# On percolation in a generalized backbend process††thanks: We thank Kumarjit
Saha for helpful discussions.

Pinaki Mandal and Souvik Roy Corresponding author: Economic Research Unit,
Indian Statistical Institute, Kolkata, India. E-mail:
<EMAIL_ADDRESS>Research Unit, Indian Statistical Institute,
Kolkata, India. E-mail<EMAIL_ADDRESS>
###### Abstract

We have generalized the idea of backbend in a nearest-neighbor oriented bond
percolation process by considering a backbend sequence
$\beta:\mathbb{Z}_{+}\to\mathbb{Z}_{+}\cup\\{\infty\\}$, and defining a
$\beta$-backbend path from the origin as a path that never retreats further
than $\beta(h)$ levels back from its record level $h$. We study the
relationship between the critical probabilities of different percolation
processes based on different backbend sequences on half-space, full-space, and
half-slabs of the $d$-dimensional ($d\geq 2$) body-centered cubic (BCC)
lattice. We also give sufficient conditions on the backbend sequences such
that there will be no percolation at the critical probabilities of the
corresponding percolation processes on half-space and full-space of the BCC
lattice.

##  1 Introduction

In this paper, we discuss an extension of the concept of nearest-neighbor
oriented bond percolation with backbend on certain sublattices of the
$d$-dimensional ($d\geq 2$) body-centered cubic (BCC) lattice. Roy et al.
(1998) studied nearest-neighbor oriented bond percolation with backbend in the
$d$-dimensional simple cubic lattice, where they worked with $b$-backbend
paths for some $b\in\mathbb{Z}_{+}$.111We denote by $\mathbb{Z}_{+}$ the set
$\mathbb{N}\cup\\{0\\}$. A $b$-backbend path never retreats further than $b$
levels back from its record level. We have generalized their idea of backbend
as follows: instead of taking a constant value $b$ for permissible backbend of
a path, we consider a backbend sequence
$\beta:\mathbb{Z}_{+}\to\mathbb{Z}_{+}\cup\\{\infty\\}$, and define a
$\beta$-backbend path from the origin as a path that never retreats further
than $\beta(h)$ levels back from its record level $h$. The generality of our
framework allows us to embed important special cases; a $\beta$-backbend path
can be an oriented path, a $b$-backbend path, or an ordinary (unoriented) path
depending on the backbend sequence $\beta$.

We study the relationship between the critical probabilities of different
percolation processes based on different backbend sequences on half-space,
full-space, and half-slabs of the BCC lattice. We also give sufficient
conditions on the backbend sequences such that there will be no percolation at
the critical probabilities of the corresponding percolation processes on half-
space and full-space of the BCC lattice.

##  2 Model

We denote the $d$-dimensional ($d\geq 2$) BCC lattice by
$(\mathbb{V},\mathbb{E})$, where the vertex set
$\mathbb{V}:=\big{\\{}x\in\mathbb{Z}^{d}:x_{i}\equiv x_{j}\mbox{ (mod 2)}$ for
all $i,j\in\\{1,\ldots,d\\}\big{\\}}$ and the edge set $\mathbb{E}$ contains
all _unordered_ pairs $\langle x,y\rangle$ of vertices $x,y\in\mathbb{V}$ with
$|x_{1}-y_{1}|=\cdots=|x_{d}-y_{d}|=1$.222If we rotate the square lattice
using the following rotational matrix  $\begin{bmatrix}1&-1\\\ 1&1\\\
\end{bmatrix},$ we will get the 2-dimensional BCC lattice. Thus,
$2$-dimensional BCC lattice is equivalent to the square lattice (see Grimmett
(2013) for the formal definition of the square lattice). Each edge is
independently _open_ with probability $p\in[0,1]$ and _closed_ with
probability $1-p$. Let $\mathbb{P}_{p}$ denote the corresponding probability
measure for the total configurations of all the edges.333More formally, we
consider the following probability space. As _sample space_ we take
$\Omega=\underset{e\in\mathbb{E}}{\prod}\hskip 2.84526pt\\{0,1\\}$, points of
which are represented as $\omega=\\{\omega(e):e\in\mathbb{E}\\}$ and called
_configurations_ ; the value $w(e)=0$ corresponds to $e$ being closed, and
$w(e)=1$ corresponds to $e$ being open. We take $\mathcal{F}$ to be the
$\sigma$-algebra of subsets of $\Omega$ generated by the finite-dimensional
cylinders. Finally, we take product measure with density $p$ on
$(\Omega,\mathcal{F})$; this is the measure
$\mathbb{P}_{p}=\underset{e\in\mathbb{E}}{\prod}\hskip 2.84526pt\mu_{e}$ where
$\mu_{e}$ is Bernoulli measure on $\\{0,1\\}$, given by
$\mu_{e}(\omega(e)=0)=1-p$ and $\mu_{e}(\omega(e)=1)=p$.

For $\widehat{\mathbb{V}}\subseteq\mathbb{V}$, a _path in
$\widehat{\mathbb{V}}$_ is a sequence $x^{0},x^{1},\ldots,x^{n}$ of _distinct_
vertices such that $x^{i}\in\widehat{\mathbb{V}}$ and $\langle
x^{i},x^{i+1}\rangle\in\mathbb{E}$. Note that paths are self-avoiding by
definition. We call a path _open_ if all of its edges are open. For a path
$\pi=(x^{0},\ldots,x^{n})$, the record level attained by the path $\pi$ till
$x^{i}$, denoted by $h^{i}(\pi)$, is defined as $\underset{j\leq
i}{\max}\hskip 2.84526ptx^{j}_{d}$. In words, the record level attained by the
path $\pi$ till $x^{i}$ is the maximum value of the $d$-th coordinates of
vertices till $x^{i}$.

A backbend sequence is a mapping
$\beta:\mathbb{Z}_{+}\to\mathbb{Z}_{+}\cup\\{\infty\\}$. For ease of
presentation, we write $\beta(i)$ as $\beta_{i}$. We introduce the notion of a
$\beta$-backbend path.

######  Definition 2.1.

For $\widehat{\mathbb{V}}\subseteq\mathbb{V}$, $x\in\widehat{\mathbb{V}}$ with
$x_{d}\geq 0$, and a backbend sequence $\beta$, we say a path
$\pi=(x^{0}=x,\ldots,x^{n})$ in $\widehat{\mathbb{V}}$ is a _$\beta$ -backbend
path in $\widehat{\mathbb{V}}$ from $x$ to $x^{n}$_ if for every $i$,

$x^{i}_{d}\geq h^{i}(\pi)-\beta_{h^{i}(\pi)}.$

In words, a $\beta$-backbend path never retreats further than $\beta_{h}$
levels back from its record level $h$. Thus, if $\beta_{i}=0$ for all
$i\in\mathbb{Z}_{+}$, then the $\beta$-backbend path is an oriented path in
the direction of the positive $d$-th coordinate axis (that is, in the
direction of $(0,\ldots,0,1)$). If $\beta_{i}=b$ for all $i\in\mathbb{Z}_{+}$
and for some $b\in\mathbb{N}$, then the $\beta$-backbend path is a
$b$-backbend path in the direction of the positive $d$-th coordinate
axis.444See Roy et al. (1998) for the definition of a $b$-backbend path in the
context of $d$-dimensional simple cubic lattice. Note that in their paper, the
orientation is in the direction of $(1,\ldots,1)$. Finally, if
$\beta_{i}=\infty$ for all $i\in\mathbb{Z}_{+}$, then the $\beta$-backbend
path is an ordinary (unoriented) path.

A backbend sequence $\beta$ is $k$-cyclic for some $k\in\mathbb{N}$, if for
all $i\in\\{0,\ldots,k-1\\}$, we have
$\beta_{kn+i}=\beta_{i}\in\mathbb{Z}_{+}$ for all $n\in\mathbb{N}$ (that is,
$\beta_{i}=\beta_{k+i}=\beta_{2k+i}=\cdots$ and $\beta_{i}\in\mathbb{Z}_{+}$).
Note that a $1$-cyclic backbend sequence is a constant sequence in
$\mathbb{Z}_{+}$. Let $\tilde{\beta}$ be a backbend sequence. We say that
$\tilde{\beta}$ converges to a $k$-cyclic backbend sequence $\beta$ for some
$k\in\mathbb{N}$, if for all $i\in\\{0,\ldots,k-1\\}$ we have
$\underset{n\rightarrow\infty}{\lim}\hskip
2.84526pt\tilde{\beta}_{kn+i}=\beta_{i}$, and say that $\tilde{\beta}$
converges from below to a $k$-cyclic backbend sequence $\beta$ for some
$k\in\mathbb{N}$, if for all $i\in\\{0,\ldots,k-1\\}$ we have
$\underset{n\rightarrow\infty}{\lim}\hskip
2.84526pt\tilde{\beta}_{kn+i}=\beta_{i}$ and
$\tilde{\beta}_{kn+i}\leq\beta_{i}$ for all $n\in\mathbb{Z}_{+}$.555Note that
$\tilde{\beta}$ converges from below to a $k$-cyclic backbend sequence does
not mean that for any $i\in\\{0,\ldots,k-1\\}$, the subsequence
$(\tilde{\beta}_{kn+i})_{n\in\mathbb{Z}_{+}}$ is monotonically increasing. For
ease of reference, we call a backbend sequence $k$-cyclic in the limit if it
converges to a $k$-cyclic backbend sequence, and $k$-cyclic in the limit from
below if it converges from below to a $k$-cyclic backbend sequence. Note that
a backbend sequence is $1$-cyclic in the limit if and only if it converges to
a finite integer.

Throughout this paper, we refer to the vertex $(0,\ldots,0)$ as the origin.
For $\widehat{\mathbb{V}}\subseteq\mathbb{V}$ and a backbend sequence $\beta$,
define the following random set of vertices:

$C^{\beta}_{\widehat{\mathbb{V}}}:=\\{x\in\widehat{\mathbb{V}}:\mbox{there is
an open $\beta$-backbend path in $\widehat{\mathbb{V}}$ from the origin to
$x$}\\}.$

For $p\in[0,1]$, $\widehat{\mathbb{V}}\subseteq\mathbb{V}$, and a backbend
sequence $\beta$, the percolation probability is defined as

$\theta^{\beta}_{\widehat{\mathbb{V}}}(p):=\mathbb{P}_{p}\big{(}|C^{\beta}_{\widehat{\mathbb{V}}}|=\infty\big{)}.$

For $\widehat{\mathbb{V}}\subseteq\mathbb{V}$ and a backbend sequence $\beta$,
the critical probability is defined as

$p^{\beta}_{c}(\widehat{\mathbb{V}}):=\sup\hskip
2.84526pt\\{p:\theta^{\beta}_{\widehat{\mathbb{V}}}(p)=0\\}.$

As we have explained earlier, by the definition of a $\beta$-backbend path,
the critical probability $p^{\beta}_{c}(\widehat{\mathbb{V}})$ becomes the
same as the critical probability of the (i) oriented percolation process in
the direction of the positive $d$-th coordinate axis when $\beta_{i}=0$ for
all $i\in\mathbb{Z}_{+}$, (ii) $b$-backbend percolation process in the
direction of the positive $d$-th coordinate axis when $\beta_{i}=b$ for all
$i\in\mathbb{Z}_{+}$ and for some $b\in\mathbb{N}$, and (iii) ordinary
(unoriented) percolation process when $\beta_{i}=\infty$ for all
$i\in\mathbb{Z}_{+}$.

For $\widehat{\mathbb{V}}\subseteq\mathbb{V}$, we denote the critical
probability of the oriented percolation process on $\widehat{\mathbb{V}}$ by
$p^{0}_{c}(\widehat{\mathbb{V}})$, and the critical probability of the
ordinary percolation process on $\widehat{\mathbb{V}}$ by
$p_{c}(\widehat{\mathbb{V}})$.

##  3 Results

Let $\mathbb{H}:=\\{x\in\mathbb{V}:x_{d}\geq 0\\}$ denote the half-space. For
all $l\in\mathbb{N}$ and all $e\in\\{2,\ldots,d\\}$, let
$\mathbb{Q}_{l}^{e}:=\\{x\in\mathbb{V}:x\in[-l,l]^{d-e}\times\mathbb{Z}^{e-1}\times\mathbb{Z}_{+}\\}$
denote an $e$-dimensional half-slab.666Note that for all $l\in\mathbb{N}$,
$\mathbb{Q}_{l}^{d}=\mathbb{H}$. For all $t\in\mathbb{N}$, let
$\mathbb{S}_{t}:=\\{x\in\mathbb{V}:0\leq x_{d}\leq t\\}$ denote a
$d-1$-dimensional slab. Furthermore, for all
$l_{1},\ldots,l_{d},r_{1},\ldots,r_{d}\in\mathbb{Z}\cup\\{-\infty,\infty\\}$
with $l_{i}\leq r_{i}$ for all $i=1,\ldots,d$, let us define

$B\Big{(}\underset{i=1}{\overset{d}{\prod}}[l_{i},r_{i}]\Big{)}:=\Big{\\{}x\in\mathbb{V}:x\in\underset{i=1}{\overset{d}{\prod}}[l_{i},r_{i}]\Big{\\}}.$

For ease of presentation, we denote
$B\Big{(}\underset{i=1}{\overset{d-1}{\prod}}[l_{i},r_{i}]\times[l,l]\Big{)}$
by $B\Big{(}\underset{i=1}{\overset{d-1}{\prod}}[l_{i},r_{i}]\times l\Big{)}$.

Furthermore, following our notational terminology, for
$\widehat{\mathbb{V}}\subseteq\mathbb{V}$,
$\mathbb{V}^{\prime}\subseteq\mathbb{H}$, and a backbend sequence $\beta$,
define the random set of vertices

$C^{\beta}_{\widehat{\mathbb{V}}}(\mathbb{V}^{\prime}):=\Big{\\{}y\in\widehat{\mathbb{V}}:\mbox{
there is an open $\beta$-backbend path in }\widehat{\mathbb{V}}\mbox{ from
some }x\in\mathbb{V}^{\prime}\mbox{ to $y$}\Big{\\}}.$

In what follows, we present a technical result which we will use in proving
our main results of the paper.

######  Proposition 3.1.

Let $\beta$ be a $\hat{k}$-cyclic backbend sequence for some
$\hat{k}\in\mathbb{N}$. Suppose $0<p<1$ and
$\theta^{\beta}_{\mathbb{H}}(p)>0$. Then, there exist $l,r\in\mathbb{N}$ and
$0<\delta<p$ such that

$\mathbb{P}_{p-\delta}\bigg{(}\Big{|}C^{\beta}_{\mathbb{Q}_{l}^{2}}\Big{(}B\big{(}[-r,r]^{d-1}\times
0\big{)}\Big{)}\Big{|}=\infty\bigg{)}>0.$

The proof of this proposition is relegated to Appendix A.

###  3.1 Results on the half-space

Our next theorem establishes a relation between percolation probabilities of a
$\tilde{\beta}$-backbend percolation process and a $\beta$-backbend
percolation process on the half-space, where $\beta$ is $k$-cyclic and
$\tilde{\beta}$ converges from below to $\beta$.

######  Theorem 3.1.

Suppose a backbend sequence $\tilde{\beta}$ converges from below to a
$k$-cyclic backbend sequence $\beta$ for some $k\in\mathbb{N}$. Then, for all
$p\in[0,1]$, we have $\theta^{\tilde{\beta}}_{\mathbb{H}}(p)>0$ if and only if
$\theta^{\beta}_{\mathbb{H}}(p)>0$.

###### Proof of Theorem 3.1.

(If part) Suppose $\theta^{\beta}_{\mathbb{H}}(p)>0$. It must be that $p>0$.
Since $\underset{n\rightarrow\infty}{\lim}\hskip
2.84526pt\tilde{\beta}_{kn+i}=\beta_{i}$ for all $i\in\\{0,\ldots,k-1\\}$,
there exists $n^{*}\in\mathbb{N}$ such that $\tilde{\beta}_{kn+i}=\beta_{i}$
for all $n\geq n^{*}$ and all $i\in\\{0,\ldots,k-1\\}$. Because $\beta$ is
$k$-cyclic, by the construction of the $\beta$-backbend percolation process,
we have for all $y\in\mathbb{H}$ with $y_{d}=kn^{*}$,

$\mathbb{P}_{p}\Big{(}\big{|}C^{\beta}_{(\mathbb{H}+y)}(\\{y\\})\big{|}=\infty\Big{)}=\theta^{\beta}_{\mathbb{H}}(p).$
(1)

Choose $y^{*}\in\mathbb{H}$ with $y^{*}_{d}=kn^{*}$ such that there exists an
oriented path $\pi$ from the origin to $y^{*}$.888For instance, $y^{*}$ can be
taken as the vertex that is connected to the origin through the following
oriented path:
$((0,\ldots,0),(1,\ldots,1),(0,\ldots,0,2),(1,\ldots,1,3),\ldots,(0,\ldots,0,kn^{*}))$
if $kn^{*}$ is even, and
$((0,\ldots,0),(1,\ldots,1),(0,\ldots,0,2),\ldots,(1,\ldots,1,kn^{*}))$ if
$kn^{*}$ is odd. As $p>0$, the probability of $\pi$ being open is
positive.999Note that since the path $\pi$ is oriented, it will have $kn^{*}$
edges, and hence $\mathbb{P}_{p}\big{(}\pi\mbox{ is open}\big{)}=p^{kn^{*}}$.

Since $\tilde{\beta}_{n}=\beta_{n}$ for all $n\geq kn^{*}$,
$y^{*}_{d}=kn^{*}$, and $\pi$ is an oriented path from the origin to $y^{*}$,
it follows that $\pi$ concatenated with any infinite $\beta$-backbend path in
$\mathbb{H}+y^{*}$ from $y^{*}$ produces an infinite $\tilde{\beta}$-backbend
path in $\mathbb{H}$ from the origin. Furthermore, by the construction of
$\pi$, the event of $\pi$ being open is independent of the event
$\Big{\\{}\omega\in\Omega:\big{|}C^{\beta}_{(\mathbb{H}+y^{*})}(\\{y^{*}\\})\big{|}=\infty\Big{\\}}$.
Combining these two facts with (1), we have
$\theta^{\tilde{\beta}}_{\mathbb{H}}(p)\geq\mathbb{P}_{p}\big{(}\pi\mbox{ is
open}\big{)}\times\theta^{\beta}_{\mathbb{H}}(p)$. Because
$\mathbb{P}_{p}\big{(}\pi\mbox{ is open}\big{)}>0$ and
$\theta^{\beta}_{\mathbb{H}}(p)>0$, this implies
$\theta^{\tilde{\beta}}_{\mathbb{H}}(p)>0$. This completes the proof of the
“if” part of Theorem 3.1.
(Only-if part) Suppose $\theta^{\tilde{\beta}}_{\mathbb{H}}(p)>0$. By the
assumptions on $\tilde{\beta}$ and $\beta$, for all $i\in\\{0,\ldots,k-1\\}$,
we have $\tilde{\beta}_{kn+i}\leq\beta_{i}$ for all $n\in\mathbb{Z}_{+}$. This
implies that every $\tilde{\beta}$-backbend path in $\mathbb{H}$ from the
origin is a $\beta$-backbend path in $\mathbb{H}$ from the origin. Therefore,
$\theta^{\beta}_{\mathbb{H}}(p)\geq\theta^{\tilde{\beta}}_{\mathbb{H}}(p)$.
Since $\theta^{\tilde{\beta}}_{\mathbb{H}}(p)>0$, this implies
$\theta^{\beta}_{\mathbb{H}}(p)>0$. This completes the proof of the “only-if”
part of Theorem 3.1. ∎

We obtain the following corollary (Corollary 3.1) from Theorem 3.1. It says
that if a backbend sequence $\tilde{\beta}$ converges from below to a
$k$-cyclic backbend sequence $\beta$, then on the half-space, the critical
probability of $\tilde{\beta}$-backbend percolation process will be equal to
that of $\beta$-backbend percolation process.

