$\mathscr{E}^{\prime\prime\prime\prime}$,
$\mathscr{E}^{\prime\prime\prime\prime\prime}_{u}$,
$\mathscr{E}^{\prime\prime\prime\prime\prime\prime}_{v}$, and
$\mathscr{E}^{\prime\prime\prime\prime\prime\prime\prime}_{\Gamma}$ introduced
above, observe that the series of modifications,

$\displaystyle\textbf{P}_{h}\big{[}e\in\mathcal{E}\cap\mathscr{E}^{\prime\prime\prime\prime}\cap\mathscr{E}^{\prime\prime\prime\prime\prime}_{u,<h}\cap\mathscr{E}^{\prime\prime\prime\prime\prime\prime}_{v,<h}\cap\mathscr{E}^{\prime\prime\prime\prime\prime\prime\prime}_{\Gamma,<h}\big{]}>0\overset{\mathrm{{}^{\prime}edge\text{
}closing^{\prime}}}{\Longleftrightarrow}\textbf{P}_{h}\big{[}e\in\mathcal{E}\cap\mathscr{E}^{\prime\prime\prime\prime}\cap\mathscr{E}^{\prime\prime\prime\prime\prime}_{u,\geq
h}\cap\mathscr{E}^{\prime\prime\prime\prime\prime\prime}_{v,\geq
h}\cap\mathscr{E}^{\prime\prime\prime\prime\prime\prime\prime}_{\Gamma,\geq
h}\big{]}>0\text{ }\text{ , }$

incorporate changes from open edges to closed edges, depending upon whether
the metric-graph GFF attains a sufficiently high level relative to the height
threshold $h$.

  * $\bullet$

Modification two (opening edges from the shortest path between $\Gamma$ and
$u$). Introduce the set of edges,

$\displaystyle\mathscr{E}^{\prime\prime\prime\prime\prime\prime\prime\prime}_{u,\Gamma,\geq
h}\equiv\mathscr{E}_{u,\Gamma,\geq
h}\equiv\big{\\{}e\in\mathcal{E}:\big{\\{}e\cap\big{\\{}u\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\Gamma\big{\\}}\neq\emptyset\big{\\}}\cup\big{\\{}d_{u,\Gamma}\big{\\}}\big{\\}}\text{
}\text{ , }$

where $d_{u,\Gamma}$ satisfies,

$\displaystyle
d_{u,\Gamma}\equiv{\underset{u\in\mathscr{U}}{\underset{\Gamma,\Gamma^{\prime}\subset\mathcal{S}}{\mathrm{inf}}}}\big{\\{}\Gamma^{\prime}\big{(}\Gamma,u,\big{\\{}\Gamma\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}u\big{\\}}\big{)}\big{\\}}\text{ }\text{ , }$

with $\Gamma^{\prime}$ satisfying the three following conditions,
simultaneously,

$\displaystyle\Gamma^{\prime}\equiv\big{\\{}\forall e\in\mathcal{E},\exists
n^{\prime\prime}\in\textbf{N}:\big{\\{}\big{\\{}e\cap\Gamma\big{\\}}\neq\emptyset\big{\\}}\cup\big{\\{}\big{\\{}u\cap\Gamma\big{\\}}\neq\emptyset\big{\\}}\cup\big{\\{}d_{u,\Gamma}\equiv
n^{\prime\prime}\big{\\}}\big{\\}}\text{ }\text{ . }$

As a result, for suitable paths $\gamma^{\prime}$,

$\displaystyle\textbf{P}_{h}\big{[}\big{\\{}e\in\mathscr{E}^{\prime\prime\prime\prime\prime\prime\prime\prime}_{u,\Gamma,\geq
h}\big{\\}}\cup\big{\\{}\gamma^{\prime}\equiv\Gamma^{\prime}\big{(}\Gamma,u,\big{\\{}\Gamma\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}u\big{\\}}\big{)}\big{\\}}\big{]}\overset{\mathrm{{}^{\prime}edge\text{
}opening^{\prime}}}{\Longleftrightarrow}\textbf{P}_{h}\big{[}\big{\\{}e\in\mathscr{E}^{\prime\prime\prime\prime\prime\prime\prime\prime}_{u,\Gamma,<h}\big{\\}}\cup\cdots$
$\displaystyle\big{\\{}\gamma^{\prime}\equiv\Gamma^{\prime}\big{(}\Gamma,u,\big{\\{}\Gamma\overset{\mathscr{Z},<h}{\longleftrightarrow}u\big{\\}}\big{)}\big{\\}}\big{]}\text{
, }$

in which, relative to the height threshold crossing $h$, edges in the
pushforward configuration $f_{h}\big{(}\omega\big{)}$ are opened, along the
shortest such path $\Gamma^{\prime}$.

  * $\bullet$

Modification three (opening edges from the shortest path between $\Gamma$ and
$v$). Lastly, introduce the set of edges,

$\displaystyle\mathscr{E}^{\prime\prime\prime\prime\prime\prime\prime\prime\prime}_{v,\Gamma,\geq
h}\equiv\mathscr{E}_{v,\Gamma,\geq
h}\equiv\big{\\{}e\in\mathcal{E}:\big{\\{}e\cap\big{\\{}v\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\Gamma\big{\\}}\neq\emptyset\big{\\}}\cup\big{\\{}d_{v,\Gamma}\big{\\}}\big{\\}}\text{
}\text{ , }$

where $d_{v,\Gamma}$ satisfies, as does $d_{u,\Gamma}$,

$\displaystyle
d_{u,\Gamma}\equiv{\underset{v\in\mathscr{V}}{\underset{\Gamma,\Gamma^{\prime\prime}\subset\mathcal{S}}{\mathrm{inf}}}}\big{\\{}\Gamma^{\prime}\big{(}\Gamma,v,\big{\\{}\Gamma\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}v\big{\\}}\big{)}\big{\\}}\text{ }\text{ , }$

with $\Gamma^{\prime}$ satisfying the three following conditions,
simultaneously,

