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# On upper tail large deviation rate function for chemical distance in
supercritical percolation

Barbara Dembin  ETH, Zurich, Switzerland  barbara.dembin(at)math.ethz.ch and
Shuta Nakajima  Meiji University, Kanagawa, Japan  njima(at)meiji.ac.jp

###### Abstract.

We consider the supercritical bond percolation on $\mathbb{Z}^{d}$ and study
the graph distance on the percolation graph called the chemical distance. It
is well-known that there exists a deterministic constant $\mu(x)$ such that
the chemical distance $\mathcal{D}(0,nx)$ between two connected points $0$ and
$nx$ grows like $n\mu(x)$. Garet and Marchand [21] proved that the probability
of the upper tail large deviation event
$\left\\{n\mu(x)(1+\varepsilon)<\mathcal{D}(0,nx)<\infty\right\\}$ decays
exponentially with respect to $n$. In this paper, we prove the existence of
the rate function for upper tail large deviation when $d\geq 3$ and
$\varepsilon>0$ is small enough. Moreover, we show that for any
$\varepsilon>0$, the upper tail large deviation event is created by space-time
cut-points (points that all paths from $0$ to $nx$ must cross after a given
time) that force the geodesics to consume more time by going in a non-optimal
direction or by wiggling considerably. This enables us to express the rate
function in regards to space-time cut-points.

###### Key words and phrases:

large deviations, chemical distance, percolation, rate function

###### 2010 Mathematics Subject Classification:

Primary 60K37; secondary 60K35; 82A51; 82D30

##  1\. Introduction and main results

The model of Bernoulli bond percolation was introduced by Broadbent and
Hammersley in 1957 to model the circulation of water in a porous medium [7].
Since then, various models of percolation have been developed, such as level-
sets of random fields, Boolean and Voronoi percolation, random interlacements,
etc. A common feature among all of these models is that they involve a random
subset of the underlying space, which is characterized by a parameter. This
random subset grows in size as the parameter increases. The primary focus is
on studying the connectivity of the random subset, specifically identifying
the parameter values at which the random subset has an unbounded connected
component. Of particular interest is the existence of a phase transition that
occurs in the parameter when transitioning from a regime where there is no
infinite connected component to a regime where one exists.

The study of these problems has uncovered numerous properties and techniques,
giving rise to the so-called percolation theory. This theory holds significant
importance in probability theory and statistical physics, especially regarding
its relationship with other statistical physics models like lattice spin
models, interacting particle systems, and random walks in random environments.
These connections enable the successful application of powerful results and
arguments from percolation theory to other fields, even in the absence of a
direct link. Refer to [23] for comprehensive background information and known
results on percolation theory.

In the model of Bernoulli bond percolation, each edge of
$(\mathbb{Z}^{d},\mathbb{E}^{d})$ is independently removed with probability
$1-p$. When $p$ increases the set of remaining edges increases. In particular,
Aizenman–Kesten–Newman [1] proved the existence of a phase transition in the
sense that there exists a special parameter $p_{c}(d)\in(0,1)$ such that below
$p_{c}(d)$, there is no infinite connected component; above $p_{c}(d)$, there
exists a unique one. In Bernoulli percolation, the subcritical regime
$p<p_{c}(d)$ and the supercritical regime $p>p_{c}(d)$ are both well-studied
and a lot of important results have been obtained rigorously. On the other
hand, the study of critical percolation (i.e. the case $p=p_{c}(d)$) have
appeared considerably more complicated. One of the major challenges in
percolation is to exclude the existence of an infinite connected component in
critical percolation. In $d=2$, Kesten [26] proved that $p_{c}(2)=1/2$ and
there is no infinite connected component at $p=p_{c}(2)$. Additionally, planar
critical percolation was hypothesized to be associated with Schramm-Loewner
evolution with $\kappa=6$. As a result, this area has been an active research
field for the past three decades. For $d\geq 11$, the absence of an infinite
cluster in critical percolation has been conclusively proven through the use
of lace expansion and mean-field approximation techniques [25, 20].
Nonetheless, for $d\in\\{3,...,10\\}$, this problem remains unsolved and is
regarded as one of the most significant conjectures in probability theory.

From the viewpoint of graph theory, it is also important to study the graph
distance in a graph since the graph distance gives even more indepth
information on geometric structures of the graph compared to connectivity
alone. In a percolation graph, the graph distance is referred to as the
chemical distance, a name coined by a physicist S. Alexander. A central theme
in this field is the asymptotic behavior of the chemical distance between two
distant endpoints. This question has been explored for various models,
including those with long-range interactions [2, 6, 19, 18, 9]. The extensive
study of the chemical distance started in the 1980s. Physicists were
particularly interested in the growth exponent for the chemical distance in
critical percolation, known as the chemical distance exponent. Despite
numerous efforts and advancements in this area, there is still no consensus on
the exact values of the chemical distance exponents even for the planar
critical percolation. Recently, Damron–Hanson–Sosoe [15] established an upper
bound, showing that the chemical distance exponent is strictly smaller that
the growth exponent for the lowest left/right crossing. This latter exponent
is related to the three-arm exponent in percolation and its value is expected
to be universal over the 2-dimensional lattices, which was, in particular,
rigorously obtained for the triangular lattice. This suggests that the
chemical distance exponent is not characterized by a well-known quantity in
connectivity. For more on the background and known results on chemical
distance, we refer the reader to [14].

In Bernoulli percolation on $\mathbb{Z}^{d}$ lattice, the properties of the
chemical distance in supercritical percolation are also of great interest both
in mathematics and physics. Garet and Marchand proved a kind of law of large
numbers for the chemical distance [28]. So, the next step is to study the
fluctuations and large deviations. However, there have been few results on the
fluctuations of the chemical distance. Recently, Dembin [16] proved super-
concentration for the chemical distance, that is, the variance of the chemical
distance incleases at most sublinearly in the distance between the endpoints.
However, as far as we know, there are no predictions in mathematical
literature regarding the expected behavior of the variance of chemical
distance. For the lower bound, the problem whether the variance diverges or
not is important but still open for any dimension larger than $1$.

Moving on to the large deviations, Antal–Pisztora and subsequently
Garet–Marchand determined the correct speed of the upper tail large deviations
[2, 21]. However, previous studies did not establish the existence of the rate
function and a precise description for upper tail large deviations. The aim of
this paper is to obtain the rate function by identifying the correct scenario
responsible for upper tail large deviations. We believe that the methods and
concepts introduced in this paper provide new insights on upper tail large
deviations for other percolation models.

###  1.1. Chemical distance in Bernoulli percolation

The model of Bernoulli percolation is formally defined as follows. Let
$\mathbb{E}^{d}$ be the set of all pairs of nearest neighbours in
$\mathbb{Z}^{d}$. We consider i.i.d. Bernoulli random variables
$(B_{e})_{e\in\mathbb{E}^{d}}$ of parameter $p\in[0,1]$. If $B_{e}=1$, then
the edge $e$ is called open; otherwise, the edge is called closed. Let
$\mathcal{G}_{p}$ be the graph of the open edges:

$\mathcal{G}_{p}:=(\mathbb{Z}^{d},\\{e\in\mathbb{E}^{d}:B_{e}=1\\})\,.$

The graph $\mathcal{G}_{p}$ is called the percolation graph. A path is said to
be open if the path consists only of open edges. We write $x\leftrightarrow y$
if $x$ and $y$ are connected in $\mathcal{G}_{p}$. This model exhibits a phase
transition. Indeed, when $d\geq 2$, there exists a critical parameter
$p_{c}(d)\in(0,1)$ such that for $p>p_{c}(d)$ (supercritical regime), there
almost surely exists a unique infinite open cluster $\mathscr{C}_{\infty}$ in
$\mathcal{G}_{p}$. In contrast, for $p<p_{c}(d)$ (subcritical regime), there
are no infinite open clusters. We refer to [23] for general backgrounds and
known results on Bernoulli percolation. Throughout the paper, we always assume
$p>p_{c}(d)$. We denote by $\mathcal{D}^{\mathcal{G}_{p}}$ the graph distance
on the graph $\mathcal{G}_{p}$, i.e. for $x,y\in\mathbb{Z}^{d}$,

(1.1) $\mathcal{D}^{\mathcal{G}_{p}}(x,y):=\inf\Big{\\{}\,|r|:\text{$r$ is a
path from $x$ to $y$ in $\mathcal{G}_{p}$}\,\Big{\\}}\,,$

where $|r|$ denotes the number of edges in the path $r$ and we use the
convention $\inf\emptyset=+\infty$. In particular, if $x$ and $y$ are not
connected in $\mathcal{G}_{p}$, then
$\mathcal{D}^{\mathcal{G}_{p}}(x,y)=\infty$. For later purposes, we extend the
chemical distance to a function of real vectors by setting
$\mathcal{D}^{\mathcal{G}_{p}}(x,y)=\mathcal{D}^{\mathcal{G}_{p}}(\lfloor
x\rfloor,\lfloor y\rfloor)$ for $x,y\in\mathbb{R}^{d}$, where
$\lfloor\cdot\rfloor$ stands for the floor function. This
$\mathcal{D}^{\mathcal{G}_{p}}$ is the so-called chemical distance. We
interpret the quantity $\mathcal{D}^{\mathcal{G}_{p}}(x,y)$ as the time to go
from $x$ to $y$ (see Section 1.4). Any path achieving the infimum in
$\mathcal{D}^{\mathcal{G}_{p}}(x,y)$ is called a geodesic. Note that any
geodesic is self-avoiding.

##### Time constant

Garet and Marchand [28] obtained an asymptotic behavior of
$\mathcal{D}^{\mathcal{G}_{p}}(0,nx)$ as $n\to\infty$: for any $p>p_{c}(d)$,
there exists a deterministic norm $\mu:\mathbb{R}^{d}\rightarrow[0,+\infty)$
such that

(1.2) $\displaystyle\forall
x\in\mathbb{R}^{d}\qquad\lim_{\begin{subarray}{c}n\rightarrow\infty\\\
0\leftrightarrow\lfloor
nx\rfloor\end{subarray}}\frac{\mathcal{D}^{\mathcal{G}_{p}}(0,nx)}{n}=\mu(x)\qquad\text{a.s.}$

The function $\mu$ is the so-called time constant. We remark that they obtain
(1.2) in a more general context of stationary integrable ergodic fields. It is
well-known that $\mu$ is a norm, in particular convex and continuous.

##### Upper tail large deviations

Our main goal is to study the upper tail large deviation event for
$x\in\mathbb{R}^{d}$:

(1.3)
$\displaystyle\\{\mu(x)(1+\xi)n<\mathcal{D}^{\mathcal{G}_{p}}(0,nx)<\infty\\},\quad\xi>0.$

We here consider the case $x=\mathbf{e}_{1}=(1,0,\dots,0)$. To better
understand the decay rate of its probability, let us give an example
configuration. Picture the configuration where all the horizontal edges along
the segment from $-\xi n\mathbf{e}_{1}$ to $0$ are open, and all edges sharing
exactly one endpoint with that segment are closed except for the horizontal
edge $\langle-(\xi n+1)\mathbf{e}_{1},-\xi n\mathbf{e}_{1}\rangle$ (see Figure
1). Moreover, we further assume that $-\xi n\mathbf{e}_{1}$ is connected to
$n\mathbf{e}_{1}$. With this configuration, the geodesic is forced to go to
$-\xi n\mathbf{e}_{1}$, hence the upper tail large deviation event occurs with
high probability. Note that the probability of such configurations decays
exponentially with respect to $n$.

Figure 1. Construction of a cut-point at $-\xi n\mathbf{e}_{1}$. The red edges
are closed; the black edges are open.

Garet and Marchand proved in [21] that the probability of the event (1.3)
decays exponentially with respect to $n$. We aim to prove the existence of the
so-called rate function $J_{x}$ such that

$\mathbb{P}(\mu(x)(1+\xi)n<\mathcal{D}^{\mathcal{G}_{p}}(0,nx)<\infty)=e^{-nJ_{x}(\xi)+o(n)},$

and give an explicit description of the rate function. In particular, we prove
that on the upper tail large deviation event, typical scenarios are similar to
the example we gave above (a cut-point where all the geodesics are forced to
pass). For the study of cut-point, we should not only record its position but
also the time when the geodesics pass through the cut-point (that is the graph
distance from $0$ to this cut-point, see (1.4) for a formal definition). We
call this a space-time cut-point. Space-time cut-points play a central role in
studying upper tail large deviation for chemical distance as we will see
later. In the example above, we created a cut-point in a non-optimal location
so that the geodesics make a detour and even go in the opposite direction
$-\mathbf{e}_{1}$. However, there may exist other scenarios where cut-points
are located in optimal directions, but the upper tail large deviation event
still occurs. For instance, picture a space-time cut-point located around $0$
whose chemical distance to $0$ is larger than $\xi n$ (see Figure 2). Hence,
we need to compare the probabilities of all the space-time cut-points leading
to the upper tail large deviations and determine the best scenario among them.

Figure 2. Construction of a cut-point $w_{n}$ close to $0$. The path
corresponds to a geodesic from $0$ to $n\mathbf{e}_{1}$.

###  1.2. Main results

We assume $p>p_{c}(d)$. Let us give a formal definition of space-time cut-
point. For $t\geq 1$ and $z\in\mathbb{R}^{d}$, define

$\mathrm{B}_{t}(z):=\\{x\in\mathbb{Z}^{d}:\mathcal{D}^{\mathcal{G}_{p}}(z,x)\leq
t\\}.$

For short, we write $\mathrm{B}_{t}$ instead of $\mathrm{B}_{t}(0)$. Let $s>0$
and $x\in\mathbb{R}^{d}$. We denote $\Lambda_{s}(x):=x+[-s,s]^{d}$. We define
$\alpha_{d}:=1-\frac{1}{6d}<1$. The value of $\alpha_{d}$ is due to technical
reasons, and its precise value is not important. We define
$\mathcal{A}_{s,x}(n)$ the event that there is a cut-point located around $nx$
whose chemical distance from $0$ is larger than or equal to $sn$:

(1.4) $\mathcal{A}_{s,x}(n):=\left\\{\exists s_{n}\geq sn\quad\exists
x_{n}\in\Lambda_{n^{\alpha_{d}}}(nx);\,{\rm B}_{s_{n}}\setminus{\rm
B}_{s_{n}-1}=\\{x_{n}\\}\right\\}.$

The most technical and innovative part of this paper is the following.

######  Theorem 1.1 (Rate function for space-time cut-point).

There exists a function $I:[0,\infty)\times\mathbb{R}^{d}\mapsto[0,+\infty)$
such that for all $(s,x)\in[0,+\infty)\times\mathbb{R}^{d}$,

$\lim_{n\rightarrow\infty}-\frac{1}{n}\log{\mathbb{P}(\mathcal{A}_{s,x}(n))}=I(s,x),$

where the convergence is locally uniform on $[0,+\infty)\times\mathbb{R}^{d}$.
For any $s>0$ and $x\in\mathbb{R}^{d}$, $I(s,x)>0$. Moreover, the function $I$
is convex and homogeneous (i.e. $I(\lambda s,\lambda x)=\lambda I(s,x)$ for
$\lambda\geq 0$).

We define the rate function $J_{x}$ for $x\in\mathbb{Z}^{d}$ as

(1.5) $J_{x}(\xi):=\inf\left\\{I(s,y):\leavevmode\nobreak\
y\in\mathbb{R}^{d},\,s\geq 0;\,s+\mu(y-x)\geq(1+\xi)\mu(x)\right\\}.$

######  Proposition 1.2.

For any $\xi>0$, $J_{x}(\xi)>0$ and the function $\xi\mapsto J_{x}(\xi)$ is
non-decreasing and continuous.

The following is our main result.

######  Theorem 1.3.

For any $x\in\mathbb{R}^{d}\setminus\\{0\\}$, there exists
$\xi_{0}=\xi_{0}(x)>0$ such that for any $\xi\in(0,\xi_{0})$,

$\lim_{n\rightarrow\infty}\frac{1}{n}\log\mathbb{P}(\mu(x)(1+\xi)n<\mathcal{D}^{\mathcal{G}_{p}}(0,nx)<\infty)=-J_{x}(\xi).$

An explicit expression of $\xi_{0}(x)$ is given in (4.2).

######  Remark 1.4.

We shall discuss the expression as defined in (1.5). Let us examine the point
$(s,y)\in[0,+\infty)\times\mathbb{R}^{d}$ such that

$s+\mu(y-x)\geq(1+\xi)\mu(x).$

Upon the occurrence of event $\mathcal{A}_{s,y}(n)$, a geodesic connecting the
origin and $n\mathbf{e}_{1}$ arrives later than time $sn$ at a cut-point in
proximity to $ny$. According to (1.2), the chemical distance from this cut-
point to $nx$ is typically about $n\mu(y-x)$. Consequently, it is not possible
for the geodesic to arrive at $nx$ earlier than $(1+\xi)\mu(x)n$.
Consequently, $J_{x}$ can be interpreted as the minimum cost across all cut-
point situations that result in the occurrence of a large deviation event in
the upper tail.

######  Remark 1.5.

In this remark, we provide a rationale for our exclusive focus on the regime
$\xi<\xi_{0}$ in the main result. We will demonstrate that the large deviation
event arises from the presence of two cut-points, each connected to one of the
endpoints, as detailed in Proposition 5.3. When $\xi\leq\xi_{0}$, the balls
associated with these cut-points are in close enough proximity to their
corresponding endpoints, ensuring that they do not intersect. Consequently,
the scenario can be simplified to a single cut-point. However, when $\xi$ is
large, the two cut-points may leverage shared closed edges to increase the
probability relative to a single cut-point. Hence, this larger $\xi$ regime
might necessitate further studies of more complicated interacting cut-points.

###  1.3. 2D v.s. 3D and higher

We expect that the upper tail large deviations behave differently in $d=2$ and
$d\geq 3$. For $d\geq 3$, only space-time cut-points can create an upper tail
large deviation event. Whereas, for $d=2$, we need to consider other scenarios
unrelated to space-time cut-points, which create the event. One possible
scenario is to add a close wall in the middle that forces the geodesic to
deviate from the horizontal line (see figure 3). In $d=2$, the cost of adding
the wall is of order $\exp(cn)$, so it is of the same order as the speed of
large deviations. However, in dimensions $d\geq 3$, the cost becomes of order
$\exp(cn^{d-1})$. According to this picture, we have the following theorem,
which will be proved in Section 2.4.

Figure 3. Construction of a closed wall. The edges in red are closed and force
the geodesic to deviate a lot from the straight line.

######  Theorem 1.6.

For any $\varepsilon>0$ and $\xi>0$, there exists $c>0$ such that for $n$
large enough,

$\displaystyle\mathbb{P}(\mathcal{D}^{\mathcal{G}_{p}}(\Lambda_{\varepsilon
n}(0),\Lambda_{\varepsilon
n}(n\mathbf{e}_{1}))>(1+\xi)\mu(\mathbf{e}_{1})n)\leq e^{-cn^{d-1}},$

where $\mathcal{D}^{\mathcal{G}_{p}}(A,B):=\inf_{x\in A,y\in
B}\mathcal{D}^{\mathcal{G}_{p}}(x,y)$ for $A,B\subset\mathbb{Z}^{d}$.

The aforementioned theorem emphasizes the distinction between dimension $2$
and dimensions $d\geq 3$. In dimensions $d\geq 3$, the upper tail large
deviation event for the chemical distance between two macroscopic boxes decays
significantly more rapidly than the upper tail large deviation event for the
chemical distance between two points. This implies that if the balls
$\mathrm{B}_{\varepsilon n}(0)$ and $\mathrm{B}_{\varepsilon
n}(n\mathbf{e}_{1})$ are both substantially large (as would be the case in a
typical supercritical percolation), the probability of the upper tail large
deviation event decays much faster than $e^{-cn}$. Consequently, during the
upper tail large deviation event, the balls at the endpoints do not exhibit
typical behavior and are noticeably small. In this context, we describe the
upper tail large deviation as local, since it only pertains to edges in close
proximity to the endpoints. However, in $d=2$, the balls may maintain their
usual volumetric properties, and both global scenarios (e.g. closed walls
mentioned earlier) and local scenarios (e.g. space-time cut-points) may
coexist, necessitating the identification of the most advantageous scenario
among them.

###  1.4. Link between First-passage percolation and chemical distance

The model of First-passage percolation (FPP) was first introduced by
Hammersley and Welsh [24] as a model for the spread of a fluid in a porous
medium. In the model, we assign a non-negative random variable $\tau_{e}$ to
each edge $e$ such that the family $(\tau_{e})_{e\in\mathbb{E}^{d}}$ is
independent and identically distributed with a distribution $F$. The random
variable $\tau_{e}$ may be interpreted as the time needed for the fluid to
cross the edge $e$. For any pair of vertices $x,y\in\mathbb{Z}^{d}$, the
random variable $T(x,y)$, called the first passage time, is the shortest time
to go from $x$ to $y$. We are interested in the asymptotic behavior of the
quantity $T(0,x)$ when $\|x\|$ goes to infinity. Under some integrability
conditions on $F$, it is proved that

(1.6)
$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{n}T(0,nx)=\inf_{n\in\mathbb{N}}\frac{1}{n}\mathbb{E}T(0,nx):=\mu_{F}(x)\qquad\text{a.s.},$

where $\mu_{F}$ is a semi-norm associated to the distribution $F$ called the
time constant for FPP. Indeed, Cox and Durrett [13] proved (1.6) under
necessary and sufficient integrability conditions on the distribution $F$ in
dimension $2$; Kesten [27] extended the result to dimensions $d\geq 2$, and
proved that $\mu_{F}$ is exactly a norm if and only if
$\mathbb{P}(\tau_{e}=0)<p_{c}(d)$. See [3] for more detailed backgrounds on
FPP.

We here mention an important correspondence between the chemical distance in
percolation and FPP. Consider the bond percolation $\mathcal{G}_{p}$
introduced in Section 1.1. Let us couple the bond percolation with the time
configuration $(\tau_{e})_{e\in\mathbb{E}^{d}}$ in the following way:

(1.7) $\displaystyle\tau_{e}:=\begin{cases}1&\text{ if $e$ is open,}\\\
\infty&\text{ if $e$ is closed.}\end{cases}$

In this setting, the infinite cluster made of the edges with passage time $1$
corresponds to the infinite cluster $\mathscr{C}_{\infty}$. For
$x,y\in\mathbb{Z}^{d}$, we have

$T(x,y):=\inf_{\gamma:x\leftrightarrow
y}\sum_{e\in\gamma}\tau_{e}=\inf_{\gamma:x\leftrightarrow
y\subset\mathcal{G}_{p}}|\gamma|=\mathcal{D}^{\mathcal{G}_{p}}(x,y),$

where the first infimum is taken over all paths from $x$ to $y$ in
$\mathbb{Z}^{d}$; the second infimum is taken over all open paths from $x$ to
$y$. Hence, the first passage time for the distribution
$G_{p}=p\delta_{1}+(1-p)\delta_{\infty}$ is the same as the chemical distance
in percolation.

###  1.5. Related work

In this section, we discuss related work on the large deviations for chemical
distance.