######  Corollary 3.1.

Suppose a backbend sequence $\tilde{\beta}$ converges from below to a
$k$-cyclic backbend sequence $\beta$ for some $k\in\mathbb{N}$. Then,
$p^{\tilde{\beta}}_{c}(\mathbb{H})=p^{\beta}_{c}(\mathbb{H})$.

######  Note 3.1.

It is worth mentioning that Corollary 3.1 does not hold if we relax its
assumption by requiring that $\tilde{\beta}$ converges (not necessarily from
below) to a $k$-cyclic backbend sequence $\beta$ (see Example 3.1 for
details). However, for this case, using the same argument as for the “if” part
of Theorem 3.1, we have $p^{\tilde{\beta}}_{c}(\mathbb{H})\leq
p^{\beta}_{c}(\mathbb{H})$.

######  Example 3.1.

Suppose $d=3$. Let $\beta$ be a $1$-cyclic backbend sequence such that
$\beta_{i}=0$ for all $i\in\mathbb{Z}_{+}$. Suppose $\tilde{\beta}$ is a
backbend sequence such that
$\tilde{\beta}_{0}=0,\tilde{\beta}_{1}=1,\tilde{\beta}_{2}=2,\tilde{\beta}_{3}=3$,
and $\tilde{\beta}_{i}=0$ for all $i\geq 4$. Clearly, $\tilde{\beta}$
converges to $\beta$. By construction, $p^{\tilde{\beta}}_{c}(\mathbb{H})\leq
p_{c}(\mathbb{S}_{3})$. Since $p_{c}(\mathbb{S}_{3})=0.21113018(38)$ (see
Gliozzi et al. (2005) for details) and
$p^{\beta}_{c}(\mathbb{H})=0.2873383(1)$ (see Perlsman and Havlin (2002) for
details), the fact $p^{\tilde{\beta}}_{c}(\mathbb{H})\leq
p_{c}(\mathbb{S}_{3})$ implies that
$p^{\tilde{\beta}}_{c}(\mathbb{H})<p^{\beta}_{c}(\mathbb{H})$.101010Error bars
in the last digit or digits are shown by numbers in parentheses. Thus,
$0.21113018(38)$ signifies $0.21113018\pm 0.00000038$.

######  Remark 3.1.

In two dimensions, Corollary 3.1 holds for the case where $\tilde{\beta}$
converges (not necessarily from below) to a $k$-cyclic backbend sequence
$\beta$ satisfying an additional property. See Proposition 3.3 for details.

The next theorem is our main result on half-space which says that if a
backbend sequence is $k$-cyclic in the limit from below, then there will be no
percolation on the half-space at the critical probability.

######  Theorem 3.2.

Suppose a backbend sequence $\tilde{\beta}$ is $k$-cyclic in the limit from
below for some $k\in\mathbb{N}$. Then,
$\theta^{\tilde{\beta}}_{\mathbb{H}}(p^{\tilde{\beta}}_{c}(\mathbb{H}))=0$.

Before we start proving Theorem 3.2, we make a remark on the critical
probability of any backbend percolation process on the half-space.

######  Remark 3.2.

For every backbend sequence $\beta$, we have $p^{\beta}_{c}(\mathbb{H})<1$. To
see this fix a backbend sequence $\beta$. By the definition of a
$\beta$-backbend path, it follows that $p^{\beta}_{c}(\mathbb{H})\leq
p^{0}_{c}(\mathbb{H})$. Using a similar argument as in Grimmett (2013), it can
be verified that $p^{0}_{c}(\mathbb{H})$ decreases as $d$ (the number of
dimensions) increases. Furthermore, since $2$-dimensional BCC lattice is
equivalent to the square lattice, it is known that in two dimensions,
$p^{0}_{c}(\mathbb{H})<1$ (see Durrett (1984), Durrett (1988), Balister et al.
(1993)). Combining all these facts, we have $p^{\beta}_{c}(\mathbb{H})<1$.

###### Proof of Theorem 3.2.

Let $\beta$ be the $k$-cyclic backbend sequence such that $\tilde{\beta}$
converges from below to $\beta$. In view of Theorem 3.1, it is enough to show
that $\theta^{\beta}_{\mathbb{H}}(p^{\beta}_{c}(\mathbb{H}))=0$.

Assume for contradiction that
$\theta^{\beta}_{\mathbb{H}}(p^{\beta}_{c}(\mathbb{H}))>0$. Since
$\theta^{\beta}_{\mathbb{H}}(p^{\beta}_{c}(\mathbb{H}))>0$, we have
$p^{\beta}_{c}(\mathbb{H})>0$. This, together with Remark 3.2, implies
$0<p^{\beta}_{c}(\mathbb{H})<1$. Moreover, since
$0<p^{\beta}_{c}(\mathbb{H})<1$ and
$\theta^{\beta}_{\mathbb{H}}(p^{\beta}_{c}(\mathbb{H}))>0$, by Proposition
3.1, there exist $l,r\in\mathbb{N}$ and $0<\delta<p^{\beta}_{c}(\mathbb{H})$
such that

$\mathbb{P}_{p^{\beta}_{c}(\mathbb{H})-\delta}\bigg{(}\Big{|}C^{\beta}_{\mathbb{Q}_{l}^{2}}\Big{(}B\big{(}[-r,r]^{d-1}\times
0\big{)}\Big{)}\Big{|}=\infty\bigg{)}>0.$ (2)

Since $\mathbb{Q}_{l}^{2}\subseteq\mathbb{H}$, (2) implies

$\mathbb{P}_{p^{\beta}_{c}(\mathbb{H})-\delta}\bigg{(}\Big{|}C^{\beta}_{\mathbb{H}}\Big{(}B\big{(}[-r,r]^{d-1}\times
0\big{)}\Big{)}\Big{|}=\infty\bigg{)}>0.$ (3)

Because $B\big{(}[-r,r]^{d-1}\times 0\big{)}$ is a finite set, (3) implies
that there exists some $x\in B\big{(}[-r,r]^{d-1}\times 0\big{)}$ such that
$\mathbb{P}_{p^{\beta}_{c}(\mathbb{H})-\delta}\Big{(}\big{|}C^{\beta}_{\mathbb{H}}(\\{x\\})\big{|}=\infty\Big{)}>0$.
Furthermore, by the construction of the $\beta$-backbend percolation process,
we have
$\mathbb{P}_{p^{\beta}_{c}(\mathbb{H})-\delta}\Big{(}\big{|}C^{\beta}_{\mathbb{H}}\big{|}=\infty\Big{)}=\mathbb{P}_{p^{\beta}_{c}(\mathbb{H})-\delta}\Big{(}\big{|}C^{\beta}_{\mathbb{H}}(\\{x\\})\big{|}=\infty\Big{)}$,
and hence

$\mathbb{P}_{p^{\beta}_{c}(\mathbb{H})-\delta}\Big{(}\big{|}C^{\beta}_{\mathbb{H}}\big{|}=\infty\Big{)}>0.$
(4)

However, since $\delta>0$, (4) contradicts the definition of
$p^{\beta}_{c}(\mathbb{H})$. This completes the proof of Theorem 3.2. ∎

###  3.2 Results on the full-space

In this subsection, we consider the full-space and a particular class of
backbend sequences. These backbend sequences have the property that they
converge from below to a $k$-cyclic backbend sequence $\beta$ satisfying the
property that
$\beta_{1}\leq\beta_{0}+1,\ldots,\beta_{k-1}\leq\beta_{k-2}+1,\mbox{ and
}\beta_{0}\leq\beta_{k-1}+1$ (that is, $\beta_{i+1}\leq\beta_{i}+1$ for all
$i\in\\{0,\ldots,k-1\\}$). For such backbend sequences, the first part of
Theorem 3.3 says that the critical probability on the full-space will be the
same as that on the half-space, and the second part of the theorem says that
there will be no percolation on the full-space at the critical probability.

######  Theorem 3.3.

Let $k\in\mathbb{N}$ and suppose that a backbend sequence $\tilde{\beta}$
converges from below to a $k$-cyclic backbend sequence $\beta$ satisfying
$\beta_{i+1}\leq\beta_{i}+1$ for all $i\in\\{0,\ldots,k-1\\}$. Then,

  1. (i)

$p^{\tilde{\beta}}_{c}(\mathbb{V})=p^{\tilde{\beta}}_{c}(\mathbb{H})$, and

  2. (ii)

$\theta^{\tilde{\beta}}_{\mathbb{V}}(p^{\tilde{\beta}}_{c}(\mathbb{V}))=0$.

###### Proof of Theorem 3.3.

We first prove a claim which we will use in the proof of the theorem.

######  Claim 3.1.

For all $p\in[0,1]$, $\theta^{\beta}_{\mathbb{V}}(p)>0$ implies
$\theta^{\beta}_{\mathbb{H}}(p)>0$.

###### Proof of Claim 3.1.

Suppose $\theta^{\beta}_{\mathbb{V}}(p)>0$. It must be that $p>0$. Let
$n^{*}\in k\mathbb{N}$ be such that
$n^{*}\geq\max\\{\beta_{0},\ldots,\beta_{k-1}\\}$.111111For $k\in\mathbb{N}$,
we denote by $k\mathbb{N}$ the set $\\{kn:n\in\mathbb{N}\\}$. Thus, $n^{*}$ is
such that any $\beta$-backbend path starting from a vertex $y$ with
$y_{d}=n^{*}$ will always be in the half-space $\mathbb{H}$. Because $\beta$
is $k$-cyclic, by the construction of the $\beta$-backbend percolation
process, we have for all $y\in\mathbb{H}$ with $y_{d}=n^{*}$,

$\mathbb{P}_{p}\Big{(}\big{|}C^{\beta}_{\mathbb{V}}(\\{y\\})\big{|}=\infty\Big{)}=\theta^{\beta}_{\mathbb{V}}(p).$
(5)

Choose $y^{*}\in\mathbb{H}$ with $y^{*}_{d}=n^{*}$ such that there exists an
oriented path $\pi$ from the origin to $y^{*}$. As $p>0$, the probability of
$\pi$ being open is positive.

For each infinite $\beta$-backbend path $\hat{\pi}$ in $\mathbb{V}$ from
$y^{*}$, we construct the path $\hat{\pi}^{*}$ by distinguishing the following
cases.

  1. (i)

Suppose $\pi$ and $\hat{\pi}$ have no common vertex other than $y^{*}$. Then,
$\hat{\pi}^{*}$ is obtained by concatenating the paths $\pi$ and $\hat{\pi}$.

  2. (ii)

Suppose $\pi$ and $\hat{\pi}$ have common vertices other than $y^{*}$. Let
$z^{*}$ be the first vertex of $\pi$ such that $z^{*}\in\hat{\pi}$. Then
$\hat{\pi}^{*}$ is obtained by concatenating $\pi_{s}$ and $\hat{\pi}_{s}$,
where $\pi_{s}$ is the sub-path of $\pi$ from the origin to $z^{*}$ and
$\hat{\pi}_{s}$ is the sub-path of $\hat{\pi}$ from $z^{*}$.

To see that $\hat{\pi}^{*}$ is indeed a path, observe that $\hat{\pi}^{*}$ is
self-avoiding by construction. We claim that for each infinite
$\beta$-backbend path $\hat{\pi}$ in $\mathbb{V}$ from $y^{*}$, the path
$\hat{\pi}^{*}$ constructed as above is an infinite $\beta$-backbend path in
$\mathbb{H}$ from the origin. Clearly, $\hat{\pi}^{*}$ is an infinite path
from the origin. Moreover, as we have mentioned earlier, by the choice of
$n^{*}$, the path $\hat{\pi}$ is in $\mathbb{H}$, and hence the path
$\hat{\pi}^{*}$ is in $\mathbb{H}$. We proceed to show that $\hat{\pi}^{*}$ is
a $\beta$-backbend path.

Assume for contradiction that $\hat{\pi}^{*}$ is not a $\beta$-backbend path.
Since $\hat{\pi}^{*}$ is not a $\beta$-backbend path, there exists a vertex
$x^{*}$ in $\hat{\pi}^{*}$ such that $x^{*}_{d}<h^{*}-\beta_{h^{*}}$, where
$h^{*}$ is the record level attained by the path $\hat{\pi}^{*}$ till $x^{*}$.
Since $\pi$ is an oriented path, by the construction of $\hat{\pi}^{*}$, it
follows that $x^{*}$ must be in $\hat{\pi}$. Let $\hat{h}$ be the record level
attained by the path $\hat{\pi}$ till $x^{*}$. Because $\hat{\pi}$ is a
$\beta$-backbend path, it must be that $x^{*}_{d}\geq\hat{h}-\beta_{\hat{h}}$.
This, together with the fact $x^{*}_{d}<h^{*}-\beta_{h^{*}}$, yields

$h^{*}-\beta_{h^{*}}>\hat{h}-\beta_{\hat{h}}.$ (6)

Since $\pi$ is an oriented path to $y^{*}$ and $\hat{\pi}$ is a path from
$y^{*}$, by the construction of $\hat{\pi}^{*}$, we have $\hat{h}\geq h^{*}$.
The assumptions on $\beta$ imply that $\beta_{l}-\beta_{m}\leq l-m$ for all
$l\geq m$. Since $\hat{h}\geq h^{*}$, this yields
$\beta_{\hat{h}}-\beta_{h^{*}}\leq\hat{h}-h^{*}$, a contradiction to (6). So,
it must be that $\hat{\pi}^{*}$ is a $\beta$-backbend path.

Since for each infinite $\beta$-backbend path $\hat{\pi}$ in $\mathbb{V}$ from
$y^{*}$, $\hat{\pi}^{*}$ is an infinite $\beta$-backbend path in $\mathbb{H}$
from the origin, by the construction of $\hat{\pi}^{*}$, we have

$\theta^{\beta}_{\mathbb{H}}(p)\geq\mathbb{P}_{p}\Big{(}\pi\mbox{ is open and
}\big{|}C^{\beta}_{\mathbb{V}}(\\{y^{*}\\})\big{|}=\infty\Big{)},$

and hence, by FKG inequality (see Grimmett (2013) for details),

$\theta^{\beta}_{\mathbb{H}}(p)\geq\mathbb{P}_{p}\big{(}\pi\mbox{ is
open}\big{)}\times\mathbb{P}_{p}\Big{(}\big{|}C^{\beta}_{\mathbb{V}}(\\{y^{*}\\})\big{|}=\infty\Big{)}.$

By (5), this gives
$\theta^{\beta}_{\mathbb{H}}(p)\geq\mathbb{P}_{p}\big{(}\pi\mbox{ is
open}\big{)}\times\theta^{\beta}_{\mathbb{V}}(p)$. Since
$\mathbb{P}_{p}\big{(}\pi\mbox{ is open}\big{)}>0$ and
$\theta^{\beta}_{\mathbb{V}}(p)>0$, we have
$\theta^{\beta}_{\mathbb{H}}(p)>0$. This completes the proof of Claim 3.1. ∎

Now, we complete the proof of Theorem 3.3. In view of Theorem 3.2, it is
enough to show that for all $p\in[0,1]$, we have
$\theta^{\tilde{\beta}}_{\mathbb{V}}(p)>0$ if and only if
$\theta^{\tilde{\beta}}_{\mathbb{H}}(p)>0$.

Since $\mathbb{H}\subset\mathbb{V}$, it follows that for all $p\in[0,1]$,
$\theta^{\tilde{\beta}}_{\mathbb{H}}(p)>0$ implies
$\theta^{\tilde{\beta}}_{\mathbb{V}}(p)>0$. We proceed to show that for all
$p\in[0,1]$, $\theta^{\tilde{\beta}}_{\mathbb{V}}(p)>0$ implies
$\theta^{\tilde{\beta}}_{\mathbb{H}}(p)>0$. Suppose
$\theta^{\tilde{\beta}}_{\mathbb{V}}(p)>0$ for some $p\in[0,1]$. By the
assumptions on $\tilde{\beta}$ and $\beta$, it follows that every
$\tilde{\beta}$-backbend path in $\mathbb{V}$ from the origin is a
$\beta$-backbend path in $\mathbb{V}$ from the origin. Therefore,
$\theta^{\tilde{\beta}}_{\mathbb{V}}(p)>0$ implies
$\theta^{\beta}_{\mathbb{V}}(p)>0$. This, together with the assumptions on
$\beta$ and Claim 3.1, yields $\theta^{\beta}_{\mathbb{H}}(p)>0$. Moreover,
since $\tilde{\beta}$ converges from below to $\beta$ and
$\theta^{\beta}_{\mathbb{H}}(p)>0$, by Theorem 3.1, we have
$\theta^{\tilde{\beta}}_{\mathbb{H}}(p)>0$. This completes the proof of
Theorem 3.3. ∎

###  3.3 Results on half-slabs

Throughout this subsection, we assume that $d\geq 3$. Our next theorem
(Theorem 3.4) establishes a relation between percolation probabilities of a
$\tilde{\beta}$-backbend percolation process and a $\beta$-backbend
percolation process on every $e$-dimensional half-slab, where $\beta$ is
$k$-cyclic and $\tilde{\beta}$ converges from below to $\beta$. The proof of
the “only-if” part of the theorem is similar to that of Theorem 3.1, and the
proof of the “if” part of the theorem follows from that of Theorem 3.1 by
additionally imposing the following requirements: (i) $n^{*}\in 2\mathbb{N}$,
(ii) $y^{*}=(0,\ldots,0,kn^{*})$, and (iii) $\pi$ is in $\mathbb{Q}_{l}^{e}$.
It can be verified that these additional requirements do not affect the logic
of the proof of the “if” part of Theorem 3.1.

######  Theorem 3.4.

Suppose a backbend sequence $\tilde{\beta}$ converges from below to a
$k$-cyclic backbend sequence $\beta$ for some $k\in\mathbb{N}$. Then, for all
$l\in\mathbb{N}$, all $e\in\\{2,\ldots,d-1\\}$, and all $p\in[0,1]$, we have
$\theta^{\tilde{\beta}}_{\mathbb{Q}_{l}^{e}}(p)>0$ if and only if
$\theta^{\beta}_{\mathbb{Q}_{l}^{e}}(p)>0$.