$\displaystyle\Gamma^{\prime\prime}\equiv\big{\\{}\forall
e\in\mathcal{E},\exists
n^{\prime\prime\prime}\in\textbf{N}:\big{\\{}\big{\\{}e\cap\Gamma\big{\\}}\neq\emptyset\big{\\}}\cup\big{\\{}\big{\\{}v\cap\Gamma\big{\\}}\neq\emptyset\big{\\}}\cup\big{\\{}d_{v,\Gamma}\equiv
n^{\prime\prime\prime}\big{\\}}\big{\\}}\text{ }\text{ . }$

As a result, for suitable paths $\gamma^{\prime}$,

$\displaystyle\textbf{P}_{h}\big{[}\big{\\{}e\in\mathscr{E}^{\prime\prime\prime\prime\prime\prime\prime\prime\prime}_{v,\Gamma,\geq
h}\big{\\}}\cup\big{\\{}\gamma^{\prime}\equiv\Gamma^{\prime\prime}\big{\\}}\big{]}\overset{\mathrm{{}^{\prime}edge\text{
}opening^{\prime}}}{\Longleftrightarrow}\textbf{P}_{h}\big{[}\big{\\{}e\in\mathscr{E}^{\prime\prime\prime\prime\prime\prime\prime\prime\prime}_{v,\Gamma,<h}\big{\\}}\cup\big{\\{}\gamma^{\prime}\equiv\Gamma^{\prime\prime}\big{\\}}\big{]}\text{
}\text{ , }$

in which, as previously argued for Modification two, implies, relative to the
height threshold crossing $h$, that edges in the pushforward $f_{h}(\omega)$
can have edges opened as demonstrated in the fashion above.

As with Modification one, Modification two, and Modification three introduced
for the case
$\overline{\big{\\{}u\big{\\}}}\equiv\overline{\big{\\{}v\big{\\}}}$, if
$\overline{\mathscr{U}}\cap\overline{\mathscr{V}}\equiv\emptyset$, another
strictly positive upper bound between the intersection of common edges between
$\omega$ and $f_{h}\big{(}\omega\big{)}$ satisfies,

$\displaystyle\big{|}\omega\cap
f_{h}\big{(}\omega\big{)}\big{|}<\mathscr{D}^{\prime}\text{ }\text{ . }$

Hence, for $\overline{\mathscr{U}}\cap\overline{\mathscr{V}}\equiv\emptyset$,
$\overline{\big{\\{}u\big{\\}}}$ can be uniquely reconstructed from
$f_{h}\big{(}\omega\big{)}$, in which,

$\displaystyle\textbf{P}_{h}\big{[}\mathscr{X}\overset{\Gamma^{\prime\prime},\geq
h}{\longleftrightarrow}\mathscr{Y}\big{]}\text{ }\text{ . }$

Across the two subcases within Case A, it has been demonstrated that each
possibility holds with $\mathscr{D}$, and with $\mathscr{D}^{\prime}$,
respectively. To proceed, consider the following, Case B.

For the following analysis of the second case, we make use of a similar lower
bound from the following intersection of events, for suitable, strictly
positive, $\mathcal{C}_{1}$,

$\displaystyle\textbf{P}_{h}\big{[}\mathcal{E}_{1}\cap\mathscr{E}_{1}\cap\mathscr{E}_{2}\cap\mathscr{E}_{3}\cap\mathscr{E}_{4}\cap\mathscr{E}^{\prime}_{5}\big{]}\leq\mathcal{C}_{1}\textbf{P}_{h}\big{[}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\mathscr{Y}\big{]}\text{ }\text{ , }$ ($\bar{\Gamma}$
Probability III)

with the exception that the last event is,

$\displaystyle\mathscr{E}^{\prime}_{5}\equiv\big{\\{}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\Gamma\big{\\}}\text{ }\text{ , }$

instead of,

$\displaystyle\mathscr{E}_{5}\equiv\big{\\{}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\not\longleftrightarrow}\Gamma\big{\\}}\text{ }\text{ . }$

In the arguments below, we very briefly summarize the points distinguishing
Case C, and Case D, each of which respectively share in the arguments for Case
A, and for Case B, in which edges are closed and/or opened.

As for Case A and for Case B, denote the map,

$\displaystyle
f_{h}:\big{\\{}\mathcal{E}_{1}\cap\mathscr{E}_{1}\cap\mathscr{E}_{2}\cap\mathscr{E}_{3}\cap\mathscr{E}_{4}\cap\mathscr{E}^{\prime}_{5}\big{\\}}\longrightarrow\big{\\{}\mathscr{X}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\mathscr{Y}\big{\\}}\text{ }\text{ , }$

in which we would like to consider the action of the pushforward
$f_{h}\big{(}\cdot\big{)}$ on $\omega$, where,

$\displaystyle\omega\in\big{\\{}\mathcal{E}_{1}\cap\mathscr{E}_{1}\cap\mathscr{E}_{2}\cap\mathscr{E}_{3}\cap\mathscr{E}_{4}\cap\mathscr{E}^{\prime}_{5}\big{\\}}\text{
}\text{ , }$

is some configuration from the preimage space of $f_{h}\big{(}\cdot\big{)}$.
For the two cases below, fix the same objects, as in previous cases, which
include $u\in\mathscr{U}$, with $\big{\\{}\mathscr{Y}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\Gamma\big{\\}}\supsetneq\big{\\{}\mathscr{Y}\overset{\mathscr{Z}\backslash\overline{\\{u\\}},\geq
h}{\longleftrightarrow}\Gamma\big{\\}}$.