####  1.5.1. Large deviations for the chemical distance in supercritical
percolation

The first upper tail large deviation bound on the chemical distance for
supercritical percolation was obtained by Antal and Pisztora [2]. They
obtained the correct speed in the regime of large $\xi$: for every
$p>p_{c}(d)$, there exists $C\geq 1$ such that

(1.8)
$\limsup_{\|x\|_{1}\rightarrow\infty}\frac{1}{\|x\|_{1}}\log\mathbb{P}\left(C\|x\|_{1}\leq\mathcal{D}^{\mathcal{G}_{p}}(0,x)<\infty\right)<0\,.$

######  Remark 1.7.

Antal and Pisztora actually proved a stronger claim [2, (4.9)]: there exists
$C\geq 1$ and $c>0$ such that

(1.9) $\forall x\in\mathbb{Z}^{d}\quad\forall l\geq
C\|x\|_{1}\quad\mathbb{P}\left(l\leq\mathcal{D}^{\mathcal{G}_{p}}(0,x)<\infty\right)\leq\exp(-cl).$

We use this claim several times in this paper.

The result was later extended to the entire regime of $\xi$ by Garet and
Marchand in [21]: for every $p>p_{c}(d)$ and $\xi>0$,

(1.10)
$\limsup_{\|x\|_{1}\rightarrow\infty}\frac{1}{\|x\|_{1}}\log\mathbb{P}\left((1+\xi)\mu(x)\leq\mathcal{D}^{\mathcal{G}_{p}}(0,x)<\infty\right)<0\,.$

In the same paper, they also proved the existence of the rate function for
lower tail large deviation: for every $p>p_{c}(d)$, $\xi>0$, and
$x\in\mathbb{R}^{d}\setminus\\{0\\}$, the following limit exists and is
negative or $-\infty$:

(1.11)
$\lim_{n\to\infty}\frac{1}{n}\log{\mathbb{P}(\mathcal{D}^{\mathcal{G}_{p}}(0,nx)<\mu(nx)(1-\xi))}.$

####  1.5.2.  Upper tail large deviations in FPP with compactly supported
distributions

Basu, Ganguly, and Sly [5] established the existence of a rate function for
upper tail large deviation when the distribution $F$ is both compactly
supported and possesses a continuous density. It is important to note the
contrast between the behavior of the upper tail large deviation for the
chemical distance and that for the FPP with a compactly supported
distribution. In the first scenario, the speed of large deviation exhibits a
linear order. Conversely, in the second scenario, increasing the first passage
time necessitates the involvement of the entire environment, resulting in a
volumetric speed, i.e. of order $n^{d}$. The foundation of the proof in [5]
lies in the presence of an underlying limiting metric accountable for the
upper tail large deviation. Although the authors did not formalize this
limiting metric, it enabled them to derive an involved subadditivity for the
upper tail large deviation event. Dembin and Théret [17] later formalized the
concept of the limiting environment for maximal flow in FPP in order to
demonstrate the upper tail large deviation principle.

####  1.5.3.  Upper tail large deviations in FPP with distributions under tail
estimates

Our approach was initially inspired by the arguments in [11]. We briefly
explain the main results and the sketch of the proof in [11] to discuss
obstacles when adapting the strategies there to chemical distance. Cosco and
Nakajima [11] considered the FPP on $\mathbb{Z}^{d}$ with distributions
satisfying

$\mathbb{P}(\tau_{e}\geq t)\asymp e^{-\alpha t^{r}}\text{ as $t\to\infty$ with
some constants }r\in(0,d],\,\alpha>0.$

They derived a specific rate function for upper tail large deviations, known
as the discrete p-capacity. In order to demonstrate this, they considered
numerous slabs and found a good slab among them where the first passage time
between any pair of points $x,y$ inside the slab is approximately equal to the
time constant $\mu_{F}(y-x)$. This suggests that on the upper tail large
deviation event, the first passage time to join one of the endpoints and this
good slab is abnormally large. This enables us to replace the large deviation
event $\\{T(0,n\mathbf{e}_{1})>\mu_{F}(\mathbf{e}_{1})(1+\xi)n\\}$ with a
local event where there are many high weights around the endpoints. To
calculate the rate function, they explicitly estimated the probability of the
local event, invoking the Laplace principle.

Nonetheless, these arguments are inapplicable to the current model. Contrary
to FPP, there are closed edges that a geodesic is unable to traverse. In FPP,
when a geodesic reaches a suitable slab, it is understood that the geodesic
moves at a typical speed. In this study, however, arriving at an appropriate
slab does not necessarily guarantee that a geodesic will maintain a typical
speed within the slab. In fact, even when the geodesic crosses a slab that
possesses beneficial connectivity within its infinite cluster, if the geodesic
avoids this infinite cluster, the slab’s advantageous connectivity cannot be
utilized.

Furthermore, the high weights around the endpoints in FPP are substituted by
the presence of space-time cut-points in our situation. Specifically, we
cannot employ the Laplace principle, so we resort to the rate function for
space-time cut-point instead.

###  1.6. Sketch of the proofs

####  1.6.1. Sketch of the proof of Theorem 1.6

The proof is inspired by the arguments in [11]. Let $N$ be a fixed large
integer. We consider disjoint $\mathbb{Z}^{2}$-slabs of order $n^{d-2}$ and of
thickness $N$ that intersect $\Lambda_{\varepsilon n}(0),\Lambda_{\varepsilon
n}(n\mathbf{e}_{1})$. Thanks to a renormalization argument, we can prove that
for each slab the probability that the time in the slab between the two boxes
$\Lambda_{\varepsilon n}(0),\Lambda_{\varepsilon n}(n\mathbf{e}_{1})$ is
larger than $(1+\xi)\mu(\mathbf{e}_{1})$ decays exponentially fast in $n$.
Thus, if $\mathcal{D}^{\mathcal{G}_{p}}(\Lambda_{\varepsilon
n}(0),\Lambda_{\varepsilon n}(n\mathbf{e}_{1}))>(1+\xi)\mu(\mathbf{e}_{1})n$,
then the time between $\Lambda_{\varepsilon n}(0),\Lambda_{\varepsilon
n}(n\mathbf{e}_{1})$ on each slab is abnormally large. Since we consider
disjoint slabs of order $n^{d-2}$, it follows that the decay of this event is
of order $\exp(-cn^{d-1})$.

####  1.6.2.  Sketch of the proof of Theorem 1.1

Consider two configurations in $\mathcal{A}_{s,x}(n)$ and
$nx+\mathcal{A}_{s,x}(m)$, where

(1.12) $\displaystyle y+\mathcal{A}_{s,x}(m):=\left\\{\exists t\geq
sm\quad\exists x_{m}\in\Lambda_{m^{\alpha_{d}}}(y+mx):\,{\rm
B}_{t}(y)\setminus{\rm B}_{s-1}(y)=\\{x_{m}\\}\right\\}.$

Let $s_{n}\geq sn,s_{m}\geq sm,$ and $w_{n},z_{m}\in\mathbb{Z}^{d}$ such that
$\mathrm{B}_{s_{n}}\setminus\mathrm{B}_{s_{n}-1}=\\{w_{n}\\}$ and
$\mathrm{B}_{s_{m}}(nx)\setminus\mathrm{B}_{s_{m}-1}(nx)=\\{z_{m}\\}$. The two
balls come from different time configurations. To build a configuration in
$\mathcal{A}_{s,x}(n+m)$, we want to join $w_{n}$ to $xn$ without changing the
metric structure so that $z_{m}$ is a still cut-point and the chemical
distance between $0$ and $z_{m}$ is larger than $s(n+m)$. A first problem is
when the two balls intersect. We can circumvent this problem by finding an
appropriate translation of $\mathrm{B}_{s_{m}}(nx)$ not far from its original
position. Such a translation can be found using the ball size control.

The main problem comes from the situations where $w_{n}$ is not connected to
infinity in $\mathbb{Z}^{d}\setminus\mathrm{B}_{s_{n}}$, or where it is
connected to infinity but a path to join $w_{n}$ and $nx$ wiggles a lot. These
scenarios may seem pathological, but we believe there are no easy ways to
discard these scenarios. Indeed, when $x=0$, we can consider the events where
the ball $\mathrm{B}_{s_{n}}$ fills most of the box $\Lambda_{n^{1/d}}(0)$
(e.g. a space-filling spiral) such that $0$ is not connected to $\infty$ in
$\mathbb{Z}^{d}\setminus\mathrm{B}_{s_{n}}$ or $0$ is connected to
$\partial\Lambda_{n^{1/d}}(0)$ but with a very large graph distance (of order
$n$). In the first scenario, it is not possible to open edges to create a
connection between the cut-point $w_{n}$ of the first ball to the origin $nx$
of the second ball without modifying the second ball structure. In the second
scenario, the cost to join $w_{n}$ and $nx$ is at least of order exponential
in $n$ (in particular, it does not vanish in the limit).

To solve this issue, we modify the ball structure to create a free line, that
is a line intersecting $\mathrm{B}_{s_{n}}$ only at $w_{n}$ (see Lemma 3.7).
The main difficulty of this step is to only slightly modify the ball structure
so that the new space-time cut-point is still close to the original one. In
particular, the metric structure of the ball has to be preserved as much as
possible to ensure that the event $\mathcal{A}_{s,x}(n)$ still occurs for the
modified environment. The technical part in this procedure is that the cost of
modifying the structure of the ball has to vanish in the limit. Creating the
free lines at $w_{n}$ and $nx$ gives more freedom to build a connection
between these two vertices.

####  1.6.3. Sketch of the proof of Theorem 1.6

Consider a couple $(s,x)\in\mathbb{R}_{+}\times\mathbb{R}^{d}$ such that

(1.13) $s+\mu(x-\mathbf{e}_{1})>(1+\xi)\mu(\mathbf{e}_{1}).$

On the event $\mathcal{A}_{s,x}(n)$, any geodesic between $0$ and
$n\mathbf{e}_{1}$ arrives at a cut-point located close to $xn$ at time larger
than $sn$. Moreover, the time needed to go from $nx$ to $n\mathbf{e}_{1}$ is
typically at least $n\mu(x-\mathbf{e}_{1})>((1+\xi)\mu(\mathbf{e}_{1})-s)n$
thanks to (1.2) and (1.13). Hence, the event
$\\{\mathcal{D}^{\mathcal{G}_{p}}(0,n\mathbf{e}_{1})>\mu(\mathbf{e}_{1})(1+\xi)n\\}$
is very likely to occur if the event $\mathcal{A}_{s,x}(n)$ occurs. Using
Theorem 1.1, it follows that

$-\liminf_{n\rightarrow\infty}\frac{1}{n}\log\mathbb{P}(\mu(\mathbf{e}_{1})(1+\xi)n<\mathcal{D}^{\mathcal{G}_{p}}(0,n\mathbf{e}_{1})<\infty)\geq
I(s,x).$

Taking the infimum over $(s,x)$ satisfying (1.13) on the right hand side, we
get

(1.14)
$\begin{split}-\liminf_{n\rightarrow\infty}\frac{1}{n}\log\mathbb{P}&(\mu(\mathbf{e}_{1})(1+\xi)n<\mathcal{D}^{\mathcal{G}_{p}}(0,n\mathbf{e}_{1})<\infty)\geq
J(\xi).\end{split}$

To get the converse inequality, we have to prove that if the upper tail large
deviation event occurs, then there exists a space-time cut-point. The idea
goes as follows. Consider first the balls $\mathrm{B}_{s}(0)$,
$\mathrm{B}_{t}(n\mathbf{e}_{1})$ at both endpoints at the moments $s$ and $t$
when their respective size reaches $n^{7/4}$ (See (5.8) for formal
definitions). Since the balls are large enough, it is very likely to find
$x\in\mathrm{B}_{s}(0)$ and $y\in\mathrm{B}_{t}(n\mathbf{e}_{1})$ such that
$\mathcal{D}^{\mathcal{G}_{p}}(x,y)\leq\mu(x-y)+o(n)$. Moreover, since the
balls are of size negligible compared to $n^{2}$, we can make $x$ and $y$ cut-
points (see Lemma 3.6). Besides, since we are on the upper tail large
deviation event, these cut-points must satisfy

$\mathcal{D}^{\mathcal{G}_{p}}(0,x)+\mu(x-y)+\mathcal{D}^{\mathcal{G}_{p}}(y,n\mathbf{e}_{1})+o(n)\geq\mu(\mathbf{e}_{1})(1+\xi)n\,.$

Hence, one expects from the Laplace principle and the continuity of $I$
(Theorem 1.1) that

$\begin{split}\limsup_{n\rightarrow\infty}\frac{1}{n}\log\mathbb{P}&(\mu(\mathbf{e}_{1})(1+\xi)n<\mathcal{D}^{\mathcal{G}_{p}}(0,n\mathbf{e}_{1})<\infty)\\\
&\leq\limsup_{n\rightarrow\infty}\frac{1}{n}\log\mathbb{P}\left(\bigcup_{\begin{subarray}{c}s,t\geq
0,\,x,y\in\mathbb{R}^{d}:\\\
s+t+\mu(x-y-\mathbf{e}_{1})\geq(1+\xi)\mu(\mathbf{e}_{1})\end{subarray}}\mathcal{A}_{s,x}(n)\cap(n\mathbf{e}_{1}+\mathcal{A}_{t,y}(n))\right)\\\
&\leq\sup_{\begin{subarray}{c}s,t\geq 0,\,x,y\in\mathbb{R}^{d}:\\\
s+t+\mu(x-y-\mathbf{e}_{1})\geq(1+\xi)\mu(\mathbf{e}_{1})\end{subarray}}\limsup_{n\rightarrow\infty}\frac{1}{n}\log\mathbb{P}\left(\mathcal{A}_{s,x}(n)\cap(n\mathbf{e}_{1}+\mathcal{A}_{t,y}(n))\right).\end{split}$

When $\xi\leq\xi_{0}$, the two space-time cut-points at $0$ and
$n\mathbf{e}_{1}$ occur disjointly (see the proof of Propostion 5.6). Using
the property of $I$ (convexity and homogeneity), we argue that the cost of
having a unique cut-point at one endpoint $\mathcal{A}_{s+t,x-y}(n)$ is
smaller than the cost of having two cut-points one at each endpoint of
$\mathcal{A}_{s,x}(n)\cap(n\mathbf{e}_{1}+\mathcal{A}_{t,y}(n))$. Therefore,
(1.6.3) is further bounded from above by

(1.15)
$\begin{split}\limsup_{n\rightarrow\infty}\frac{1}{n}\log\mathbb{P}\left(\bigcup_{\begin{subarray}{c}s\geq
0,\,x\in\mathbb{R}^{d}:\\\
s+\mu(x-\mathbf{e}_{1})\geq(1+\xi)\mu(\mathbf{e}_{1})\end{subarray}}\mathcal{A}_{s,x}(n)\right)\leq-J(\xi).\end{split}$

The result follows combining (1.14) and (1.15).

###  1.7. Notations and terminologies

In this section, we collect useful notations, terminologies, and claims. For
$m\geq 1$, set $[m]:=\\{1,\cdots,m\\}$.

Distances. We denote by $\|\cdot\|_{1},\|\cdot\|_{\infty},\|\cdot\|_{2}$
respectively the $\ell_{1}$, $\ell_{\infty}$ and $\ell_{2}$ norms. For
$A,B\subset\mathbb{R}^{d}$ and $i\in\\{1,2,\infty\\}$, we denote

$\mathrm{d}_{i}(A,B):=\inf_{x\in A,y\in B}\|x-y\|_{i}\,.$

Diameter. For a set $S\subset\mathbb{Z}^{d}$ and $i\in[d]$, we define

$\operatorname{Diam}_{i}(S):=\max_{x,y\in
S}|x_{i}-y_{i}|\qquad\text{and}\qquad\operatorname{Diam}(S)=\max_{1\leq i\leq
d}\operatorname{Diam}_{i}(S).$

$\mathbb{Z}^{d}$-path. We say that a sequence
$\gamma=(v_{0},\dots,v_{n})\in\mathbb{Z}^{d}$ is a $\mathbb{Z}^{d}$-path if
for all $i\in[n]$, $\|v_{i}-v_{i-1}\|_{1}=1$. Note that a
$\mathbb{Z}^{d}$-path may also be seen as a set of edges $\\{\langle
v_{i},v_{i+1}\rangle,\,0\leq i\leq n-1\\}$. In what follows, a path will
implicitly mean a $\mathbb{Z}^{d}$-path.

Cluster. Given $x,y\in\mathbb{Z}^{d}$ and $A\subset\mathbb{Z}^{d}$, if there
exists a $\mathbb{Z}^{d}$-path $\gamma\subset A$ from $x$ to $y$ in
$\mathcal{G}_{p}$, then we write $x\overset{A}{\leftrightarrow}y.$ When
$A=\mathbb{Z}^{d}$, we simply write $x\leftrightarrow y$. Given
$x\in\mathbb{Z}^{d}$, we define the open cluster containing $x$ as

(1.16)
$\displaystyle\mathcal{C}(x):=\\{y\in\mathbb{Z}^{d}:\leavevmode\nobreak\
x\leftrightarrow y\\}.$

Paths concatenation. Given two paths $\gamma^{1},\gamma^{2}$ such that the end
point of $\gamma^{1}$ is the same as the starting point of $\gamma^{2}$, we
denote by $\gamma^{1}\oplus\gamma^{2}$ the concatenation of $\gamma^{1}$ and
$\gamma^{2}$. Inductively, we can define the concatenation of $n$ paths
$\gamma^{1}\oplus\cdots\oplus\gamma^{n}$.

Interior and exterior boundary. For a set $\Gamma\subset\mathbb{Z}^{d}$, we
denote by $\mathrm{Int}(\Gamma)$ the set of points in $\mathbb{Z}^{d}$
enclosed by $\Gamma$:

$\mathrm{Int}(\Gamma):=\\{x\in\mathbb{Z}^{d}\setminus\Gamma:\text{ $x$ is not
connected to $\infty$ in $\mathbb{Z}^{d}\setminus\Gamma$}\\}.$

We define the outer boundary of $\Gamma$ as

$\partial^{\rm ext}\Gamma:=\\{x\in\mathbb{Z}^{d}\setminus\Gamma:x\text{ is
connected to infinity in $\mathbb{Z}^{d}\setminus\Gamma$ and has a neighbour
in $\Gamma$}\\}.$

There exists a constant $\kappa_{d}>1$ such that for every connected set
$\Gamma\subset\mathbb{Z}^{d}$, we have the following discrete isoperimetric
inequality (see, for instance, [12, Theorem 1])

(1.17) $|\Gamma|\leq\kappa_{d}|\partial^{\rm ext}\Gamma|^{d/(d-1)}.$

$*$-connected and lattice animals. We say that $x$ and $y$ are $*$-neighbor if
$\|x-y\|_{\infty}=1$. We say that a sequence
$\gamma=(v_{0},\dots,v_{n})\subset\mathbb{Z}^{d}$ is a $*$-path from $x$ to
$y$ in $\Gamma\subset\mathbb{Z}^{d}$ if $v_{0}=x\in\Gamma$, $v_{n}=y$ and for
all $i\in[n]$, $v_{i}\in\Gamma$ and $\|v_{i}-v_{i-1}\|_{\infty}=1$. We say
that $x$ and $y$ are $*$-connected in $\Gamma$ if such a path exists. We say
that $\Gamma$ is $*$-connected if for any $x,y\in\Gamma$, $x$ and $y$ are
$*$-connected in $\Gamma$. We denote by $\textbf{Animal}_{\mathbf{i}}^{k}$ the
set of $*$-connected components of $\mathbb{Z}^{d}$ of size $k$ containing
$\mathbf{i}\in\mathbb{Z}^{d}$. We have (see for instance Grimmett [23, p85])

(1.18) $|\textbf{Animal}_{\mathbf{i}}^{k}|\leq 7^{dk}.$

Lines and hyperplanes. Given $w\in\mathbb{Z}^{d}$, $n\geq 1$ and $i\in[d]$, we
define

$\displaystyle\mathrm{L}_{i}(w):=\\{w+k\mathbf{e}_{i}:\leavevmode\nobreak\
k\in\mathbb{Z}\\},$
$\displaystyle\mathrm{L}_{i}^{n}(w):=\\{w+k\mathbf{e}_{i}:\leavevmode\nobreak\
|k|\leq n\\},$ $\displaystyle\mathrm{H}_{i}(w):=\\{w+\textstyle\sum_{j\neq
i}k_{j}\mathbf{e}_{j}:\leavevmode\nobreak\ k_{j}\in\mathbb{Z}\\}.$

Let $P_{i}$ be the projection on the hyperplane $\mathrm{H}_{i}(0)$.

###  1.8. Organisation of the paper

In Section 2, we introduce a renormalization procedure and prove Theorem 1.6.
In Section 3, we prove the existence of the rate function for space-time cut-
point (Theorem 1.1). Finally, in Section 5, we prove that on the upper tail
large deviation event, there exist space-time cut-points and deduce the
existence of the rate function for upper tail large deviation (Theorem 1.3).
As before, we always assume $p>p_{c}(d)$ in the rest of the paper.

Acknowledgement. The authors are grateful to FIM, for the hospitality during
the second author’s stay at ETH where the present work was completed. The
research of the first author has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
program (grant agreement No 851565). The second author is supported by JSPS
KAKENHI 22K20344.

##  2\. Renormalization

For simplicity, we write $\Lambda_{s}:=\Lambda_{s}(0)=[-s,s]^{d}$.

###  2.1. Preliminary on percolation

The following estimate controls the probability that the density of the
infinite cluster is atypically small.

######  Theorem 2.1.

[29, Theorem 2] Let $\varepsilon>0$. There exists $c>0$ such that for $n$
large enough

$\mathbb{P}\left(\frac{|\mathscr{C}_{\infty}\cap\Lambda_{n}(w)|}{|\Lambda_{n}(w)|}\leq\theta(p)-\varepsilon\right)\leq
e^{-cn^{d-1}},$

where $\theta(p):=\mathbb{P}(0\in\mathscr{C}_{\infty})$ corresponds to the
density of the infinite cluster.

The following theorem may be seen as a corollary of the previous theorem. It
controls the probability that the infinite cluster does not intersect a given
box.

######  Theorem 2.2 (Holes).

There exists $c>0$ such that for $n\in\mathbb{N}$ large enough,

$\mathbb{P}(\mathscr{C}_{\infty}\cap\Lambda_{n}=\emptyset)\leq e^{-cn^{d-1}}.$

The following theorem enables us to control the probability of the existence
of two large disjoint clusters (e.g. see [23, Lemma 7.104] for a reference).

######  Theorem 2.3 (Distinct clusters).

For any $\varepsilon>0$, there exists $c>0$ such that for $n\in\mathbb{N}$
large enough,

$\mathbb{P}(\exists\text{two disjoint open clusters of diameter at least
$\varepsilon n$ in $\Lambda_{n}$})\leq e^{-cn}.$

###  2.2. Macroscopic lattice

We define the $N$-box and the enlarged box for $\mathbf{k}\in\mathbb{Z}^{d}$:

$\underline{\Lambda}_{N}(\mathbf{k}):=[-N,N)^{d}\cap\mathbb{Z}^{d}+2\mathbf{k}N,\quad\underline{\Lambda}_{N}^{\prime}(\mathbf{k}):=\bigcup_{\|\mathbf{i}-\mathbf{k}\|_{\infty}\leq
1}\underline{\Lambda}_{N}(\mathbf{i}).$

Then, we have the decomposition of $\mathbb{Z}^{d}$ as
$\mathbb{Z}^{d}=\bigsqcup_{\mathbf{i}\in\mathbb{Z}^{d}}\underline{\Lambda}_{N}(\mathbf{i})$
where $\sqcup$ denotes the disjoint union. Thus, for any $x\in\mathbb{Z}^{d}$,
there exists a unique $\mathbf{i}\in\mathbb{Z}^{d}$ such that
$x\in\underline{\Lambda}_{N}(\mathbf{i})$. With abuse of notation, we use the
notation $\underline{\Lambda}_{N}(x)$ for the box
$\underline{\Lambda}_{N}(\mathbf{i})$ containing $x$. The sites corresponding
to the boxes are the so-called macroscopic lattice of sidelength $N$; whereas
the standard vertices in $\mathbb{Z}^{d}$ correspond to the microcopic
lattice.