We obtain the following corollary (Corollary 3.2) from Theorem 3.4. It says
that if a backbend sequence $\tilde{\beta}$ converges from below to a
$k$-cyclic backbend sequence $\beta$, then on every $e$-dimensional half-slab,
the critical probability of $\tilde{\beta}$-backbend percolation process will
be equal to that of $\beta$-backbend percolation process.

######  Corollary 3.2.

Suppose a backbend sequence $\tilde{\beta}$ converges from below to a
$k$-cyclic backbend sequence $\beta$ for some $k\in\mathbb{N}$. Then, for all
$l\in\mathbb{N}$ and all $e\in\\{2,\ldots,d-1\\}$, we have
$p^{\tilde{\beta}}_{c}(\mathbb{Q}_{l}^{e})=p^{\beta}_{c}(\mathbb{Q}_{l}^{e})$.

Our next theorem says that if a backbend sequence is $k$-cyclic in the limit
from below, then for all $e\in\\{2,\ldots,d-1\\}$, the limit of the critical
probability on the $e$-dimensional half-slab $\mathbb{Q}_{l}^{e}$ as $l$ goes
to infinity will be equal to the critical probability on the half-space.

######  Theorem 3.5.

Suppose a backbend sequence $\tilde{\beta}$ is $k$-cyclic in the limit from
below for some $k\in\mathbb{N}$. Then, for all $e\in\\{2,\ldots,d-1\\}$, we
have $\underset{l\rightarrow\infty}{\lim}\hskip
2.84526ptp^{\tilde{\beta}}_{c}(\mathbb{Q}_{l}^{e})=p^{\tilde{\beta}}_{c}(\mathbb{H})$.

###### Proof of Theorem 3.5.

Fix an arbitrary $e\in\\{2,\ldots,d-1\\}$. Let $\beta$ be the $k$-cyclic
backbend sequence such that $\tilde{\beta}$ converges from below to $\beta$.
In view of Corollary 3.1 and Corollary 3.2, it is enough to show that
$\underset{l\rightarrow\infty}{\lim}\hskip
2.84526ptp^{\beta}_{c}(\mathbb{Q}_{l}^{e})=p^{\beta}_{c}(\mathbb{H})$.

Since $\mathbb{Q}_{l}^{e}\subset\mathbb{Q}_{l+1}^{e}\subset\mathbb{H}$ for all
$l\in\mathbb{N}$, we have $\underset{l\rightarrow\infty}{\lim}\hskip
2.84526ptp^{\beta}_{c}(\mathbb{Q}_{l}^{e})\geq p^{\beta}_{c}(\mathbb{H})$.
Assume for contradiction that $\underset{l\rightarrow\infty}{\lim}\hskip
2.84526ptp^{\beta}_{c}(\mathbb{Q}_{l}^{e})>p^{\beta}_{c}(\mathbb{H})$. Let
$p\in[0,1]$ be such that
$p^{\beta}_{c}(\mathbb{H})<p<\underset{l\rightarrow\infty}{\lim}\hskip
2.84526ptp^{\beta}_{c}(\mathbb{Q}_{l}^{e})$. It must be that $0<p<1$ and
$\theta^{\beta}_{\mathbb{H}}(p)>0$. Since $0<p<1$ and
$\theta^{\beta}_{\mathbb{H}}(p)>0$, by Proposition 3.1, there exist
$l^{*},r^{*}\in\mathbb{N}$ and $0<\delta<p$ such that

$\mathbb{P}_{p-\delta}\bigg{(}\Big{|}C^{\beta}_{\mathbb{Q}_{l^{*}}^{2}}\Big{(}B\big{(}[-r^{*},r^{*}]^{d-1}\times
0\big{)}\Big{)}\Big{|}=\infty\bigg{)}>0,$

and hence

$\mathbb{P}_{p}\bigg{(}\Big{|}C^{\beta}_{\mathbb{Q}_{l^{*}}^{2}}\Big{(}B\big{(}[-r^{*},r^{*}]^{d-1}\times
0\big{)}\Big{)}\Big{|}=\infty\bigg{)}>0.$ (7)

Because $B\big{(}[-r^{*},r^{*}]^{d-1}\times 0\big{)}$ is a finite set, (7)
implies that there exists some $x\in B\big{(}[-r^{*},r^{*}]^{d-1}\times
0\big{)}$ such that
$\mathbb{P}_{p}\Big{(}\big{|}C^{\beta}_{(x+\mathbb{Q}_{l^{*}+r^{*}}^{2})}(\\{x\\})\big{|}=\infty\Big{)}>0$.
Furthermore, by the construction of the $\beta$-backbend percolation process,
we have
$\mathbb{P}_{p}\Big{(}\big{|}C^{\beta}_{\mathbb{Q}_{l^{*}+r^{*}}^{2}}\big{|}=\infty\Big{)}=\mathbb{P}_{p}\Big{(}\big{|}C^{\beta}_{(x+\mathbb{Q}_{l^{*}+r^{*}}^{2})}(\\{x\\})\big{|}=\infty\Big{)}$,
which implies
$\mathbb{P}_{p}\Big{(}\big{|}C^{\beta}_{\mathbb{Q}_{l^{*}+r^{*}}^{2}}\big{|}=\infty\Big{)}>0$.
Since $\mathbb{Q}_{l^{*}+r^{*}}^{2}\subseteq\mathbb{Q}_{l^{*}+r^{*}}^{e}$,
this gives

$\mathbb{P}_{p}\Big{(}\big{|}C^{\beta}_{\mathbb{Q}_{l^{*}+r^{*}}^{e}}\big{|}=\infty\Big{)}>0.$
(8)

However, since $p<\underset{l\rightarrow\infty}{\lim}\hskip
2.84526ptp^{\beta}_{c}(\mathbb{Q}_{l}^{e})$ and
$p^{\beta}_{c}(\mathbb{Q}_{l}^{e})$ decreases as $l$ increases, it must be
that $p<p^{\beta}_{c}(\mathbb{Q}_{l^{*}+r^{*}}^{e})$, a contradiction to (8).
This completes the proof of Theorem 3.5. ∎

######  Note 3.2.

As we have noted (Note 3.1) for Corollary 3.1, Theorem 3.5 also does not hold
in general. More precisely, as Example 3.2 shows, if we relax the assumption
of Theorem 3.5 by requiring that $\tilde{\beta}$ converges (not necessarily
from below) to a $k$-cyclic backbend sequence $\beta$, then Theorem 3.5 does
not hold on two-dimensional half-slabs when $\beta$ satisfies the condition
introduced in Subsection 3.2 (that is, $\beta_{i+1}\leq\beta_{i}+1$ for all
$i\in\\{0,\ldots,k-1\\}$).

We use the following proposition in Example 3.2. It provides an upper bound
and a lower bound of the critical probability of a $\tilde{\beta}$-backbend
percolation process on a two-dimensional half-slab when $\tilde{\beta}$
converges to a $k$-cyclic backbend sequence $\beta$ satisfying
$\beta_{i+1}\leq\beta_{i}+1$ for all $i\in\\{0,\ldots,k-1\\}$.

######  Proposition 3.2.

Let $k\in\mathbb{N}$ and suppose that a backbend sequence $\tilde{\beta}$
converges to a $k$-cyclic backbend sequence $\beta$ satisfying
$\beta_{i+1}\leq\beta_{i}+1$ for all $i\in\\{0,\ldots,k-1\\}$. Then, for all
$l\in\mathbb{N}$, we have $p^{\beta}_{c}(\mathbb{Q}_{2l}^{2})\leq
p^{\tilde{\beta}}_{c}(\mathbb{Q}_{l}^{2})\leq
p^{\beta}_{c}(\mathbb{Q}_{l}^{2})$.

The proof of this proposition is relegated to Appendix B.

We are now ready to present our counter example.

######  Example 3.2.

Suppose $d=3$. Consider the backbend sequences $\beta$ and $\tilde{\beta}$
given in Example 3.1, where $\beta$ is $1$-cyclic and $\tilde{\beta}$
converges to $\beta$. We have already shown that
$p^{\tilde{\beta}}_{c}(\mathbb{H})<p^{\beta}_{c}(\mathbb{H})$ in Example 3.1.
By the assumption on $\beta$ and $\tilde{\beta}$, it follows from Proposition
3.2 that $\underset{l\rightarrow\infty}{\lim}\hskip
2.84526ptp^{\tilde{\beta}}_{c}(\mathbb{Q}_{l}^{2})=\underset{l\rightarrow\infty}{\lim}\hskip
2.84526ptp^{\beta}_{c}(\mathbb{Q}_{l}^{2})$. Furthermore, by the assumption on
$\beta$, it follows from Theorem 3.5 that
$\underset{l\rightarrow\infty}{\lim}\hskip
2.84526ptp^{\beta}_{c}(\mathbb{Q}_{l}^{2})=p^{\beta}_{c}(\mathbb{H})$. The
facts $p^{\tilde{\beta}}_{c}(\mathbb{H})<p^{\beta}_{c}(\mathbb{H})$,
$\underset{l\rightarrow\infty}{\lim}\hskip
2.84526ptp^{\tilde{\beta}}_{c}(\mathbb{Q}_{l}^{2})=\underset{l\rightarrow\infty}{\lim}\hskip
2.84526ptp^{\beta}_{c}(\mathbb{Q}_{l}^{2})$, and
$\underset{l\rightarrow\infty}{\lim}\hskip
2.84526ptp^{\beta}_{c}(\mathbb{Q}_{l}^{2})=p^{\beta}_{c}(\mathbb{H})$ together
imply $\underset{l\rightarrow\infty}{\lim}\hskip
2.84526ptp^{\tilde{\beta}}_{c}(\mathbb{Q}_{l}^{2})>p^{\tilde{\beta}}_{c}(\mathbb{H})$.

In the following, we present a result on the half-space in two dimensions (as
we have mentioned in Remark 3.1). The proof of this result follows by using
similar logic as for the proof of Proposition 3.2.

######  Proposition 3.3.

Suppose $d=2$. Let $k\in\mathbb{N}$ and suppose that a backbend sequence
$\tilde{\beta}$ converges to a $k$-cyclic backbend sequence $\beta$ satisfying
$\beta_{i+1}\leq\beta_{i}+1$ for all $i\in\\{0,\ldots,k-1\\}$. Then,
$p^{\tilde{\beta}}_{c}(\mathbb{H})=p^{\beta}_{c}(\mathbb{H})$.

##  Appendix A Proof of Proposition 3.1

We first introduce some notations and make a remark to facilitate the proof.

For all $l,t\in\mathbb{Z}_{+}\cup\\{\infty\\}$, all $u\in\\{-1,1\\}^{d-1}$,
and all $v\in\\{-1,1\\}^{d-2}$, let us define

$\displaystyle T(l,t):=\Big{\\{}x\in
B\big{(}[-l,l]^{d-1}\times[0,t]\big{)}:x_{d}=t\Big{\\}},$ $\displaystyle
F(l,t):=\Big{\\{}x\in B\big{(}[-l,l]^{d-1}\times[0,t]\big{)}:|x_{i}|=l\mbox{
for some }i\in\\{1,\ldots,d-1\\}\Big{\\}},$ $\displaystyle
F_{(d-1)^{+}}(l,t):=\Big{\\{}x\in F(l,t):x_{d-1}=l\Big{\\}},$ $\displaystyle
T^{u}(l,t):=\Big{\\{}x\in T(l,t):0\leq x_{i}u_{i}\leq l\mbox{ for all
}i\in\\{1,\ldots,d-1\\}\Big{\\}},\mbox{ and}$ $\displaystyle
F^{v}_{(d-1)^{+}}(l,t):=\Big{\\{}x\in F_{(d-1)^{+}}(l,t):0\leq x_{i}v_{i}\leq
l\mbox{ for all }i\in\\{1,\ldots,d-2\\}\Big{\\}}.$

The following remark follows from basic probability.

######  Remark A.1.

Let $\alpha>0$ and let $A_{1},A_{2},A_{3}$ be events such that $A_{2}$ and
$A_{3}$ are disjoint with $\mathbb{P}_{p}(A_{2})>0$ and
$\mathbb{P}_{p}(A_{3})>0$.

  1. (a)

Suppose $\mathbb{P}_{p}(A_{1}\mid A_{2})<\alpha$ and $\mathbb{P}_{p}(A_{1}\mid
A_{3})<\alpha$. Then, $\mathbb{P}_{p}\big{(}A_{1}\bigm{|}(A_{2}\cup
A_{3})\big{)}<\alpha$.

  2. (b)

Suppose $\mathbb{P}_{p}(A_{1}\mid A_{2})>\alpha$ and $\mathbb{P}_{p}(A_{1}\mid
A_{3})>\alpha$. Then, $\mathbb{P}_{p}\big{(}A_{1}\bigm{|}(A_{2}\cup
A_{3})\big{)}>\alpha$.

  3. (c)

Suppose $\mathbb{P}_{p}(A_{1}\mid A_{2})=\alpha$ and $\mathbb{P}_{p}(A_{1}\mid
A_{3})=\alpha$. Then, $\mathbb{P}_{p}\big{(}A_{1}\bigm{|}(A_{2}\cup
A_{3})\big{)}=\alpha$.

###  A.1 The proof

Let $\eta^{*}\in(0,1)$ be sufficiently small. Choose $i^{*}\in\mathbb{N}$ with
$i^{*}>10$, and $\epsilon^{*}\in(0,1)$ such that
$(1-\epsilon^{*})^{2i^{*}}>1-\eta^{*}$. We prove Proposition 3.1 by
distinguishing the following two cases.
Case 1: Suppose $\theta^{\beta}_{\mathbb{S}_{t}}(p)=0$ for all
$t\in\mathbb{N}$.

Since $\theta^{\beta}_{\mathbb{H}}(p)>0$, by a standard argument (see Liggett
(2012), Theorem 1.10(d) in Chapter VI for details), there exists $r^{*}\in
2\hat{k}\mathbb{N}$ with
$2r^{*}\geq\max\\{\beta_{0},\ldots,\beta_{\hat{k}-1}\\}$ such that

$\mathbb{P}_{p}\Bigg{(}\Big{|}C^{\beta}_{\mathbb{H}}\Big{(}B\big{(}[-r^{*},r^{*}]^{d-1}\times
0\big{)}\Big{)}\Big{|}=\infty\Bigg{)}>1-\frac{1}{2^{2^{d-1}}}\bigg{(}\frac{\epsilon^{*}}{5}\bigg{)}^{d2^{d-1}}.$
(A.1)

For ease of presentation, let us denote $B\big{(}[-r^{*},r^{*}]^{d-1}\times
0\big{)}$ by $D^{*}$. We define the following terms involving $\epsilon^{*}$
and $r^{*}$ for our next lemma.

  1. $\bullet$

Let $\alpha^{*}:=\min\left\\{\begin{aligned}
&\mathbb{P}_{p}\Big{(}B\big{(}[-r^{*},r^{*}]^{d-1}\times[2r^{*},2r^{*}+2\hat{k}]\big{)}\subseteq
C^{0}_{B\big{(}[-r^{*},r^{*}]^{d-1}\times[0,2r^{*}+2\hat{k}]\big{)}}\Big{)},\\\
&\mathbb{P}_{p}\Big{(}B\big{(}[0,2r^{*}]\times[-r^{*},r^{*}]^{d-2}\times[2r^{*},2r^{*}+2\hat{k}]\big{)}\subseteq
C^{0}_{B\big{(}[0,2r^{*}]\times[-r^{*},r^{*}]^{d-2}\times[0,2r^{*}+2\hat{k}]\big{)}}\Big{)}\end{aligned}\right\\}$.121212By
$C^{0}_{\widehat{\mathbb{V}}}$, we denote $C^{\beta}_{\widehat{\mathbb{V}}}$
where $\beta_{i}=0$ for all $i\in\mathbb{Z}_{+}$. Clearly, $\alpha^{*}>0$.

  2. $\bullet$

Let $m^{*}\in\mathbb{N}$ be such that
$(1-\alpha^{*})^{m^{*}}<\frac{\epsilon^{*}}{5}$. The implication of $m^{*}$ is
that if one performs at least $m^{*}$ independent trials with the probability
of success in each trial being $\alpha^{*}$, then the probability that there
is at least one success exceeds $1-\frac{\epsilon^{*}}{5}$.

  3. $\bullet$

Let $n^{*}\in\mathbb{N}$ be such that $n^{*}\geq
m^{*}(8r^{*}+8\hat{k}+1)^{d}$. The purpose of $n^{*}$ is to ensure that in any
subset of $\mathbb{V}$ having size $n^{*}$ or larger, there are at least
$m^{*}$ vertices such the $L^{\infty}$-distance between any two of them is at
least $(4r^{*}+4\hat{k}+1)$.131313$L^{\infty}(x,y):=\underset{i}{\max}\hskip
2.84526pt\big{\\{}|x_{i}-y_{i}|\big{\\}}$.

######  Lemma A.1.

There exist $l^{*},t^{*}\in 2\hat{k}\mathbb{N}$ with $l^{*}\geq r^{*}$ such
that

  1. (i)

$\mathbb{P}_{p}\bigg{(}\Big{|}C^{\beta}_{B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,t^{*}]\big{)}}(D^{*})\cap
T^{u}(l^{*},t^{*})\Big{|}\geq n^{*}\bigg{)}>1-\frac{\epsilon^{*}}{5}$ for all
$u\in\\{-1,1\\}^{d-1}$, and

  2. (ii)

$\mathbb{P}_{p}\bigg{(}\Big{|}C^{\beta}_{B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,t^{*}]\big{)}}(D^{*})\cap
F^{v}_{(d-1)^{+}}(l^{*},t^{*})\Big{|}\geq
n^{*}\bigg{)}>1-\frac{\epsilon^{*}}{5}$ for all $v\in\\{-1,1\\}^{d-2}$.

The proof of this lemma is relegated to Appendix A.1.1.

Suppose $l^{*}$ and $t^{*}$ are such that the statement of Lemma A.1 holds.
Let $s^{*}:=t^{*}+2r^{*}+2\hat{k}$. Since
$2r^{*}\geq\max\\{\beta_{0},\ldots,\beta_{\hat{k}-1}\\}$ and $r^{*},t^{*}\in
2\hat{k}\mathbb{N}$, by the construction of $s^{*}$, we have $s^{*}\in
2\hat{k}\mathbb{N}$ and $s^{*}>\max\\{\beta_{0},\ldots,\beta_{\hat{k}-1}\\}$.

######  Lemma A.2.

There exists $0<\delta<p$ such that for all $x\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times 0\times 0\big{)}$, we have

$\mathbb{P}_{p-\delta}\left(\begin{aligned} &\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-l^{*},3l^{*}]\times[0,z_{d}]\big{)}}(D^{*}+x)\\\
&\mbox{ for some }z\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[l^{*},2l^{*}]\times[s^{*},2s^{*}]\big{)}\mbox{
with }z_{d}\in 2\hat{k}\mathbb{N}\end{aligned}\right)>1-\epsilon^{*}.$

The proof of this lemma is relegated to Appendix A.1.2.