Case C

  * $\bullet$

Modification one (closing edges of the metric-graph GFF). The sequence of
arguments slightly differs from that presented in Case A. Namely, from the
event given in the preimage space, introduce the set of edges,

$\displaystyle\mathscr{E}^{6}\equiv\mathscr{E}_{\overline{\\{u\\}},\geq
h}\equiv\big{\\{}\forall e\in\mathcal{E},\exists
e\cap\overline{\big{\\{}u\big{\\}}}\neq\emptyset:\textbf{P}_{h}\big{[}\big{\\{}\Gamma\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\overline{\\{u\\}}\big{\\}}\cup\big{\\{}\Gamma\cap
e\equiv\emptyset\big{\\}}\big{]}>0\big{\\}}\text{ }\text{ , }$

corresponding to the edges of the three-dimensional environment with nonempty
intersection with $\overline{\big{\\{}u\big{\\}}}$. Equipped with
$\mathscr{E}_{6}$, additionally introduce,

$\displaystyle\big{(}\mathscr{E}^{6}\big{)}^{\prime}\equiv\mathscr{E}^{\prime}_{\overline{\\{u\\}},\geq
h}\equiv\big{\\{}\forall e\in\mathcal{E},\exists
e\cap\overline{\\{u\\}}\neq\emptyset:e\cap\overline{\\{u\\}}\cap\pi_{u}\equiv\pi_{u},\textbf{P}_{h}\big{[}\overline{\\{u\\}}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\pi_{u}\big{]}\equiv 0\big{\\}}\text{ }\text{ , }$

and,

$\displaystyle\big{(}\mathscr{E}^{6}\big{)}^{\prime\prime}\equiv\mathscr{E}^{\prime\prime}_{\overline{\\{u\\}},\geq
h}\equiv\big{\\{}\forall e\in\mathcal{E},\exists
e\cap\overline{\\{u\\}}\neq\emptyset:e\cap\overline{\\{u\\}}\cap\Gamma\equiv\Gamma,\textbf{P}_{h}\big{[}\overline{\\{u\\}}\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\Gamma\big{]}\equiv 0\big{\\}}\text{ }\text{ . }$

Concluding, performing the ’edge closing’ procedure in from the edge
constraints of $\mathcal{E}$ provided from $\mathscr{E}_{6}$,
$\mathscr{E}^{\prime}_{6}$ and $\mathscr{E}^{\prime\prime}_{6}$ yields,

$\displaystyle\textbf{P}_{h}\big{[}e\in\mathscr{E}_{\overline{\\{u\\}},<h}\cap\mathscr{E}^{\prime}_{\overline{\\{u\\}},<h}\cap\mathscr{E}^{\prime\prime}_{\overline{\\{u\\}},<h}\big{]}$
$\displaystyle\overset{\mathrm{{}^{\prime}edge\text{
}closing^{\prime}}}{\Longleftrightarrow}$
$\displaystyle\textbf{P}_{h}\big{[}e\in\mathscr{E}_{\overline{\\{u\\}},\geq
h}\cap\mathscr{E}^{\prime}_{\overline{\\{u\\}},\geq
h}\cap\mathscr{E}^{\prime\prime}_{\overline{\\{u\\}},\geq h}\big{]}\text{
}\text{ . }$

  * $\bullet$

Modification two (opening edges of the metric-graph GFF after closing edges of
the metric-graph GFF). Introduce the set of edges,

$\displaystyle\mathscr{E}_{7}\equiv\mathscr{E}_{7}\big{(}e,\Gamma,\overline{\\{u\\}}\big{)}\equiv\mathscr{E}_{7}\big{(}e,\overline{\\{u\\}}\big{)}\equiv\big{\\{}e\in\mathcal{E}\big{(}\overline{\\{u\\}}\big{)}:e\in\mathcal{E}\big{(}u\cap\Gamma\big{)}\cap\mathcal{E}\big{(}\overline{\\{u\\}}\big{)}\big{\\}}\text{
}\text{ , }$

where,

$\displaystyle\mathcal{E}\supsetneq\mathcal{E}\big{(}u\cap\Gamma\big{)}\equiv\big{\\{}e\in\mathcal{E}:e\cap
u\cap\Gamma\neq\emptyset\big{\\}}\text{ }\text{ , }$

which, from previous observation,

$\displaystyle\mathscr{E}^{7}_{u,\Gamma,\geq
h}\equiv\mathscr{E}_{u,\Gamma,\geq
h}\equiv\big{\\{}e\in\mathcal{E}:\big{\\{}e\cap\big{\\{}u\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\Gamma\big{\\}}\neq\emptyset\big{\\}}\cup\big{\\{}d_{u,\Gamma}\big{\\}}\big{\\}}\text{
}\text{ , }$

for the smallest length of the path realized by the same infimum
$d_{u,\Gamma}$. Concluding,

$\displaystyle\textbf{P}_{h}\big{[}e\in\mathscr{E}_{u,\Gamma,\geq
h}\big{]}\overset{\mathrm{{}^{\prime}edge\text{
}opening^{\prime}}}{\Longleftrightarrow}\textbf{P}_{h}\big{[}e\in\mathscr{E}_{u,\Gamma,<h}\big{]}\text{
}\text{ . }$

As a result,

$\displaystyle\big{|}\omega\cap
f_{h}\big{(}\omega\big{)}\big{|}<\mathscr{D}^{\prime\prime}\text{ }\text{ , }$

for suitable, strictly positive $\mathscr{D}^{\prime\prime}$. Hence
$\overline{\\{u\\}}$ can be reconstructed from $f_{h}\big{(}\omega\big{)}$ as
previously described.