######  Definition 2.4.

Let $\varepsilon\in(0,{\frac{1}{2}})$ and $N\in\mathbb{N}$. We say that a site
$\mathbf{i}\in\mathbb{Z}^{d}$ is $\varepsilon$-good if the following hold:

  1. (1)

There exists a unique open cluster, denoted by $\mathscr{C}(\mathbf{i})$, in
$\underline{\Lambda}^{\prime}_{N}(\mathbf{i})$ with diameter at least
$\frac{N}{2}$;

  2. (2)

This cluster $\mathscr{C}(\mathbf{i})$ intersects with every subbox in
$\underline{\Lambda}^{\prime}_{N}(\mathbf{i})$ of sidelength $\varepsilon N$;

  3. (3)

For all
$x,y\in\underline{\Lambda}^{\prime}_{N}(\mathbf{i})\cap\mathscr{C}(\mathbf{i})$,
we have

$\mathcal{D}^{\mathcal{G}_{p}}(x,y)\leq\mu(x-y)+\varepsilon N.$

Otherwise, if at least one of the three conditions does not hold, we say that
the site $\mathbf{i}$ is $\varepsilon$-bad.

######  Remark 2.5.

Thanks to our definition of $\varepsilon$-good, for any $*$-neighbours
$\mathbf{i}$ and $\mathbf{j}$ that are $\varepsilon$-good, the clusters
$\mathscr{C}(\mathbf{i})$ and $\mathscr{C}(\mathbf{j})$ are connected in
$\underline{\Lambda}^{\prime}_{N}(\mathbf{j})$. Indeed, thanks to Property
(2), $\mathscr{C}(\mathbf{i})\cap\underline{\Lambda}^{\prime}_{N}(\mathbf{j})$
has a connected component of diameter at least $N/2$ and the claim follows
from Property (1) for the site $\mathbf{j}$.

When there is no confusion, we say good instead of $\varepsilon$-good. The
states of the boxes have a short-range dependence. To see this, set

(2.1) $\rho:=\lfloor 10d\mu(\mathbf{e}_{1})\rfloor.$

Thanks to the definition of $\varepsilon$-good, for
$\mathbf{i,j}\in\mathbb{Z}^{d}$ with
$\|\mathbf{i}-\mathbf{j}\|_{\infty}\geq\rho$, the states of the sites
$\mathbf{i}$ and $\mathbf{j}$ are independent. Finally, set

$\underline{\Lambda}_{N}^{\rho}(\mathbf{i}):=\bigcup_{\mathbf{j}:\|\mathbf{i}-\mathbf{j}\|_{\infty}\leq\rho}\underline{\Lambda}_{N}(\mathbf{j}).$

######  Proposition 2.6.

Let $\varepsilon>0$. For $p>p_{c}(d)$, there exists $c>0$ such that for
$N\in\mathbb{N}$ large enough,

$\mathbb{P}(\underline{\Lambda}_{N}({\bf 0})\text{ is $\varepsilon$-bad})\leq
e^{-cN}\,.$

###### Proof.

Thanks to Theorems 2.2,
$\mathbb{P}(\mathscr{C}_{\infty}\cap\underline{\Lambda}_{N}(\mathbf{0})=\emptyset)\leq
e^{-cN}$. Hence, we suppose
$\mathscr{C}_{\infty}\cap\underline{\Lambda}_{N}(\mathbf{0})\neq\emptyset.$
Then, we can take a connected component $\mathscr{C}(\mathbf{0})$ in
$\mathscr{C}_{\infty}\cap\underline{\Lambda}^{\prime}_{N}(\mathbf{0})$ with
diameter at least $N/2$. Thanks to Theorem 2.3, this is a unique connected
component of diameter at least $N/2$ with probability at least $1-e^{-cN/2}$.
The result follows from the union bound with (1.9), (1.10), and Theorem 2.2. ∎

The following lemma controls the number of bad boxes.

######  Lemma 2.7.

Let $\varepsilon>0$ and $N=N_{n}:=\log^{2}n:=\lfloor(\log{n})^{2}\rfloor$. For
$n$ large enough,

(2.2)
$\displaystyle\mathbb{P}\left(\\#\\{\mathbf{i}\in\Lambda_{n^{2}}\cap\mathbb{Z}^{d}:\mathbf{i}\text{
is $\varepsilon$-bad}\\}>n\right)\leq e^{-n\log{n}}.$

###### Proof.

By the union bound, we have

$\mathbb{P}(\\#\\{\mathbf{i}\in\Lambda_{n^{2}}\cap\mathbb{Z}^{d}:\mathbf{i}\text{
is
$\varepsilon$-bad}\\}>n)\leq\sum_{I\subset\Lambda_{n^{2}}\cap\mathbb{Z}^{d}:|I|>n}\mathbb{P}(\forall\mathbf{i}\in
I\quad\mathbf{i}\text{ is $\varepsilon$-bad}).$

If $\|\mathbf{i}-\mathbf{j}\|_{\infty}\geq\rho$, then the states of
$\mathbf{i}$ and $\mathbf{j}$ are independent. It is easy to see (e.g. [10,
Lemma 4.3]) that there exists $J\subset I$ such that $|J|\geq|I|/(4\rho)^{d}$
and for all $\textbf{i}\neq\textbf{j}\in J$,
$\|\textbf{i}-\textbf{j}\|_{\infty}\geq\rho$. It follows from Proposition 2.6
that for $n$ large enough,

$\displaystyle\mathbb{P}(\\#\\{\mathbf{i}\in\Lambda_{n^{2}}\cap\mathbb{Z}^{d}:\mathbf{i}\text{
is
$\varepsilon$-bad}\\}>n)\leq\sum_{\begin{subarray}{c}I\subset\Lambda_{n^{2}}\cap\mathbb{Z}^{d}:\\\
|I|>n\end{subarray}}\exp\left(-c\frac{|I|}{(4\rho)^{d}}\log^{2}n\right)$ (2.3)
$\displaystyle\leq\sum_{k>n}(4n^{2})^{dk}\exp\left(-c\frac{k}{(4\rho)^{d}}\log^{2}n\right)\leq
e^{-n\log{n}}.$

∎

###  2.3. Construction of microscopic path from macroscopic path

Let $\varepsilon\in(0,1/2)$ and $N\in\mathbb{N}$. Consider the macroscopic
lattice of sidelength $N$. For $\mathbf{w}\in\mathbb{Z}^{d}$, denote by
$\mathfrak{C}(\mathbf{w})$ the $\varepsilon$-bad $\mathbb{Z}^{d}$-cluster of
$\mathbf{w}$ in the macroscopic lattice, that is, the set of all
$\varepsilon$-bad sites connected to $\mathbf{w}$ by a macroscopic
$\mathbb{Z}^{d}$-path of bad sites. We define
$\mathfrak{C}(\mathbf{w}):=\emptyset$ if $\mathbf{w}$ is $\varepsilon$-good.

######  Lemma 2.8.

Let $\delta,\varepsilon>0$. If $N$ is large enough, then for any set of
macroscopic sites $\Gamma$,

$\mathbb{P}\left(\left|\bigcup_{\mathbf{w}\in\Gamma}\mathfrak{C}(\mathbf{w})\right|\geq\delta|\Gamma|\right)\leq\exp(-\delta|\Gamma|/2).$

###### Proof.

Let $(\mathfrak{C}_{i})_{i\geq 1}$ be a family of independent cluster
following the law of $\mathfrak{C}(\mathbf{0})$. We have

(2.4)
$\mathbb{P}\left(\left|\bigcup_{\mathbf{w}\in\Gamma}\mathfrak{C}(\mathbf{w})\right|\geq\delta|\Gamma|\right)\leq\mathbb{P}\left(\sum_{i=1}^{|\Gamma|}|\mathfrak{C}_{i}|\geq\delta|\Gamma|\right)\leq\mathbb{E}(\exp(|\mathfrak{C}(\mathbf{0})|-\delta))^{|\Gamma|},$

where we reference the proof of [22, Lemma 3.6] for the first inequality,
while the second inequality employs the exponential Markov inequality. For $N$
large enough, using the same arguments as in (2.2), by (1.18),

(2.5) $\mathbb{P}(|\mathfrak{C}(\mathbf{0})|\geq k)\leq\sum_{\ell\geq
k}\sum_{A\in\textbf{Animal}_{\textbf{0}}^{\ell}}\mathbb{P}(\forall\mathbf{i}\in
A\quad\mathbf{i}\text{ is $\varepsilon$-bad})\leq\sum_{\ell\geq
k}7^{d\ell}\,e^{-c\ell N/(4\rho)^{d}}\leq\exp(-ckN/(8\rho)^{d}).$

If $N$ is large enough depending on $\rho,c$ and $\delta$, then we have

$\mathbb{E}(\exp(|\mathfrak{C}(\textbf{0})|))\leq 1+\sum_{k\geq
1}e^{k}\mathbb{P}(|\mathfrak{C}(\textbf{0})|\geq k)\leq 1+\sum_{k\geq
1}\exp(k(1-cN/(8\rho)^{d}))\leq\exp(\delta/2).$

Combined with (2.4), this yields the claim. ∎

We will also need the following lemma that is an easy adaptation of [4, Lemma
3.2].

######  Lemma 2.9.

Let $\Gamma$ be a $*$-connected set of $\varepsilon$-good sites. Let
$x\in\mathscr{C}(\mathbf{j}){\cap\underline{\Lambda}_{N}(\mathbf{j})}$ and
$y\in\mathscr{C}(\mathbf{k}){\cap\underline{\Lambda}_{N}(\mathbf{k})}$ with
$\mathbf{j},\,\mathbf{k}\in\Gamma$. Then, we can find a microscopic, open path
joining $x$ and $y$ of length at most $2d\mu(\mathbf{e}_{1})N|\Gamma|$ in
$\cup_{\mathbf{i}\in\Gamma}\underline{\Lambda}_{N}^{\rho}(\mathbf{i})$.

We say that $S$ is a macroscopic slab, if there exist $\mathrm{I}\subset[d]$
with $|\mathrm{I}|\leq d-2$ and
$(k_{i})_{i\in\mathrm{I}}\subset\mathbb{Z}^{|I|}$ such that

$S=\\{(k_{1},k_{2},\dots,k_{d}):\leavevmode\nobreak\
(k_{i})_{i\in[d]\setminus\mathrm{I}}\subset\mathbb{Z}\\}.$

We denote by $\mathscr{C}_{\infty}(S)$ the infinite $\mathbb{Z}^{d}$-connected
component of $\varepsilon$-good sites contained in $S$. Let

$\underline{S}:=\bigcup_{\mathbf{i}\in
S}\underline{\Lambda}_{N}^{\rho}(\mathbf{i}).$

Given $A\subset\mathbb{Z}^{d}$, we define the chemical distance constrained on
$A$: for $B,C\subset A$,

$\displaystyle\mathcal{D}^{\mathcal{G}_{p}}_{A}(B,C):=\inf\\{|\gamma|:\leavevmode\nobreak\
x\in B,\,y\in C,\,\gamma:x\overset{A}{\leftrightarrow}y\\}.$

We write
$\mathcal{D}^{\mathcal{G}_{p}}_{A}(x,C):=\mathcal{D}^{\mathcal{G}_{p}}_{A}(\\{x\\},C)$,
$\mathcal{D}^{\mathcal{G}_{p}}_{A}(x,y):=\mathcal{D}^{\mathcal{G}_{p}}_{A}(\\{x\\},\\{y\\})$.

######  Proposition 2.10.

Let $\varepsilon>0$. There exist $c>0$ and $N\in\mathbb{N}$ such that for any
macroscopic slab $S$ and $\mathbf{x,y}\in S\cap\mathbb{Z}^{d}$ satisfying
$\mathbf{x}-\mathbf{y}\in\mathbb{Z}\mathbf{e}_{k}$ with some $k\in[d]$,

(2.6) $\mathbb{P}(\mathbf{x,y}\in\mathscr{C}_{\infty}(S),\leavevmode\nobreak\
\exists
x\in\mathscr{C}(\mathbf{x})\cap\underline{\Lambda}_{N}(\mathbf{x})\,\exists
y\in\mathscr{C}(\mathbf{y})\cap\underline{\Lambda}_{N}(\mathbf{y});\mathcal{D}^{\mathcal{G}_{p}}_{\underline{S}}(x,y)\geq(1+\varepsilon)\mu(x-y))\leq
e^{-c\|\mathbf{x}-\mathbf{y}\|_{1}}.$

###### Proof.

Without loss of generality, we assume $\mathbf{y}=\mathbf{x}+K\mathbf{e}_{1}$
with some $K\in\mathbb{N}$. Let
$\varepsilon_{0}:=\varepsilon/(2\varepsilon+4d\mu(\mathbf{e}_{1}))^{3}\in(0,1/2)$.
We take $N=N(\varepsilon_{0})\in\mathbb{N}$ large enough as in Lemma 2.8 with
$\varepsilon_{0}$ in place of $\varepsilon$ and $\delta$. We consider the
macroscopic lattice of sidelength $N$ with $\varepsilon_{0}$-good. Since the
probability in the claim is always less than $1$, we can assume
$\|\textbf{x}-\textbf{y}\|_{\infty}\geq 2N$ by taking $c=c(\varepsilon_{0},N)$
small enough. Let $\Gamma\subset\mathbb{Z}^{d}$ be the set of macroscopic
sites intersecting the line between $\mathbf{x}$ and $\mathbf{y}$, that is

$\Gamma:=\\{\mathbf{x}+i\mathbf{e_{1}}:i\in\\{0,\dots,K\\}\\}.$

Our aim is to prove

(2.7) $\\{\textbf{x,y}\in\mathscr{C}_{\infty}(S),\,\exists
x\in\mathscr{C}(\mathbf{x}),\,\exists
y\in\mathscr{C}(\mathbf{y}),\,\mathcal{D}^{\mathcal{G}_{p}}_{\underline{S}}(x,y)\geq(1+\varepsilon)\mu(x-y)\\}\subset\left\\{\left|\bigcup_{\mathbf{w}\in\Gamma}\mathfrak{C}(\mathbf{w})\right|\geq\varepsilon_{0}|\Gamma|\right\\}.$

Note that (2.7) implies the claim for sufficiently large
$N=N(\varepsilon_{0})$ and sufficiently small $c=c(\varepsilon_{0},N)$ since
by Lemma 2.8, the LHS of (2.6) is bounded from above by

$\exp{(-\varepsilon_{0}|\Gamma|/2)}\leq\exp{(-c\|\textbf{x}-\textbf{y}\|_{1})}.$

To prove the contrapositive of (2.7), we suppose
$\left|\bigcup_{\mathbf{w}\in\Gamma}\mathfrak{C}(\mathbf{w})\right|<\varepsilon_{0}|\Gamma|$,
$\textbf{x,y}\in\mathscr{C}_{\infty}(S),$ and
$x\in\mathscr{C}(\mathbf{x}),\,y\in\mathscr{C}(\mathbf{y})$. Denote by
$\mathfrak{C}_{S}(\mathbf{w})$ the set of all $\varepsilon$-bad sites
connected to $\mathbf{w}$ by a macroscopic $\mathbb{Z}^{d}$-path of bad sites
included in $S$. Since
$\mathfrak{C}_{S}(\mathbf{w})\subset\mathfrak{C}(\mathbf{w})$,
$\left|\bigcup_{\mathbf{w}\in\Gamma}\mathfrak{C}_{S}(\mathbf{w})\right|<\varepsilon_{0}|\Gamma|.$

Let $x\in\mathscr{C}(\mathbf{x})$ and $y\in\mathscr{C}(\mathbf{y})$ and denote
by $\mathrm{L}$ the line joining $x$ and $y$. Since
$\|\textbf{x}-\textbf{y}\|_{\infty}=\|\textbf{x}-\textbf{y}\|_{1}\geq 2N$, we
can take $(x_{k})_{k=0}^{K}\subset\mathrm{L}$ such that $x_{0}=x$, $x_{K}=y$,
and $\|x_{i}-x_{i-1}\|_{\infty}\in[N,2N]$ and
$x_{i}\in\underline{\Lambda}_{N}(\mathbf{x}_{i})$ where
$\mathbf{x}_{i}:=\mathbf{x}+i\mathbf{e}_{1}\in\Gamma$. To each good site
$\mathbf{x}_{k}\in\Gamma$, choose the closest point $x_{k}^{\prime}$ from
$x_{k}$ in $\mathscr{C}(\mathbf{x}_{k})$ in a deterministic rule breaking
ties. Thanks to Property (2) of $\varepsilon_{0}$-good, we have
$\|x_{k}-x^{\prime}_{k}\|_{\infty}\leq\varepsilon_{0}N.$ We decompose $\Gamma$
into portions of paths consisting of only good boxes as follows: We define

$\tau_{in}(1):=1\,,\qquad\tau_{out}(1):=\max\big{\\{}\,j\geq\tau_{in}(1):\,\forall
i\in\\{\tau_{in}(1),\dots,j\\},\quad\mathbf{x}_{i}\text{ is
good}\,\big{\\}}\,.$

Suppose that $\tau_{in}(1),\dots,\tau_{in}(k)$ and
$\tau_{out}(1),\dots,\tau_{out}(k)$ have been defined. We define

$\displaystyle\tau_{in}(k+1)$
$\displaystyle:=\min\big{\\{}\,j\geq\tau_{out}(k):\,\forall i\geq
j,\quad\mathbf{x}_{i}\notin\mathfrak{C}_{S}(\mathbf{x}_{\tau_{out}(k)+1})\,\big{\\}}\,,$
$\displaystyle\tau_{out}(k+1)$
$\displaystyle:=\max\big{\\{}\,j\geq\tau_{in}(k+1):\,\forall
i\in\\{\tau_{in}(k+1),\dots,j\\},\quad\mathbf{x}_{i}\text{ is
good}\,\big{\\}}\,.$

We stop this procedure once $\tau_{out}(k+1)=K$. By construction, using
$\textbf{x},\textbf{y}\in\mathscr{C}_{\infty}(S)$, we have
$\mathbf{x}_{\tau_{out}(k)},\mathbf{x}_{\tau_{in}(k+1)}\in\partial^{\rm
ext}_{S}\mathfrak{C}_{S}(\mathbf{x}_{\tau_{out}(k)+1})$, where

$\partial^{\rm ext}_{S}A:=\\{\mathbf{x}\in S\setminus A:\mathbf{x}\text{ is
connected to infinity in $S\setminus A$ and has a neighbor in $A$}\\}.$

Since $\partial^{\rm ext}_{S}\mathfrak{C}_{S}(\mathbf{x}_{\tau_{out}(k)+1})$
consists only of good boxes and is $*$-connected in $S$ by [30, Lemma 2],
there exists a $\varepsilon_{0}$-good $*$-path from
$\mathbf{x}_{\tau_{out}(k)}$ to $\mathbf{x}_{\tau_{in}(k+1)}$ in
$\partial^{\rm ext}_{S}\mathfrak{C}_{S}(\mathbf{x}_{\tau_{out}(k)+1})$. By
Lemma 2.9, we can therefore build a microscopic, open path between
$x^{\prime}_{\tau_{out}(k)}$ and $x^{\prime}_{\tau_{in}(k+1)}$ of length at
most

$2dN\mu(\mathbf{e}_{1})|\partial^{\rm
ext}_{S}\mathfrak{C}_{S}(\mathbf{x}_{\tau_{out}(k)+1})|\leq(2d)^{2}N\mu(\mathbf{e}_{1})|\mathfrak{C}_{S}(\mathbf{x}_{\tau_{out}(k)+1})|.$

Between $x^{\prime}_{\tau_{in}(k)}$ and $x^{\prime}_{\tau_{out}(k)}$, thanks
to Property (3) of $\varepsilon_{0}$-good, we can build a microscopic, open
path of length at most

$\begin{split}\sum_{j=\tau_{in}(k)}^{\tau_{out}(k)-1}[\mu(x^{\prime}_{j+1}-x^{\prime}_{j})+\varepsilon_{0}N]&\leq\sum_{j=\tau_{in}(k)}^{\tau_{out}(k)-1}[\mu(x_{j+1}-x_{j})+4d\mu(\mathbf{e}_{1})\varepsilon_{0}N]\\\
&\leq\mu(x_{\tau_{out}(k)}-x_{\tau_{in}(k)})+4d\mu(\mathbf{e}_{1})\varepsilon_{0}N(\tau_{out}(k)-\tau_{in}(k)),\end{split}$

where we have used $x_{k}\in L\cap\underline{\Lambda}_{N}(\mathbf{x}_{k})$ in
the last inequality.

Finally, by the assumption
$\left|\bigcup_{\mathbf{w}\in\Gamma}\mathfrak{C}_{S}(\mathbf{w})\right|<\varepsilon_{0}|\Gamma|$,

$\sum_{k}|\mathfrak{C}_{S}(\mathbf{x}_{\tau_{out}(k)+1})|\leq\varepsilon_{0}|\Gamma|,\quad\sum_{k}(\tau_{out}(k)-\tau_{in}(k))\leq|\Gamma|.$

Therefore, since $\|\textbf{x}-\textbf{y}\|_{\infty}\geq 2N$ and $|\Gamma|\leq
2d\mu(x-y)/N$, and
$\varepsilon_{0}=\varepsilon/(2\varepsilon+4d\mu(\mathbf{e}_{1}))^{3}$, we can
build a microscopic, open path from $x$ to $y$ of length at most

$\mu(x-y)+(4d)^{2}\mu(\mathbf{e}_{1})\varepsilon_{0}N|\Gamma|\leq(1+\varepsilon)\mu(x-y).$

Therefore, we have (2.7). ∎

###  2.4. Proof of Theorem 1.6

To prove Theorem 1.6, we need the following proposition.