Suppose $\delta^{*}$ is such that the statement of Lemma A.2 holds.

######  Lemma A.3.

For all $x\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[-2l^{*},2l^{*}]\times[0,2s^{*}]\big{)}$
with $x_{d}\in 2\hat{k}\mathbb{Z}_{+}$, we have

$\mathbb{P}_{p-\delta^{*}}\left(\begin{aligned}
&\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-3l^{*},4l^{*}]\times[x_{d},z_{d}]\big{)}}(D^{*}+x)\\\
&\mbox{ for some }z\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[-l^{*},3l^{*}]\times[2s^{*},4s^{*}]\big{)}\mbox{
with }z_{d}\in 2\hat{k}\mathbb{N}\end{aligned}\right)>(1-\epsilon^{*})^{2}.$

The proof of this lemma is relegated to Appendix A.1.3.

For all $i\in\mathbb{N}$, we denote the set of vertices
$B\big{(}[-l^{*},l^{*}]^{d-2}\times[(i-2)l^{*},(i+2)l^{*}]\times[2is^{*},2(i+1)s^{*}]\big{)}$
by $B^{+}_{i}$. For all $i\in\mathbb{N}$ and all $z\in B^{+}_{i}$, let us
define

$\mathcal{R}^{+}(z):=\left\\{y\in\mathbb{H}:\begin{aligned} &\hskip
2.84526pt-2l^{*}\leq y_{j}\leq 2l^{*}\mbox{ for all
}j\in\\{1,\ldots,d-2\\},\\\ &\hskip
2.84526pt-5l^{*}+\frac{l^{*}}{2s^{*}}y_{d}\leq y_{d-1}\leq
5l^{*}+\frac{l^{*}}{2s^{*}}y_{d},\mbox{ and }0\leq y_{d}\leq
z_{d}\end{aligned}\right\\}.$

For all $i\in\mathbb{N}$ and all $x\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[-2l^{*},2l^{*}]\times[0,2s^{*}]\big{)}$
with $x_{d}\in 2\hat{k}\mathbb{Z}_{+}$, let us define the following event:

$G_{i}^{+}(x):=\left\\{\omega\in\Omega:\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{\mathcal{R}^{+}(z)}(D^{*}+x)\mbox{ for some }z\in B^{+}_{i}\mbox{
with }z_{d}\in 2\hat{k}\mathbb{N}\right\\}.$

######  Lemma A.4.

For all $i\in\mathbb{N}$ and all $x\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[-2l^{*},2l^{*}]\times[0,2s^{*}]\big{)}$
with $x_{d}\in 2\hat{k}\mathbb{Z}_{+}$, we have
$\mathbb{P}_{p-\delta^{*}}\big{(}G_{i}^{+}(x)\big{)}>(1-\epsilon^{*})^{2i}$.

The proof of this lemma is relegated to Appendix A.1.4.

For all $i\in\mathbb{N}$, we denote the set of vertices
$B\big{(}[-l^{*},l^{*}]^{d-2}\times[(-i-2)l^{*},(-i+2)l^{*}]\times[2is^{*},2(i+1)s^{*}]\big{)}$
by $B^{-}_{i}$. For all $i\in\mathbb{N}$ and all $z\in B^{-}_{i}$, let us
define

$\mathcal{R}^{-}(z):=\left\\{y\in\mathbb{H}:\begin{aligned} &\hskip
2.84526pt-2l^{*}\leq y_{j}\leq 2l^{*}\mbox{ for all
}j\in\\{1,\ldots,d-2\\},\\\ &\hskip
2.84526pt-5l^{*}-\frac{l^{*}}{2s^{*}}y_{d}\leq y_{d-1}\leq
5l^{*}-\frac{l^{*}}{2s^{*}}y_{d},\mbox{ and }0\leq y_{d}\leq
z_{d}\end{aligned}\right\\}.$

For all $i\in\mathbb{N}$ and all $x\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[-2l^{*},2l^{*}]\times[0,2s^{*}]\big{)}$
with $x_{d}\in 2\hat{k}\mathbb{Z}_{+}$, let us define the following event:

$G_{i}^{-}(x):=\left\\{\omega\in\Omega:\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{\mathcal{R}^{-}(z)}(D^{*}+x)\mbox{ for some }z\in B^{-}_{i}\mbox{
with }z_{d}\in 2\hat{k}\mathbb{N}\right\\}.$

By the assumptions on $i^{*}$ and $\epsilon^{*}$, and the construction of the
$\beta$-backbend percolation process, Lemma A.4 implies that for all $x\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[-2l^{*},2l^{*}]\times[0,2s^{*}]\big{)}$
with $x_{d}\in 2\hat{k}\mathbb{Z}_{+}$,

$\mathbb{P}_{p-\delta^{*}}(G_{i^{*}}^{+}(x))>1-\eta^{*}\mbox{ and
}\mathbb{P}_{p-\delta^{*}}(G_{i^{*}}^{-}(x))>1-\eta^{*}.$ (A.2)

######  Lemma A.5.

$\mathbb{P}_{p-\delta^{*}}\bigg{(}\Big{|}C^{\beta}_{\mathbb{Q}_{2l^{*}}^{2}}(D^{*})\Big{|}=\infty\bigg{)}>0$.

Since $\beta$ is $\hat{k}$-cyclic,
$s^{*}>\max\\{\beta_{0},\ldots,\beta_{\hat{k}-1}\\}$, $i^{*}>10$, and
$\eta^{*}$ is sufficiently small, the proof of Lemma A.5 follows from (A.2) by
using a similar logic as for the proof of Lemma 21 in Bezuidenhout et al.
(1990).

Now, the proof of Proposition 3.1 for Case 1 follows from Lemma A.5.

Case 2: Suppose $\theta^{\beta}_{\mathbb{S}_{t}}(p)>0$ for some
$t\in\mathbb{N}$.
Fix $\hat{t}\in\mathbb{N}$ such that
$\theta^{\beta}_{\mathbb{S}_{\hat{t}}}(p)>0$. By a standard argument (see
Liggett (2012), Theorem 1.10(d) in Chapter VI for details), this implies that
there exists $r^{*}\in 2\hat{k}\mathbb{N}$ with
$2r^{*}\geq\max\\{\hat{t},\beta_{0},\ldots,\beta_{\hat{k}-1}\\}$ such that

$\mathbb{P}_{p}\Bigg{(}\Big{|}C^{\beta}_{\mathbb{S}_{\hat{t}}}\Big{(}B\big{(}[-r^{*},r^{*}]^{d-1}\times
0\big{)}\Big{)}\Big{|}=\infty\Bigg{)}>1-\bigg{(}\frac{\epsilon^{*}}{2}\bigg{)}^{(d-1)2^{d-1}}.$
(A.3)

For ease of presentation, we denote $B\big{(}[-r^{*},r^{*}]^{d-1}\times
0\big{)}$ by $D^{*}$. Let $t^{*}:=2r^{*}$. Since $r^{*}\in 2\hat{k}\mathbb{N}$
and $2r^{*}\geq\max\\{\hat{t},\beta_{0},\ldots,\beta_{\hat{k}-1}\\}$, this
implies $t^{*}\in 2\hat{k}\mathbb{N}$ and
$t^{*}\geq\max\\{\hat{t},\beta_{0},\ldots,\beta_{\hat{k}-1}\\}$. Furthermore,
since $t^{*}\geq\hat{t}$, $\theta^{\beta}_{\mathbb{S}_{\hat{t}}}(p)>0$ implies
$\theta^{\beta}_{\mathbb{S}_{t^{*}}}(p)>0$ and (A.3) implies

$\mathbb{P}_{p}\Bigg{(}\Big{|}C^{\beta}_{\mathbb{S}_{t^{*}}}(D^{*})\Big{|}=\infty\Bigg{)}>1-\bigg{(}\frac{\epsilon^{*}}{2}\bigg{)}^{(d-1)2^{d-1}}.$
(A.4)

Let us define the following terms involving $\epsilon^{*}$ and $r^{*}$ for our
next lemma.

  1. $\bullet$

Let
$\alpha^{*}:=\mathbb{P}_{p}\Big{(}B\big{(}[0,2r^{*}]\times[-r^{*},r^{*}]^{d-2}\times[2r^{*}+2\hat{k},2r^{*}+4\hat{k}]\big{)}\subseteq
C^{0}_{B\big{(}[0,2r^{*}]\times[-r^{*},r^{*}]^{d-2}\times[0,2r^{*}+4\hat{k}]\big{)}}\Big{)}$.
Clearly, $\alpha^{*}>0$.

  2. $\bullet$

Let $m^{*}\in\mathbb{N}$ be such that
$(1-\alpha^{*})^{m^{*}}<\frac{\epsilon^{*}}{2}$. The implication of $m^{*}$ is
that if one performs at least $m^{*}$ independent trials with probability of
success in each trial being $\alpha^{*}$, the probability that there is at
least one success (out of all trials) exceeds $1-\frac{\epsilon^{*}}{2}$.

  3. $\bullet$

Let $n^{*}\in\mathbb{N}$ be such that $n^{*}\geq
m^{*}(8r^{*}+16\hat{k}+1)^{d}$. The purpose of $n^{*}$ is to ensure that in
any subset of $\mathbb{V}$ having size $n^{*}$ or larger, there are at least
$m^{*}$ vertices such the $L^{\infty}$-distance between any two of them is at
least $(4r^{*}+8\hat{k}+1)$.

######  Lemma A.6.

There exists $l^{*}\in 2\hat{k}\mathbb{N}$ with $l^{*}\geq r^{*}$ such that
for all $v\in\\{-1,1\\}^{d-2}$, we have

$\mathbb{P}_{p}\bigg{(}\Big{|}C^{\beta}_{B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,t^{*}]\big{)}}(D^{*})\cap
F^{v}_{(d-1)^{+}}(l^{*},t^{*})\Big{|}\geq
n^{*}\bigg{)}>1-\frac{\epsilon^{*}}{2}.$

The proof of this lemma is relegated to Appendix A.1.5.

Suppose $l^{*}$ is such that the statement of Lemma A.6 holds. Let
$s^{*}:=t^{*}+2\hat{k}$. Since $t^{*}\in 2\hat{k}\mathbb{N}$ and
$t^{*}\geq\max\\{\beta_{0},\ldots,\beta_{\hat{k}-1}\\}$, by the construction
of $s^{*}$, we have $s^{*}\in 2\hat{k}\mathbb{N}$ and
$s^{*}>\max\\{\beta_{0},\ldots,\beta_{\hat{k}-1}\\}$.

######  Lemma A.7.

There exists $0<\delta<p$ such that for all $x\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times 0\times 0\big{)}$, we have

$\mathbb{P}_{p-\delta}\left(\begin{aligned} &\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-l^{*},3l^{*}]\times[0,z_{d}]\big{)}}(D^{*}+x)\\\
&\mbox{ for some }z\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[l^{*},2l^{*}]\times[s^{*},2s^{*}]\big{)}\mbox{
with }z_{d}\in 2\hat{k}\mathbb{N}\end{aligned}\right)>1-\epsilon^{*}.$

The proof of this lemma is relegated to Appendix A.1.6.

Suppose $\delta^{*}$ is such that the statement of Lemma A.7 holds. Then, the
proof of Proposition 3.1 for Case 2 follows from Lemma A.7 by using an
argument similar to the one which we have used to complete the proof of
Proposition 3.1 for Case 1 from Lemma A.2.

####  A.1.1 Proof of Lemma A.1

######  Claim A.1.

Let $N\in\mathbb{N}$. Then,

$\mathbb{P}_{p}\bigg{(}\big{|}C^{\beta}_{\mathbb{H}}(D^{*})\big{|}=\infty\mbox{
and }\Big{|}\Big{(}C^{\beta}_{\mathbb{S}_{t}}(D^{*})\cap
T(\infty,t)\Big{)}\Big{|}<N\mbox{ for infinitely many
}t\in\mathbb{N}\bigg{)}=0.$

###### Proof of Claim A.1..

Define the following events:

$\displaystyle
B_{1}:=\bigg{\\{}\omega\in\Omega:\big{|}C^{\beta}_{\mathbb{H}}(D^{*})\big{|}=\infty\mbox{
and }\Big{|}\Big{(}C^{\beta}_{\mathbb{S}_{t}}(D^{*})\cap
T(\infty,t)\Big{)}\Big{|}<N\mbox{ for infinitely many
}t\in\mathbb{N}\bigg{\\}},$ $\displaystyle
B_{2}:=\left\\{\omega\in\Omega:\begin{aligned} &\hskip
2.84526pt\Big{|}\Big{(}C^{\beta}_{\mathbb{S}_{t^{\prime}}}(D^{*})\cap
T(\infty,t^{\prime})\Big{)}\Big{|}>0\hskip 2.84526pt\forall\hskip
2.84526ptt^{\prime}\in\mathbb{N}\mbox{ and}\\\ &\hskip
2.84526pt\Big{|}\Big{(}C^{\beta}_{\mathbb{S}_{t}}(D^{*})\cap
T(\infty,t)\Big{)}\Big{|}<N\mbox{ for infinitely many
}t\in\mathbb{N}\end{aligned}\right\\},\mbox{ and}$ $\displaystyle
B_{3}:=\bigg{\\{}\omega\in\Omega:\big{|}C^{\beta}_{\mathbb{H}}(D^{*})\big{|}=\infty\mbox{
and }\Big{|}\Big{(}C^{\beta}_{\mathbb{S}_{t^{\prime}}}(D^{*})\cap
T(\infty,t^{\prime})\Big{)}\Big{|}=0\mbox{ for some
}t^{\prime}\in\mathbb{N}\bigg{\\}}.$

By the constructions of $B_{1},B_{2}$, and $B_{3}$, along with the
construction of the $\beta$-backbend percolation process, we have
$B_{1}=B_{2}\cup B_{3}$ and $B_{2}\cap B_{3}=\emptyset$.

First, we show that $\mathbb{P}_{p}\big{(}B_{3}\big{)}=0$. For all
$t\in\mathbb{N}$, define

$B_{3}^{t}:=\bigg{\\{}\omega\in\Omega:\big{|}C^{\beta}_{\mathbb{H}}(D^{*})\big{|}=\infty\mbox{
and }\Big{|}\Big{(}C^{\beta}_{\mathbb{S}_{t}}(D^{*})\cap
T(\infty,t)\Big{)}\Big{|}=0\bigg{\\}}.$

By the assumption for Case 1, we have $\theta^{\beta}_{\mathbb{S}_{t}}(p)=0$
for all $t\in\mathbb{N}$. Since $D^{*}$ is a finite set of vertices, by the
construction of the $\beta$-backbend percolation process, this implies
$\mathbb{P}_{p}\big{(}B_{3}^{t}\big{)}=0$ for all $t\in\mathbb{N}$. By
construction, we have $B_{3}=\underset{t\in\mathbb{N}}{\cup}\hskip
2.84526ptB_{3}^{t}$. Since $\mathbb{P}_{p}\big{(}B_{3}^{t}\big{)}=0$ for all
$t\in\mathbb{N}$, this implies $\mathbb{P}_{p}\big{(}B_{3}\big{)}=0$.

Next, we show that $\mathbb{P}_{p}\big{(}B_{2}\big{)}=0$. For all
$t\in\mathbb{Z}_{+}$, let $\mathcal{F}_{t}$ be the $\sigma$-algebra generated
by $\Big{\\{}\big{|}\big{(}C^{\beta}_{\mathbb{S}_{t^{\prime}}}(D^{*})\cap
T(\infty,t^{\prime})\big{)}\big{|}:0\leq t^{\prime}\leq t\Big{\\}}$.141414Note
that $\big{|}\big{(}C^{\beta}_{\mathbb{S}_{t}}(D^{*})\cap
T(\infty,t)\big{)}\big{|}$ is a random variable for all $t\in\mathbb{Z}_{+}$.
Fix $t\in\mathbb{N}$ and $F\in\mathcal{F}_{t}$. For a vertex $x\in
T(\infty,t)$, define the following event:

$A_{x}:=\bigg{\\{}\omega\in\Omega:\langle x,y\rangle\mbox{ is closed whenever
}y\in T(\infty,t+1)\bigg{\\}}.$

Clearly, $\mathbb{P}_{p}(A_{x})=(1-p)^{2^{d-1}}$ for all $x\in T(\infty,t)$.
Fix $S\subseteq T(\infty,t)$ such that $|S|<N$. Then,

$\displaystyle\hskip
2.84526pt\mathbb{P}_{p}\Bigg{(}\Big{|}\Big{(}C^{\beta}_{\mathbb{S}_{t+1}}(D^{*})\cap
T(\infty,t+1)\Big{)}\Big{|}=0\hskip 5.69054pt\biggm{|}\hskip 5.69054ptF\mbox{
and }\Big{(}C^{\beta}_{\mathbb{S}_{t}}(D^{*})\cap
T(\infty,t)\Big{)}=S\Bigg{)}$ $\displaystyle=$ $\displaystyle\hskip
2.84526pt\mathbb{P}_{p}\bigg{(}\underset{x\in S}{\cap}\hskip
2.84526ptA_{x}\bigg{)}\hskip 28.45274pt\mbox{(by the construction of the
$\beta$-backbend percolation process)}$ $\displaystyle=$ $\displaystyle\hskip
2.84526pt\underset{x\in S}{\prod}\hskip
2.84526pt\mathbb{P}_{p}\big{(}A_{x}\big{)}\hskip 34.1433pt\mbox{(since each
edge is independent)}$ $\displaystyle>$ $\displaystyle\hskip
2.84526pt(1-p)^{2^{d-1}N}.$

This, along with Remark A.1, yields

$\mathbb{P}_{p}\Bigg{(}\Big{|}\Big{(}C^{\beta}_{\mathbb{S}_{t+1}}(D^{*})\cap
T(\infty,t+1)\Big{)}\Big{|}>0\hskip 5.69054pt\biggm{|}\hskip 5.69054ptF\mbox{
and }\Big{|}\Big{(}C^{\beta}_{\mathbb{S}_{t}}(D^{*})\cap
T(\infty,t)\Big{)}\Big{|}<N\Bigg{)}<1-(1-p)^{2^{d-1}N}.$ (A.5)

Define the sequence of stopping times $(\tau_{m})_{m\in\mathbb{N}}$ as
follows. For all $\omega\in\Omega$,

  1. (i)

$\tau_{1}(\omega):=\min\hskip
2.84526pt\bigg{\\{}t\in\mathbb{N}:\Big{|}\Big{(}C^{\beta}_{\mathbb{S}_{t}}(D^{*})\cap
T(\infty,t)\Big{)}\Big{|}<N\bigg{\\}}$ and