Case D

    * $\bullet$

Modification one (closing edges of the metric-graph GFF configuration). For
the final case, we refer to Modification one for Case C included in the case
above first. With the same objects introduced for Modification one, also
introduce the following two objects, the first of which is given by,

$\displaystyle\mathscr{E}^{7}_{u}\equiv\mathscr{E}^{7}\big{(}e,\Gamma,\pi_{u}\big{)}\equiv\mathscr{E}^{7}\big{(}e,\pi_{u}\big{)}\equiv\big{\\{}e\in\mathcal{E}:e\in\mathcal{E}\big{(}\pi_{u}\cap\Gamma\big{)}\cap\mathcal{E}\big{(}u\big{)}\big{\\}}\text{
}\text{ , }$

and the second of which is similarly given by the edge-set,

$\displaystyle\mathscr{E}^{7}_{v}\equiv\mathscr{E}^{7}\big{(}e,\Gamma,\pi_{v}\big{)}\equiv\mathscr{E}^{7}\big{(}e,\pi_{v}\big{)}\equiv\big{\\{}e\in\mathcal{E}:e\in\mathcal{E}\big{(}\pi_{v}\cap\Gamma\big{)}\cap\mathcal{E}\big{(}v\big{)}\big{\\}}\text{
}\text{ , }$

implying,

$\displaystyle\textbf{P}_{h}\big{[}e\in\mathscr{E}^{7}_{u}\cap\mathscr{E}^{7}_{v}\cap\mathscr{E}^{7}:\big{\\{}e\overset{\mathscr{Z},<h}{\longleftrightarrow}\overline{\\{u\\}}\big{\\}}\big{]}>0\overset{\mathrm{{}^{\prime}edge\text{
}closing^{\prime}}}{\Longleftrightarrow}\textbf{P}_{h}\big{[}e\in\mathscr{E}^{7}_{u}\cap\mathscr{E}^{7}_{v}\cap\mathscr{E}^{7}:\big{\\{}e\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\overline{\\{u\\}}\big{\\}}\big{]}>0\text{ }\text{ , }$

with the intersection of three edge sets,
$\mathscr{E}^{7}_{u}\cap\mathscr{E}^{7}_{u}\cap\mathscr{E}^{7}$,

$\displaystyle\mathscr{E}^{7}_{u}\cap\mathscr{E}^{7}_{u}\cap\mathscr{E}^{7}\equiv\big{\\{}e\in\mathcal{E}\big{(}\mathcal{S}\big{)}:e\in\mathcal{E}\big{(}\pi_{u}\cap\Gamma\big{)}\cap\mathcal{E}\big{(}\pi_{v}\cap\Gamma\big{)}\cap\mathcal{E}\big{(}u\big{)}\cap\mathcal{E}\big{(}v\big{)}\big{\\}}\text{
}\text{ . }$

    * $\bullet$

Modification two (opening edges from the shortest path of the metric-graph GFF
configuration after opening edges). From Modification two introduced in Case
C, introduce the following modification to the edge set,

$\displaystyle\mathscr{E}^{8}_{u}\equiv\mathscr{E}^{7}_{u}\text{ }\text{ , }$

in addition to the definition of the following path,

$\displaystyle\rho_{u,\mathcal{S}}\equiv\rho_{u}\equiv\underset{\Gamma\subset\mathcal{S}}{\underset{u\in\mathcal{V}\big{(}\mathcal{S}\big{)}}{\mathrm{inf}}}\big{\\{}\rho_{\mathcal{S}}\big{(}u,\Gamma\big{)}\big{\\}}\text{
}\text{ , }$

which for the shortest such path, implies that Modification two takes the
form, for path realizations $\gamma$ of the shortest path between $u$ and
$\Gamma$,

$\displaystyle\textbf{P}_{h}\big{[}e\in\mathcal{E}^{8}_{u}:\big{\\{}\gamma\equiv\rho_{u}\big{\\}}\cup\big{\\{}u\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\Gamma\big{\\}}\big{]}\overset{\mathrm{{}^{\prime}edge\text{
}opening^{\prime}}}{\Longleftrightarrow}\textbf{P}_{h}\big{[}e\in\mathcal{E}^{8}_{u}:\big{\\{}\gamma\equiv\rho_{u}\big{\\}}\cup\big{\\{}u\overset{\mathscr{Z},<h}{\longleftrightarrow}\Gamma\big{\\}}\big{]}\text{
}\text{ , }$

as opening the edges for the metric-graph GFF entails the change to the level-
set crossing event,

$\displaystyle\big{\\{}u\overset{\mathscr{Z},<h}{\longleftrightarrow}\Gamma\big{\\}}\text{
}\text{ , }$

a modification to the height of the metric-graph GFF crossing event.

    * $\bullet$

Modification three (opening edges of the metric-graph GFF configuration once
more). Introducing the following modification to edge sets used in
Modification two of Case D implies,

$\displaystyle\textbf{P}_{h}\big{[}e\in\mathcal{E}^{8}_{u}:\big{\\{}\gamma\equiv\rho^{\prime}_{v}\big{\\}}\cup\big{\\{}u\overset{\mathscr{Z},\geq
h}{\longleftrightarrow}\Gamma\big{\\}}\big{]}\overset{\mathrm{{}^{\prime}edge\text{
}opening^{\prime}}}{\Longleftrightarrow}\textbf{P}_{h}\big{[}e\in\mathcal{E}^{8}_{u}:\big{\\{}\gamma\equiv\rho^{\prime}_{v}\big{\\}}\cup\big{\\{}u\overset{\mathscr{Z},<h}{\longleftrightarrow}\Gamma\big{\\}}\big{]}\text{
}\text{ , }$

instead for the distance minimizing path $\gamma$ between $v$ and
$\Gamma\cup\rho$, as,

$\displaystyle\rho^{\prime}_{v,\mathcal{S}}\equiv\rho^{\prime}_{v}\equiv\underset{\Gamma\cup\rho\subset\mathcal{S}}{\underset{v\in\mathcal{V}\big{(}\mathcal{S}\big{)}}{\mathrm{inf}}}\big{\\{}\rho_{\mathcal{S}}\big{(}v,\Gamma\cup\rho\big{)}\big{\\}}\text{
}\text{ . }$

As a result,

$\displaystyle\big{|}\omega\cap
f_{h}\big{(}\omega\big{)}\big{|}<\mathscr{D}^{\prime\prime\prime}\text{
}\text{ , }$

for suitable, strictly positive $\mathscr{D}^{\prime\prime\prime}$. Hence
$\overline{\\{u\\}}$ can be reconstructed from $f_{h}\big{(}\omega\big{)}$ as
previously described.