######  Proposition 2.11.

Suppose $d\geq 3$. Let $S=\mathbb{Z}^{2}\times\\{0\\}^{d-2}$. We denote

$\Lambda_{\varepsilon,\rho}(n;N):=[-\varepsilon n,\varepsilon
n]^{2}\times[-{\rho N},{\rho N}]^{d-2}.$

For any $\xi,\varepsilon>0$, there exist $N\in\mathbb{N}$ and $c>0$ such that
for $n$ large enough,

$\mathbb{P}\left(\mathcal{D}^{\mathcal{G}_{p}}_{\underline{S}}(\Lambda_{\varepsilon,\rho}(n;N),n\mathbf{e}_{1}+\Lambda_{\varepsilon,\rho}(n;N))>(\mu+\xi)n\right)\leq
e^{-cn}.$

###### Proof.

Let $\xi,\varepsilon>0$, and $N\in\mathbb{N}$ large enough. Let
$\mathcal{F}_{n}$ be the following event

$\mathcal{F}_{n}:=\begin{array}[]{c}\\{\mathscr{C}_{\infty}(S)\cap\\{(k_{1},0,0,\dots,0):k_{1}\in[-\varepsilon
n/(2N),\varepsilon n/(2N)]\\}=\emptyset\\}\\\
\cup\\{\mathscr{C}_{\infty}(S)\cap\\{(\lfloor
n/N\rfloor+k_{1},0,\dots,0):k_{1}\in[-\varepsilon n/(2N),\varepsilon
n/(2N)]\\}=\emptyset\\}\end{array}\,.$

If the event occurs, then there exists a $*$-connected component of bad sites
in $S$ such that it encloses $\\{(k_{1},0,0,\dots,0):k_{1}\in[-\varepsilon
n/(2N),\varepsilon n/(2N)]\cap\mathbb{Z}\\}$ or $\\{(\lfloor
n/N\rfloor+k_{1},0,0,\dots,0):k_{1}\in[-\varepsilon n/(2N),\varepsilon
n/(2N)]\cap\mathbb{Z}\\}$. In particular, this $*$-connected component is of
size at least $\varepsilon n/N$. By a similar computation as in (2.5), there
exists $c=c(\varepsilon,\rho,N,\kappa_{2})>0$ such that for $n$ large enough,
$\mathbb{P}(\mathcal{F}_{n})\leq e^{-cn}.$

On the event $\mathcal{F}_{n}^{c}$, there exist
$\mathbf{x}\in\mathscr{C}_{\infty}(S)\cap\\{(k_{1},0,0,\dots,0):k_{1}\in[-\varepsilon
n/(2N),\varepsilon n/(2N)]\\}$ and
$\mathbf{y}\in\mathscr{C}_{\infty}(S)\cap\\{(\lfloor
n/N\rfloor+k_{1},0,0,\dots,0):k_{1}\in[-\varepsilon n/(2N),\varepsilon
n/(2N)]\\}$. The result follows by using Proposition 2.10.

∎

###### Proof of Theorem 1.6.

Let $\xi,\varepsilon>0$. Let $N=N(\xi,\varepsilon)\in\mathbb{N}$ be as in
Proposition 2.11. For $z\in 2{\rho}\mathbb{Z}^{d-2}$, denote
$S(z):=\mathbb{Z}^{2}\times\\{z\\}$. Note that the slabs
$(\underline{S}(z):z\in 2{\rho}\mathbb{Z}^{d-2})$ are all disjoint. Let
$\bar{z}:=(0,0,z)$. We define

$\mathcal{E}(z):=\left\\{\mathcal{D}^{\mathcal{G}_{p}}_{\underline{S}(z)}(\bar{z}+\Lambda_{\varepsilon,\rho}(n;N),\bar{z}+n\mathbf{e}_{1}+\Lambda_{\varepsilon,\rho}(n;N))>(\mu+\xi)n\right\\}.$

Notice that

$\\{\mathcal{D}^{\mathcal{G}_{p}}(\Lambda_{2\varepsilon
n}(0),\Lambda_{2\varepsilon
n}(n\mathbf{e}_{1}))>(\mu+\xi)n\\}\subset\bigcap_{z\in{2\rho}\mathbb{Z}^{d-2}\cap[-\varepsilon
n/(2N),\varepsilon n/(2N)]^{d-2}}\mathcal{E}(z).$

Since the events on the right hand side are independent, using Proposition
2.11, we get

$\mathbb{P}(\mathcal{D}^{\mathcal{G}_{p}}(\Lambda_{2\varepsilon
n}(0),\Lambda_{2\varepsilon
n}(n\mathbf{e}_{1}))>(\mu+\xi)n)\leq\exp\left(-cn\left(\frac{\varepsilon
N}{4N\rho}\right)^{d-2}\right).$

∎

##  3\. Space-time cut-points

###  3.1. Rate function for space-time cut-point

The proof of Theorem 1.1 relies on the following key lemma. We postpone its
proof until Section 3.5.

######  Lemma 3.1.

Let $s_{0}>0$. There exist $c=c(s_{0})>0$ and
$n_{0}=n_{0}(s_{0})\in\mathbb{N}$ such that the following holds. For any
$s,s^{\prime}\in[0,s_{0}]$, $x,x^{\prime}\in[-s_{0},s_{0}]^{d}$ and $n\geq
m\geq n_{0}$,

$\mathbb{P}(\mathcal{A}_{s,x}(n))\mathbb{P}(\mathcal{A}_{s^{\prime},x^{\prime}}(m))\leq
e^{cn^{\alpha_{d}}}\mathbb{P}(\mathcal{A}_{\frac{n}{n+m}s+\frac{m}{n+m}s^{\prime},\frac{n}{n+m}x+\frac{m}{n+m}x^{\prime}}(n+m)).$

Moreover, when $n=m$, we have

$\mathbb{P}(\mathcal{A}_{s,x}(n))\mathbb{P}(\mathcal{A}_{s^{\prime},x^{\prime}}(n))\leq
e^{cn^{\alpha_{d}}}\mathbb{P}(\mathcal{A}_{s+s^{\prime},x+x^{\prime}}(n)).$

######  Remark 3.2.

Note that $\mathcal{A}_{s/2,x/2}(2n)\neq\mathcal{A}_{s,x}(n)$ in general.
Hence, the second inequality does not directly follow from the first
inequality in Lemma 3.1 by setting $n=m$.

We have the following corollary:

######  Corollary 3.3.

Let $s_{0}>0$. There exist $c=c(s_{0})>0$ and
$n_{0}=n_{0}(s_{0})\in\mathbb{N}$ such that the following holds. For any
$\delta>0$, $s,s^{\prime}\in[0,s_{0}]$ and $x,x^{\prime}\in[-s_{0},s_{0}]^{d}$
with $|s-s^{\prime}|\leq\delta$ and $\|x-x^{\prime}\|_{1}\leq\delta$, and for
any $n\geq n_{0}$,

$\mathbb{P}(\mathcal{A}_{s,x}(n))\leq e^{c(\delta
n+n^{\alpha_{d}})}\mathbb{P}(\mathcal{A}_{s^{\prime},x^{\prime}}(n)).$

###### Proof.

We first suppose $s\leq s^{\prime}$. Thanks to Lemma 3.1, we have

$\mathbb{P}(\mathcal{A}_{s,x}(n))\mathbb{P}(\mathcal{A}_{s^{\prime}-s,x^{\prime}-x}(n))\leq\mathbb{P}(\mathcal{A}_{s^{\prime},x^{\prime}}(n))e^{c^{\prime}n^{\alpha_{d}}},$

with some constant $c^{\prime}>0$ independent of $\delta,n$. Since
$|s^{\prime}-s|+\|x^{\prime}-x\|_{1}\leq 2\delta$, one can check that (as in
(3.5) below), we have

$\mathbb{P}(\mathcal{A}_{s^{\prime}-s,x^{\prime}-x}(n))\geq
e^{-c^{\prime\prime}\delta n}$

with some constant $c^{\prime\prime}>0$ independent of $\delta,n$. Therefore,
with $c:=c^{\prime}+c^{\prime\prime}$,

$\mathbb{P}(\mathcal{A}_{s,x}(n))\leq e^{c(\delta
n+n^{\alpha_{d}})}\mathbb{P}(\mathcal{A}_{s^{\prime},x^{\prime}}(n)).$

If $s>s^{\prime}$, by the result above, then

$\mathbb{P}(\mathcal{A}_{s,x}(n))\leq\mathbb{P}(\mathcal{A}_{s^{\prime},x}(n))\leq
e^{c(\delta
n+n^{\alpha_{d}})}\mathbb{P}(\mathcal{A}_{s^{\prime},x^{\prime}}(n)).$

∎

Let us see how this lemma and corollary imply Theorem 1.1.

###### Proof of Theorem 1.1.

Let $s\geq 0$ and $x\in\mathbb{R}^{d}$. Let $n\geq m\geq 1$. Thanks to Lemma
3.1,

$\mathbb{P}(\mathcal{A}_{s,x}(n))\mathbb{P}(\mathcal{A}_{s,x}(m))\leq
e^{cn^{\alpha_{d}}}\mathbb{P}(\mathcal{A}_{s,x}(n+m)),$

with some constant $c>0$ independent of $n,m$. It follows that

$-\log\mathbb{P}(\mathcal{A}_{s,x}(n+m))\leq-\log\mathbb{P}(\mathcal{A}_{s,x}(n))-\log\mathbb{P}(\mathcal{A}_{s,x}(m))+cn^{\alpha_{d}}.$

Using deBruijn and Erdôs’s subadditive lemma [8], the following limit exists:

(3.1)
$I(s,x):=\lim_{n\rightarrow\infty}-\frac{1}{n}{\log}\,\mathbb{P}(\mathcal{A}_{s,x}(n))\in[0,+\infty)\,.$

By Corollary 3.3, $I$ is continuous. In particular, $I$ is uniformly
continuous on a compact set.

Let $K\subset[0,\infty)\times\mathbb{R}^{d}$ be a compact set and $s_{0}:={\rm
diam}(K)+1$. Let $\delta\in(0,1/(4ds_{0}))$ arbitrary. Since the cardinality
of $K_{\delta}:=(\delta\mathbb{Z})^{d+1}\cap K$ is finite, the convergence
(3.1) is uniform over $K_{\delta}$. We take $(s,x)\in K$ arbitrary. Let
$(s^{\prime},x^{\prime})$ be the closest point from $(s,x)$ in $K_{\delta}$
satisfying $|s^{\prime}-s|+\|x^{\prime}-x\|_{1}\leq 4d\delta$ with a
deterministic rule breaking ties. By Corollary 3.3 and the uniform convergence
over $K_{\delta}$, since $I$ is uniformly continuous on $K$, we have

$\displaystyle\mathbb{P}(\mathcal{A}_{s,x}(n))\leq
e^{o_{\delta}(n)}\mathbb{P}(\mathcal{A}_{s^{\prime},x^{\prime}}(n))\leq
e^{o_{\delta}(n)}e^{-I(s^{\prime},x^{\prime})n}\leq
e^{-I(s,x)n+o_{\delta}(n)},$

where $o_{\delta}(n)$ are some positive constants such that $o_{\delta}(n)/n$
converges to $0$ uniformly over $(s,x)\in K$ when $n\to\infty$, and then
$\delta\to 0$. Similarly, we have

$\mathbb{P}(\mathcal{A}_{s,x}(n))\geq
e^{-o_{\delta}(n)}\mathbb{P}(\mathcal{A}_{s^{\prime},x^{\prime}}(n))\geq
e^{-o_{\delta}(n)}e^{-I(s^{\prime},x^{\prime})n}\geq
e^{-I(s,x)n-o_{\delta}(n)}.$

Thus, letting $\delta\to 0$, we find that the convergence (3.1) is uniform on
$K$.

Let $s_{0}>0$. We take $c>0$ and $n_{0}\in\mathbb{N}$ as in Lemma 3.1. Let
$s,s^{\prime}\in[0,s_{0}]$, $x,x^{\prime}\in[-s_{0},s_{0}]^{d}$, and Let
$\lambda\in(0,1)\cap\mathbb{Q}$. Let $(u_{n})_{n\geq 1}$ be a sequence of
integer larger than $n_{0}/\min(\lambda,1-\lambda)$ such that $\lambda
u_{n}\in\mathbb{N}$ and $u_{n}\to\infty$ as $n\to\infty$. Thanks to Lemma 3.1,
we have

$\mathbb{P}(\mathcal{A}_{s,x}(\lambda
u_{n}))\mathbb{P}(\mathcal{A}_{s^{\prime},x^{\prime}}((1-\lambda)u_{n}))\leq
e^{cu_{n}^{\alpha_{d}}}\mathbb{P}(\mathcal{A}_{\lambda
s+(1-\lambda)s^{\prime},\lambda x+(1-\lambda)x^{\prime}}(u_{n})).$

By passing to the limit using (3.1) with $u_{n}$ in place of $n$, we get

(3.2) $I(\lambda s+(1-\lambda)s^{\prime},\lambda
x+(1-\lambda)x^{\prime})\leq\lambda
I(s,x)+(1-\lambda)I(s^{\prime},x^{\prime}).$

This together with the continuity of $I$ implies (3.2) for general $\lambda$.
Moreover, thanks to Lemma 3.1 with $n=m$, we have

$\mathbb{P}(\mathcal{A}_{s,x}(n))\mathbb{P}(\mathcal{A}_{s^{\prime},x^{\prime}}(n))\leq
e^{cn^{\alpha_{d}}}\mathbb{P}(\mathcal{A}_{s+s^{\prime},x+x^{\prime}}(n)).$

By passing to the limit using (3.1), we get

(3.3) $I(s+s^{\prime},x+x^{\prime})\leq I(s,x)+I(s^{\prime},x^{\prime}).$

Next, we consider the homogeneity. Let $k,m\in\mathbb{N}$, $s\geq 0$, and
$x\in\mathbb{R}^{d}$. By (3.3), $I(ks,kx)\leq kI(s,x)$. On the other hand, the
opposite inequality is trivial because of
$\mathcal{A}_{ks,kx}(n)\subset\mathcal{A}_{s,x}(kn)$. Therefore, we have
$I(ks,kx)=kI(s,x)$. This implies

(3.4) $I(ks/m,kx/m)=mI(ks/m,kx/m)/m=I(ks,kx)/m=kI(s,x)/m.$

For general $\lambda\geq 0$, consider a sequence
$\lambda_{n}\in[0,\infty)\cap\mathbb{Q}$ whose limit is $\lambda$. The
homogeneity follows by the continuity of $I$ and (3.4) with $k/m=\lambda_{n}$.

Finally, we prove $I(s,x)>0$ for any $s>0$ and $x\in\mathbb{R}^{d}$. Note that
by (3.3)

$I(2s,0)\leq I(s,x)+I(s,-x)=2I(s,x).$

Moreover, thanks to (1.9), $I(2s,0)>0$. Thus, we have $I(s,x)>0$. ∎

###  3.2. Size of the ball corresponding to cut-point

For $K>0$, define

$\mathcal{A}_{s,x}^{K}(n):=\left\\{\begin{array}[]{l}\exists s_{n}\geq
sn,\,\exists
w_{n}\in\Lambda_{n^{\alpha_{d}}}(nx),\,\exists\Gamma\subset\mathbb{Z}^{d};\\\
\,{\rm B}_{s_{n}}\setminus{\rm B}_{s_{n}-1}=\\{w_{n}\\},\,|\Gamma|\leq
Kn,\,\mathrm{B}_{s_{n}}\subset\mathrm{Int}(\Gamma)\end{array}\right\\}.$

######  Proposition 3.4.

For any $s_{0}>0$, there exist $n_{0}=n_{0}(s_{0})\in\mathbb{N}$ and
$K=K(s_{0})>0$ such that for any $n\geq n_{0}$,

$\mathbb{P}\left(\bigcup_{s\in[0,s_{0}],x\in[-s_{0},s_{0}]^{d}}(\mathcal{A}_{s,x}(n)\setminus\mathcal{A}_{s,x}^{K}(n))\right)\leq
e^{-n}\min_{s\in[0,s_{0}],x\in[-s_{0},s_{0}]^{d}}\mathbb{P}(\mathcal{A}_{s,x}(n)).$

###### Proof.

Let $s\in[0,s_{0}]$, $x\in[-s_{0},s_{0}]^{d}$, and $n\geq 1$. Let
$w_{n}\in\Lambda_{n^{\alpha_{d}}}(nx){\setminus\\{0\\}}$ and $\gamma$ be a
deterministic self-avoiding path from $0$ to $w_{n}$ such that
$s_{0}n\leq|\gamma|\leq 2{d}s_{0}n$. Let us define

$\mathcal{E}:=\left\\{\forall e\in\gamma,\text{ $e$ is open},\quad\forall
e\not\in\gamma\text{ with $|e\cap\gamma|=1$, $e$ is closed}\right\\}.$

By construction, we have $\mathcal{E}\subset\mathcal{A}_{s,x}(n)$, and

$\mathbb{P}(\mathcal{E})\geq
p^{|\gamma|}(1-p)^{2d|\gamma|}\geq(p(1-p)^{2d})^{2ds_{0}n}.$

Hence, there exists $C>0$ depending on $s_{0},\,p$, and $d$ such that for all
$n\geq 1$,

(3.5) $\mathbb{P}(\mathcal{A}_{s,x}(n))\geq e^{-Cn}.$

Let $s_{n}\geq sn$ be such that ${\rm B}_{s_{n}}\setminus{\rm
B}_{{s_{n}}-1}=\\{w_{n}\\}$. Set $\varepsilon=1/2$. Let $N,K\in\mathbb{N}$ be
chosen later. Let $\Gamma_{N}$ be the set of boxes intersecting
$\mathrm{B}_{s_{n}}$:

$\Gamma_{N}:=\\{\mathbf{i}\in\mathbb{Z}^{d}:\underline{\Lambda}_{N}(\mathbf{i})\cap\mathrm{B}_{s_{n}}\neq\emptyset\\}.$

Note that $\Gamma_{N}$ is connected in $\mathbb{Z}^{d}$. Let us consider the
exterior boundary of $\Gamma_{N}$ as

$\partial^{\rm
ext}\Gamma_{N}:=\\{\mathbf{i}\in\mathbb{Z}^{d}\setminus\Gamma_{N}:\leavevmode\nobreak\
\exists\mathbf{j}\in\Gamma_{N};\leavevmode\nobreak\
\|\mathbf{i}-\mathbf{j}\|_{1}=1,\,\text{$\mathbf{i}$ is connected to infinity
in $\mathbb{Z}^{d}\setminus\Gamma_{N}$}\\}.$

Note that $\partial^{\rm ext}\Gamma_{N}$ is $*$-connected by [30, Lemma 2].
Let $\mathbf{i}\in\Gamma_{N}$ and $\mathbf{j}\in\partial^{\rm ext}\Gamma_{N}$
be such that $\|\mathbf{i}-\mathbf{j}\|_{1}=1$ and
$w\notin\underline{\Lambda}^{\prime}_{N}(\mathbf{i})$. Let us prove that
$\mathbf{i}$ or $\mathbf{j}$ is $\varepsilon$-bad. By definition of
$\Gamma_{N}$ and $\partial^{\rm ext}\Gamma_{N}$, we have
$\underline{\Lambda}_{N}(\mathbf{j})\cap\mathrm{B}_{s_{n}}=\emptyset$. Let us
assume that $\mathbf{i}$ and $\mathbf{j}$ are both good. Let
$x\in\mathrm{B}_{s_{n}}\cap\underline{\Lambda}_{N}(\mathbf{i})$. Note that $x$
is connected to $w\notin\underline{\Lambda}_{N}^{\prime}(\mathbf{i})$ by a
microscopic, open path inside $\mathrm{B}_{s_{n}}$. By definition of good box,
this yields that
$x\in\mathscr{C}(\mathbf{i})\cap\mathrm{B}_{s_{n}}\cap\underline{\Lambda}_{N}(\mathbf{i})$.
Since $\underline{\Lambda}_{N}(\mathbf{j})\cap\mathrm{B}_{s_{n}}=\emptyset$,
and $\mathbf{i}$ and $\mathbf{j}$ are good, there exists
$y\in{\mathscr{C}(\mathbf{j})\cap\mathscr{C}(\mathbf{i})\setminus\mathrm{B}_{s_{n}}}$.
Thus, $x$ and $y$ are connected by a microscopic, open path in
$\mathscr{C}(\mathbf{i})\subset\underline{\Lambda}^{\prime}_{N}(\mathbf{i})$.
However, by definition of cut-point, any microscopic, open path from
$x\in\mathrm{B}_{s_{n}}$ to $y\notin\mathrm{B}_{s_{n}}$ must contain $w_{n}$,
which derives a contradiction since
$w\notin\underline{\Lambda}^{\prime}_{N}(\mathbf{i})$. It follows that any
$\mathbf{j}\in\partial^{\rm ext}\Gamma_{N}$ is $\varepsilon$-bad, or has a
$\varepsilon$-bad neighbor, or has a neighbor whose enlarged box contains
$w_{n}$. Let $A^{+}:=\\{\mathbf{i}\in\mathbb{Z}^{d}:\leavevmode\nobreak\
\exists\mathbf{j}\in A;\leavevmode\nobreak\ \|\mathbf{i}-\mathbf{j}\|_{1}\leq
1\\}$ for $A\subset Z^{d}$. Therefore, $\partial^{\rm ext}\Gamma_{N}$ is
$*$-connected, its interior contains $\Gamma_{N}$, and

$|\\{\mathbf{i}\in(\partial^{\rm ext}\Gamma_{N})^{+}:\leavevmode\nobreak\
\mathbf{i}\text{ is bad}\\}|\geq\frac{|\partial^{\rm
ext}\Gamma_{N}|}{(4d)^{d}}-(4d)^{d}.$

Let $\Gamma:=\bigcup_{\mathbf{j}\in\partial^{\rm
ext}\Gamma_{N}}\underline{\Lambda}_{N}(\mathbf{j}).$ Since the interior of
$\partial^{\rm ext}\Gamma_{N}$ contains $\Gamma_{N}$,
$\mathrm{B}_{s_{n}}\subset\mathrm{Int}(\Gamma).$

Let $\kappa=1/(10d)^{d}$ and $K\geq(10d)^{10d}$. Denote by $\mathcal{F}$ the
event that there exists a macroscopic, $*$-connected set $A$ such that
$|A|\geq Kn$, $|\\{\mathbf{i}\in A^{+}:\leavevmode\nobreak\ \mathbf{i}\text{
is bad}\\}|\geq\kappa|A|$, and its interior contains $0$. To compute its
probability, we first fix such a $*$-connected set $A$. By pigeon-hole
principle as in Proposition 2.6, there exists a subset of $A^{+}$ containing
at least $\frac{\kappa}{(4\rho)^{d}}|A|$ bad sites at $\ell_{\infty}$ distance
at least $\rho$ from each other (recall the notation $\rho$ from (2.1)). In
particular, the states of these sites are independent. By Proposition 2.6, for
$N$ large enough depending on $d,p$,

$\mathbb{P}(|\\{\mathbf{x}\in A^{+}:\mathbf{x}\text{ is
$\varepsilon$-bad}\\}|\geq\kappa|A|)\leq
2^{|A^{+}|}\exp\left(-cN\frac{\kappa}{(4\rho)^{d}}|A|\right).$

This yields

$\displaystyle\mathbb{P}(\mathcal{F})$ $\displaystyle\leq\sum_{k\geq
Kn}\sum_{y\in[-k,k]^{d}\cap\mathbb{Z}^{d}}\sum_{A\in\textbf{Animal}^{k}_{y}}\mathbb{P}(|\\{\mathbf{x}\in
A^{+}:\mathbf{x}\text{ is $\varepsilon$-bad}\\}|\geq\kappa|A|)$
$\displaystyle\leq\sum_{k\geq
Kn}7^{dk}(2k+1)^{d}2^{4^{d}k}\exp\left(-cN\frac{\kappa}{(4\rho)^{d}}k\right){\leq\exp\left(-cN\frac{\kappa}{(8\rho)^{d}}Kn\right)},$

where we have used (1.18) in the last line. Hence, thanks to (3.5), for $K$
large enough depending on $s_{0}$, $p$, and $d$, we have for all
$n\in\mathbb{N}$,

(3.6) $\mathbb{P}(\mathcal{F})\leq e^{-(1+C)n}\leq
e^{-n}{\min_{s\in[0,s_{0}],x\in[-s_{0},s_{0}]^{d}}\mathbb{P}(\mathcal{A}_{s,x}(n))}.$

On the event $\mathcal{F}^{c}$, $|\Gamma|\leq(2N)^{d}|\partial^{\rm
ext}\Gamma_{N}|\leq(2N)^{d}Kn=:K^{\prime}n\,.$ Therefore, we obtain
$\mathcal{A}_{s,x}(n)\setminus\mathcal{A}_{s,x}^{K^{\prime}}(n)\subset\mathcal{F}$,
and by using (3.6), it follows that

$\mathbb{P}\left(\bigcup_{s\in[0,s_{0}],x\in[-s_{0},s_{0}]^{d}}(\mathcal{A}_{s,x}(n)\setminus\mathcal{A}_{s,x}^{K^{\prime}}(n))\right)\leq\mathbb{P}(\mathcal{F})\leq
e^{-n}\min_{s\in[0,s_{0}],x\in[-s_{0},s_{0}]^{d}}\mathbb{P}(\mathcal{A}_{s,x}(n)).$

∎

###  3.3. Resampling arguments

######  Lemma 3.5 (Resampling lemma).

Let $A\in\mathbb{N}$. Recall $\tau=(\tau_{e})_{e\in\mathbb{E}^{d}}$ from
(1.7). Let $E_{0}=E_{0}(\tau)$, $E_{1}=E_{1}(\tau)$ be random sets of edges
inside $[-n^{2},n^{2}]^{d}$ depending on $\tau$ such that

$E_{0}\cap E_{1}=\emptyset,\quad\max\\{|E_{0}|,|E_{1}|\\}\leq
A\quad\text{a.s.}$

Then, there exists another random configuration $(\tau_{e}^{\rm
r})_{e\in\mathbb{E}^{d}}\in\\{1,\infty\\}^{\mathbb{E}^{d}}$ with the same law
as $\tau$ such that for any event
$\mathcal{E}\subset\\{1,\infty\\}^{\mathbb{E}^{d}}$ and
$n\geq\max\\{4d,(p(1-p))^{-1}\\}$,

$\displaystyle\mathbb{P}(\tau\in\mathcal{E};\,\forall e\in
E_{0}(\tau),\,\tau_{e}^{\rm r}=+\infty;\,\forall e\in
E_{1}(\tau),\,\tau_{e}^{\rm r}=1;\,\forall e\notin E_{0}(\tau)\cup
E_{1}(\tau),\,\tau^{\rm r}_{e}=\tau_{e})$ $\displaystyle\geq e^{-8d\,A\log
n}\mathbb{P}(\tau\in\mathcal{E}).$

The configuration $\tau^{\rm r}=(\tau_{e}^{\rm r})_{e\in\mathbb{E}^{d}}$ is
called the resampled configuration.