  2. (ii)

for all $m\in\mathbb{N}\setminus\\{1\\}$, $\tau_{m}(\omega):=\min\hskip
2.84526pt\bigg{\\{}t>\tau_{m-1}(\omega):\Big{|}\Big{(}C^{\beta}_{\mathbb{S}_{t}}(D^{*})\cap
T(\infty,t)\Big{)}\Big{|}<N\bigg{\\}}$,

where $\min\hskip 2.84526pt\emptyset=\infty$. By the definition of
$\\{\tau_{m}\\}_{m\in\mathbb{N}}$, along with (A.5) and Remark A.1, for all
$m\in\mathbb{N}$ and all $F\in\mathcal{F}_{\tau_{m}}$, we have

$\mathbb{P}_{p}\bigg{(}\Big{|}\Big{(}C^{\beta}_{\mathbb{S}_{\tau_{m}+1}}(D^{*})\cap
T(\infty,\tau_{m}+1)\Big{)}\Big{|}>0\hskip 5.69054pt\Bigm{|}\hskip
5.69054ptF\mbox{ and }\tau_{m}<\infty\bigg{)}<1-(1-p)^{2^{d-1}N}.$ (A.6)

Now,

$\displaystyle\mathbb{P}_{p}(B_{2})\hskip 2.84526pt=$ $\displaystyle\hskip
2.84526pt\mathbb{P}_{p}\bigg{(}\Big{|}\Big{(}C^{\beta}_{\mathbb{S}_{t^{\prime}}}(D^{*})\cap
T(\infty,t^{\prime})\Big{)}\Big{|}>0\hskip 2.84526pt\forall\hskip
2.84526ptt^{\prime}\in\mathbb{N}\mbox{ and }\tau_{m}<\infty\hskip
2.84526pt\forall\hskip 2.84526ptm\in\mathbb{N}\bigg{)}$ (A.7) $\displaystyle=$
$\displaystyle\hskip
2.84526pt\mathbb{P}_{p}\big{(}\tau_{1}<\infty\big{)}\times\mathbb{P}_{p}\bigg{(}\Big{|}\Big{(}C^{\beta}_{\mathbb{S}_{\tau_{1}+1}}(D^{*})\cap
T(\infty,\tau_{1}+1)\Big{)}\Big{|}>0\hskip 5.69054pt\biggm{|}\hskip
5.69054pt\tau_{1}<\infty\bigg{)}$
$\displaystyle\times\mathbb{P}_{p}\Bigg{(}\tau_{2}<\infty\hskip
5.69054pt\biggm{|}\hskip
5.69054pt\tau_{1}<\infty,\Big{|}\Big{(}C^{\beta}_{\mathbb{S}_{\tau_{1}+1}}(D^{*})\cap
T(\infty,\tau_{1}+1)\Big{)}\Big{|}>0\Bigg{)}$
$\displaystyle\times\mathbb{P}_{p}\left(\Big{|}\Big{(}C^{\beta}_{\mathbb{S}_{\tau_{2}+1}}(D^{*})\cap
T(\infty,\tau_{2}+1)\Big{)}\Big{|}>0\hskip 9.95845pt\vrule\hskip
9.95845pt\begin{aligned}
&\tau_{1}<\infty,\Big{|}\Big{(}C^{\beta}_{\mathbb{S}_{\tau_{1}+1}}(D^{*})\cap
T(\infty,\tau_{1}+1)\Big{)}\Big{|}>0,\\\ &\tau_{2}<\infty\\\
\end{aligned}\right)$ $\displaystyle\times\cdots.$

Since $p<1$, we have $1-(1-p)^{2^{d-1}N}<1$. This, together with (A.6) and
(A.7), implies $\mathbb{P}_{p}(B_{2})=0$.

Since $B_{1}=B_{2}\cup B_{3}$ and $B_{2}\cap B_{3}=\emptyset$, the facts
$\mathbb{P}_{p}(B_{2})=0$ and $\mathbb{P}_{p}(B_{3})=0$ together imply
$\mathbb{P}_{p}(B)=0$. This completes the proof of Claim A.1. ∎

Let $k^{*}\in\mathbb{N}$ be such that
$\Big{(}1-(1-p^{2\hat{k}})^{k^{*}}\Big{)}^{n^{*}}>1-\frac{\epsilon^{*}}{10}$.151515Since
$p$, $n^{*}$, and $\epsilon^{*}$ are already fixed, the facts $p>0$ and
$1-\Big{(}1-\frac{\epsilon^{*}}{10}\Big{)}^{\frac{1}{n^{*}}}>0$ together imply
$k^{*}$ is well-defined. Define
$A:=\Big{\\{}\omega\in\Omega:\big{|}C^{\beta}_{\mathbb{H}}(D^{*})\big{|}=\infty\Big{\\}}$
and
$A_{k}:=\bigg{\\{}\omega\in\Omega:\Big{|}\Big{(}C^{\beta}_{\mathbb{S}_{t}}(D^{*})\cap
T(\infty,t)\Big{)}\Big{|}\geq 2^{d-1}k^{*}n^{*}\hskip 2.84526pt\forall\hskip
2.84526ptt\geq k\bigg{\\}}$ for all $k\in\mathbb{N}$. By construction, we have

$A_{k}\underset{k\rightarrow\infty}{\uparrow}\Bigg{(}A\setminus\bigg{\\{}\omega:\big{|}C^{\beta}_{\mathbb{H}}(D^{*})\big{|}=\infty\mbox{
and }\Big{|}\Big{(}C^{\beta}_{\mathbb{S}_{t}}(D^{*})\cap
T(\infty,t)\Big{)}\Big{|}<2^{d-1}k^{*}n^{*}\mbox{ for infinitely many
}t\in\mathbb{N}\bigg{\\}}\Bigg{)},$

which, together with Claim A.1, yields
$\mathbb{P}_{p}(A_{k})\underset{k\rightarrow\infty}{\uparrow}\mathbb{P}_{p}(A)$.
This, along with (A.1), implies that there exists $t_{1}\in
2\hat{k}\mathbb{N}$ such that
$\mathbb{P}_{p}\big{(}A_{t_{1}}\big{)}>1-\frac{1}{2^{2^{d-1}}}\Big{(}\frac{\epsilon^{*}}{5}\Big{)}^{d2^{d-1}}$.
This, together with the construction of $A_{t_{1}}$, yields

$\mathbb{P}_{p}\bigg{(}\Big{|}\Big{(}C^{\beta}_{\mathbb{S}_{t_{1}}}(D^{*})\cap
T(\infty,t_{1})\Big{)}\Big{|}\geq
2^{d-1}k^{*}n^{*}\bigg{)}>1-\frac{1}{2^{2^{d-1}}}\bigg{(}\frac{\epsilon^{*}}{5}\bigg{)}^{d2^{d-1}}.$
(A.8)

Furthermore, by the assumption for Case 1, we have
$\theta^{\beta}_{\mathbb{S}_{t_{1}}}(p)=0$. Since $D^{*}$ is a finite set of
vertices, this implies

$\mathbb{P}_{p}\bigg{(}\Big{|}\Big{(}C^{\beta}_{B\big{(}[-l,l]^{d-1}\times[0,t_{1}]\big{)}}(D^{*})\cap
T(l,t_{1})\Big{)}\Big{|}\geq
2^{d-1}k^{*}n^{*}\bigg{)}\underset{l\rightarrow\infty}{\uparrow}\mathbb{P}_{p}\bigg{(}\Big{|}\Big{(}C^{\beta}_{\mathbb{S}_{t_{1}}}(D^{*})\cap
T(\infty,t_{1})\Big{)}\Big{|}\geq 2^{d-1}k^{*}n^{*}\bigg{)}.$

Since
$\Big{(}\frac{\epsilon^{*}}{10}\Big{)}^{2^{d-1}}>\frac{1}{2^{2^{d-1}}}\Big{(}\frac{\epsilon^{*}}{5}\Big{)}^{d2^{d-1}}$,
this, along with (A.8), implies that there exists $l(t_{1})\in
2\hat{k}\mathbb{N}$ such that

$\mathbb{P}_{p}\bigg{(}\Big{|}\Big{(}C^{\beta}_{B\big{(}[-l(t_{1}),l(t_{1})]^{d-1}\times[0,t_{1}]\big{)}}(D^{*})\cap
T(l(t_{1}),t_{1})\Big{)}\Big{|}\geq
2^{d-1}k^{*}n^{*}\bigg{)}>1-\bigg{(}\frac{\epsilon^{*}}{10}\bigg{)}^{2^{d-1}}.$

Define

$s(l(t_{1}),t_{1}):=\underset{s\in 2\hat{k}\mathbb{N}}{\min}\hskip
2.84526pt\Bigg{\\{}s\geq
t_{1}:\mathbb{P}_{p}\bigg{(}\Big{|}\Big{(}C^{\beta}_{B\big{(}[-l(t_{1}),l(t_{1})]^{d-1}\times[0,s]\big{)}}(D^{*})\cap
T(l(t_{1}),s)\Big{)}\Big{|}\geq 2^{d-1}k^{*}n^{*}\bigg{)}\leq
1-\bigg{(}\frac{\epsilon^{*}}{10}\bigg{)}^{2^{d-1}}\Bigg{\\}},$

where $\min\hskip 2.84526pt\emptyset=\infty$.161616Note that by construction,
$s(l(t_{1}),t_{1})>t_{1}$. Since $p<1$ and $B(l(t_{1}),\infty)$ is a one-
dimensional cylinder, we have $s(l(t_{1}),t_{1})\in 2\hat{k}\mathbb{N}$.

Construct three sequences of positive integers $(l_{k})_{k\in\mathbb{N}}$,
$(t_{k})_{k\in\mathbb{N}}$, and $(s_{k})_{k\in\mathbb{N}}$ as follows. Let
$l_{1}=\max\big{\\{}l(t_{1}),r^{*}\big{\\}}$ and $s_{1}=s(l_{1},t_{1})$.
Suppose $k\geq 1$ and suppose that $l_{1},\ldots,l_{k}$, $t_{1},\ldots,t_{k}$
and $s_{1},\ldots,s_{k}$ are constructed. Choose $t_{k+1}=s_{k}+2\hat{k}$,
$l_{k+1}=\max\big{\\{}l(t_{k+1}),(l_{k}+2\hat{k})\big{\\}}$, and
$s_{k+1}=s(l_{k+1},t_{k+1})$. Note that by construction, for all
$k\in\mathbb{N}$, we have $l_{k},s_{k}\in 2\hat{k}\mathbb{N}$ with
$l_{k+1}\geq l_{k}+1$ and $s_{k+1}>s_{k}+1$.171717Since
$s_{k+1}>t_{k+1}=s_{k}+2\hat{k}$. Furthermore, note that for all
$k\in\mathbb{N}$,

$\mathbb{P}_{p}\bigg{(}\Big{|}\Big{(}C^{\beta}_{B\big{(}[-l_{k},l_{k}]^{d-1}\times[0,t_{k}]\big{)}}(D^{*})\cap
T(l_{k},t_{k})\Big{)}\Big{|}\geq
2^{d-1}k^{*}n^{*}\bigg{)}>1-\bigg{(}\frac{\epsilon^{*}}{10}\bigg{)}^{2^{d-1}},\mbox{
and}$ (A.9a)
$\mathbb{P}_{p}\bigg{(}\Big{|}\Big{(}C^{\beta}_{B\big{(}[-l_{k},l_{k}]^{d-1}\times[0,s_{k}]\big{)}}(D^{*})\cap
T(l_{k},s_{k})\Big{)}\Big{|}\geq 2^{d-1}k^{*}n^{*}\bigg{)}\leq
1-\bigg{(}\frac{\epsilon^{*}}{10}\bigg{)}^{2^{d-1}}.$ (A.9b)

For all $k\in\mathbb{N}$, define the random variable
$N_{k}:\Omega\longrightarrow\mathbb{N}$ such that for all $\omega\in\Omega$,

$N_{k}(\omega)=\Big{|}\Big{(}C^{\beta}_{B\big{(}[-l_{k},l_{k}]^{d-1}\times[0,s_{k}]\big{)}}(D^{*})\cap\big{(}T(l_{k},s_{k})\cup
F(l_{k},s_{k})\big{)}\Big{)}\Big{|}.$

######  Claim A.2.

There exists $k_{0}\in\mathbb{N}$ such that

$\mathbb{P}_{p}\Big{(}N_{k_{0}}\geq
2^{d-1}k^{*}n^{*}+2(d-1)2^{d-2}n^{*}\Big{)}>1-\frac{1}{2^{2^{d-1}}}\bigg{(}\frac{\epsilon^{*}}{5}\bigg{)}^{d2^{d-1}}.$

###### Proof of Claim A.2..

For all $k\in\mathbb{N}$, let $\mathcal{G}_{k}$ be the $\sigma$-algebra
generated by $\big{\\{}N_{u}:1\leq u\leq k\big{\\}}$. Fix $k\in\mathbb{N}$ and
$G\in\mathcal{G}_{k}$. For a vertex $x\in\big{(}T(l_{k},s_{k})\cup
F(l_{k},s_{k})\big{)}$, define the following event:

$A_{x}:=\bigg{\\{}\omega\in\Omega:\langle x,y\rangle\mbox{ is closed whenever
}y\in\Big{(}T\big{(}l_{k}+1,s_{k}+1\big{)}\cup
F\big{(}l_{k}+1,s_{k}+1\big{)}\Big{)}\bigg{\\}}.$

Clearly, $\mathbb{P}_{p}(A_{x})\geq(1-p)^{2^{d}}$ for all
$x\in\big{(}T(l_{k},s_{k})\cup F(l_{k},s_{k})\big{)}$. Fix
$S\subseteq\big{(}T(l_{k},s_{k})\cup F(l_{k},s_{k})\big{)}$ such that
$|S|<\big{[}2^{d-1}k^{*}n^{*}+2(d-1)2^{d-2}n^{*}\big{]}$. Recall that for all
$k\in\mathbb{N}$, we have $l_{k+1}\geq l_{k}+1$ and $s_{k+1}>s_{k}+1$. It
follows that

$\displaystyle\hskip 2.84526pt\mathbb{P}_{p}\Bigg{(}N_{k+1}=0\hskip
5.69054pt\biggm{|}\hskip 5.69054ptG\mbox{ and
}\Big{(}C^{\beta}_{B\big{(}[-l_{k},l_{k}]^{d-1}\times[0,s_{k}]\big{)}}(D^{*})\cap\big{(}T(l_{k},s_{k})\cup
F(l_{k},s_{k})\big{)}\Big{)}=S\Bigg{)}$ $\displaystyle\geq$
$\displaystyle\hskip 2.84526pt\mathbb{P}_{p}\bigg{(}\underset{x\in
S}{\cap}\hskip 2.84526ptA_{x}\bigg{)}\hskip 28.45274pt\mbox{(by the
construction of the $\beta$-backbend percolation process)}$ $\displaystyle=$
$\displaystyle\hskip 2.84526pt\underset{x\in S}{\prod}\hskip
2.84526pt\mathbb{P}_{p}\big{(}A_{x}\big{)}\hskip 34.1433pt\mbox{(since each
edge is independent)}$ $\displaystyle>$ $\displaystyle\hskip
2.84526pt(1-p)^{2^{2d-1}(k^{*}n^{*}+(d-1)n^{*})}.$

This, along with Remark A.1, yields

$\mathbb{P}_{p}\bigg{(}N_{k+1}>0\hskip 5.69054pt\Bigm{|}\hskip
5.69054ptG\mbox{ and
}N_{k}<\big{[}2^{d-1}k^{*}n^{*}+2(d-1)2^{d-2}n^{*}\big{]}\bigg{)}<1-(1-p)^{2^{2d-1}(k^{*}n^{*}+(d-1)n^{*})}.$
(A.10)

Since $p<1$, we have $1-(1-p)^{2^{2d-1}(k^{*}n^{*}+(d-1)n^{*})}<1$. This,
together with (A.10) and an argument similar to the one which we use to
complete the proof of Claim A.1 from (A.5), yields

$\mathbb{P}_{p}\Big{(}\big{|}C^{\beta}_{\mathbb{H}}(D^{*})\big{|}=\infty\mbox{
and }N_{k}<\big{[}2^{d-1}k^{*}n^{*}+2(d-1)2^{d-2}n^{*}\big{]}\mbox{ for
infinitely many }k\in\mathbb{N}\Big{)}=0.$ (A.11)

The proof of Claim A.2 follows from (A.1) and (A.11) by using an argument
similar to the one which we use to obtain (A.8) from Claim A.1. ∎

By Claim A.2, along with FKG inequality, we have

$\displaystyle\frac{1}{2^{2^{d-1}}}\bigg{(}\frac{\epsilon^{*}}{5}\bigg{)}^{d2^{d-1}}\hskip
2.84526pt\geq$ $\displaystyle\hskip
2.84526pt\mathbb{P}_{p}\bigg{(}\Big{|}\Big{(}C^{\beta}_{B\big{(}[-l_{k_{0}},l_{k_{0}}]^{d-1}\times[0,s_{k_{0}}]\big{)}}(D^{*})\cap
T(l_{k_{0}},s_{k_{0}})\Big{)}\Big{|}<2^{d-1}k^{*}n^{*}\bigg{)}$
$\displaystyle\times\mathbb{P}_{p}\bigg{(}\Big{|}\Big{(}C^{\beta}_{B\big{(}[-l_{k_{0}},l_{k_{0}}]^{d-1}\times[0,s_{k_{0}}]\big{)}}(D^{*})\cap
F(l_{k_{0}},s_{k_{0}})\Big{)}\Big{|}<2(d-1)2^{d-2}n^{*}\bigg{)},$

which, together with (A.9b), yields

$\bigg{(}\frac{\epsilon^{*}}{5}\bigg{)}^{2(d-1)2^{d-2}}>\mathbb{P}_{p}\bigg{(}\Big{|}\Big{(}C^{\beta}_{B\big{(}[-l_{k_{0}},l_{k_{0}}]^{d-1}\times[0,s_{k_{0}}]\big{)}}(D^{*})\cap
F(l_{k_{0}},s_{k_{0}})\Big{)}\Big{|}<2(d-1)2^{d-2}n^{*}\bigg{)}.$

By FKG inequality and the construction of the $\beta$-backbend percolation
process, this implies

$\mathbb{P}_{p}\bigg{(}\Big{|}\Big{(}C^{\beta}_{B\big{(}[-l_{k_{0}},l_{k_{0}}]^{d-1}\times[0,s_{k_{0}}]\big{)}}(D^{*})\cap
F^{v}_{(d-1)^{+}}(l_{k_{0}},s_{k_{0}})\Big{)}\Big{|}<n^{*}\bigg{)}<\frac{\epsilon^{*}}{5}\mbox{
for all }v\in\\{-1,1\\}^{d-2}.$ (A.12)

Furthermore, by the construction of $l_{k_{0}}$, $t_{k_{0}}$, and $s_{k_{0}}$,
we have