This concludes the arguments to demonstrate that the upper bound displayed in
($\bar{\Gamma}$ Probability II) holds, from which we conclude the argument for
establishing quasi-multiplicativity.

##  4 The Villain model, and its cable system counterpart, akin to the metric-
graph extension of the GFF

We now turn towards defining another model of interest, in which the authors
of [12] propose that it can be of interest co construct IIC-type limits also
for the Villain model, from the expectation value, for $k\equiv 1$, $k\equiv
2$, and some $\theta>0$,

$\displaystyle\langle\mathrm{exp}\big{(}ik\theta_{0}\big{)}\rangle_{\mathrm{IIC}_{k}\propto\textbf{P}^{\mathscr{V}}}\text{
}\text{ , }$

of an observable, for $k\equiv 1$, with respect to the the incipient infinite
cluster measure that is proportional to the base measure
$\textbf{P}^{\mathscr{V}}\big{[}\cdot\big{]}$ for the Villain model which will
be introduced formally below. Denote the finite subgraph of $\textbf{Z}^{d}$
with connected $\mathcal{G}\equiv G\equiv\big{(}V,E\big{)}$. As with the
metric-graph extension for the GFF in three dimensions presented at the
beginning of the paper, denote the metric-graph from $\mathcal{G}$ with
$\widetilde{\mathcal{G}}\equiv\widetilde{G}\equiv\big{(}\widetilde{V},\widetilde{E}\big{)}$.

###  4.1 Villain objects

Definition 14 (Villain model Hamiltonian). Introduce,

$\displaystyle\mathcal{H}\big{(}\theta\big{)}\equiv\underset{n\in\textbf{Z}}{\prod}\big{(}-\beta\big{(}\theta_{x}-\theta_{y}+2\pi
n\big{)}^{2}\big{)}\text{ }\text{ , }$

corresponding to the Hamiltonian of the Villain model, for the following
configurations drawn from spin space U,

$\displaystyle\big{(}u_{x}\big{)}_{x\in
V}\equiv\big{(}\mathrm{exp}\big{(}i\theta_{x}\big{)}\big{)}_{x\in
V}\in\Omega^{\mathrm{Villain}}\simeq\frac{\textbf{R}}{2\pi\textbf{Z}}\simeq\textbf{U}\text{
}\text{ , }$

at inverse temperature $\beta>0$, for $\theta_{x},\theta_{y}>0$.

Definition 15 (Villain model probability measure from the Hamiltonian).
Introduce,

$\displaystyle\big{(}\textbf{P}^{\mathscr{V}}_{\mathcal{G}}\big{[}\omega\big{]}\big{)}^{\theta}\equiv\textbf{P}^{\mathscr{V}}_{\mathcal{G}}\big{[}\omega\big{]}\equiv\mathcal{Z}^{-1}\mathrm{exp}\big{(}-\mathcal{H}\big{(}\theta\big{)}\big{)}\underset{x\in
V}{\prod}\mathrm{d}\theta_{x}\equiv\mathcal{Z}^{-1}\underset{\\{xy\\}\in
E}{\prod}\mathrm{exp}\big{\\{}-\underset{n\in\textbf{Z}}{\prod}\big{(}-\beta\big{(}\theta_{x}-\theta_{y}+2\pi
n\big{)}^{2}\big{)}\big{\\}}\underset{x\in V}{\prod}\mathrm{d}\theta_{x}$
$\displaystyle\overset{\beta\longrightarrow+\infty}{\approx}\mathcal{Z}^{-1}\underset{\\{xy\\}\in
E}{\prod}\mathrm{exp}\big{\\{}-\underset{n\in\textbf{Z}}{\prod}\big{(}-\frac{\beta}{2}\big{(}\mathrm{cos}\big{(}\theta_{x}-\theta_{y}\big{)}\big{)}\big{\\}}\underset{x\in
V}{\prod}\mathrm{d}\theta_{x}\text{ }\text{ , }$

corresponding to the probability of the Villain model, with configurations
from sample space $\Omega^{\mathrm{Villain}}$ provided in the previous
definition, with partition function,

$\displaystyle\mathcal{Z}^{\mathscr{V}}_{\mathcal{G}}\big{(}\theta\equiv
1,\theta_{x},\theta_{y},\beta\big{)}\equiv\mathcal{Z}^{\mathscr{V}}_{\mathcal{G}}\big{(}\theta_{x},\theta_{y},\beta\big{)}\equiv\mathcal{Z}\big{(}\theta_{x},\theta_{y},\beta\big{)}\equiv\mathcal{Z}\equiv\underset{x,y\in\mathcal{G}}{\sum}\underset{\\{xy\\}\in
E}{\prod}\mathrm{exp}\big{\\{}-\underset{n\in\textbf{Z}}{\prod}\big{(}-\beta\big{(}\theta_{x}-\theta_{y}+2\pi
n\big{)}^{2}\big{)}\big{\\}}\underset{x\in V}{\prod}\mathrm{d}\theta_{x}\text{
}\text{ , }$

for boundary conditions $\theta\equiv 1$.