###### Proof.

Let $\tau^{*}=(\tau_{e}^{*})_{e\in\mathbb{E}^{d}}$ be an independent copy of
$\tau$. Let $\mathfrak{E}_{0}$ and $\mathfrak{E}_{1}$ be two independent
random variables distributed uniformly on the set

$\\{E\subset\mathbb{E}^{d}:\quad\forall e\in E\quad
e\subset[-n^{2},n^{2}]^{d},\quad|E|\leq A\\}.$

In particular, we have

$\mathbb{P}(\mathfrak{E}_{0}=E_{0},\,\mathfrak{E}_{1}=E_{1})\geq e^{-5d\,A\log
n}.$

We define the resampled configuration $\tau^{\rm r}$ as

$\tau^{\rm r}_{e}=\begin{cases}\tau_{e}^{*}&\text{ if
$e\in\mathfrak{E}_{0}\cup\mathfrak{E}_{1}$},\\\ \tau_{e}&\text{
otherwise.}\end{cases}$

Recall that $n\geq\max\\{4d,(p(1-p))^{-1}\\}$. We conclude

$\displaystyle e^{-8d\,A\log n}\,\mathbb{P}(\tau\in\mathcal{E})$
$\displaystyle\leq\mathbb{P}(\tau\in\mathcal{E},\,\forall
e\in\mathfrak{E}_{1}\,\,\tau_{e}^{\rm r}=+\infty\,,\,\forall
e\in\mathfrak{E}_{0}\,\,\tau_{e}^{\rm
r}=1\,,\,\mathfrak{E}_{0}=E_{0},\mathfrak{E}_{1}=E_{1})$
$\displaystyle\leq\mathbb{P}(\tau\in\mathcal{E};\,\forall e\in
E_{0}(\tau),\,\tau_{e}^{\rm r}=+\infty;\,\forall e\in
E_{1}(\tau),\,\tau_{e}^{\rm r}=1;\,\forall e\notin E_{0}(\tau)\cup
E_{1}(\tau),\,\tau^{\rm r}_{e}=\tau_{e}).$

∎

We say that an event $\mathcal{E}$ is decreasing if for any
$\tau\in\mathcal{E}$ and $\tau^{\prime}$ satisfying for all
$e\in\mathbb{E}^{d}$ $\tau^{\prime}_{e}\geq\tau_{e}$,
$\tau^{\prime}\in\mathcal{E}$. The terminology decreasing is here related to
percolation. In other words, the less open edges there are, the more likely
the event occurs.

######  Lemma 3.6 (Making a cut-point).

Let $t_{0}\in\mathbb{N}$. Consider a family of decreasing events
$(\mathcal{E}(w),w\in\mathbb{Z}^{d})$,
$\mathcal{E}(w)\subset\\{1,\infty\\}^{\mathbb{E}^{d}}$. For any
$k\geq(4d+(p(1-p))^{-1})^{2}$, $t_{0}\geq\sqrt{k}$, and
$\Lambda\subset\mathbb{Z}^{d}$,

$\displaystyle e^{-(8d)^{2}\sqrt{k}\log{k}}\mathbb{P}(\exists t\geq
t_{0}\quad\exists x\in\Lambda:|\mathrm{B}_{t}|\leq
k,\,x\in\mathrm{B}_{t}\backslash\mathrm{B}_{t-1},\,\tau\in\mathcal{E}(x))$
$\displaystyle\qquad\leq\mathbb{P}(\exists t\geq t_{0}\quad\exists
x\in\Lambda:\,\mathrm{B}_{t}\setminus\mathrm{B}_{t-1}=\\{x\\},\leavevmode\nobreak\
\tau\in\mathcal{E}(x)).$

###### Proof.

Set

$\mathcal{E}:=\\{\tau\in\\{1,\infty\\}^{\mathbb{E}^{d}}:\leavevmode\nobreak\
\exists t\geq t_{0}\quad\exists x\in\Lambda:|\mathrm{B}_{t}|\leq
k,\,x\in\mathrm{B}_{t}\backslash\mathrm{B}_{t-1},\,\tau\in\mathcal{E}(x)\\}.$

Given $\tau\in\mathcal{E}$, let $t=t(\tau)\geq t_{0}$ and
$x=x(\tau)\in\Lambda$ be such that $|\mathrm{B}_{t}|\leq
k,\,x\in\mathrm{B}_{t}\backslash\mathrm{B}_{t-1},\,\tau\in\mathcal{E}(x)$. We
take a geodesic $\gamma_{x}$ from $0$ to $x$. In case there are several
choices, we pick one of them according to a deterministic rule. Since the sets
$(\mathrm{B}_{r}\backslash\mathrm{B}_{r-1},r=1\dots t)$ are disjoint, we have

$\sum_{r=t-\lceil\sqrt{k}\rceil}^{t}|\mathrm{B}_{r}\backslash\mathrm{B}_{r-1}|\leq|\mathrm{B}_{t}|\leq
k.$

Hence, there exists $r\in[t]\backslash[t-\lceil\sqrt{k}\rceil]$ such that
$|\mathrm{B}_{r}\backslash\mathrm{B}_{r-1}|\leq\sqrt{k}$. Let
$E_{1}(\tau)=\emptyset$ and

$E_{0}(\tau):=\\{\langle
v,w\rangle\in\mathbb{E}^{d}\setminus\gamma_{x}:\leavevmode\nobreak\
v\in\gamma_{x}\setminus\mathrm{B}_{r-1}\\}\cup\\{e\in\mathbb{E}^{d}\setminus\gamma_{x}:\leavevmode\nobreak\
e\text{ connects $\mathrm{B}_{r}\backslash\mathrm{B}_{r-1}$ and
$\mathrm{B}_{r-1}$}\\}.$

Note that $|E_{0}(\tau)|\leq 4d\sqrt{k}$ and $E_{0}(\tau)\subset[-k,k]^{d}$.
Hence, we can take a resampled configuration $\tau^{\rm r}=(\tau_{e}^{\rm
r})_{e\in\mathbb{E}^{d}}$ as in Lemma 3.5 with
$\mathcal{E}\subset\\{1,\infty\\}^{\mathbb{E}^{d}}$ so that $\tau_{e}^{\rm
r}\geq\tau_{e}$. Denote by $\mathrm{B}^{\rm r}_{s}$ the ball of radius $s$
centered at $0$ for $\tau^{\rm r}$. Since $\mathcal{E}(x)$ is decreasing,
$\tau^{\rm r}\in\mathcal{E}(x)$. Thus, we get

$\displaystyle e^{-(8d)^{2}\sqrt{k}\log{k}}\mathbb{P}(\exists t\geq
t_{0}\quad\exists x\in\Lambda:|\mathrm{B}_{t}|\leq
k,\,x\in\mathrm{B}_{t}\backslash\mathrm{B}_{t-1},\,\tau\in\mathcal{E}(x))$
$\displaystyle\leq\mathbb{P}(\exists t\geq t_{0}\quad\exists
x\in\Lambda:|\mathrm{B}_{t}|\leq
k,\,x\in\mathrm{B}_{t}\backslash\mathrm{B}_{t-1},\,\forall e\in
E_{0}(\tau)\quad\tau_{e}^{\rm r}=\infty\,,\,\tau\in\mathcal{E}(x))$
$\displaystyle\leq\mathbb{P}(\exists t\geq t_{0}\quad\exists
x\in\Lambda:|\mathrm{B}_{t}^{\rm r}|\leq k,\,\mathrm{B}_{t}^{\rm
r}\setminus\mathrm{B}_{t-1}^{\rm r}=\\{x\\},\,\tau^{\rm r}\in\mathcal{E}(x))$
$\displaystyle=\mathbb{P}(\exists t\geq t_{0}\quad\exists
x\in\Lambda:|\mathrm{B}_{t}|\leq
k\,,\,\mathrm{B}_{t}\setminus\mathrm{B}_{t-1}=\\{x\\},\,\tau\in\mathcal{E}(x)).$

∎

###  3.4. Drilling free line

We define the number of lines in direction $\mathbf{e}_{i}$ intersecting
$A\subset\mathbb{Z}^{d}$ as

(3.7) $\mathrm{N}_{i}(A):=|\\{z\in\mathbb{Z}^{d}:\leavevmode\nobreak\
z_{i}=0,\,\mathrm{L}_{i}(z)\cap A\neq\emptyset\\}|.$

Given $s>0$ and $x\in\mathbb{R}^{d}$, let us define

(3.10) $\displaystyle\mathcal{A}^{\rm
free}_{s,x}(n):=\left\\{\begin{array}[]{c}\exists i\neq j\in[d],\quad\exists
s_{n}\geq sn-3n^{\alpha_{d}},\quad\exists
w_{n}\in\Lambda_{4n^{\alpha_{d}}}(nx);\,\mathrm{B}_{s_{n}}\setminus\mathrm{B}_{s_{n}-1}=\\{w_{n}\\},\\\
\mathrm{L}_{i}(w_{n})\cap\mathrm{B}_{s_{n}}=\\{w_{n}\\},\,\forall k\neq
j\quad\mathrm{N}_{k}(\mathrm{H}_{j}(w_{n})\cap\mathrm{B}_{s_{n}})\leq
n^{\alpha_{d}},\,|\mathrm{B}_{s_{n}}|\leq n^{7/4}\end{array}\right\\}.$

The following is a key result of this section, which is the most technical
part of this paper.

######  Lemma 3.7 (Drilling free line at cut-point).

Let $s_{0}>0$ and $K=K(s_{0})>0$ be as in Proposition 3.4. There exists
$n_{0}=n(s_{0})\in\mathbb{N}$ such that the following holds. For any
$s\in[0,s_{0}]$ and $x\in[-s_{0},s_{0}]^{d}$, and $n\geq n_{0}$,

$e^{-n^{\alpha_{d}}}\mathbb{P}(\mathcal{A}_{s,x}^{K}(n))\leq\mathbb{P}\left(\mathcal{A}^{\rm
free}_{s,x}(n)\right).$

We will need the following deterministic lemma. We postpone its proof until
the appendix.

######  Lemma 3.8.

Suppose $d\geq 2$. Let $S\subset\mathbb{Z}^{d}$. We define

$m(d,S):=\left({\frac{|S|}{2^{d-1}\operatorname{Diam}(S)}}\right)^{\frac{1}{d-1}}.$

There exist $i\neq j\in[d]$ and $S^{\prime}\subset S$ with $|S^{\prime}|\geq
m(d,S)$ such that $z_{i}\neq z^{\prime}_{i}$ and $z_{j}\neq z_{j}^{\prime}$
for any $z\neq z^{\prime}\in S^{\prime}$.

From now on, if it is clear from context, we simply write $n^{a}$ instead of
$\lfloor n^{a}\rfloor$ for $a>0$. On the event $\mathcal{A}_{s,x}^{K}(n)$, let
$s_{n}\geq sn$ and $w_{n}\in\Lambda_{n^{\alpha_{d}}}(nx)$ such that
$\mathrm{B}_{s_{n}}\setminus\mathrm{B}_{s_{n}-1}=\\{w_{n}\\}$. We take
$\gamma_{0,w_{n}}=(w^{1},\cdots,w^{s_{n}})$ to be a geodesic between $0$ and
$w_{n}$ with a deterministic rule breaking ties. Recall
$\alpha_{d}=1-\frac{1}{6d}$. Set $\beta_{d}=\frac{5\alpha_{d}-4}{{5}(d-1)}$.
In particular, we have $1-\beta_{d}<\alpha_{d}$.

If $sn<3n^{\alpha_{d}}$ and $x\in{\Lambda_{4n^{\alpha_{d}}}}(0)$, then
$\mathcal{A}^{\rm free}_{s,x}(n)$ occurs with $s_{n}=0$ and $w_{n}=0$, in
particular

$\mathbb{P}(\mathcal{A}_{s,x}^{K}(n))\leq 1=\mathbb{P}(\mathcal{A}^{\rm
free}_{s,x}(n)).$

Otherwise, then since $s_{n}\geq 3n^{\alpha_{d}}$ or
$w_{n}\notin\Lambda_{3n^{\alpha_{d}}}(0)$, we eventually have $s_{n}\geq
3n^{\alpha_{d}}$ that we assume from now on. Let

$S:=\\{w^{s_{n}-2n^{\alpha_{d}}},\cdots,w^{s_{n}-n^{\alpha_{d}}}\\}.$

We consider the events:

$\displaystyle\mathcal{A}^{[1]}_{n}$
$\displaystyle:=\mathcal{A}_{s,x}^{K}(n)\cap\left\\{\operatorname{Diam}(S)\geq
2^{-(d-1)}\,n^{\frac{4}{5}}\right\\},$ $\displaystyle\mathcal{A}^{[2]}_{n}$
$\displaystyle:=\mathcal{A}_{s,x}^{K}(n)\cap\\{\exists i\neq j\in[d],\,\exists
S^{\prime}\subset S;\leavevmode\nobreak\ |S^{\prime}|=n^{\beta_{d}},\,\forall
z\neq z^{\prime}\in S^{\prime}\,z_{i}\neq z^{\prime}_{i},\,z_{j}\neq
z^{\prime}_{j},\,\\}.$

By Lemma 3.8,
$\mathcal{A}_{s,x}^{K}(n)\subset\mathcal{A}^{[1]}_{n}\cup\mathcal{A}^{[2]}_{n}$.
Hence, Lemma 3.7 follows from the following.

######  Lemma 3.9.

For $n\in\mathbb{N}$ large enough,

(3.11) $\displaystyle\mathbb{P}(\mathcal{A}_{n}^{[1]})$
$\displaystyle\leq\frac{1}{2}e^{n^{\alpha_{d}}}\mathbb{P}\left(\mathcal{A}^{\rm
free}_{s,x}(n)\right),$ (3.12)
$\displaystyle\mathbb{P}(\mathcal{A}_{n}^{[2]})$
$\displaystyle\leq\frac{1}{2}e^{n^{\alpha_{d}}}\mathbb{P}\left(\mathcal{A}^{\rm
free}_{s,x}(n)\right).$

The proof of Lemma 3.9 involves modifying the initial environment and
adjusting the time of the edges such that a new ball in the resampled
environment possesses a free line at its cut-point, meaning a line that
intersects the ball solely at the cut-point. To achieve this, we drill a line
near the end of geodesic $\gamma_{0,w_{n}}$, which is in close proximity to
the cut-point $w_{n}$. The primary challenge in this process is maintaining
the overall structure of the ball, ensuring the new space-time cut-point
remains close to the original one in both position and time. We need to be
cautious in order to prevent the creation of a shortcut within the ball during
the resampling process.

The first case (3.11) is simpler, as it pertains to situations where the
geodesic $\gamma_{0,w_{n}}$ does not wiggle too much (i.e. the diameter of $S$
is sufficiently large). In this case, we can identify a hyperplane
intersecting $S$ where the size of the ball’s intersection with the hyperplane
is negligible compared to $n$. As a result, we only incur a minimal cost to
maintain the connections after the shift and avoid a specific line. However,
the second case (3.12) is considerably more difficult. This case applies to
instances where the path has significant fluctuations around the endpoint, and
the intersections of the ball with the hyperplane are not negligible. In this
scenario, the shifting procedure differs; we employ two hyperplanes and shift
each quadrant. It is not feasible to preserve all connections within the ball,
as the expense may become exponentially high. Instead, we maintain the
connections of the geodesic while blocking connections around bad boxes and
the ball’s boundary. We will show that this resampling process does not
generate a shortcut inside the ball.

###### Proof of (3.11).

We suppose the event $\mathcal{A}_{n}^{[1]}$ occurs. Without loss of
generality, we assume $\operatorname{Diam}_{1}(S)\geq
n^{\frac{4}{5}}/2^{d-1}$. Then, there exists $S^{\prime}\subset
S=\\{w^{s_{n}-2n^{\alpha_{d}}},\cdots,w^{s_{n}-n^{\alpha_{d}}}\\}$ such that
$|S^{\prime}|=n^{4/5}/2^{d-1}$ and the hyperplanes $(\mathrm{H}_{1}(z),\,z\in
S^{\prime})$ are disjoint. Let $\Gamma\subset\mathbb{Z}^{d}$ be such that
$|\Gamma|\leq Kn$ and $\mathrm{B}_{s_{n}}\subset\mathrm{Int}(\Gamma)$. Then,
there exists $w^{*}=w^{s_{*}}\in S^{\prime}$ such that
$|(\mathrm{H}_{1}(w^{*}-\mathbf{e}_{1})\cup\mathrm{H}_{1}(w^{*}))\cap\Gamma|\leq
2^{d}Kn^{\frac{1}{5}}.$ Hence, by the isoperimetry (1.17), we have

(3.13)
$|(\mathrm{H}_{1}(w^{*}-\mathbf{e}_{1})\cup\mathrm{H}_{1}(w^{*}))\cap\mathrm{B}_{s_{n}}|\leq
Cn^{2/5}$

with some $C=C(d,K)\in\mathbb{N}$. We write
$\overline{\mathrm{H}}_{1}^{*}:=\mathrm{H}_{1}(w^{*}-\mathbf{e}_{1})\cup\mathrm{H}_{1}(w^{*})$
and $\mathrm{H}_{1}^{*}:=\mathrm{H}_{1}(w^{*})$.

Recall $(\tau_{e})_{e\in\mathbb{E}^{d}}$ from (1.7). Let $(\tau_{e}^{*})$ be
an independent copy of $(\tau_{e})$, and $\mathbf{X}=(X_{i})_{i=1}^{d}$ a
uniform random variable on $[-Kn,Kn]^{d}\cap\mathbb{Z}^{d}$ independent both
from $(\tau_{e}),(\tau^{*}_{e})$. Let $\wp_{n}=2Cn^{2/5}\in 2\mathbb{N}$. We
consider the following shift:

$\mathfrak{s}(x)=\mathfrak{s}_{\mathbf{X}}(x):=\begin{cases}x-(\wp_{n}-1)\mathbf{e}_{1}&\text{if
$x_{1}<X_{1}$},\\\ x+\wp_{n}\mathbf{e}_{1}&\text{if $x_{1}\geq
X_{1}$}.\end{cases}$

Then, we define the resampled configuration $(\tau_{e}^{\rm r})$ as follows:
for $e=\langle x,y\rangle$,

$\displaystyle\tau_{e}^{\rm r}:=\left\\{\begin{array}[]{ll}\tau_{e}^{*}&\text{
if $x_{1}$ or $y_{1}\in(X_{1}-\wp_{n},X_{1}+\wp_{n})$},\\\
\tau_{\langle\mathfrak{s}^{-1}(x),\mathfrak{s}^{-1}(y)\rangle}&\text{
otherwise}.\end{array}\right.$

In the proof, we simply write $\mathcal{D}=\mathcal{D}^{\mathcal{G}_{p}}$ for
the chemical distance for the configuration $\tau$. Denote by
$\mathcal{D}^{\rm r}$ the chemical distance for the configuration $\tau^{\rm
r}$. We need the following claim.

######  Claim 3.10.

Let $n\in\mathbb{N}$ and
$(x^{i})_{i=1}^{n},(y^{i})_{i=1}^{n}\subset\mathbb{Z}^{d}$ such that

$2n=-x_{1}^{i}=y^{i}_{1},\,x_{j}^{i}=y^{i}_{j}\,\forall j\geq 2.$

Then, there exist disjoint paths $(\gamma^{i})_{i=1}^{n}$ such that

$\gamma^{i}:x^{i}\to
y^{i},\,\gamma^{i}\subset(-2n,2n)\times\mathbb{Z}^{d-1}\cup\\{x^{i},y^{i}\\},\,\gamma^{i}\cap\mathrm{L}_{3}(0)=\emptyset,\,|\gamma^{i}|\leq
8n.$

###### Proof.

Without loss of generality, we can suppose $x^{1}_{2}\leq x^{2}_{2}\leq\cdots
x^{n}_{2}$. We denote by $L(x,y)$ the straight line between $x$ and $y$. By
abuse of notation, we can see $L(x,y)$ as a path by considering all the edges
$e$ such that $e\subset L(x,y)$. Set

$\ell:=\inf\\{k\geq 1:x_{2}^{k}\geq 0\\}.$

For $i<\ell$, we take $\gamma^{i}=L(x^{i},y^{i})$; for $i\geq\ell$, we take
$\gamma^{i}$ to be

$\displaystyle L(x^{i},x^{i}+2(n-i)\mathbf{e}_{1})\oplus
L(x^{i}+2(n-i)\mathbf{e}_{1},x^{i}+2(n-i)\mathbf{e}_{1}+\mathbf{e}_{2})$
$\displaystyle\qquad\oplus
L(x^{i}+2(n-i)\mathbf{e}_{1}+\mathbf{e}_{2},y^{i}-2(n-i)\mathbf{e}_{1}+\mathbf{e}_{2})\oplus
L(y^{i}-2(n-i)\mathbf{e}_{1}+\mathbf{e}_{2},y^{i}-2(n-i)\mathbf{e}_{1})$
$\displaystyle\qquad\qquad\oplus L(y^{i}-2(n-i)\mathbf{e}_{1},y^{i}).$

From the definition, $(\gamma^{i})_{i=1}^{n}$ are disjoint paths and do not
intersect $\mathrm{L}_{3}(0)$ (See Figure 4).