$\bigg{(}\frac{\epsilon^{*}}{10}\bigg{)}^{2^{d-1}}>\mathbb{P}_{p}\bigg{(}\Big{|}\Big{(}C^{\beta}_{B\big{(}[-l_{k_{0}},l_{k_{0}}]^{d-1}\times[0,s_{k_{0}}-2\hat{k}]\big{)}}(D^{*})\cap
T(l_{k_{0}},s_{k_{0}}-2\hat{k})\Big{)}\Big{|}<2^{d-1}k^{*}n^{*}\bigg{)}$

By FKG inequality and the construction of the $\beta$-backbend percolation
process, this implies

$\mathbb{P}_{p}\bigg{(}\Big{|}\Big{(}C^{\beta}_{B\big{(}[-l_{k_{0}},l_{k_{0}}]^{d-1}\times[0,s_{k_{0}}-2\hat{k}]\big{)}}(D^{*})\cap
T^{u}(l_{k_{0}},s_{k_{0}}-2\hat{k})\Big{)}\Big{|}<k^{*}n^{*}\bigg{)}<\frac{\epsilon^{*}}{10}\mbox{
for all }u\in\\{-1,1\\}^{d-1}.$ (A.13)

######  Claim A.3.

For all $u\in\\{-1,1\\}^{d-1}$, we have

$\mathbb{P}_{p}\bigg{(}\Big{|}\Big{(}C^{\beta}_{B\big{(}[-l_{k_{0}},l_{k_{0}}]^{d-1}\times[0,s_{k_{0}}]\big{)}}(D^{*})\cap
T^{u}(l_{k_{0}},s_{k_{0}})\Big{)}\Big{|}\geq
n^{*}\bigg{)}>1-\frac{\epsilon^{*}}{5}.$

###### Proof of Claim A.3..

Fix $u\in\\{-1,1\\}^{d-1}$. For a vertex $x\in
T^{u}(l_{k_{0}},s_{k_{0}}-2\hat{k})$, define the following event:

$A_{x}:=\left\\{\omega\in\Omega:\begin{aligned} &\mbox{ there is an open path
}(x^{0}=x,\ldots,x^{2\hat{k}})\mbox{ in
}B\big{(}[-l_{k_{0}},l_{k_{0}}]^{d-1}\times[0,s_{k_{0}}]\big{)}\mbox{ from
}x,\\\ &\mbox{ where }x^{2i}=x+(0,\ldots,0,2i)\mbox{ for all
}i\leq\hat{k}\end{aligned}\right\\}.$

By construction, $\\{A_{x}\\}_{x\in T^{u}(l_{k_{0}},s_{k_{0}}-2\hat{k})}$ are
independent events. Furthermore, note that $\mathbb{P}_{p}(A_{x})\geq
p^{2\hat{k}}$ for all $x\in T^{u}(l_{k_{0}},s_{k_{0}}-2\hat{k})$. Fix
$S\subseteq T^{u}(l_{k_{0}},s_{k_{0}}-2\hat{k})$ such that $|S|\geq
k^{*}n^{*}$. Make a partition $\big{\\{}S_{1},\ldots,S_{n^{*}}\big{\\}}$ of
$S$ such that $|S_{i}|\geq k^{*}$ for all $i=1,\ldots,n^{*}$. For all
$i=1,\ldots,n^{*}$, define

$B_{i}:=\underset{x\in S_{i}}{\cup}\hskip 2.84526ptA_{x}.$

Clearly, $\mathbb{P}_{p}(B_{i})\geq 1-(1-p^{2\hat{k}})^{k^{*}}$ for all
$i=1,\ldots,n^{*}$.181818Note that since $\\{A_{x}\\}_{x\in
T^{u}(l_{k_{0}},s_{k_{0}}-2\hat{k})}$ are independent events,
$\\{A^{c}_{x}\\}_{x\in T^{u}(l_{k_{0}},s_{k_{0}}-2\hat{k})}$ are also
independent events. Now,

$\displaystyle\hskip 2.84526pt\mathbb{P}_{p}\left(\begin{aligned}
\Big{|}\Big{(}C^{\beta}_{B\big{(}[-l_{k_{0}},l_{k_{0}}]^{d-1}\times[0,s_{k_{0}}]\big{)}}(D^{*})\cap&T^{u}(l_{k_{0}},s_{k_{0}})\Big{)}\Big{|}\\\
&\geq n^{*}\end{aligned}\hskip 9.95845pt\vrule\hskip 9.95845pt\begin{aligned}
\Big{(}C^{\beta}_{B\big{(}[-l_{k_{0}},l_{k_{0}}]^{d-1}\times[0,s_{k_{0}}-2\hat{k}]\big{)}}(D^{*})\cap&T^{u}(l_{k_{0}},s_{k_{0}}-2\hat{k})\Big{)}\\\
&=S\end{aligned}\right)$ $\displaystyle\geq$ $\displaystyle\hskip
2.84526pt\mathbb{P}_{p}\bigg{(}\underset{i=1}{\overset{n^{*}}{\cap}}\hskip
2.84526ptB_{i}\bigg{)}\hskip 28.45274pt\mbox{(by the construction of the
$\beta$-backbend percolation process)}$ $\displaystyle\geq$
$\displaystyle\hskip 2.84526pt\underset{i=1}{\overset{n^{*}}{\prod}}\hskip
2.84526pt\mathbb{P}_{p}(B_{i})\hskip 34.1433pt\mbox{(by FKG inequality)}$
$\displaystyle\geq$ $\displaystyle\hskip
2.84526pt\underset{i=1}{\overset{n^{*}}{\prod}}\hskip
2.84526pt\Big{(}1-(1-p^{2\hat{k}})^{k^{*}}\Big{)}$ $\displaystyle>$
$\displaystyle\hskip 2.84526pt1-\frac{\epsilon^{*}}{10}.\hskip
34.1433pt\mbox{(by the definition of $k^{*}$)}$

This, together with Remark A.1, implies

$\displaystyle\mathbb{P}_{p}\left(\begin{aligned}
\Big{|}\Big{(}C^{\beta}_{B\big{(}[-l_{k_{0}},l_{k_{0}}]^{d-1}\times[0,s_{k_{0}}]\big{)}}(D^{*})\cap&T^{u}(l_{k_{0}},s_{k_{0}})\Big{)}\Big{|}\\\
&\geq n^{*}\end{aligned}\hskip 11.38109pt\vrule\hskip
11.38109pt\begin{aligned}
\Big{|}\Big{(}C^{\beta}_{B\big{(}[-l_{k_{0}},l_{k_{0}}]^{d-1}\times[0,s_{k_{0}}-2\hat{k}]\big{)}}(D^{*})\cap&T^{u}(l_{k_{0}},s_{k_{0}}-2\hat{k})\Big{)}\Big{|}\\\
&\geq k^{*}n^{*}\end{aligned}\right)$ $\displaystyle>\hskip
2.84526pt1-\frac{\epsilon^{*}}{10},$

which, along with (A.13), yields

$\mathbb{P}_{p}\bigg{(}\Big{|}\Big{(}C^{\beta}_{B\big{(}[-l_{k_{0}},l_{k_{0}}]^{d-1}\times[0,s_{k_{0}}]\big{)}}(D^{*})\cap
T^{u}(l_{k_{0}},s_{k_{0}})\Big{)}\Big{|}\geq
n^{*}\bigg{)}>1-\frac{\epsilon^{*}}{5}.$

This completes the proof of Claim A.3. ∎

Let $l^{*}=l_{k_{0}}$ and $t^{*}=s_{k_{0}}$. By construction, we have
$l^{*},t^{*}\in 2\hat{k}\mathbb{N}$ with $l^{*}\geq r^{*}$.191919By
construction, $l^{*}=l_{k_{0}}\geq l_{1}\geq r^{*}$. This, along with (A.12)
and Claim A.3, completes the proof of Lemma A.1. ∎

####  A.1.2 Proof of Lemma A.2

Fix an arbitrary $x\in B\big{(}[-l^{*},l^{*}]^{d-2}\times 0\times 0\big{)}$.
Define the random variable $\tau:\Omega\longrightarrow
2\hat{k}\mathbb{N}\cup\\{\infty\\}$ such that for all $\omega\in\Omega$,

$\tau(\omega):=\min\left\\{t\in 2\hat{k}\mathbb{N}:\begin{aligned} &\hskip
2.84526pt(D^{*}+z)\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-l^{*},l^{*}+2r^{*}]\times[0,t]\big{)}}(D^{*}+x)\\\
&\mbox{ for some }z\in B\big{(}[-l^{*},l^{*}]^{d-2}\times(l^{*}+r^{*})\times
t\big{)}\end{aligned}\right\\},$

where $\min\hskip 2.84526pt\emptyset=\infty$.

######  Claim A.4.

$\mathbb{P}_{p}\Big{(}\tau\in[0,s^{*}]\Big{)}>\Big{(}1-\frac{\epsilon^{*}}{5}\Big{)}^{2}$.

###### Proof of Claim A.4..

Choose $v\in\\{-1,1\\}^{d-2}$ such that $x_{i}v_{i}\leq 0$ for all
$i=1,\ldots,d-2$.202020Note that there may be more than one such $v$. Define
the following events:

$\displaystyle A_{1}:=\left\\{\omega\in\Omega:\begin{aligned} &\hskip
2.84526pt\exists\hskip
2.84526ptS_{1}\subseteq\Big{(}C^{\beta}_{\big{(}B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,t^{*}]\big{)}+x\big{)}}(D^{*}+x)\cap\big{(}F^{v}_{(d-1)^{+}}(l^{*},t^{*})+x\big{)}\Big{)}\mbox{
with }|S_{1}|\geq m^{*}\\\ &\mbox{ such that for all distinct }y_{1},y_{2}\in
S_{1},L^{\infty}(y_{1},y_{2})\geq 4r^{*}+4\hat{k}+1\end{aligned}\right\\},$
$\displaystyle A_{2}:=\left\\{\omega\in\Omega:\begin{aligned} &\hskip
2.84526pt\exists\hskip
2.84526pty\in\Big{(}C^{\beta}_{\big{(}B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,t^{*}]\big{)}+x\big{)}}(D^{*}+x)\cap\big{(}F^{v}_{(d-1)^{+}}(l^{*},t^{*})+x\big{)}\Big{)}\mbox{
such that}\\\
&\Big{(}B\big{(}[-r^{*},r^{*}]^{d-2}\times[0,2r^{*}]\times[2r^{*},2r^{*}+2\hat{k}]\big{)}+y\Big{)}\subseteq
C^{0}_{\big{(}B\big{(}[-r^{*},r^{*}]^{d-2}\times[0,2r^{*}]\times[0,2r^{*}+2\hat{k}]\big{)}+y\big{)}}(\\{y\\})\end{aligned}\right\\}.$

By the construction of $v$,
$\big{(}F^{v}_{(d-1)^{+}}(l^{*},t^{*})+x\big{)}\subseteq
B\big{(}[-l^{*},l^{*}]^{d-2}\times l^{*}\times[0,t^{*}]\big{)}$. Since
$2r^{*}\geq\max\\{\beta_{0},\ldots,\beta_{\hat{k}-1}\\}$,
$s^{*}=t^{*}+2r^{*}+2\hat{k}$, and
$\big{(}F^{v}_{(d-1)^{+}}(l^{*},t^{*})+x\big{)}\subseteq
B\big{(}[-l^{*},l^{*}]^{d-2}\times l^{*}\times[0,t^{*}]\big{)}$, it follows
from the construction of $A_{2}$ that
$A_{2}\subseteq\big{\\{}\omega\in\Omega:\tau(\omega)\in[0,s^{*}]\big{\\}}$.
So, we have

$\displaystyle\mathbb{P}_{p}\Big{(}\tau\in[0,s^{*}]\Big{)}\geq$
$\displaystyle\mathbb{P}_{p}\Big{(}A_{2}\Big{)}$ (A.14) $\displaystyle\geq$
$\displaystyle\hskip 2.84526pt\mathbb{P}_{p}\Big{(}A_{2}\mid
A_{1}\Big{)}\times\mathbb{P}_{p}\Big{(}A_{1}\Big{)}$ $\displaystyle\geq$
$\displaystyle\hskip 2.84526pt\mathbb{P}_{p}\Big{(}A_{2}\mid A_{1}\Big{)}$
$\displaystyle\times\mathbb{P}_{p}\bigg{(}A_{1}\hskip 5.69054pt\biggm{|}\hskip
5.69054pt\Big{|}\Big{(}C^{\beta}_{\big{(}B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,t^{*}]\big{)}+x\big{)}}(D^{*}+x)\cap\big{(}F^{v}_{(d-1)^{+}}(l^{*},t^{*})+x\big{)}\Big{)}\Big{|}\geq
n^{*}\bigg{)}$
$\displaystyle\times\mathbb{P}_{p}\bigg{(}\Big{|}\Big{(}C^{\beta}_{\big{(}B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,t^{*}]\big{)}+x\big{)}}(D^{*}+x)\cap\big{(}F^{v}_{(d-1)^{+}}(l^{*},t^{*})+x\big{)}\Big{)}\Big{|}\geq
n^{*}\bigg{)}.$

Define the mapping
$S_{1}:A_{1}\longrightarrow\mathcal{P}\big{(}\mathbb{H}\big{)}$ such that for
all $\omega\in A_{1}$,

  1. (i)

$S_{1}(\omega)\subseteq\Big{(}C^{\beta}_{\big{(}B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,t^{*}]\big{)}+x\big{)}}(D^{*}+x)\cap\big{(}F^{v}_{(d-1)^{+}}(l^{*},t^{*})+x\big{)}\Big{)}$,
and

  2. (ii)

$|S_{1}(\omega)|\geq m^{*}$, where $L^{\infty}(y_{1},y_{2})\geq
4r^{*}+4\hat{k}+1$ for all distinct $y_{1},y_{2}\in S_{1}(\omega)$.

By the definition of $m^{*}$, $A_{1}$, $S_{1}$ and $A_{2}$, along with Remark
A.1 and the construction of the $\beta$-backbend percolation process, we have

$\displaystyle\mathbb{P}_{p}\Big{(}A_{2}\mid A_{1}\Big{)}$ (A.15)
$\displaystyle\geq$ $\displaystyle\mathbb{P}_{p}\left(\begin{aligned}
&\exists\hskip 2.84526pty\in S_{1}\mbox{ such that }\\\
&\Big{(}B\big{(}[-r^{*},r^{*}]^{d-2}\times[0,2r^{*}]\times[2r^{*},2r^{*}+2\hat{k}]\big{)}+y\Big{)}\subseteq
C^{0}_{\big{(}B\big{(}[-r^{*},r^{*}]^{d-2}\times[0,2r^{*}]\times[0,2r^{*}+2\hat{k}]\big{)}+y\big{)}}(\\{y\\})\end{aligned}\hskip
9.95845pt\vrule\hskip 9.95845ptA_{1}\right)$ $\displaystyle>$ $\displaystyle
1-\frac{\epsilon^{*}}{5}.$

By the definition of $A_{1}$ and $n^{*}$,

$\mathbb{P}_{p}\bigg{(}A_{1}\hskip 5.69054pt\biggm{|}\hskip
5.69054pt\Big{|}\Big{(}C^{\beta}_{\big{(}B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,t^{*}]\big{)}+x\big{)}}(D^{*}+x)\cap\big{(}F^{v}_{(d-1)^{+}}(l^{*},t^{*})+x\big{)}\Big{)}\Big{|}\geq
n^{*}\bigg{)}=1,$ (A.16)

and by Lemma A.1 and the construction of the $\beta$-backbend percolation
process,

$\mathbb{P}_{p}\bigg{(}\Big{|}\Big{(}C^{\beta}_{\big{(}B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,t^{*}]\big{)}+x\big{)}}(D^{*}+x)\cap\big{(}F^{v}_{(d-1)^{+}}(l^{*},t^{*})+x\big{)}\Big{)}\Big{|}\geq
n^{*}\bigg{)}>1-\frac{\epsilon^{*}}{5}.$

This, together with (A.14), (A.15), and (A.16), completes the proof of Claim
A.4. ∎

Define the mapping $\zeta:\Omega\longrightarrow
B\big{(}[-l^{*},l^{*}]^{d-2}\times(l^{*}+r^{*})\times
2\hat{k}\mathbb{N}\big{)}\cup\\{\infty\\}$ such that for all
$\omega\in\Omega$,

$\zeta(\omega):=\begin{cases}z&\mbox{ if }\tau(\omega)<\infty\mbox{ and }\\\
&(D^{*}+z)\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-l^{*},l^{*}+2r^{*}]\times[0,z_{d}]\big{)}}(D^{*}+x)\mbox{
with }z_{d}=\tau(\omega);\vspace{9pt}\\\ \infty&\mbox{ if
}\tau(\omega)=\infty\end{cases}$

######  Claim A.5.

$\mathbb{P}_{p}\left(\begin{aligned} &\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{\big{(}B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,s^{*}]\big{)}+\zeta\big{)}}\big{(}D^{*}+\zeta\big{)}\\\
&\mbox{for some }z\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[l^{*},2l^{*}]\times(\tau+s^{*})\big{)}\end{aligned}\hskip
9.95845pt\vrule\hskip
9.95845pt\tau\in[0,s^{*}]\right)>\Big{(}1-\frac{\epsilon^{*}}{5}\Big{)}^{2}.$

###### Proof of Claim A.5..

Define the mapping $u:\Omega\longrightarrow\\{-1,1\\}^{d-1}\cup\\{\infty\\}$
as follows.

  1. (i)

For all $\omega\in\Omega$ with $\zeta(\omega)\in\mathbb{H}$,
$u_{d-1}(\omega)=1$ and $\zeta_{i}(\omega)u_{i}(\omega)\leq 0$ for all
$i=1,\ldots,d-2$.

  2. (ii)

For all $\omega\in\Omega$ with $\zeta(\omega)=\infty$, $u(\omega)=\infty$.