Definition 16 (transition probability for Brownian motion over the Villain
model sample space for the cable system representation). Introduce an
exponential with a similar Hamiltonian to that provided in Definition 14,
namely,

$\displaystyle
p_{t}\big{(}\theta_{1},\theta_{2}\big{)}\equiv\frac{1}{\sqrt{2\pi
t}}\underset{n\in\textbf{Z}}{\sum}\mathrm{exp}\big{\\{}-\big{(}2t\big{)}^{-1}\big{(}\theta_{1}-\theta_{2}+2\pi
n\big{)}^{2}\big{)}\big{\\}}\text{ }\text{ , }$

corresponding to the probability transition function for Brownian motion over
$\Omega^{\mathrm{Villain}}$, for $t<1$, or $t>1$.

Definition 17 (cable system representation of the Villain model Hamiltonian).
Introduce,

$\displaystyle\mathcal{H}^{\mathrm{C}-\mathrm{S}}\big{(}\theta\big{)}\equiv\mathcal{H}\big{(}\theta\big{)}\equiv\mathrm{ln}\big{\\{}\big{(}2\pi
t\big{)}^{\frac{|V|}{2}}\underset{\\{xy\\}\in\widetilde{E}}{\prod}p_{t}\big{(}\theta_{1},\theta_{2}\big{)}\big{\\}}\text{
}\text{ , }$

corresponding to the representation of the Hamiltonian for the Villain model
under the cable system.

Definition 18 (cable system probability measure for the extended Villain
model). Introduce,

$\displaystyle\big{(}\widetilde{\textbf{P}^{\mathscr{V}}_{\mathcal{\widetilde{G}}}}\big{[}\widetilde{\omega}\big{]}\big{)}^{\widetilde{\theta}}\equiv\widetilde{\textbf{P}^{\mathscr{V}}_{\mathcal{\widetilde{G}}}}\big{[}\widetilde{\omega}\big{]}\equiv\widetilde{\mathcal{Z}^{-1}}\mathrm{exp}\big{\\{}-\mathcal{H}\big{(}\theta\big{)}\big{\\}}\equiv\widetilde{\mathcal{Z}^{-1}}\big{(}2\pi
t\big{)}^{\frac{|V|}{2}}\underset{\\{xy\\}\in
E}{\prod}p_{t}\big{(}\theta_{x},\theta_{y}\big{)}\text{ }\text{ , }$

corresponding to the probability measure for the extended Villain model, with
partition function,

$\displaystyle\widetilde{\mathcal{Z}^{\mathscr{V}}_{\widetilde{\mathcal{G}}}}\big{(}\widetilde{\theta}\equiv
1,\theta_{1},\theta_{2},t\big{)}\equiv\widetilde{\mathcal{Z}^{\mathscr{V}}_{\widetilde{\mathcal{G}}}}\big{(}\theta_{1},\theta_{2},t\big{)}\equiv\widetilde{\mathcal{Z}^{\mathscr{V}}_{\widetilde{\mathcal{G}}}}\equiv\widetilde{\mathcal{Z}}\equiv\underset{x,y\in\widetilde{\mathcal{G}}}{\sum}\big{(}2\pi
t\big{)}^{\frac{|V|}{2}}\underset{\\{xy\\}\in
E}{\prod}p_{t}\big{(}\theta_{x},\theta_{y}\big{)}\text{ }\text{ , }$

for boundary conditions $\widetilde{\theta}\equiv 1$.

From these objects, we proceed to also introduce connectivity events for the
extended Villain model.

Definition 19 (connectivity events). Introduce, for $S\subset\textbf{U}$, and
$\theta\in C\big{(}\widetilde{\mathcal{G}},\textbf{U}\big{)}$,

$\displaystyle\big{\\{}x\overset{S}{\longleftrightarrow}y\big{\\}}\text{
}\text{ , }$

corresponding to a connectivity event, between $x,y\in\widetilde{\mathcal{G}}$
along some path $\gamma$, so that $\theta\big{(}\gamma\big{)}\subset S$.

Additionally, besides the Villain crossing event denoted above, we also
introduce Villain clusters for the extended model.

Definition 20 (extended Villain clusters). Introduce, for
$\Lambda\subsetneq\textbf{Z}^{d}$ with boundary $\partial\Lambda$,

$\displaystyle\mathcal{C}^{\mathrm{Villain}}\equiv\mathcal{C}\equiv\big{\\{}y\in\widetilde{\mathcal{G}}:y\overset{S}{\longleftrightarrow}\partial\Lambda\big{\\}}\text{
}\text{ , }$

corresponding to clusters in the extended Villain model, within $S$.

Definition 21 (linear heat kernel in the extended Villain model with absorbing
boundary). Introduce,

$\displaystyle
p^{[-\frac{\pi}{2},\frac{\pi}{2}]}_{t}\big{(}\theta_{1},\theta_{2}\big{)}\equiv\frac{1}{\sqrt{2\pi
t}}\underset{n\in\textbf{Z}}{\sum}\bigg{(}\mathrm{exp}\bigg{(}-\big{(}2t\big{)}^{-1}\big{(}\theta_{1}-\theta_{2}+2\pi
n\big{)}^{2}\bigg{)}-\mathrm{exp}\bigg{(}-\big{(}2t\big{)}^{-1}\big{(}\theta_{1}+\theta_{2}+\big{(}2n-1\big{)}\pi\big{)}^{2}\bigg{)}\bigg{)}\text{
}\text{ , }$

corresponding to the heat kernel of the extended Villain model, obtained from
repeated reflection principle as discussed in [12].

From the linear heat kernel, also introduce the following result.