Figure 4. Construction of $(\gamma^{i})$ colored in red.

∎

From now on, we suppose $\mathbf{X}=w^{*}$. Since
$|\overline{\mathrm{H}}_{1}^{*}\cap\mathrm{B}_{s_{n}}|\leq\wp_{n}/2$, by Claim
3.10 with $\wp_{n}/2$ in place of $n$, there exist disjoint paths
$\bar{\gamma}_{x,y}=\bar{\gamma}_{y,x}:x-\wp_{n}\mathbf{e}_{1}\to
x+\wp_{n}\mathbf{e}_{1}$ for each pair
$x,y\in\overline{\mathrm{H}}_{1}^{*}\cap\mathrm{B}_{s_{n}}$ with
$y=x-\mathbf{e}_{1}$ such that

$\bar{\gamma}_{x,y}\backslash\\{x-\wp_{n}\mathbf{e}_{1},x+\wp_{n}\mathbf{e}_{1}\\}\subset\mathbb{Z}^{d}\backslash(\mathfrak{s}(\mathbb{Z}^{d})\cup\mathrm{L}_{3}(w^{*})),$

and $|\bar{\gamma}_{x,y}|\leq 4\wp_{n}$. We say that $e\in\mathbb{E}^{d}$
crosses $\mathrm{H}^{*}_{1}$ if there exists $z\in\mathrm{H}_{1}^{*}$ such
that $e=\langle z-\mathbf{e}_{i},z\rangle$. Note that here we make no
distinction on the orientation of the crossing. Let $(i_{j})_{j=1}^{k}$ be
such that $\langle w^{i_{j}},w^{i_{j}+1}\rangle$ is the j-th crossing of
$\gamma_{0,w_{n}}$ over $\mathrm{H}^{*}_{1}$ until $s_{*}$. Let us define

$\displaystyle\mathbf{E}_{1}$
$\displaystyle:=\left\\{e\in\mathbb{E}^{d}:\leavevmode\nobreak\
e\in\bar{\gamma}_{w^{i_{j}},w^{i_{j}+1}}\text{ with }j\leq k,\text{ or
}e\subset L(w^{*},\mathfrak{s}(w^{*}))\right\\},$
$\displaystyle\mathbf{E}_{2}$
$\displaystyle:=\\{e\in\mathbb{E}^{d}\backslash\mathbf{E}_{1}:\leavevmode\nobreak\
|e\cap e^{\prime}|=1\text{ with some $e^{\prime}\in\mathbf{E}_{1}$ or
$e\cap\mathfrak{s}(\mathrm{B}_{s_{n}}\cap\overline{\mathrm{H}}_{1}^{*})\neq\emptyset$}\\}.$

Then, we consider the event

$\displaystyle\mathcal{A}^{\rm
r}:=\left\\{\mathbf{X}=w^{*};\,\tau^{*}_{e}=1\,\text{ for
}e\in\mathbf{E}_{1};\,\tau^{*}_{e}=\infty\,\text{ for
}e\in\mathbf{E}_{2}\right\\}.$

Since $|\mathbf{E}_{1}\cup\mathbf{E}_{2}|\leq 4d\wp_{n}^{2}\leq
16C^{2}dn^{4/5}$ on $\mathcal{A}^{[1]}_{n}$, for $n$ large enough depending on
$p$, $d$, and $s_{0}$,

(3.14) $\begin{split}\mathbb{P}(\mathcal{A}^{\rm
r}\cap\mathcal{A}^{[1]}_{n})&=\mathbb{E}[\mathbb{P}(\mathbf{X}=w^{*},\,\tau^{*}_{e}=1\,\text{
for }e\in\mathbf{E}_{1};\,\tau^{*}_{e}=\infty\,\text{ for
}e\in\mathbf{E}_{2}|\leavevmode\nobreak\
\mathcal{A}^{[1]}_{n})\,\mathbf{1}_{\mathcal{A}^{[1]}_{n}}]\\\ \
&\geq\mathbb{E}[(2Kn+1)^{-d}p^{|\mathbf{E}_{1}|}(1-p)^{|\mathbf{E}_{2}|}\mathbf{1}_{\mathcal{A}^{[1]}_{n}}]\geq
e^{-c_{0}n^{4/5}}\mathbb{P}(\mathcal{A}^{[1]}_{n}),\end{split}$

where $c_{0}$ is a positive constant depending on $d$, $s_{0}$ and $p$. Set
$\widetilde{0}:=\mathfrak{s}_{\mathbf{X}}(0)$ and $t_{*}:=\mathcal{D}^{\rm
r}(\widetilde{0},w^{*})$. We will prove that on $\mathcal{A}^{\rm
r}\cap\mathcal{A}^{[1]}_{n}$, the following occur:

  1. (i)

$t_{*}\in[sn-3n^{\alpha_{d}},\infty)$ and
$w^{*}\in\Lambda_{4n^{\alpha_{d}}}(nx+\widetilde{0})$;

  2. (ii)

$\mathrm{L}_{3}(w^{*})\cap{\rm B}^{\rm r}_{t_{*}}(\widetilde{0})=\\{w^{*}\\}$;

  3. (iii)

$\forall k\neq 1\quad\mathrm{N}_{k}(\mathrm{H}_{1}^{*}\cap\mathrm{B}^{\rm
r}_{t_{*}}(\widetilde{0}))\leq C_{0}n^{4/5}$ where $C_{0}:=16dC^{2}$;

  4. (iv)

$|\mathrm{B}_{t_{*}}^{\rm r}(\widetilde{0})|\leq n^{7/4}$.

Since $(\tau^{\rm r}_{e+\widetilde{0}})\overset{\rm law}{=}(\tau_{e})$,
together with (3.14), this will imply

$\displaystyle\mathbb{P}\left(\begin{array}[]{c}\exists i\neq
j\in[d],\quad\exists t\geq sn-3n^{\alpha_{d}},\quad\exists
w_{n}\in\Lambda_{4n^{\alpha_{d}}}(nx);\\\ w_{n}\in{\rm B}_{t}\setminus{\rm
B}_{t-1},\,\mathrm{L}_{i}(w_{n})\cap\mathrm{B}_{t}=\\{w_{n}\\},\,\mathrm{N}_{k}(\mathrm{H}_{j}(w_{n})\cap\mathrm{B}_{t})\leq
C_{0}n^{4/5}\,\forall k\neq j,|\mathrm{B}_{t}|\leq n^{7/4}\end{array}\right)$
$\displaystyle\geq\mathbb{P}\left(\begin{array}[]{c}\exists i\neq
j\in[d],\quad\exists t\geq sn-3n^{\alpha_{d}},\quad\exists
w_{n}\in\Lambda_{4n^{\alpha_{d}}}(nx+\widetilde{0});\,w_{n}\in{\rm B}^{\rm
r}_{t}(\widetilde{0})\setminus{\rm B}^{\rm r}_{t-1}(\widetilde{0}),\\\
\mathrm{L}_{i}(w_{n})\cap B^{\rm
r}_{t}(\widetilde{0})=\\{w_{n}\\},\,\mathrm{N}_{k}(\mathrm{H}_{j}(w_{n})\cap\mathrm{B}^{\rm
r}_{t}(\widetilde{0}))\leq C_{0}n^{4/5}\,\forall k\neq j,|\mathrm{B}_{t}^{\rm
r}(\widetilde{0})|\leq n^{7/4}\end{array}\right)$ $\displaystyle\geq
e^{-c_{0}n^{4/5}}\mathbb{P}(\mathcal{A}^{[1]}_{n}).$

Together with Lemma 3.6, this yields the claim. Let us now prove the
conditions (i)–(iv).

Since $\|\widetilde{0}-0\|_{\infty}\leq\wp_{n}$ and
$\|w^{*}-w_{n}\|_{\infty}\leq 2n^{\alpha_{d}}$, for $n$ large enough depending
on $d,s_{0}$, we have $w^{*}\in\Lambda_{4n^{\alpha_{d}}}(nx+\widetilde{0})$.
We define a modified path:

$\displaystyle\widetilde{\gamma}$
$\displaystyle:=(\mathfrak{s}(w^{1}),\mathfrak{s}(w^{2}),\cdots,\mathfrak{s}(w^{i_{1}}))\oplus\bar{\gamma}_{w^{i_{1}},w^{i_{1}+1}}\oplus(\mathfrak{s}(w^{i_{1}+1}),\cdots,\mathfrak{s}(w^{i_{2}}))$
$\displaystyle\qquad\qquad\oplus\bar{\gamma}_{w^{i_{2}},w^{i_{2}+1}}\oplus\cdots\oplus(\mathfrak{s}(w^{i_{k}+1}),\cdots,\mathfrak{s}(w^{*}))\oplus
L(\mathfrak{s}(w^{*}),w^{*}).$

On the event $\mathcal{A}^{\rm r}\cap\mathcal{A}_{n}^{[1]}$, by (3.13),
$\widetilde{\gamma}$ is an open path from $\widetilde{0}=\mathfrak{s}(w^{1})$
to $w^{*}$ for $\tau^{\rm r}$ such that

$|\widetilde{\gamma}|\leq
s_{*}+|\overline{\mathrm{H}}_{1}^{*}\cap\mathrm{B}_{s_{n}}|\times
8\wp_{n}+\wp_{n}\leq s_{*}+5\wp_{n}^{2}.$

Hence, since $s_{*}\leq s_{n}-n^{\alpha_{d}}$, for $n$ large enough depending
on $p,d,s_{0}$,

(3.15) $\displaystyle\max\\{\mathcal{D}^{\rm
r}(\widetilde{0},\mathfrak{s}(w^{*})),\mathcal{D}^{\rm
r}(\widetilde{0},w^{*})\\}\leq s_{*}+20C^{2}n^{4/5}<s_{n}.$

Next, we will prove that

(3.16) $\displaystyle t_{*}:=\mathcal{D}^{\rm r}(\widetilde{0},w^{*})\geq
s_{*}-\wp_{n}.$

To this end, we take a geodesic $\gamma^{\rm r}=(v^{i})_{i=1}^{\ell}$ from
$\widetilde{0}$ to $\mathfrak{s}(w^{*})$ for $\tau^{\rm r}.$ Let

$\displaystyle i_{1}$ $\displaystyle:=\inf\\{k\geq 1:\leavevmode\nobreak\
\,v^{k}_{1}\in(w^{*}_{1}-\wp_{n},w^{*}_{1}+\wp_{n})\\}-1,$ $\displaystyle
j_{1}$ $\displaystyle:=\inf\\{k\geq i_{1}:\leavevmode\nobreak\
\,v^{k}_{1}\in(w^{*}_{1}-\wp_{n},w^{*}_{1}+\wp_{n})^{c}\\}.$

Assuming that $i_{1},j_{1},\dots,i_{l-1},j_{l-1}$ have been defined, we define

$\displaystyle i_{l}$ $\displaystyle:=\inf\\{k>j_{l-1}:\leavevmode\nobreak\
\,v^{k}_{1}\in(w^{*}_{1}-\wp_{n},w^{*}_{1}+\wp_{n})\\}-1,$ $\displaystyle
j_{l}$ $\displaystyle:=\inf\\{k\geq i_{l}:\leavevmode\nobreak\
\,v^{k}_{1}\in(w^{*}_{1}-\wp_{n},w^{*}_{1}+\wp_{n})^{c}\\}.$

We continue this procedure until $i_{l}=\infty$ (using the convention
$\inf\emptyset=\infty$). Let $m$ be the first number such that
$i_{m+1}=\infty$. Note that $i_{l+1}>j_{l}$ for any $l<m$, since $\gamma^{\rm
r}$ is self-avoiding. Let us prove by induction that for all ${l}\leq m$,

(3.17)
$\mathfrak{s}^{-1}(v^{i_{l}})\in\mathrm{B}_{s_{n}-1},\,\mathfrak{s}^{-1}(v^{j_{l}})\in\mathrm{B}_{s_{n}},\,\tau_{\langle\mathfrak{s}^{-1}(v^{i_{l}}),\mathfrak{s}^{-1}(v^{j_{l}})\rangle}=1,\text{
and }\mathcal{D}^{\rm
r}(\widetilde{0},v^{i_{l}})\geq\mathcal{D}(0,\mathfrak{s}^{-1}(v^{i_{l}})).$

We first note that the path
$(\mathfrak{s}^{-1}(v^{1}),\dots,\mathfrak{s}^{-1}(v^{i_{1}}))$ is open for
$\tau$ by definition of the resampled environment and
$(v^{1},\cdots,v^{i_{1}})\subset\mathfrak{s}(\mathbb{Z}^{d})$. Hence, by
(3.15),

$\displaystyle s_{n}>\mathcal{D}^{\rm
r}(\widetilde{0},v^{i_{1}})\geq\mathcal{D}(0,\mathfrak{s}^{-1}(v^{i_{l}})),$

which implies $\mathfrak{s}^{-1}(v^{i_{1}})\in\mathrm{B}_{s_{n}-1}$. Let us
assume $\mathfrak{s}^{-1}(v^{i_{l}})\in\mathrm{B}_{s_{n}-1}$ with $l<m$. Since
$\mathfrak{s}^{-1}(v^{i_{l}})\in\overline{\mathrm{H}}_{1}^{*}$,
$\mathfrak{s}^{-1}(v^{i_{l}})\in\mathrm{B}_{s_{n}}\cap\overline{\mathrm{H}}_{1}^{*}$.
On the event $\mathcal{A}^{\rm r}\cap\mathcal{A}_{n}^{[1]}$, we have
necessarily that
$\langle\mathfrak{s}^{-1}(v^{i_{l}}),\mathfrak{s}^{-1}(v^{j_{l}})\rangle$ is
an edge colinear to $\mathbf{e}_{1}$ and corresponds to a crossing of
$\gamma_{0,w_{n}}$ over $\mathrm{H}^{*}_{1}$. In particular, the edge
$\langle\mathfrak{s}^{-1}(v^{i_{l}}),\mathfrak{s}^{-1}(v^{j_{l}})\rangle$ is
open for $\tau$, and it follows that
$\mathfrak{s}^{-1}(v^{j_{l}})\in\mathrm{B}_{s_{n}}$. The path
$(\mathfrak{s}^{-1}(v^{j_{l}}),\dots,\mathfrak{s}^{-1}(v^{i_{l+1}}))$ is open
for $\tau$ by definition of the resampled environment. Recalling
$0=\mathfrak{s}^{-1}(v^{1})$, by induction hypothesis and (3.15),

$\begin{split}s_{n}>\mathcal{D}^{\rm
r}(\widetilde{0},v^{i_{l+1}})&=\mathcal{D}^{\rm
r}(\widetilde{0},v^{i_{l}})+\mathcal{D}^{\rm
r}(v^{i_{l}},v^{j_{l}})+\mathcal{D}^{\rm r}(v^{j_{l}},v^{i_{l+1}})\\\
&\geq\mathcal{D}(0,\mathfrak{s}^{-1}(v^{i_{l}}))+1+\mathcal{D}(\mathfrak{s}^{-1}(v^{j_{l}}),\mathfrak{s}^{-1}(v^{i_{l+1}}))\\\
&\geq\mathcal{D}(0,\mathfrak{s}^{-1}(v^{i_{l+1}})).\end{split}$

It follows that $\mathfrak{s}^{-1}(v^{i_{l+1}})\in\mathrm{B}_{s_{n}-1}$. Since
$\langle\mathfrak{s}^{-1}(v^{i_{l+1}}),\mathfrak{s}^{-1}(v^{j_{l+1}})\rangle$
is an open edge for $\tau$ on $\mathcal{A}^{\rm r}\cap\mathcal{A}_{n}^{[1]}$
as before, $\mathfrak{s}^{-1}(v^{j_{l+1}})\in\mathrm{B}_{s_{n}}$. This
concludes the induction.

Since $\mathfrak{s}(w^{*})\in\mathfrak{s}(\mathbb{Z}^{d})$, by maximality of
$m$, the path $(v^{j_{m}},\dots,v^{\ell})$ stays in
$\mathfrak{s}(\mathbb{Z}^{d})$. In particular, by definition of the
resampling, the path
$(\mathfrak{s}^{-1}(v^{j_{m}}),\cdots,\mathfrak{s}^{-1}(v^{\ell}))$ is open
for $\tau$. Thanks to (3.17),

(3.18) $\displaystyle\mathcal{D}^{\rm r}(\widetilde{0},\mathfrak{s}(w^{*}))$
$\displaystyle=\mathcal{D}^{\rm r}(\widetilde{0},v^{i_{m}})+\mathcal{D}^{\rm
r}(v^{i_{m}},v^{j_{m}})+\mathcal{D}^{\rm r}(v^{j_{m}},\mathfrak{s}(w^{*}))$
$\displaystyle\geq\mathcal{D}(0,\mathfrak{s}^{-1}(v^{i_{m}}))+1+\mathcal{D}(\mathfrak{s}^{-1}(v^{j_{m}}),w^{*})\geq\mathcal{D}(0,w^{*})=s_{*}.$

Thus, we obtain for $n$ large enough depending on $s_{0},p,d$,

$\displaystyle t_{*}$ $\displaystyle=\mathcal{D}^{\rm
r}(\widetilde{0},w^{*})\geq\mathcal{D}^{\rm
r}(\widetilde{0},\mathfrak{s}(w^{*}))-\mathcal{D}^{\rm
r}(w^{*},\mathfrak{s}(w^{*}))$ $\displaystyle\geq s_{*}-\wp_{n}\geq
sn-2n^{\alpha_{d}}-2Cn^{2/5}\geq sn-3n^{\alpha_{d}}.$

This concludes the proof of (i).

Next, we will prove that

(3.19) $\displaystyle\mathrm{B}^{\rm
r}_{t_{*}}(\widetilde{0})\subset\mathfrak{s}(\mathrm{B}_{s_{n}})\cup\bigcup_{j=1}^{k}\bar{\gamma}_{w^{i_{j}},w^{i_{j}+1}}\cup
L(w^{*},\mathfrak{s}(w^{*})).$

The proof is similar to that of (3.16). To this end, we take
$v\in\mathrm{B}^{\rm r}_{t_{*}}(\widetilde{0})$ arbitrary and take a geodesic
$\gamma^{\rm r}=(v^{i})_{i=1}^{\ell}$ from $\widetilde{0}$ to $v$. We define
$(i_{l},j_{l})_{l=1}^{m}$ as in (3.16) with $v$ in place of
$\mathfrak{s}(w^{*})$. If $v\in\mathfrak{s}(\mathbb{Z}^{d})$, then by (3.15),
the same argument as in (3.18) shows

$s_{n}>t_{*}\geq\mathcal{D}^{\rm
r}(\widetilde{0},v)\geq\mathcal{D}(0,\mathfrak{s}^{-1}(v)).$

Hence, $v\in\mathfrak{s}(\mathrm{B}_{s_{n}})$. Let us assume
$v\notin\mathfrak{s}(\mathbb{Z}^{d})$. Thanks to (3.17), we have
$\mathfrak{s}^{-1}(v^{i_{m}})\in\mathrm{B}_{s_{n}}\cap\mathrm{H}^{*}_{1}$.
Thus, on the event $\mathcal{A}^{\rm r}\cap\mathcal{A}_{n}^{[1]}$,
$v\in\bar{\gamma}_{w^{i_{j}},w^{i_{j}+1}}$ with some $j\leq k$ or $v\in
L(w^{*},\mathfrak{s}(w^{*}))$, and (3.19) follows. By the definition of
$\bar{\gamma}_{w^{i_{j}},w^{i_{j}+1}}$ and isoperimetry (1.17), since
$\mathfrak{s}$ is injective, one can check that (iv) follows from (3.19).

Finally, by $|\mathrm{B}_{s_{n}}\cap\mathrm{H}^{*}_{1}|\leq\wp_{n}/2$, (3.19),
and $\mathrm{H}_{1}^{*}\cap\mathfrak{s}(\mathbb{Z}^{d})=\emptyset$, we have

$|\mathrm{H}_{1}^{*}\cap{\rm B}^{\rm
r}_{t_{*}}(\widetilde{0})|\leq\left|\bigcup_{j=1}^{k}\bar{\gamma}_{w^{i_{j}},w^{i_{j}+1}}\cup
L(w^{*},\mathfrak{s}(w^{*}))\right|\leq 4d\wp_{n}^{2}\leq 16dC^{2}n^{4/5}.$

Thus, we have $\mathrm{N}_{k}(\mathrm{H}_{1}^{*}\cap\mathrm{B}^{\rm
r}_{t}(\widetilde{0}))\leq 16dC^{2}n^{4/5}$ for $k\neq 1$. Therefore, we
obtain (iii) for $n$ large enough depending on $s_{0},p,d$. This concludes the
proof. ∎

###### Proof of (3.12).

Without loss of generality, we assume $\mathcal{A}^{[2]}_{n}$ occurs with
$i=1,\,j=2$. Thanks to (1.9), there exists $c\in(0,1)$ depending only on
$s_{0},p,d$ such that

$\mathbb{P}(\exists w\in\mathbb{Z}^{d};\,\|w\|_{1}\leq 2s_{0}n,\,n\log
n\leq\mathcal{D}^{\mathcal{G}_{p}}(0,w)<\infty)\leq(5s_{0}n)^{d}e^{-cn\log
n}.$

Let $\varepsilon:=1/2$. Set $N:=\log^{2}{n}$ and

$\mathbf{V}_{n}:=\\{x\in[-n^{2},n^{2}]^{d}{\cap\mathbb{Z}^{d}}:\leavevmode\nobreak\
x\in\underline{\Lambda}_{N}(\mathbf{i})\text{ with $\mathbf{i}$:
$\varepsilon$-bad}\\}.$

Thanks to Lemma 2.7, we have

(3.20) $\mathbb{P}(|\mathbf{V}_{n}|\geq(2N)^{d}n)\leq e^{-n\log n}.$

Therefore, we have

$\mathbb{P}(\mathcal{A}_{n}^{[2]})\leq\mathbb{P}(\mathcal{A}_{n}^{[2]},|\mathbf{V}_{n}|\leq(2N)^{d}n,s_{n}\leq
n\log n)+2(5s_{0}n)^{d}e^{-cn\log n}.$

We distinguish two cases, either $\mathbb{P}(\mathcal{A}_{n}^{[2]})\leq
e^{-\frac{c}{2}n\log n}$ or
$\mathbb{P}(\mathcal{A}_{n}^{[2]})>e^{-\frac{c}{2}n\log n}$. There is no need
to examine the first case since it is negligible compared to
$\mathbb{P}(\mathcal{A}_{n}^{[1]})$ by (3.5). For the second case, we have for
$n$ large enough,

$\mathbb{P}(\mathcal{A}_{n}^{[2]})\leq
2\mathbb{P}(\mathcal{A}_{n}^{[2]},|\mathbf{V}_{n}|\leq(2N)^{d}n,s_{n}\leq
n\log n).$