Since $\beta$ is $\hat{k}$-cyclic, it follows from the definition of $\tau$
and $\zeta$, the construction of the $\beta$-backbend percolation process,
Lemma A.1, and Remark A.1 that

$\mathbb{P}_{p}\bigg{(}\Big{|}\Big{(}C^{\beta}_{\big{(}B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,t^{*}]\big{)}+\zeta\big{)}}(D^{*}+\zeta)\cap\big{(}T^{u}(l^{*},t^{*})+\zeta\big{)}\Big{)}\Big{|}\geq
n^{*}\hskip 5.69054pt\biggm{|}\hskip
5.69054pt\tau\in[0,s^{*}]\bigg{)}>1-\frac{\epsilon^{*}}{5}.$ (A.17)

Define the following event:

$B_{1}:=\left\\{\omega\in\Omega:\begin{aligned} &\hskip 2.84526pt\exists\hskip
2.84526ptS_{2}\subseteq\Big{(}C^{\beta}_{\big{(}B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,t^{*}]\big{)}+\zeta\big{)}}(D^{*}+\zeta)\cap\big{(}T^{u}(l^{*},t^{*})+\zeta\big{)}\Big{)}\mbox{
with }|S_{2}|\geq m^{*}\\\ &\mbox{ such that for all distinct }y_{1},y_{2}\in
S_{2},L^{\infty}(y_{1},y_{2})\geq 4r^{*}+4\hat{k}+1\end{aligned}\right\\}.$

Now, we have

$\displaystyle\mathbb{P}_{p}\Big{(}B_{1}\hskip 5.69054pt\Bigm{|}\hskip
5.69054pt\tau\in[0,s^{*}]\Big{)}$ (A.18) $\displaystyle\geq$
$\displaystyle\mathbb{P}_{p}\Bigg{(}B_{1}\hskip 5.69054pt\biggm{|}\hskip
5.69054pt\tau\in[0,s^{*}]\mbox{ and
}\Big{|}\Big{(}C^{\beta}_{\big{(}B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,t^{*}]\big{)}+\zeta\big{)}}(D^{*}+\zeta)\cap\big{(}T^{u}(l^{*},t^{*})+\zeta\big{)}\Big{)}\Big{|}\geq
n^{*}\Bigg{)}$
$\displaystyle\times\mathbb{P}_{p}\bigg{(}\Big{|}\Big{(}C^{\beta}_{\big{(}B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,t^{*}]\big{)}+\zeta\big{)}}(D^{*}+\zeta)\cap\big{(}T^{u}(l^{*},t^{*})+\zeta\big{)}\Big{)}\Big{|}\geq
n^{*}\hskip 5.69054pt\biggm{|}\hskip 5.69054pt\tau\in[0,s^{*}]\bigg{)}.$

By the definition of $B_{1}$ and $n^{*}$, together with Remark A.1, we have

$\mathbb{P}_{p}\Bigg{(}B_{1}\hskip 5.69054pt\biggm{|}\hskip
5.69054pt\tau\in[0,s^{*}]\mbox{ and
}\Big{|}\Big{(}C^{\beta}_{\big{(}B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,t^{*}]\big{)}+\zeta\big{)}}(D^{*}+\zeta)\cap\big{(}T^{u}(l^{*},t^{*})+\zeta\big{)}\Big{)}\Big{|}\geq
n^{*}\Bigg{)}=1.$

This, together with (A.17) and (A.18), implies

$\mathbb{P}_{p}\Big{(}B_{1}\hskip 5.69054pt\Bigm{|}\hskip
5.69054pt\tau\in[0,s^{*}]\Big{)}>1-\frac{\epsilon^{*}}{5}.$ (A.19)

Define the mapping
$S_{2}:B_{1}\longrightarrow\mathcal{P}\big{(}\mathbb{H}\big{)}$ such that for
all $\omega\in B_{1}$,

  1. (i)

$S_{2}(\omega)\subseteq\Big{(}C^{\beta}_{\big{(}B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,t^{*}]\big{)}+\zeta\big{)}}(D^{*}+\zeta)\cap\big{(}T^{u}(l^{*},t^{*})+\zeta\big{)}\Big{)}$,
and

  2. (ii)

$|S_{2}(\omega)|\geq m^{*}$, where $L^{\infty}(y_{1},y_{2})\geq
4r^{*}+4\hat{k}+1$ for all distinct $y_{1},y_{2}\in S_{2}(\omega)$.

By the definition of $\tau$, $\zeta$, $u$, and $B_{1}$, we have
$\big{(}T^{u}(l^{*},t^{*})+\zeta\big{)}\subseteq
B\big{(}[-l^{*},l^{*}]^{d-2}\times[l^{*}+r^{*},2l^{*}+r^{*}]\times(\tau+t^{*})\big{)}$
for all $\omega\in B_{1}$. Define the mappings
$\hat{S}_{2},\bar{S}_{2}:B_{1}\longrightarrow\mathcal{P}\big{(}\mathbb{H}\big{)}$
such that for all $\omega\in B_{1}$, $\hat{S}_{2}(\omega):=\\{y\in
S_{2}(\omega):y_{d-1}>2l^{*}\\}$, and $\bar{S}_{2}(\omega):=\\{y\in
S_{2}(\omega):y_{d-1}\leq 2l^{*}\\}$. Define the following sub-events of
$B_{1}$:

$\displaystyle\hat{B}_{1}:=\left\\{\omega\in B_{1}:\begin{aligned} &\hskip
2.84526pt\exists\hskip 2.84526pty\in\hat{S}_{2}(\omega)\mbox{ such that}\\\
&\hskip
2.84526pt\Big{(}D^{*}+y+(0,\ldots,0,-r^{*},2r^{*}+2\hat{k})\Big{)}\subseteq
C^{0}_{\big{(}B\big{(}[-r^{*},r^{*}]^{d-2}\times[-2r^{*},0]\times[0,2r^{*}+2\hat{k}]\big{)}+y\big{)}}(\\{y\\})\end{aligned}\right\\},\mbox{and}$
$\displaystyle\bar{B}_{1}:=\left\\{\omega\in B_{1}:\begin{aligned} &\hskip
2.84526pt\exists\hskip 2.84526pty\in\bar{S}_{2}(\omega)\mbox{ such that}\\\
&\hskip 2.84526pt\Big{(}D^{*}+y+(0,\ldots,0,2r^{*}+2\hat{k})\Big{)}\subseteq
C^{0}_{\big{(}B\big{(}[-r^{*},r^{*}]^{d-1}\times[0,2r^{*}+2\hat{k}]\big{)}+y\big{)}}(\\{y\\})\end{aligned}\right\\}.$

By the definition of $m^{*}$, $B_{1}$, $\hat{B}_{1}$, and $\bar{B}_{1}$, along
with Remark A.1 and the construction of the $\beta$-backbend percolation
process, we have

$\mathbb{P}_{p}\Big{(}\hat{B}_{1}\cup\bar{B}_{1}\hskip 5.69054pt\Bigm{|}\hskip
5.69054ptB_{1}\mbox{ and }\tau\in[0,s^{*}]\Big{)}>1-\frac{\epsilon^{*}}{5}.$
(A.20)

Since $l^{*}\geq r^{*}$ (see Lemma A.1 for details) and
$s^{*}=t^{*}+2r^{*}+2\hat{k}$, we have

$\displaystyle\hskip 2.84526pt\mathbb{P}_{p}\left(\begin{aligned}
&\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{\big{(}B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,s^{*}]\big{)}+\zeta\big{)}}\big{(}D^{*}+\zeta\big{)}\\\
&\mbox{for some }z\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[l^{*},2l^{*}]\times(\tau+s^{*})\big{)}\end{aligned}\hskip
9.95845pt\vrule\hskip 9.95845pt\tau\in[0,s^{*}]\right)$ $\displaystyle\geq$
$\displaystyle\hskip 2.84526pt\mathbb{P}_{p}\left(\begin{aligned}
&\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{\big{(}B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,s^{*}]\big{)}+\zeta\big{)}}\big{(}D^{*}+\zeta\big{)}\\\
&\mbox{for some }z\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[l^{*},2l^{*}]\times(\tau+s^{*})\big{)}\end{aligned}\hskip
9.95845pt\vrule\hskip 9.95845ptB_{1}\mbox{ and
}\tau\in[0,s^{*}]\right)\times\mathbb{P}_{p}\Big{(}B_{1}\hskip
5.69054pt\bigm{|}\hskip 5.69054pt\tau\in[0,s^{*}]\Big{)}$ $\displaystyle\geq$
$\displaystyle\hskip
2.84526pt\mathbb{P}_{p}\bigg{(}\hat{B}_{1}\cup\bar{B}_{1}\hskip
5.69054pt\biggm{|}\hskip 5.69054ptB_{1}\mbox{ and
}\tau\in[0,s^{*}]\bigg{)}\times\mathbb{P}_{p}\Big{(}B_{1}\hskip
5.69054pt\bigm{|}\hskip 5.69054pt\tau\in[0,s^{*}]\Big{)}.$

This, together with (A.19) and (A.20), completes the proof of Claim A.5. ∎

By $A_{x}$, we denote the event
$\bigg{\\{}\omega\in\Omega:\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-l^{*},3l^{*}]\times[0,z_{d}]\big{)}}(D^{*}+x)\mbox{
for some }z\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[l^{*},2l^{*}]\times[s^{*},2s^{*}]\big{)}\mbox{
with }z_{d}\in 2\hat{k}\mathbb{N}\bigg{\\}}$. Since $l^{*}\geq r^{*}$, by the
definition of $\tau$, $\zeta$, and $A_{x}$, the construction of the lattice
structure, and the construction of the $\beta$-backbend percolation process,
we have

$\displaystyle\hskip 2.84526pt\mathbb{P}_{p}\big{(}A_{x}\big{)}$ (A.21)
$\displaystyle\geq$ $\displaystyle\hskip
2.84526pt\mathbb{P}_{p}\Big{(}A_{x}\hskip 5.69054pt\Bigm{|}\hskip
5.69054pt\tau\in[0,s^{*}]\Big{)}\times\mathbb{P}_{p}\Big{(}\tau\in[0,s^{*}]\Big{)}$
$\displaystyle\geq$ $\displaystyle\hskip
2.84526pt\mathbb{P}_{p}\left(\begin{aligned} &\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{\big{(}B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,s^{*}]\big{)}+\zeta\big{)}}\big{(}D^{*}+\zeta\big{)}\\\
&\mbox{for some }z\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[l^{*},2l^{*}]\times(\tau+s^{*})\big{)}\end{aligned}\hskip
9.95845pt\vrule\hskip
9.95845pt\tau\in[0,s^{*}]\right)\times\mathbb{P}_{p}\Big{(}\tau\in[0,s^{*}]\Big{)}$
$\displaystyle>$ $\displaystyle\hskip
2.84526pt\Big{(}1-\frac{\epsilon^{*}}{5}\Big{)}^{4}\hskip 28.45274pt\mbox{(by
Claim \ref{claim tau is less than S probability} and Claim \ref{claim
translation probability})}$ $\displaystyle>$ $\displaystyle\hskip
2.84526pt1-\epsilon^{*}.$

Note that the event $A_{x}$ depends only on the edges in the set
$\Big{(}B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-l^{*},3l^{*}]\times[0,2s^{*}]\big{)}\Big{)}^{2}$.
Since this set is finite, it follows that $\mathbb{P}_{p}(A_{x})$ is a
continuous function of $p$. Moreover, by construction, $\mathbb{P}_{p}(A_{x})$
is an increasing function of $p$. Since $\mathbb{P}_{p}(A_{x})$ is an
increasing continuous function of $p$, (A.21) implies there exists
$\delta_{x}>0$ with $\delta_{x}<p$ such that
$\mathbb{P}_{p-\delta_{x}}(A_{x})>1-\epsilon^{*}$.

Let

$\delta:=\min\hskip 2.84526pt\Big{\\{}\delta_{x}:x\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times 0\times 0\big{)}\Big{\\}}.$

Since $B\big{(}[-l^{*},l^{*}]^{d-2}\times 0\times 0\big{)}$ is a finite set of
vertices, by the construction of $\delta$, we have $0<\delta<p$. Furthermore,
for all $x\in B\big{(}[-l^{*},l^{*}]^{d-2}\times 0\times 0\big{)}$, since
$\mathbb{P}_{p}(A_{x})$ is an increasing continuous function of $p$ and
$\mathbb{P}_{p-\delta_{x}}(A_{x})>1-\epsilon^{*}$, it follows from the
construction of $\delta$ that $\mathbb{P}_{p-\delta}(A_{x})>1-\epsilon^{*}$
for all $x\in B\big{(}[-l^{*},l^{*}]^{d-2}\times 0\times 0\big{)}$. This
completes the proof of Lemma A.2. ∎

####  A.1.3 Proof of Lemma A.3

Since $\beta$ is $\hat{k}$-cyclic, Lemma A.2, together with the construction
of the $\beta$-backbend percolation process, implies that for all $y\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times\mathbb{Z}\times\mathbb{Z}_{+}\big{)}$ with
$y_{d}\in 2\hat{k}\mathbb{Z}_{+}$,

$\mathbb{P}_{p-\delta^{*}}\left(\begin{aligned}
&\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-l^{*}+y_{d-1},3l^{*}+y_{d-1}]\times[y_{d},z_{d}]\big{)}}(D^{*}+y)\\\
&\mbox{ for some }z\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[l^{*}+y_{d-1},2l^{*}+y_{d-1}]\times[s^{*}+y_{d},2s^{*}+y_{d}]\big{)}\\\
&\mbox{ with }z_{d}\in 2\hat{k}\mathbb{N}\end{aligned}\right)>1-\epsilon^{*},$
(A.22a) $\mathbb{P}_{p-\delta^{*}}\left(\begin{aligned}
&\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-3l^{*}+y_{d-1},l^{*}+y_{d-1}]\times[y_{d},z_{d}]\big{)}}(D^{*}+y)\\\
&\mbox{ for some }z\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[-2l^{*}+y_{d-1},-l^{*}+y_{d-1}]\times[s^{*}+y_{d},2s^{*}+y_{d}]\big{)}\\\
&\mbox{ with }z_{d}\in 2\hat{k}\mathbb{N}\end{aligned}\right)>1-\epsilon^{*}.$
(A.22b)

Fix an arbitrary $x\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[-2l^{*},2l^{*}]\times[0,2s^{*}]\big{)}$
with $x_{d}\in 2\hat{k}\mathbb{Z}_{+}$. By $A_{x}$, we denote the event
$\Big{\\{}\omega\in\Omega:\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-3l^{*},4l^{*}]\times[x_{d},z_{d}]\big{)}}(D^{*}+x)\mbox{
for some }z\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[-l^{*},3l^{*}]\times[2s^{*},4s^{*}]\big{)}\mbox{
with }z_{d}\in 2\hat{k}\mathbb{N}\Big{\\}}$. To prove Lemma A.3, we
distinguish the following five cases.
Case 1: Suppose $x\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[-2l^{*},l^{*}]\times[s^{*},2s^{*}]\big{)}$.
It follows from (A.22a) that
$\mathbb{P}_{p-\delta^{*}}\big{(}A_{x}\big{)}>1-\epsilon^{*}$.
Case 2: Suppose $x\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[l^{*},2l^{*}]\times[s^{*},2s^{*}]\big{)}$.
It follows from (A.22b) that
$\mathbb{P}_{p-\delta^{*}}\big{(}A_{x}\big{)}>1-\epsilon^{*}$.
Case 3: Suppose $x\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[-2l^{*},0]\times[0,s^{*}]\big{)}$.
It follows from (A.22a) that

$\mathbb{P}_{p-\delta^{*}}\left(\begin{aligned}
&\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-3l^{*},3l^{*}]\times[x_{d},z_{d}]\big{)}}(D^{*}+x)\\\
&\mbox{ for some }z\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[-l^{*},2l^{*}]\times[s^{*},3s^{*}]\big{)}\mbox{
with }z_{d}\in 2\hat{k}\mathbb{N}\end{aligned}\right)>1-\epsilon^{*}.$ (A.23)

Define the following events:

$\displaystyle A_{1}:=\left\\{\omega\in\Omega:\begin{aligned} &\hskip
2.84526pt\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-3l^{*},3l^{*}]\times[x_{d},z_{d}]\big{)}}(D^{*}+x)\\\
&\mbox{ for some }z\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[-l^{*},2l^{*}]\times[2s^{*},3s^{*}]\big{)}\mbox{
with }z_{d}\in 2\hat{k}\mathbb{N}\end{aligned}\right\\},$ $\displaystyle
A_{2}:=\left\\{\omega\in\Omega:\begin{aligned} &\hskip
2.84526pt\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-3l^{*},3l^{*}]\times[x_{d},z_{d}]\big{)}}(D^{*}+x)\\\
&\mbox{ for some }z\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[-l^{*},l^{*}]\times[s^{*},2s^{*}]\big{)}\mbox{
with }z_{d}\in 2\hat{k}\mathbb{N}\end{aligned}\right\\}\setminus A_{1},\mbox{
and}$ $\displaystyle A_{3}:=\left\\{\omega\in\Omega:\begin{aligned} &\hskip
2.84526pt\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-3l^{*},3l^{*}]\times[x_{d},z_{d}]\big{)}}(D^{*}+x)\\\
&\mbox{ for some }z\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[l^{*},2l^{*}]\times[s^{*},2s^{*}]\big{)}\mbox{
with }z_{d}\in 2\hat{k}\mathbb{N}\end{aligned}\right\\}\setminus(A_{1}\cup
A_{2}).$

Clearly $A_{1},A_{2}$, and $A_{3}$ are disjoint. By the construction of
$A_{1},A_{2}$, and $A_{3}$, (A.23) implies
$\mathbb{P}_{p-\delta^{*}}\Big{(}A_{1}\cup A_{2}\cup
A_{3}\Big{)}>1-\epsilon^{*}$. Since $A_{1},A_{2}$, and $A_{3}$ are disjoint,
$\mathbb{P}_{p-\delta^{*}}\Big{(}A_{1}\cup A_{2}\cup
A_{3}\Big{)}>1-\epsilon^{*}$ implies

$\mathbb{P}_{p-\delta^{*}}(A_{1})+\mathbb{P}_{p-\delta^{*}}(A_{2})+\mathbb{P}_{p-\delta^{*}}(A_{3})>1-\epsilon^{*}.$
(A.24)

Define the mappings $z^{\prime}:A_{2}\longrightarrow
B\big{(}[-l^{*},l^{*}]^{d-2}\times[-l^{*},l^{*}]\times[s^{*},2s^{*}]\big{)}$
and $z^{\prime\prime}:A_{3}\longrightarrow
B\big{(}[-l^{*},l^{*}]^{d-2}\times[l^{*},2l^{*}]\times[s^{*},2s^{*}]\big{)}$
as follows.

  1. (i)

For all $\omega\in A_{2}$, $\big{(}D^{*}+z^{\prime}(\omega)\big{)}\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-3l^{*},3l^{*}]\times[x_{d},z^{\prime}_{d}(\omega)]\big{)}}(D^{*}+x)$
with $z^{\prime}_{d}(\omega)\in 2\hat{k}\mathbb{N}$.

  2. (ii)

For all $\omega\in A_{3}$,
$\big{(}D^{*}+z^{\prime\prime}(\omega)\big{)}\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-3l^{*},3l^{*}]\times[x_{d},z^{\prime\prime}_{d}(\omega)]\big{)}}(D^{*}+x)$
with $z^{\prime\prime}_{d}(\omega)\in 2\hat{k}\mathbb{N}$.