Definition 22 (extended connectivity event probability). Given
$x,y\in\widetilde{\mathcal{G}}$, and $z\in\\{xy\\}$, as introduced in
Definition 19, introduce,

$\displaystyle\big{\\{}\mathrm{exp}\big{(}i\theta_{x}\big{)}\equiv
u\big{(}x\big{)}\overset{\\{\pm
i\\}^{c}}{\longleftrightarrow}u\big{(}y\big{)}\equiv\mathrm{exp}\big{(}i\theta_{y}\big{)}\big{\\}}\text{
}\text{ , }$

corresponding to the connectivity event between $u(x)$ and $u(y)$ in $\\{\pm
i\\}^{c}$.

Finally, introduce the following Lemma, from [12], for relating the
probability of the connectivity event above, namely,

$\displaystyle\widetilde{\textbf{P}^{\mathscr{V}}_{\mathcal{\widetilde{G}}}}\big{[}u\big{(}x\big{)}\overset{\\{\pm
i\\}^{c}}{\longleftrightarrow}u\big{(}y\big{)}\big{]}\text{ }\text{ , }$

with the linear heat kernel.

Lemma 3.1 (equating the probability of an extended connectivity event
occurring with a normalized linear heat kernel, ([12], Lemma 3.1)). Given
parameters specified in Definition 22, the probability of an extended
connectivity event occurring between $u(x)$ and $u(y)$ is,

$\displaystyle 0\text{ if
}\mathrm{cos}\big{(}\theta_{x}\big{)}\mathrm{cos}\big{(}\theta_{y}\big{)}\leq
0$
$\displaystyle\frac{p^{[-\frac{\pi}{2},\frac{\pi}{2}]}_{t}\big{(}\theta_{x},\theta_{y}\big{)}}{p_{t}\big{(}\theta_{x},\theta_{y}\big{)}}\text{
}\text{ if }\text{
}\theta_{x},\theta_{y}\in\big{(}-\frac{\pi}{2},\frac{\pi}{2}\big{)}\text{
}\text{ , }$
$\displaystyle\frac{p^{[-\frac{\pi}{2},\frac{\pi}{2}]}_{t}\big{(}\theta_{x}-\pi,\theta_{y}-\pi\big{)}}{p_{t}\big{(}\theta_{x},\theta_{y}\big{)}}\text{
}\text{ if }\text{
}\theta_{x},\theta_{y}\in\big{(}\frac{\pi}{2},\frac{3\pi}{2}\big{)}\text{
}\mathrm{mod}\big{(}2\pi\big{)}\text{ }\text{ , }$

depending upon the conditions enforced on $\theta_{x}$ and $\theta_{y}$ from
each of the three possible values of the extended connectivity event
occurring.

As with the GFF which satisfies (FKG), the Villain model, as a variant of the
XY model, also satisfies (FKG), the statement of which is provided below.

Proposition Villain (FKG property of the Villain model from FKG property of
the XY model, ([33], Proposition A.1). The XY probability measures
$\rho\big{[}\cdot\big{]}$ are FKG.

From the probability of an extended connectivity event occurring from the
linear heat kernel, the conditional probability that
$\big{\\{}x\overset{\\{\pm i\\}^{c}}{\longleftrightarrow}y\big{\\}}$ occurs
also admits a particular representation which is given below.

Proposition Villain-1 (conditional probability of a connectivity event
occurring, [12]). Conditionally upon $\theta|_{\mathcal{G}}$, the probability,

$\displaystyle\widetilde{\textbf{P}^{\mathscr{V}}_{\mathcal{\widetilde{G}}}}\big{[}u\big{(}x\big{)}\overset{\\{\pm
i\\}^{c}}{\longleftrightarrow}u\big{(}y\big{)}\text{ }|\text{
}\theta|_{\mathcal{G}}\big{]}\text{ }\text{ , }$

is equivalent to,

$\displaystyle\mathrm{max}\big{(}0,p+q-1\big{)}\leq
c\leq\mathrm{min}\big{(}p,q\big{)}\text{ }\text{ , }$

for parameters $c,p,q>0$.

Before presenting further arguments, under previously stated assumptions,

$\displaystyle\big{(}\textbf{P}^{\mathscr{V}}_{\mathcal{G}}\big{[}\big{(}u_{x}\big{)}_{x\in
V}\big{]}\big{)}^{\theta}\equiv\big{(}\textbf{P}^{\mathscr{V}}_{\mathcal{G}}\big{[}\big{(}\overline{u}_{x}\big{)}_{x\in
V}\big{]}\big{)}^{\theta}\text{ }\text{ , }$

for boundary conditions $\theta$.

Finally, it is also worth mentioning that, as is the case for the metric-graph
GFF, that the Villain model also satisfies a version of the strong Markov
property (see Lemma 2.1, from [12]).

###  4.2 Formulating an IIC-type limit in the Villain context

From prerequisites in the previous section, we also introduce the following
two items.

Proposition Villain-2 ([12], Proposition 3.2). For each $t_{0}>0$, there
exists $\delta\equiv\delta\big{(}t_{0},\mathrm{deg}(x)\big{)}>0$, such that,
if $t\big{(}xy\big{)}\geq t_{0}$ for all neighbors $y\sim x$, then,

$\displaystyle\delta\leq\frac{\langle\mathrm{cos}\big{(}\theta_{x}\big{)}\rangle}{\textbf{P}\big{[}x\overset{\\{\pm
i\\}^{c}}{\longleftrightarrow}\partial\big{]}}\leq 1\text{ }\text{ , }$

for $\partial\equiv\partial V$, and
$\textbf{P}_{\mathscr{G}}\big{[}\cdot\big{]}\equiv\textbf{P}\big{[}\cdot\big{]}$.