From now on, with abuse of notation, we write $\mathcal{A}_{n}^{[2]}$ instead
of
$\mathcal{A}_{n}^{[2]}\cap\\{|\mathbf{V}_{n}|\leq(2N)^{d}n\\}\cap\\{s_{n}\leq
n\log n\\}$. Let $S^{\prime}\subset S$ and $\Gamma\subset\mathbb{Z}^{d}$ be
such that $|\Gamma|\leq Kn$ and
$\mathrm{B}_{s_{n}}\subset\mathrm{Int}(\Gamma)$ as in the event
$\mathcal{A}_{n}^{[2]}$. Recall that $\gamma_{0,w_{n}}$ is a geodesic from $0$
to $w_{n}$. Since the hyperplanes $(\mathrm{H}_{1}(z),z\in S^{\prime})$ are
disjoint by definition of $S^{\prime}$, there exist
$(z^{k})_{k=1}^{n^{\beta_{d}}/2}\subset S^{\prime}$ such that for any
$k\in[n^{\beta_{d}}/2]$,

$|({\rm H}_{1}(z^{k})\cup{\rm
H}_{1}(z^{k}-\mathbf{e}_{1}))\cap(\gamma_{0,w_{n}}\cup\mathbf{V}_{n}\cup\Gamma)|\leq\frac{5|\gamma_{0,w_{n}}\cup\mathbf{V}_{n}\cup\Gamma|}{n^{\beta_{d}}}.$

Similarly, there exists $w^{*}\in(z^{k})_{k=1}^{n^{\beta_{d}}/2}$ such that

$|({\rm H}_{2}(w^{*})\cup{\rm
H}_{2}(w^{*}-\mathbf{e}_{2}))\cap(\gamma_{0,w_{n}}\cup\mathbf{V}_{n}\cup\Gamma)|\leq\frac{5|\gamma_{0,w_{n}}\cup\mathbf{V}_{n}\cup\Gamma|}{n^{\beta_{d}}}.$

Therefore,

$\displaystyle|({\rm H}_{1}(w^{*})\cup{\rm
H}_{1}(w^{*}-\mathbf{e}_{1})\cup{\rm H}_{2}(w^{*})\cup{\rm
H}_{2}(w^{*}-\mathbf{e}_{2}))\cap(\gamma_{0,w_{n}}\cup\mathbf{V}_{n}\cup\Gamma)|$
$\displaystyle\quad\leq|({\rm H}_{1}(w^{*})\cup{\rm
H}_{1}(w^{*}-\mathbf{e}_{1}))\cap(\gamma_{0,w_{n}}\cup\mathbf{V}_{n}\cup\Gamma)|+|({\rm
H}_{2}(w^{*})\cup{\rm
H}_{2}(w^{*}-\mathbf{e}_{2}))\cap(\gamma_{0,w_{n}}\cup\mathbf{V}_{n}\cup\Gamma)|$
$\displaystyle\quad\leq\frac{10|\gamma_{0,w_{n}}\cup\mathbf{V}_{n}\cup\Gamma|}{n^{\beta_{d}}}.$

Recall that $s_{n}\leq n\log n$ on $\mathcal{A}_{n}^{[2]}$ and so
$|\gamma_{0,w_{n}}|\leq n\log n$. Thus, on the event $\mathcal{A}_{n}^{[2]}$,
we have $|\gamma_{0,w_{n}}\cup\mathbf{V}_{n}\cup\Gamma|\leq 4^{d}nN^{d}$. Let
$s_{*}\in\\{s_{n}-2n^{\alpha_{d}}+1,\cdots,s_{n}-n^{\alpha_{d}}\\}$ be such
that $w^{*}=w^{s_{*}}$. Set $\mathrm{H}^{*}:={\rm H}_{1}(w^{*})\cup{\rm
H}_{2}(w^{*})\cup{\rm H}_{1}(w^{*}-\mathbf{e}_{1})\cup{\rm
H}_{2}(w^{*}-\mathbf{e}_{2}).$ It yields

(3.21)
$\displaystyle|\mathrm{H}^{*}\cap(\gamma_{0,w_{n}}\cup\mathbf{V}_{n}\cup\Gamma)|\leq
40^{d}n^{1-\beta_{d}}N^{d}.$

Let $(\tau_{e}^{*})$ be an independent copy of $(\tau_{e})$, and
$\mathbf{X}=(X_{i})_{i=1}^{d}$ an independent uniform random variable on
$[-Kn,Kn]^{d}\cap\mathbb{Z}^{d}$. We consider the following shift: for
$x=(x_{i})_{i=1}^{d}\in\mathbb{Z}^{d}$,

$\mathfrak{s}(x)=\mathfrak{s}_{\mathbf{X}}(x):=\begin{cases}x+\mathbf{e}_{1}+\mathbf{e}_{2}&\text{if
$x_{1}\geq X_{1}$ and $x_{2}\geq X_{2}$},\\\
x-\mathbf{e}_{1}+\mathbf{e}_{2}&\text{if $x_{1}<X_{1}$ and $x_{2}\geq
X_{2}$},\\\ x-\mathbf{e}_{1}-\mathbf{e}_{2}&\text{if $x_{1}<X_{1}$ and
$x_{2}<X_{2}$},\\\ x+\mathbf{e}_{1}-\mathbf{e}_{2}&\text{if $x_{1}\geq X_{1}$
and $x_{2}<X_{2}$}.\end{cases}$

Figure 5. Grey region represents $\mathfrak{s}_{\mathbf{X}}(\mathbb{Z}^{d})$.
Red lines represent $\mathrm{H}(\mathbf{X})$

We note that for any $x\in\mathbb{Z}^{d}$,
$\mathfrak{s}_{\mathbf{X}}(x)_{i}\neq X_{i}$ for $i=1,2$,
$\mathrm{H}^{*}=\mathbb{Z}^{d}\setminus\mathfrak{s}_{\mathbf{X}}(\mathbb{Z}^{d})$
and $\mathfrak{s}_{\mathbf{X}}$ is injective. We define the resampled
configuration $(\tau_{e}^{\rm r})$: for $e=\langle x,y\rangle$,

$\displaystyle\tau^{\rm
r}_{e}=\left\\{\begin{array}[]{ll}\tau_{\langle\mathfrak{s}^{-1}(x),\mathfrak{s}^{-1}(y)\rangle}&\text{
if $x,y\in\mathfrak{s}(\mathbb{Z}^{d})$},\\\ \tau_{e}^{*},&\text{
otherwise}.\end{array}\right.$

From now on, we assume $\mathbf{X}=w^{*}$. We write
$\mathrm{H}_{1}^{*}=\mathrm{H}_{1}(w^{*}),\mathrm{H}_{2}^{*}=\mathrm{H}_{2}(w^{*})$.
For any $x\in\mathrm{H}^{*}_{1}$, we consider the straight path
$\bar{\gamma}_{x-\mathbf{e}_{1},x}=\bar{\gamma}_{x,x-\mathbf{e}_{1}}$ between
$\mathfrak{s}(x)$ and $\mathfrak{s}(x-\mathbf{e}_{1})$; for any
$x\in\mathrm{H}^{*}_{2}$, we consider the straight path
$\bar{\gamma}_{x-\mathbf{e}_{2},x}=\bar{\gamma}_{x,x-\mathbf{e}_{2}}$ between
$\mathfrak{s}(x)$ and $\mathfrak{s}(x-\mathbf{e}_{2})$. In particular, these
paths are made of three edges. We say that $e\in\mathbb{E}^{d}$ crosses
$\mathrm{H}^{*}$ if there exist $i\in\\{1,2\\}$ and $z\in\mathrm{H}^{*}_{i}$
such that $e=\langle z-\mathbf{e}_{i},z\rangle$. Let $(i_{j})_{j=1}^{k}$ be
such that $\langle w^{i_{j}},w^{i_{j}+1}\rangle$ is the $j$-th crossing of
$\gamma_{0,w_{n}}$ over $\mathrm{H}^{*}$ until $s_{*}$. Since
$\gamma_{0,w_{n}}$ is self-avoiding and $w^{s_{*}}=w^{*}$, we have
$w^{i_{j}}\neq w^{s_{*}}$. Moreover, there exists a path
$\bar{\gamma}_{\mathfrak{s}(w^{*}),w^{*}}$ between $\mathfrak{s}(w^{*})$ and
$w^{*}$ with $|\bar{\gamma}_{\mathfrak{s}(w^{*}),w^{*}}|=2$ that do not have
any edge in common with $\bar{\gamma}_{w^{i_{j}},w^{i_{j}+1}}$ $j\leq k$;
namely

$\bar{\gamma}_{\mathfrak{s}(w^{*}),w^{*}}:=\left\\{\begin{array}[]{ll}(\mathfrak{s}_{w^{*}}(w^{*}),\mathfrak{s}_{w^{*}}(w^{*})-\mathbf{e}_{1},w^{*})&\mbox{if
$w^{s_{*}-1}=w^{s_{*}}-\mathbf{e}_{2}$},\\\
(\mathfrak{s}_{w^{*}}(w^{*}),\mathfrak{s}_{w^{*}}(w^{*})-\mathbf{e}_{2},w^{*})&\mbox{otherwise.}\end{array}\right.$

Define

$\displaystyle\mathbf{E}_{1}$
$\displaystyle:=\\{e\in\mathbb{E}^{d}:\leavevmode\nobreak\
e\in\bar{\gamma}_{w^{i_{j}},w^{i_{j}+1}}\text{ with }j\leq k,\,\text{ or
}e\in\bar{\gamma}_{\mathfrak{s}(w^{*}),w^{*}}\\},$
$\displaystyle\mathbf{E}_{2}$
$\displaystyle:=\\{e\in\mathbb{E}^{d}\setminus\mathbf{E}_{1}:\leavevmode\nobreak\
\mathrm{d}_{\infty}(e,(\gamma_{0,w_{n}}\cup\mathbf{V}_{n}\cup\Gamma)\cap\mathrm{H}^{*})\leq
2N\\},$ $\displaystyle\mathbf{E}_{3}$
$\displaystyle:=\\{e\in\mathbb{E}^{d}\setminus\mathbf{E}_{1}:\leavevmode\nobreak\
|e\cap\mathrm{L}_{3}(w^{*})|=1,\,\mathrm{d}_{\infty}(e,\mathrm{Int}(\Gamma)\cap\mathrm{H}^{*})\leq
N\\}.$

Let $\overline{\rm H}_{1}^{*}={\rm H}_{1}(w^{*})\cup{\rm
H}_{1}(w^{*}-\mathbf{e}_{1})$ and $\overline{{\rm H}}_{2}^{*}={\rm
H}_{2}(w^{*})\cup{\rm H}_{2}(w^{*}-\mathbf{e}_{2})$. We now prove

(3.22) $\displaystyle|\mathbf{E}_{3}|\leq
12dN(\mathrm{N}_{2}(\mathrm{Int}(\Gamma)\cap\overline{\rm
H}_{1}^{*})+\mathrm{N}_{1}(\mathrm{Int}(\Gamma)\cap\overline{\rm
H}_{2}^{*})).$

Let us take $e=\langle u,v\rangle\in\mathbf{E}_{3}$ with
$u=u^{e}\in\mathrm{L}_{3}(w^{*})$. Note that $u_{3}=v_{3}$. Then, we select
$z^{e}\in\mathrm{Int}(\Gamma)\cap\mathrm{H}^{*}$ such that
$|z^{e}_{3}-u_{3}|\leq N$. If
$z^{e}\in\mathrm{Int}(\Gamma)\cap\overline{\mathrm{H}}_{1}^{*}$, setting
$g^{e}_{2}=0$ and $g_{i}^{e}=z_{i}^{e}$ for $i\neq 2$, we have
$z^{e}\in\mathrm{L}_{2}(g^{e})\cap\mathrm{Int}(\Gamma)\cap\overline{\rm
H}_{1}^{*}$, in particular
$\mathrm{L}_{2}(g^{e})\cap\mathrm{Int}(\Gamma)\cap\overline{\rm
H}_{1}^{*}\neq\emptyset$. Note that
$u^{e}_{3}\in[g_{3}^{e}-N,g_{3}^{e}+N]\cap\mathbb{Z}$. Similarly, if
$z^{e}\in\mathrm{Int}(\Gamma)\cap\mathrm{H}_{2}^{*}$, setting $r^{e}_{1}=0$
and $r_{i}^{e}=z_{i}^{e}$ for $i\neq 1$, we have
$\mathrm{L}_{1}(r^{e})\cap\mathrm{Int}(\Gamma)\cap\overline{\rm
H}_{2}^{*}\neq\emptyset$. Note that
$u^{e}_{3}\in[r_{3}^{e}-N,r_{3}^{e}+N]\cap\mathbb{Z}$. Therefore, (3.22)
follows from the fact that $|A|\leq|B|\max_{y\in B}|f^{-1}(y)|$ for any map
$f:A\to B$, and for any $g=(g_{i})\in\mathbb{Z}^{d}$ and
$r=(r_{i})\in\mathbb{Z}^{d}$ with $g_{2}=0$ and $r_{1}=0$,

$\max\\{|\\{e\in\mathbf{E}_{3}:\,\leavevmode\nobreak\
g=g^{e}\\}|,|\\{e\in\mathbf{E}_{3}:\,\leavevmode\nobreak\ r=r^{e}\\}|\\}\leq
12dN.$

Next we estimate $\mathrm{N}_{k}(\\{z\in\mathbb{Z}^{d}:\leavevmode\nobreak\
\mathrm{d}_{\infty}(z,\mathrm{Int}(\Gamma)\cap\overline{\rm H}_{1}^{*})\leq
N\\})$ for $k\neq 1$. Note that we here consider a stronger estimate for a
later purpose. Let $z\in\mathbb{Z}^{d}$ be such that $z_{k}=0$ and
$\mathrm{L}_{k}(z)\cap\\{z^{\prime}\in\mathbb{Z}^{d}:\leavevmode\nobreak\
\mathrm{d}_{\infty}(z^{\prime},\mathrm{Int}(\Gamma)\cap\overline{\rm
H}_{1}^{*})\leq N\\}\neq\emptyset$. Then, there exists
$z^{0}=(z^{0}_{i})\in\mathbb{Z}^{d}$ such that $\|z-z^{0}\|_{\infty}\leq N$,
$z^{0}_{k}=0$, and
$\mathrm{L}_{k}(z^{0})\cap\mathrm{Int}(\Gamma)\cap\overline{\rm
H}_{1}^{*}\neq\emptyset$. It follows that
$\mathrm{L}_{k}(z^{0})\cap\Gamma\cap\overline{\rm H}_{1}^{*}\neq\emptyset$.
Recalling that $|\Gamma\cap\mathrm{H}^{*}|\leq 40^{d}n^{1-\beta_{d}}N^{d}$,
this yields

(3.23)
$\begin{split}\mathrm{N}_{k}&(\\{z\in\mathbb{Z}^{d}:\leavevmode\nobreak\
\mathrm{d}_{\infty}(z,\mathrm{Int}(\Gamma)\cap\overline{\rm H}_{1}^{*})\leq
N\\})\\\ &\hskip 28.45274pt\leq|\\{z^{0}\in\mathbb{Z}^{d}:\leavevmode\nobreak\
\|z^{0}\|_{\infty}\leq N\\}|\times|\Gamma\cap\overline{\rm H}_{1}^{*}|\leq
10^{2d}n^{1-\beta_{d}}N^{2d}.\end{split}$

By the same argument, we have for $k\neq 2$,

$\mathrm{N}_{k}(\\{z\in\mathbb{Z}^{d}:\leavevmode\nobreak\
\mathrm{d}_{\infty}(z,\mathrm{Int}(\Gamma)\cap\overline{\rm H}_{2}^{*})\leq
N\\})\leq 10^{2d}n^{1-\beta_{d}}N^{2d}.$

Combined with (3.22) and (3.23), this gives

(3.24) $|\mathbf{E}_{3}|\leq 10^{3d}n^{1-\beta_{d}}N^{2d+1}.$

We consider the event

$\displaystyle\mathcal{A}^{\rm
r}:=\left\\{\mathbf{X}=w^{*};\,\tau^{*}_{e}=1\,\text{ for
}e\in\mathbf{E}_{1};\,\tau^{*}_{e}=\infty\,\text{ for
}e\in\mathbf{E}_{2}\cup\mathbf{E}_{3}\right\\}.$

Thanks to (3.21),
$|\mathbf{E}_{1}\cup\mathbf{E}_{2}|\leq{10^{3d}}n^{1-\beta_{d}}N^{2d}$. Recall
that $1-\beta_{d}<\alpha_{d}$. Combining this with (3.24), we have for $n$
large enough depending on $s_{0},d,p$,

$\displaystyle\mathbb{P}(\mathcal{A}^{\rm r}\cap\mathcal{A}^{[2]}_{n})$
$\displaystyle=\mathbb{E}[\mathbb{P}(\mathcal{A}^{\rm r}|\leavevmode\nobreak\
\mathcal{A}^{[2]}_{n})\mathbf{1}_{\mathcal{A}^{[2]}_{n}}]$
$\displaystyle\geq\mathbb{E}[(2Kn+1)^{-d}p^{|\mathbf{E}_{1}|}(1-p)^{|\mathbf{E}_{2}\cup\mathbf{E}_{3}|}\mathbf{1}_{\mathcal{A}^{[2]}_{n}}]\geq
e^{-n^{\alpha_{d}}}\mathbb{P}(\mathcal{A}^{[2]}_{n}).$

Let $\widetilde{0}:=\mathfrak{s}_{\mathbf{X}}(0)$ and $t_{*}:=\mathcal{D}^{\rm
r}(\widetilde{0},w^{*})$. We will prove that on $\mathcal{A}^{\rm
r}\cap\mathcal{A}^{[2]}_{n}$, the following occur:

  1. (i)

$t_{*}\in[sn-3n^{\alpha_{d}},\infty)$ and
$w^{*}\in\Lambda_{4n^{\alpha_{d}}}(nx+\widetilde{0})$;

  2. (ii)

$\mathrm{L}_{3}(w^{*})\cap{\rm B}^{\rm r}_{t_{*}}(\widetilde{0})=\\{w^{*}\\}$;

  3. (iii)

$\forall k\neq 1\quad\mathrm{N}_{k}(\mathrm{H}_{1}^{*}\cap\mathrm{B}^{\rm
r}_{t_{*}}(\widetilde{0}))\leq n^{\alpha_{d}}$;

  4. (iv)

$|\mathrm{B}_{t_{*}}^{\rm r}(\widetilde{0})|\leq n^{7/4}$.

Since $(\tau^{\rm r}_{e+\widetilde{0}})\overset{\rm law}{=}(\tau_{e})$, this
implies

$\displaystyle\mathbb{P}\left(\begin{array}[]{c}\exists i\neq
j\in[d],\quad\exists t\geq sn-3n^{\alpha_{d}},\quad\exists
w_{n}\in\Lambda_{4n^{\alpha_{d}}}(nx);\\\ w_{n}\in{\rm B}_{t}\setminus{\rm
B}_{t-1},\,\mathrm{L}_{i}(w_{n})\cap\mathrm{B}_{t}=\\{w_{n}\\},\,\mathrm{N}_{k}(\mathrm{H}_{j}(w_{n})\cap\mathrm{B}_{t})\leq
n^{\alpha_{d}}\,\forall k\neq j,\,|\mathrm{B}_{t}|\leq
n^{7/4}\end{array}\right)$
$\displaystyle\geq\mathbb{P}\left(\begin{array}[]{c}\exists i\neq
j\in[d],\quad\exists t\geq sn-3n^{\alpha_{d}},\quad\exists
w_{n}\in\Lambda_{4n^{\alpha_{d}}}(nx+\widetilde{0});\,w_{n}\in{\rm B}^{\rm
r}_{t}(\widetilde{0})\setminus{\rm B}^{\rm r}_{t-1}(\widetilde{0}),\\\
\mathrm{L}_{i}(w_{n})\cap\mathrm{B}^{\rm
r}_{t}(\widetilde{0})=\\{w_{n}\\},\,\mathrm{N}_{k}(\mathrm{H}_{j}(w_{n})\cap\mathrm{B}^{\rm
r}_{t}(\widetilde{0}))\leq n^{\alpha_{d}}\,\forall k\neq
j,\,|\mathrm{B}_{t}^{\rm r}(\widetilde{0})|\leq n^{7/4}\end{array}\right)$
$\displaystyle\geq e^{-n^{\alpha_{d}}}\mathbb{P}(\mathcal{A}^{[2]}_{n}).$

Combined with Lemma 3.6, this yields the claim.

We prepare some claims to prove (i)–(iv). Given $x\neq y\in\mathrm{H}^{*}$, we
define

$x\overset{\rm H}{\sim}y\overset{\rm
def}{\Leftrightarrow}\exists\gamma:x\leftrightarrow
y\subset\mathrm{H}^{*}\setminus(\gamma_{0,w_{n}}\cup\Gamma\cup\mathbf{V}_{n}),$

with the convention $x\overset{\rm H}{\sim}x$. Let
$\mathcal{C}_{\mathrm{H}}(x)$ be the set of all points connected to
$x\in\mathrm{H}^{*}$ for the relation $\overset{\rm H}{\sim}$. Note that this
relation only depends on the configuration $\tau$. Note that

(3.25)
$x\in\gamma_{0,w_{n}}\Rightarrow\mathcal{C}_{\mathrm{H}}(x)=\\{x\\},\quad\langle
x,y\rangle\in\gamma_{0,w_{n}}\Rightarrow y\notin\mathcal{C}_{H}(x).$

######  Claim 3.11.

Assume $\mathcal{A}^{\rm r}\cap\mathcal{A}^{[2]}_{n}$. Let
$a,b\in\mathfrak{s}(\mathrm{H}^{*})$ and a self-avoiding open path
$\gamma:a\leftrightarrow b$ for $\tau^{\rm r}$ such that
$\gamma\backslash\\{a,b\\}\subset\mathrm{H}^{*}$. Then
$\mathfrak{s}^{-1}(a)\overset{\rm H}{\sim}\mathfrak{s}^{-1}(b)$ or
$\langle\mathfrak{s}^{-1}(a),\mathfrak{s}^{-1}(b)\rangle\in\gamma_{0,w_{n}}$.

###### Proof.

Note that
$\mathfrak{s}^{-1}(a)\oplus(\gamma\setminus\\{a,b\\})\oplus\mathfrak{s}^{-1}(b)$
is a $\mathbb{Z}^{d}$-path inside $\mathrm{H}^{*}$. We extract a self-avoiding
path $\gamma^{\prime}$ from this path. If
$\gamma^{\prime}\cap({\gamma_{0,w_{n}}\cup}\Gamma\cup\mathbf{V}_{n})=\emptyset$,
then by definition we have $\mathfrak{s}^{-1}(a)\overset{\rm
H}{\sim}\mathfrak{s}^{-1}(b)$. Otherwise, there exists
$e\in\gamma\setminus\\{a,b\\}$ at $\ell_{1}$-distance less than $1$ from
${\gamma_{0,w_{n}}\cup}\Gamma\cup\mathbf{V}_{n}$. On the event
$\mathcal{A}^{\rm r}\cap\mathcal{A}_{n}^{[2]}$, this implies that
$e\in\mathbf{E}_{1}$ since otherwise the edge would be closed for $\tau^{\rm
r}$. On the event $\mathcal{A}^{\rm r}\cap\mathcal{A}_{n}^{[2]}$, since the
edges with one endpoint in common with the set $\mathbf{E}_{1}$ are closed,
the path $\gamma\setminus\\{a,b\\}$ has to be included in
$\bar{\gamma}_{\mathfrak{s}(w^{*}),w^{*}}$ or
$\bar{\gamma}_{w^{i_{j}},w^{i_{j}+1}}$ for some $j$. However, since $\gamma$
is a path between $a,b\in\mathfrak{s}(\mathrm{H}^{*})$, it cannot be included
in $\bar{\gamma}_{\mathfrak{s}(w^{*}),w^{*}}$. Therefore, $a$ and $b$ are the
extremities of $\bar{\gamma}_{w^{i_{j}},w^{i_{j}+1}}$ for some $j$ and
$\langle\mathfrak{s}^{-1}(a),\mathfrak{s}^{-1}(b)\rangle\in\gamma_{0,w_{n}}$.