Proposition Villain-3 ([12], Proposition 3.3). For each $t_{0}>0$, there
exists $\delta\equiv\delta\big{(}t_{0},\mathrm{deg}(x)\big{)}>0$, such that,
if $t\big{(}xy\big{)}\geq t_{0}$ for all neighbors $y\sim x$, then,

$\displaystyle\delta\leq\frac{\langle\mathrm{cos}\big{(}2\theta_{x}\big{)}\rangle}{\textbf{P}\big{[}x\overset{\\{\pm\xi\\}^{c}}{\longleftrightarrow}\partial,x\overset{\\{\pm\bar{\xi}\\}^{c}}{\longleftrightarrow}\partial\big{]}}\leq
1\text{ }\text{ , }$

for $\partial\equiv\partial V$, and
$\textbf{P}_{\mathscr{G}}\big{[}\cdot\big{]}\equiv\textbf{P}\big{[}\cdot\big{]}$.

From each Proposition above, introduce the two limits for the Villain model
IIC-type limit which will be shown to coincide.

Definition 23 (cylinder events for the Villain model). Akin to cylinder events
for Bernoulli percolation in two-dimensions, and for the metric-graph GFF in
three-dimensions, a cylinder event for the Villain model is dependent upon a
finite number of vertices over $\textbf{Z}^{2}$, in addition to the $\theta$
interaction, as,

$\displaystyle\mathscr{E}\big{(}\theta\big{)}\equiv\mathscr{E}\text{ }\text{ ,
}$

for $\theta\in[0,2\pi]$.

Definition 24 (Villain model equivalent of the conditional crossing limit for
Bernoulli at criticality from Definition 6). Introduce, for $\theta$
interactions coinciding with $\\{\pm i\\}^{c}$, and $w\in V$, the probability,

$\displaystyle\mathcal{P}^{\prime}_{1}\equiv\textbf{P}\big{[}x\overset{\\{\pm
i\\}^{c}}{\longleftrightarrow}\partial\big{|}|\mathcal{C}\big{(}w\big{)}|=+\infty\big{]}\equiv\textbf{P}\big{[}\mathscr{E}\big{(}\pm\frac{\pi}{2}\big{)}\big{|}|\mathcal{C}\big{(}w\big{)}|=+\infty\big{]}\text{
}\text{ , }$

corresponding to the first limiting ratio of the IIC-type limit for the
Villain model.

Definition 25 (Villain model equivalent of the limit of conditionally defined
crossing probabilities for Bernoulli percolation at supercriticality from
Definition 7). Introduce, for $\theta$ interactions coinciding with
$\\{\pm\xi\\}^{c}$, and $\\{\pm\bar{\xi}\\}^{c}$, as $\xi,\bar{\xi}\searrow
i$, and $w\in V$, the intersection probability,

$\displaystyle\mathcal{P}^{\prime\prime\prime}_{2}\equiv\underset{\xi,\bar{\xi}\searrow
i}{\mathrm{lim}}\mathcal{P}^{\prime\prime}_{2}\equiv\underset{\xi,\bar{\xi}\searrow
i}{\mathrm{lim}}\textbf{P}\big{[}x\overset{\\{\pm\xi\\}^{c}}{\longleftrightarrow}\partial,x\overset{\\{\pm\bar{\xi}\\}^{c}}{\longleftrightarrow}\partial\big{|}|\mathcal{C}\big{(}w\big{)}|=+\infty\big{]}\equiv\underset{\xi,\bar{\xi}\searrow
i}{\mathrm{lim}}\textbf{P}\big{[}\mathscr{E}\big{(}\pm
i^{-1}\mathrm{ln}\big{\\{}\xi\big{\\}}\big{)},\mathscr{E}\big{(}\pm
i^{-1}\mathrm{ln}\big{\\{}\bar{\xi}\big{\\}}\big{)}\big{|}|\mathcal{C}\big{(}w\big{)}|=+\infty\big{]}\text{
, }$

corresponding to the second limiting ratio of the IIC-type limit for the
Villain model.

From Definition 23 and Definition 24, the next item below asserts the
existence of the IIC-type limit for the Villain model.

Theorem 7 (establishing equivalence between the two limiting ratio of the
Villain IIC-type limit, analogous to Theorem 3 for Bernoulli percolation, and
to Theorem 5 for the metric-graph GFF). Given the assumptions provided above,
the two limits, respectively from Definition 23, and 24, satisfy,

$\displaystyle\mathcal{P}^{\prime}_{1}\equiv\mathcal{P}^{\prime\prime\prime}_{2}\text{
}\text{ , }$

which, by Kolmogorov extension, implies,

$\displaystyle\nu_{\theta}\big{[}\forall w\in
V,\exists!\mathcal{C}\big{(}w\big{)}:|\mathcal{C}\big{(}w\big{)}|=+\infty\big{]}\equiv
1\text{ }\text{ , }$

upon denoting the common value between $\mathcal{P}^{\prime}_{1}$ and
$\mathcal{P}^{\prime\prime\prime}_{2}$ with,

$\displaystyle\nu_{\theta}\big{[}\mathscr{E}\big{]}\text{ }\text{ . }$

To prove Theorem 7 pertaining to the Villain IIC-type limits, we appeal to
arguments provided in the streamlined presentation of Kesten’s seminal
arguments, from [24], in [3].

Proof of Theorem 7. From arguments originally introduced for Bernoulli
percolation in two-dimensions, which were adapted in Theorem 5 for the three-
dimensional metric-graph GFF, we introduce the following, namely,

$\displaystyle\textbf{P}\big{[}\mathscr{E}\text{ }|\text{
}x\overset{\\{\pm\xi\\}^{c}}{\longleftrightarrow}\partial\big{]}\longrightarrow\nu_{\theta}\big{[}\mathscr{E}\big{]}\text{
}\text{ , }$

corresponding to the Kolmogorov extension result provided in $(\textit{1})$.
Corresponding to $(\textit{2})$, introducing an analogous quantity yields,