∎

Recall that $\mathscr{C}(\mathbf{a})$ denotes the largest open cluster of the
$\varepsilon$-good box $\underline{\Lambda}^{\prime}_{N}(\mathbf{a})$.

######  Claim 3.12.

Let $\mathbf{a},\mathbf{b}\in\mathbb{Z}^{d}$ be good macroscopic sites. Let
$a\in\mathrm{H}^{*}\cap\mathscr{C}(\mathbf{a})$ and
$b\in\mathscr{C}(\mathbf{b})\cap\mathcal{C}_{\mathrm{H}}(a)$. It holds that

$\mathcal{D}(a,b)\leq
8d^{2}\mu(\mathbf{e}_{1})N|\\{e\in\mathbb{E}^{d}:\leavevmode\nobreak\
e\subset\mathrm{H}^{*},\,|e\cap\mathcal{C}_{\mathrm{H}}(a)|=1\\}|.$

###### Proof.

We claim that there exists a $\mathbb{Z}^{d}$-path
$\gamma\subset\mathcal{C}_{H}(a)$ from $a$ to $b$ in $\mathrm{H}^{*}$ such
that

$|\gamma|\leq{4d}|\\{e\in\mathbb{E}^{d}:\leavevmode\nobreak\
e\subset\mathrm{H}^{*},\,|e\cap\mathcal{C}_{\mathrm{H}}(a)|=1\\}|.$

Indeed, consider a $\mathbb{Z}^{d}$-path $\gamma^{\prime}$ from $a$ to $b$
such that $\gamma^{\prime}\subset\mathrm{H}^{*}$ and
$|\gamma^{\prime}|=\|a-b\|_{1}$; we arbitrarily take a path $\gamma$ from $a$
to $b$ in

$(\gamma^{\prime}\cap\mathcal{C}_{\mathrm{H}}(a))\cup\\{x\in\mathcal{C}_{\mathrm{H}}(a):\exists
y\in{\mathrm{H}^{*}\setminus}\mathcal{C}_{\mathrm{H}}(a)\,\,\|x-y\|_{\infty}=1\\}.$

Note that the set $\\{x\in\mathcal{C}_{\mathrm{H}}(a):\exists
y\in{\mathrm{H}^{*}\setminus}\mathcal{C}_{\mathrm{H}}(a)\,\,\|x-y\|_{\infty}=1\\}$
is $\mathbb{Z}^{d}$-connected (see for instance [27, Lemma 2.23]). Moreover,
we have

$|\\{x\in\mathcal{C}_{\mathrm{H}}(a):\exists
y\in{\mathrm{H}^{*}\setminus}\mathcal{C}_{\mathrm{H}}(a)\,\,\|x-y\|_{\infty}=1\\}|\leq
2d|\\{e\in\mathbb{E}^{d}:\leavevmode\nobreak\
e\subset\mathrm{H}^{*},\,|e\cap\mathcal{C}_{\mathrm{H}}(a)|=1\\}|.$

Note that since $b\in\mathcal{C}_{\mathrm{H}}(a)$,

$\|a-b\|_{1}\leq|\\{x\in\mathcal{C}_{\mathrm{H}}(a):\exists
y\in{\mathrm{H}^{*}\setminus}\mathcal{C}_{\mathrm{H}}(a)\,\,\|x-y\|_{1}=1\\}|\leq|\\{e\in\mathbb{E}^{d}:\leavevmode\nobreak\
e\subset\mathrm{H}^{*},\,|e\cap\mathcal{C}_{\mathrm{H}}(a)|=1\\}|.$

This yields the existence of such a path $\gamma$. By construction, $\gamma$
only intersects good boxes since
$\mathcal{C}_{\mathrm{H}}(a)\cap(\gamma_{0,w_{n}}\cup\Gamma\cup\mathbf{V}_{n})=\emptyset$
and $\gamma\subset\mathcal{C}_{\mathrm{H}}(a)$. The path $\gamma$ intersects
at most $|\gamma|$ boxes and all these boxes are good by construction and
connected. By Lemma 2.9, there exists an open path from $a$ to $b$ of length
at most $2d\mu(\mathbf{e}_{1})N|\gamma|$ and

$\displaystyle\mathcal{D}(a,b)$ $\displaystyle\leq
2d\mu(\mathbf{e}_{1})N|\gamma|\leq
8d^{2}\mu(\mathbf{e}_{1})N|\\{e\in\mathbb{E}^{d}:\leavevmode\nobreak\
e\subset\mathrm{H}^{*},\,|e\cap\mathcal{C}_{\mathrm{H}}(a)|=1\\}|.$

∎

Proof of (i). Note that $\|0-\widetilde{0}\|_{\infty}\leq 1$ and
$\|w^{*}-w_{n}\|_{\infty}<{2n^{\alpha_{d}}}$. It follows from
$w_{n}\in\Lambda_{n^{\alpha_{d}}}(nx)$ that
$w^{*}\in\Lambda_{4n^{\alpha_{d}}}(\widetilde{0}+nx)\,.$ We define a modified
path as

$\displaystyle\widetilde{\gamma}$
$\displaystyle:=(\widetilde{0},\mathfrak{s}(w^{2})\cdots,\mathfrak{s}(w^{i_{1}}))\oplus\bar{\gamma}_{w^{i_{1}},w^{i_{1}+1}}\oplus(\mathfrak{s}(w^{i_{1}+1}),\cdots,\mathfrak{s}(w^{i_{2}}))\oplus\bar{\gamma}_{w^{i_{2}},w^{i_{2}+1}}$
$\displaystyle\qquad\oplus\cdots\oplus(\mathfrak{s}(w^{i_{k}+1}),\cdots,\mathfrak{s}(w^{*}))\oplus\bar{\gamma}_{\mathfrak{s}(w^{*}),w^{*}}.$

Recall that $s_{*}\leq s_{n}-n^{\alpha_{d}}$. On the event $\mathcal{A}^{\rm
r}$, $\widetilde{\gamma}$ is still an open path from $\widetilde{0}$ to
$w^{*}$ for $\tau^{\rm r}$. Hence, we have using (3.21) for $n$ large enough
depending on $d,s_{0},p$,

(3.26) $\displaystyle\mathcal{D}^{\rm
r}(\widetilde{0},w^{*})\leq|\widetilde{\gamma}|\leq|\gamma|+3|\mathrm{H}^{*}\cap\gamma_{0,w_{n}}|+{|\bar{\gamma}_{\mathfrak{s}(w^{*}),w^{*}}|}\leq
s_{*}+10^{3d}n^{1-\beta_{d}}N^{d}<s_{n}-\frac{1}{2}n^{\alpha_{d}}.$

Let $v\in\mathrm{B}^{\rm r}_{s_{n}-n^{\alpha_{d}}/2}$. Let us take a geodesic
$\gamma^{\rm r}:\widetilde{0}\to v$ with $\gamma^{\rm r}=(v^{i})_{i=1}^{\ell}$
for $\tau^{\rm r}$. We define $i_{1},j_{1},\dots,i_{m},j_{m}$ inductively as
follows: Let

$\displaystyle i_{1}$ $\displaystyle:=\inf\\{k:\leavevmode\nobreak\
\,v^{k}\in\mathfrak{s}(\mathrm{H}^{*}),v^{k+1}\in\mathrm{H}^{*}\\},$
$\displaystyle j_{1}$ $\displaystyle:=\sup\\{k\geq i_{1}:\leavevmode\nobreak\
v^{k}\in\mathfrak{s}(\mathcal{C}_{\mathrm{H}}(\mathfrak{s}^{-1}(v^{i_{1}})))\text{
or
}\langle\mathfrak{s}^{-1}(v^{i_{1}}),\mathfrak{s}^{-1}(v^{k})\rangle\in\gamma_{0,w_{n}}\\}.$

Assuming that $i_{1},j_{1},\dots,i_{l-1},j_{l-1}$ have been defined, we define

$\displaystyle i_{l}$ $\displaystyle:=\inf\\{k\geq
j_{l-1}:\leavevmode\nobreak\
\,v^{k}\in\mathfrak{s}(\mathrm{H}^{*}){\setminus\mathfrak{s}(\mathcal{C}_{H}(\mathfrak{s}^{-1}(v^{i_{l-1}}))}),v^{k+1}\in\mathrm{H}^{*}\\},$
$\displaystyle j_{l}$ $\displaystyle:=\sup\\{k\geq i_{l}:\leavevmode\nobreak\
v^{k}\in\mathfrak{s}(\mathcal{C}_{\mathrm{H}}(\mathfrak{s}^{-1}(v^{i_{l}})))\text{
or
}\langle\mathfrak{s}^{-1}(v^{i_{l}}),\mathfrak{s}^{-1}(v^{k})\rangle\in\gamma_{0,w_{n}}\\}.$

Let $m$ be the smallest integer such that $i_{m+1}{=\infty}$. For $k\leq m$,
we set

$x^{k}:=\mathfrak{s}^{-1}(v^{i_{k}})\quad\text{and}\quad
y^{k}:=\mathfrak{s}^{-1}(v^{j_{k}}).$

Note that if $i_{l+1}<\infty$, by Claim 3.11, then
$v^{k}\in\mathfrak{s}(\mathcal{C}_{\mathrm{H}}(\mathfrak{s}^{-1}(v^{i_{l}})))\text{
or
}\langle\mathfrak{s}^{-1}(v^{i_{l}}),\mathfrak{s}^{-1}(v^{k})\rangle\in\gamma_{0,w_{n}}$
with $k:=\min\\{j>i_{l}:\leavevmode\nobreak\
v^{j}\in\mathfrak{s}(\mathbb{Z}^{d})\\}$, thus we have $i_{l}<j_{l}$.
Therefore, $m$ is finite.

######  Claim 3.13.

The paths $(v^{j_{k}},\cdots,v^{i_{k+1}})$ for $k<m$ are contained in
$\mathfrak{s}(\mathbb{Z}^{d})$. In particular, the paths
$(\mathfrak{s}^{-1}(v^{j_{m}}),\cdots,\mathfrak{s}^{-1}(v^{\ell}))$ for $k<m$
are open for $\tau$. Moreover, the path $(v^{j_{m}+1},\cdots,v^{\ell})$ is
either included in $\mathfrak{s}(\mathbb{Z}^{d})$ or included in
$\mathrm{H}^{*}$.

###### Proof.

Let $k<m$. Note that $j_{k}>i_{k}$ mentioned above, and thus $x^{k}\neq
y^{k}$. If $j_{k}=i_{k+1}$, then the claim is trivial. Hence, we assume
$j_{k}<i_{k+1}.$ To prove the claim, we assume the contrary, i.e. there exists
$i\in(j_{k},i_{k+1})$ such that $v^{i}\in\mathrm{H}^{*}$. Consider the
smallest such $i.$ If $i>j_{k}+1$, then the edge $\langle
v^{i-1},v^{i}\rangle$ crosses $\mathfrak{s}(\mathrm{H}^{*})$ before $i_{k+1}$
and
$v^{i-1}\in\mathfrak{s}(\mathrm{H}^{*})\setminus\mathfrak{s}(\mathcal{C}_{H}(x^{k}))$
due to $i-1>j_{k}$, which contradicts the minimality of $i_{k+1}$. Hence, we
have $i=j_{k}+1$, i.e. $v^{j_{k}+1}\in\mathrm{H}^{*}$. Note that, by (3.25),
$\langle x^{k},y^{k}\rangle\in\gamma_{0,w_{n}}$ implies
$y^{k}\notin\mathcal{C}_{\rm H}(x^{k})$ and $j_{k}=i_{k+1}$, which contradicts
the assumption $j_{k}<i_{k+1}$. Thus, it holds $x^{k}\overset{H}{\sim}y^{k}$.
Let $j:=\min\\{i^{\prime}\geq i:\leavevmode\nobreak\
v^{i^{\prime}}\in\mathfrak{s}(\mathbb{Z}^{d})\\}$. Since $\langle
y^{k},\mathfrak{s}^{-1}(v^{j})\rangle\in\gamma_{0,w_{n}}$ again implies
$j_{k}=i_{k+1}$ by (3.25), we have
$v^{j}\in\mathfrak{s}(\mathcal{C}_{H}(y^{k}))=\mathfrak{s}(\mathcal{C}_{H}(x^{k}))$
by Claim 3.11, which contradicts the maximality of $j_{k}$. Therefore the path
$(v^{j_{k}},\cdots,v^{i_{k+1}})$ is contained in
$\mathfrak{s}(\mathbb{Z}^{d})$. Hence, by definition of $\tau^{\rm r}$, the
path $(\mathfrak{s}^{-1}(v^{j_{k}}),\cdots,\mathfrak{s}^{-1}(v^{i_{k+1}}))$ is
open for $\tau$.

We assume $(v^{j_{m}+1},\cdots,v^{\ell})$ is not included in
$\mathfrak{s}(\mathbb{Z}^{d})$, i.e. there exists
$i\in\\{j_{m}+1,\dots,\ell\\}$ such that $v^{i}\in\mathrm{H}^{*}$. We take the
smallest such $i$. If $i>j_{m}+1$, then by the same reasoning as above, it
contradicts the maximality of $m$. Hence, we have $i=j_{m}+1$. We further
suppose that $(v^{j_{m}+1},\cdots,v^{\ell})$ is not included in
$\mathrm{H}^{*}$. Define $j=\min\\{j>j_{m}:\leavevmode\nobreak\
v^{j}\in\mathfrak{s}(\mathbb{Z}^{d})\\}$. If
$y^{m}\in\mathcal{C}_{\mathrm{H}}(x^{m})$, then by (3.25) and Claim 3.11,
$\langle x^{m},\mathfrak{s}^{-1}(v^{j})\rangle=\langle
y^{m},\mathfrak{s}^{-1}(v^{j})\rangle\in\gamma_{0,w_{n}}$ or
$\mathfrak{s}^{-1}(v^{j})\in\mathcal{C}_{\mathrm{H}}(y^{m})=\mathcal{C}_{\mathrm{H}}(x^{m})$,
which contradicts the maximality of $j_{m}.$ If
$y^{m}\notin\mathcal{C}_{\mathrm{H}}(x^{m})$, then we have $j_{m}=i_{m+1}$ by
definition of $i_{k}$, which contradicts the maximality of $m.$ Therefore,
$(v^{j_{m}+1},\cdots,v^{\ell})$ is included in $\mathrm{H}^{*}$. ∎

Let us prove by induction that

(3.27) $\forall k<m$, $x^{k},y^{k}\in\mathrm{B}_{s_{n}-1}$ and
$x^{m}\in\mathrm{B}_{s_{n}-1}$.

Let $k\leq m$ and suppose that we have proved the claim for any $i<k$, i.e.
$x^{i},y^{i}\in\mathrm{B}_{s_{n}-1}$. Thanks to Claim 3.13, we have

(3.28)
$\displaystyle\mathcal{D}(0,x^{1})+\sum_{i=1}^{k-1}\mathcal{D}(y^{i},x^{i+1})\leq\mathcal{D}^{\rm
r}(\widetilde{0},\mathfrak{s}(x^{1}))+\sum_{k=1}^{k-1}\mathcal{D}^{\rm
r}(\mathfrak{s}(y^{i}),\mathfrak{s}(x^{i+1}))\leq\mathcal{D}^{\rm
r}(\widetilde{0},v^{i_{k}})\leq\ell.$

Let $C:=10^{3d}\mu(\mathbf{e}_{1})$. Next, let us prove

(3.29) $\sum_{i=1}^{k-1}\mathcal{D}(x^{i},y^{i})\leq CN^{d+1}n^{1-\beta_{d}}.$

For any $i<k$, either (1) $\langle x^{i},y^{i}\rangle\in\gamma_{0,w_{n}}$, or
(2) $y^{i}\in\mathcal{C}_{\rm H}(x^{i})$ holds. For case (1), we have
$\mathcal{D}(x^{i},y^{i})=1$. For case (2), by induction hypothesis
$x^{i},y^{i}\in\mathrm{B}_{s_{n}-1}$, they belong to the largest open cluster
of their respective good boxes. Thanks to Claim 3.12, we have

$\mathcal{D}(x^{i},y^{i})\leq
8d^{2}\mu(\mathbf{e}_{1})N|\\{e\in\mathbb{E}^{d}:\leavevmode\nobreak\
e\subset\mathrm{H}^{*},\,|e\cap\mathcal{C}_{\mathrm{H}}(x^{i})|=1\\}|.$

Finally, since $|\\{i\leq k:\leavevmode\nobreak\
|e\cap\mathcal{C}_{\mathrm{H}}(x^{i})|=1\\}|\leq 2$ for any
$e\subset\mathrm{H}^{*}$, using (3.21)

(3.30) $\begin{split}\sum_{i=1}^{k-1}\mathcal{D}(x^{i},y^{i})&\leq
8d^{2}\mu(\mathbf{e}_{1})N\sum_{i=1}^{k-1}|\\{e\in\mathbb{E}^{d}:\leavevmode\nobreak\
e\subset\mathrm{H}^{*},\,|e\cap\mathcal{C}_{\mathrm{H}}(x^{i})|=1\\}|+|\mathrm{H}^{*}\cap\gamma_{0,w_{n}}|\\\
&\leq{32d^{3}\mu(\mathbf{e}_{1})N}\mu(\mathbf{e}_{1})|\mathrm{H}^{*}\cap(\mathbf{V}_{n}\cup\Gamma\cup\gamma_{0,w_{n}})|\leq
CN^{d+1}n^{1-\beta_{d}}.\end{split}$

Therefore, since $\ell\leq s_{n}-\frac{1}{2}n^{\alpha_{d}}$ by
$v\in\mathrm{B}^{\rm r}_{s_{n}-\frac{1}{2}n^{\alpha_{d}}}$, (3.28) and (3.29)
imply

(3.31) $\displaystyle\mathcal{D}(0,x^{k})$
$\displaystyle{\leq\mathcal{D}(0,x^{1})+\sum_{i=1}^{k-1}\mathcal{D}(y^{i},x^{i+1})+\sum_{i=1}^{k-1}\mathcal{D}(x^{i},y^{i})}$
$\displaystyle\leq\ell+\sum_{i=1}^{k-1}\mathcal{D}(x^{i},y^{i})\leq\ell+CN^{d+1}n^{1-\beta_{d}}\leq
s_{n}-\frac{1}{4}n^{\alpha_{d}}.$

This, in particular, yields $x^{k}\in\mathrm{B}_{s_{n}-2}$.

It remains to prove
$y^{k}=\mathfrak{s}^{-1}(v^{j_{k}})\in\mathrm{B}_{s_{n}-1}$ for $k<m$. Let us
assume $y^{k}\notin\mathrm{B}_{s_{n}-1}$. Then, since
$\mathcal{D}(x^{k},y^{k})\geq 2$, $\langle
x^{k},y^{k}\rangle\notin\gamma_{0,w_{n}}$ and thus
$y^{k}\in\mathcal{C}_{\mathrm{H}}(x^{k})\setminus\\{x^{k}\\}$. In particular,
the boxes containing $x^{k}$ and $y^{k}$ are good, and the two points $x^{k}$
and $y^{k}$ must be connected by a path in
$\mathrm{H}^{*}\setminus(\gamma_{0,w_{n}}\cup\Gamma\cup\mathbf{V}_{n})$. If
$y^{k}$ is in the largest open cluster, since $x^{k}\in\mathrm{B}_{s_{n}}$ is
in the largest open cluster of its good box, by the same argument as in
(3.30), we have $\mathcal{D}(x^{k},y^{k})\leq CN^{d+1}n^{1-\beta_{d}}$, which
implies $y^{k}\in\mathrm{B}_{s_{n}-1}$ by (3.31). Otherwise, if $y^{k}$ is not
in the largest open cluster, since by Claim 3.13, the path
$(y^{k},\mathfrak{s}^{-1}(v^{j_{k}+1}),\cdots,\mathfrak{s}^{-1}(v^{i_{k+1}-1}),x^{k+1})$
is open for $\tau$. Since there are no two distinct large open clusters in
$\underline{\Lambda}^{\prime}_{N}(y^{k})$,
$\|y^{k}-x^{k+1}\|_{\infty}\leq\frac{N}{2}$. Moreover, since
$x^{k+1}\in\gamma_{0,w_{n}}$ implies that $y^{k}$ must be in the largest open
cluster, $x^{k+1}=\mathfrak{s}^{-1}(v^{i_{k+1}})\notin\gamma_{0,w_{n}}$. Thus,
the edge $\langle v^{i_{k+1}},v^{i_{k+1}+1}\rangle$ does not belong to
$\mathbf{E}_{1}$. Moreover, since the edge $\langle
v^{i_{k+1}},v^{i_{k+1}+1}\rangle$ crosses $\mathfrak{s}(\mathrm{H}^{*})$ and
all the edges in $\mathbf{E}_{2}$ are closed, ${\rm
d}_{\infty}(x^{k+1},\mathrm{H}^{*}\cap(\gamma_{0,w_{n}}\cup\Gamma\cup{\rm
V}_{n}))>2N$. Thus, there exists a path from $y^{k}$ to $x^{k+1}$ inside
$\mathrm{H}^{*}$ not intersecting with
$\mathrm{H}^{*}\cap(\gamma_{0,w_{n}}\cup\Gamma\cup{\rm V}_{n})$, and
$x^{k+1}\in\mathcal{C}_{\mathrm{H}}(y^{k}){=\mathcal{C}_{\mathrm{H}}(x^{k})}$,
which derives a contradiction. Therefore, we have (3.27) with $k<m$.

We can also conclude from what is above that either
$y^{m}\in\mathrm{B}_{s_{n}-1}$ or
$y^{m}\in\mathcal{C}_{\mathrm{H}}(x^{m})\setminus\\{x^{m}\\}$. In both cases,
we have that $y^{m}\in\mathrm{Int}(\Gamma)\cap\mathrm{H}^{*}$. When
$y^{m}\in\mathrm{B}_{s_{n}-1}$, by the same argument as in (3.28) and (3.29),
we have

(3.32) $\mathcal{D}(0,y^{m})\leq\mathcal{D}^{\rm
r}(\widetilde{0},\mathfrak{s}(y^{m}))+CN^{d+1}n^{1-\beta_{d}}.$

If $y^{m}\notin\mathrm{B}_{s_{n}-1}$, then $y^{m}$ does not belong to the
largest cluster of its good box, as we proved its contrapositive. If, in
addition, $v\in\mathfrak{s}(\mathbb{Z}^{d})$, by Claim 3.13, the path
$(v^{j_{m}},\dots,v^{\ell})$ is included in $\mathfrak{s}(\mathbb{Z}^{d})$.
Since there are no two distinct large open clusters in
$\underline{\Lambda}^{\prime}_{N}(y^{m})$,
$(\mathfrak{s}^{-1}(v^{j_{m}}),\dots,\mathfrak{s}^{-1}(v^{\ell}))$ is open for
$\tau$ and inside $\Lambda_{N/2}(y)$. In particular,