By the definition of $z^{\prime}$ and $z^{\prime\prime}$, along with Remark
A.1, (A.22a), and (A.22b), we have

$\displaystyle\mathbb{P}_{p-\delta^{*}}\left(\begin{aligned}
&\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-3l^{*},4l^{*}]\times[z^{\prime}_{d},z_{d}]\big{)}}(D^{*}+z^{\prime})\\\
&\mbox{ for some }z\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[-l^{*},3l^{*}]\times[2s^{*},4s^{*}]\big{)}\mbox{
with }z_{d}\in 2\hat{k}\mathbb{N}\end{aligned}\hskip 9.95845pt\vrule\hskip
9.95845ptA_{2}\right)>1-\epsilon^{*},\mbox{ and}$ (A.25)
$\displaystyle\mathbb{P}_{p-\delta^{*}}\left(\begin{aligned}
&\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-3l^{*},4l^{*}]\times[z^{\prime\prime}_{d},z_{d}]\big{)}}(D^{*}+z^{\prime\prime})\\\
&\mbox{ for some }z\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[-l^{*},3l^{*}]\times[2s^{*},4s^{*}]\big{)}\mbox{
with }z_{d}\in 2\hat{k}\mathbb{N}\end{aligned}\hskip 9.95845pt\vrule\hskip
9.95845ptA_{3}\right)>1-\epsilon^{*}.$

So, we have

$\displaystyle\hskip 2.84526pt\mathbb{P}_{p-\delta^{*}}\big{(}A_{x}\big{)}$
$\displaystyle\geq$ $\displaystyle\hskip
2.84526pt\mathbb{P}_{p-\delta^{*}}\Big{(}A_{x}\cap\big{(}A_{1}\cup A_{2}\cup
A_{3}\big{)}\Big{)}$ $\displaystyle=$ $\displaystyle\hskip
2.84526pt\mathbb{P}_{p-\delta^{*}}\big{(}A_{x}\cap
A_{1}\big{)}+\mathbb{P}_{p-\delta^{*}}\big{(}A_{x}\cap
A_{2}\big{)}+\mathbb{P}_{p-\delta^{*}}\big{(}A_{x}\cap A_{3}\big{)}\hskip
28.45274pt\mbox{(since $A_{1},A_{2}$, and $A_{3}$ are disjoint)}$
$\displaystyle=$ $\displaystyle\hskip
2.84526pt\mathbb{P}_{p-\delta^{*}}\big{(}A_{1}\big{)}+\Big{(}\mathbb{P}_{p-\delta^{*}}\big{(}A_{x}\mid
A_{2}\big{)}\times\mathbb{P}_{p-\delta^{*}}\big{(}A_{2}\big{)}\Big{)}+\Big{(}\mathbb{P}_{p-\delta^{*}}\big{(}A_{x}\mid
A_{3}\big{)}\times\mathbb{P}_{p-\delta^{*}}\big{(}A_{3}\big{)}\Big{)}\hskip
28.45274pt\mbox{(since $A_{1}\subseteq A_{x}$)}$ $\displaystyle\geq$
$\displaystyle\hskip 2.84526pt\mathbb{P}_{p-\delta^{*}}\big{(}A_{1}\big{)}+$
$\displaystyle\hskip
2.84526pt\left[\mathbb{P}_{p-\delta^{*}}\left(\begin{aligned}
&\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-3l^{*},4l^{*}]\times[z^{\prime}_{d},z_{d}]\big{)}}(D^{*}+z^{\prime})\\\
&\mbox{ for some }z\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[-l^{*},3l^{*}]\times[2s^{*},4s^{*}]\big{)}\mbox{
with }z_{d}\in 2\hat{k}\mathbb{N}\end{aligned}\hskip 9.95845pt\vrule\hskip
9.95845ptA_{2}\right)\times\mathbb{P}_{p-\delta^{*}}\big{(}A_{2}\big{)}\right]+$
$\displaystyle\hskip
2.84526pt\left[\mathbb{P}_{p-\delta^{*}}\left(\begin{aligned}
&\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-3l^{*},4l^{*}]\times[z^{\prime\prime}_{d},z_{d}]\big{)}}(D^{*}+z^{\prime\prime})\\\
&\mbox{ for some }z\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[-l^{*},3l^{*}]\times[2s^{*},4s^{*}]\big{)}\mbox{
with }z_{d}\in 2\hat{k}\mathbb{N}\end{aligned}\hskip 9.95845pt\vrule\hskip
9.95845ptA_{3}\right)\times\mathbb{P}_{p-\delta^{*}}\big{(}A_{3}\big{)}\right]$
$\displaystyle>$ $\displaystyle\hskip
2.84526pt\mathbb{P}_{p-\delta^{*}}\big{(}A_{1}\big{)}+\Big{(}\big{(}1-\epsilon^{*}\big{)}\times\mathbb{P}_{p-\delta^{*}}\big{(}A_{2}\big{)}\Big{)}+\Big{(}\big{(}1-\epsilon^{*}\big{)}\times\mathbb{P}_{p-\delta^{*}}\big{(}A_{3}\big{)}\Big{)}\hskip
28.45274pt\mbox{(by \eqref{equation transition equation 2 final})}$
$\displaystyle>$ $\displaystyle\hskip
2.84526pt\big{(}1-\epsilon^{*}\big{)}\times\Big{(}\mathbb{P}_{p-\delta^{*}}\big{(}A_{1}\big{)}+\mathbb{P}_{p-\delta^{*}}\big{(}A_{2}\big{)}+\mathbb{P}_{p-\delta^{*}}\big{(}A_{3}\big{)}\Big{)}$
$\displaystyle>$ $\displaystyle\hskip
2.84526pt\big{(}1-\epsilon^{*}\big{)}^{2}.\hskip 28.45274pt\mbox{(by
\eqref{equation transition case 3 equation 2})}$

Case 4: Suppose $x\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[0,l^{*}]\times[0,s^{*}]\big{)}$.
It follows from (A.22a) that

$\mathbb{P}_{p-\delta^{*}}\left(\begin{aligned}
&\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-l^{*},4l^{*}]\times[x_{d},z_{d}]\big{)}}(D^{*}+x)\\\
&\mbox{ for some }z\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[l^{*},3l^{*}]\times[s^{*},3s^{*}]\big{)}\mbox{
with }z_{d}\in 2\hat{k}\mathbb{N}\end{aligned}\right)>1-\epsilon^{*}.$ (A.26)

Define the following events:

$\displaystyle A_{1}:=\left\\{\omega\in\Omega:\begin{aligned} &\hskip
2.84526pt\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-l^{*},4l^{*}]\times[x_{d},z_{d}]\big{)}}(D^{*}+x)\\\
&\mbox{ for some }z\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[l^{*},3l^{*}]\times[2s^{*},3s^{*}]\big{)}\mbox{
with }z_{d}\in 2\hat{k}\mathbb{N}\end{aligned}\right\\},\mbox{ and}$
$\displaystyle A_{2}:=\left\\{\omega\in\Omega:\begin{aligned} &\hskip
2.84526pt\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-l^{*},4l^{*}]\times[x_{d},z_{d}]\big{)}}(D^{*}+x)\\\
&\mbox{ for some }z\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[l^{*},3l^{*}]\times[s^{*},2s^{*}]\big{)}\mbox{
with }z_{d}\in 2\hat{k}\mathbb{N}\end{aligned}\right\\}\setminus A_{1}.$

Clearly $A_{1}$ and $A_{2}$ are disjoint. By the construction of $A_{1}$ and
$A_{2}$, (A.26) implies $\mathbb{P}_{p-\delta^{*}}\big{(}A_{1}\cup
A_{2}\big{)}>1-\epsilon^{*}$. Since $A_{1}$ and $A_{2}$ are disjoint,
$\mathbb{P}_{p-\delta^{*}}\big{(}A_{1}\cup A_{2}\big{)}>1-\epsilon^{*}$
implies

$\mathbb{P}_{p-\delta^{*}}(A_{1})+\mathbb{P}_{p-\delta^{*}}(A_{2})>1-\epsilon^{*}.$
(A.27)

Define the mapping $z^{\prime}:A_{2}\longrightarrow
B\big{(}[-l^{*},l^{*}]^{d-2}\times[l^{*},3l^{*}]\times[s^{*},2s^{*}]\big{)}$
such that for all $\omega\in A_{2}$, we have
$\big{(}D^{*}+z^{\prime}(\omega)\big{)}\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-l^{*},4l^{*}]\times[x_{d},z^{\prime}_{d}(\omega)]\big{)}}(D^{*}+x)$
with $z^{\prime}_{d}(\omega)\in 2\hat{k}\mathbb{N}$. By the definition of
$z^{\prime}$, together with Remark A.1 and (A.22b), we have

$\mathbb{P}_{p-\delta^{*}}\left(\begin{aligned}
&\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-3l^{*},4l^{*}]\times[z^{\prime}_{d},z_{d}]\big{)}}(D^{*}+z^{\prime})\\\
&\mbox{ for some }z\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[-l^{*},3l^{*}]\times[2s^{*},4s^{*}]\big{)}\mbox{
with }z_{d}\in 2\hat{k}\mathbb{N}\end{aligned}\hskip 9.95845pt\vrule\hskip
9.95845ptA_{2}\right)>1-\epsilon^{*}.$ (A.28)

Since $A_{1}\subseteq A_{x}$, and $A_{1}$ and $A_{2}$ are disjoint, (A.27) and
(A.28), along with an argument similar to the one which we use to complete the
proof for Case 3 of this lemma from (A.25), together imply
$\mathbb{P}_{p-\delta^{*}}\big{(}A_{x}\big{)}>\big{(}1-\epsilon^{*}\big{)}^{2}$.
Case 5: Suppose $x\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[l^{*},2l^{*}]\times[0,s^{*}]\big{)}$.
It follows from (A.22b) that

$\mathbb{P}_{p-\delta^{*}}\left(\begin{aligned}
&\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-2l^{*},3l^{*}]\times[x_{d},z_{d}]\big{)}}(D^{*}+x)\\\
&\mbox{ for some }z\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[-l^{*},l^{*}]\times[s^{*},3s^{*}]\big{)}\mbox{
with }z_{d}\in 2\hat{k}\mathbb{N}\end{aligned}\right)>1-\epsilon^{*}.$ (A.29)

Define the following events:

$\displaystyle A_{1}:=\left\\{\omega\in\Omega:\begin{aligned} &\hskip
2.84526pt\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-2l^{*},3l^{*}]\times[x_{d},z_{d}]\big{)}}(D^{*}+x)\\\
&\mbox{ for some }z\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[-l^{*},l^{*}]\times[2s^{*},3s^{*}]\big{)}\mbox{
with }z_{d}\in 2\hat{k}\mathbb{N}\end{aligned}\right\\},\mbox{ and}$
$\displaystyle A_{2}:=\left\\{\omega\in\Omega:\begin{aligned} &\hskip
2.84526pt\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-2l^{*},3l^{*}]\times[x_{d},z_{d}]\big{)}}(D^{*}+x)\\\
&\mbox{ for some }z\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[-l^{*},l^{*}]\times[s^{*},2s^{*}]\big{)}\mbox{
with }z_{d}\in 2\hat{k}\mathbb{N}\end{aligned}\right\\}\setminus A_{1}.$

Clearly $A_{1}$ and $A_{2}$ are disjoint. By the construction of $A_{1}$ and
$A_{2}$, (A.29) implies $\mathbb{P}_{p-\delta^{*}}\big{(}A_{1}\cup
A_{2}\big{)}>1-\epsilon^{*}$. Since $A_{1}$ and $A_{2}$ are disjoint,
$\mathbb{P}_{p-\delta^{*}}\big{(}A_{1}\cup A_{2}\big{)}>1-\epsilon^{*}$
implies

$\mathbb{P}_{p-\delta^{*}}(A_{1})+\mathbb{P}_{p-\delta^{*}}(A_{2})>1-\epsilon^{*}.$
(A.30)

Define the mapping $z^{\prime}:A_{2}\longrightarrow
B\big{(}[-l^{*},l^{*}]^{d-2}\times[-l^{*},l^{*}]\times[s^{*},2s^{*}]\big{)}$
such that for all $\omega\in A_{2}$, we have
$\big{(}D^{*}+z^{\prime}(\omega)\big{)}\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-2l^{*},3l^{*}]\times[x_{d},z^{\prime}_{d}(\omega)]\big{)}}(D^{*}+x)$
with $z^{\prime}_{d}(\omega)\in 2\hat{k}\mathbb{N}$. By the definition of
$z^{\prime}$, together with Remark A.1 and (A.22a), we have

$\displaystyle\mathbb{P}_{p-\delta^{*}}\left(\begin{aligned}
&\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-3l^{*},4l^{*}]\times[z^{\prime}_{d},z_{d}]\big{)}}(D^{*}+z^{\prime})\\\
&\mbox{ for some }z\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[-l^{*},3l^{*}]\times[2s^{*},4s^{*}]\big{)}\mbox{
with }z_{d}\in 2\hat{k}\mathbb{N}\end{aligned}\hskip 9.95845pt\vrule\hskip
9.95845ptA_{2}\right)>1-\epsilon^{*}.$ (A.31)

Since $A_{1}\subseteq A_{x}$, and $A_{1}$ and $A_{2}$ are disjoint, (A.30) and
(A.31), along with an argument similar to the one which we use to complete the
proof for Case 3 of this lemma from (A.25), together imply
$\mathbb{P}_{p-\delta^{*}}\big{(}A_{x}\big{)}>\big{(}1-\epsilon^{*}\big{)}^{2}$.
This completes the proof of Lemma A.3. ∎

####  A.1.4 Proof of Lemma A.4

Fix $i\in\mathbb{N}$ and $x\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[-2l^{*},2l^{*}]\times[0,2s^{*}]\big{)}$
with $x_{d}\in 2\hat{k}\mathbb{Z}_{+}$. For all $z\in B^{+}_{i}$, define

$\mathcal{S}^{+}(z):=\bigg{(}\underset{j=0}{\overset{i-1}{\bigcup}}\hskip
2.84526ptB\big{(}[-2l^{*},2l^{*}]^{d-2}\times[jl^{*}-3l^{*},jl^{*}+4l^{*}]\times[2js^{*},2js^{*}+4s^{*}]\big{)}\bigg{)}\cap\mathcal{R}^{+}(z).$

Since $\beta$ is $\hat{k}$-cyclic, using the result in Lemma A.3 repeatedly,
it follows that

$\mathbb{P}_{p-\delta^{*}}\bigg{(}\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{\mathcal{S}^{+}(z)}(D^{*}+x)\mbox{ for some }z\in B^{+}_{i}\mbox{
with }z_{d}\in 2\hat{k}\mathbb{N}\bigg{)}>(1-\epsilon^{*})^{2i}.$ (A.32)

By the definition of $\mathcal{S}^{+}(z)$, we have
$\mathcal{S}^{+}(z)\subseteq\mathcal{R}^{+}(z)$ for all $z\in B^{+}_{i}$.
This, together with (A.32), implies
$\mathbb{P}_{p-\delta^{*}}\big{(}G_{i}^{+}(x)\big{)}>(1-\epsilon^{*})^{2i}$
which completes the proof of Lemma A.4. ∎

####  A.1.5 Proof of Lemma A.6

Using a similar argument as for Claim A.2 in Lemma A.1, it follows from (A.4)
that there exists $l^{*}\in 2\hat{k}\mathbb{N}$ with $l^{*}\geq r^{*}$ such
that

$\mathbb{P}_{p}\bigg{(}\Big{|}\Big{(}C^{\beta}_{B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,t^{*}]\big{)}}(D^{*})\cap
F(l^{*},t^{*})\Big{)}\Big{|}\geq
2(d-1)2^{d-2}n^{*}\bigg{)}>1-\bigg{(}\frac{\epsilon^{*}}{2}\bigg{)}^{(d-1)2^{d-1}},$

which, in particular, means

$\mathbb{P}_{p}\bigg{(}\Big{|}\Big{(}C^{\beta}_{B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,t^{*}]\big{)}}(D^{*})\cap
F(l^{*},t^{*})\Big{)}\Big{|}<2(d-1)2^{d-2}n^{*}\bigg{)}<\bigg{(}\frac{\epsilon^{*}}{2}\bigg{)}^{2(d-1)2^{d-2}}.$

By FKG inequality and the construction of the $\beta$-backbend percolation
process, this implies that

$\mathbb{P}_{p}\bigg{(}\Big{|}\Big{(}C^{\beta}_{B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,t^{*}]\big{)}}(D^{*})\cap
F^{v}_{(d-1)^{+}}(l^{*},t^{*})\Big{)}\Big{|}<n^{*}\bigg{)}<\frac{\epsilon^{*}}{2}\mbox{
for all }v\in\\{-1,1\\}^{d-2},$

which completes the proof of Lemma A.6. ∎

####  A.1.6 Proof of Lemma A.7

Fix an arbitrary $x\in B\big{(}[-l^{*},l^{*}]^{d-2}\times 0\times 0\big{)}$.
By $A_{x}$, we denote the event
$\bigg{\\{}\omega\in\Omega:\big{(}D^{*}+z\big{)}\subseteq
C^{\beta}_{B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-l^{*},3l^{*}]\times[0,z_{d}]\big{)}}(D^{*}+x)\mbox{
for some }z\in
B\big{(}[-l^{*},l^{*}]^{d-2}\times[l^{*},2l^{*}]\times[s^{*},2s^{*}]\big{)}\mbox{
with }z_{d}\in 2\hat{k}\mathbb{N}\bigg{\\}}$. Choose $v\in\\{-1,1\\}^{d-2}$
such that $x_{i}v_{i}\leq 0$ for all $i=1,\ldots,d-2$.212121Note that there
may be more than one such $v$. Define the following events:

$\displaystyle A_{x}^{1}:=\left\\{\omega\in\Omega:\begin{aligned} &\hskip
2.84526pt\exists\hskip
2.84526ptS_{1}\subseteq\Big{(}C^{\beta}_{\big{(}B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,t^{*}]\big{)}+x\big{)}}(D^{*}+x)\cap\big{(}F^{v}_{(d-1)^{+}}(l^{*},t^{*})+x\big{)}\Big{)}\mbox{
with }|S_{1}|\geq m^{*}\\\ &\mbox{ such that for all distinct }y_{1},y_{2}\in
S_{1},L^{\infty}(y_{1},y_{2})\geq 4r^{*}+8\hat{k}+1\end{aligned}\right\\},$
$\displaystyle A_{x}^{2}:=\left\\{\omega\in\Omega:\begin{aligned} &\hskip
2.84526pt\exists\hskip
2.84526pty\in\Big{(}C^{\beta}_{\big{(}B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,t^{*}]\big{)}+x\big{)}}(D^{*}+x)\cap\big{(}F^{v}_{(d-1)^{+}}(l^{*},t^{*})+x\big{)}\Big{)}\mbox{
such that}\\\
&\Big{(}B\big{(}[-r^{*},r^{*}]^{d-2}\times[0,2r^{*}]\times[2r^{*}+2\hat{k},2r^{*}+4\hat{k}]\big{)}+y\Big{)}\subseteq
C^{0}_{B\big{(}[-r^{*},r^{*}]^{d-2}\times[0,2r^{*}]\times[0,2r^{*}+4\hat{k}]\big{)}+y\big{)}}(\\{y\\})\end{aligned}\right\\}.$

By the construction of $v$,
$\big{(}F^{v}_{(d-1)^{+}}(l^{*},t^{*})+x\big{)}\subseteq
B\big{(}[-l^{*},l^{*}]^{d-2}\times l^{*}\times[0,t^{*}]\big{)}$. Since
$2r^{*}\geq\max\\{\beta_{0},\ldots,\beta_{\hat{k}-1}\\}$, $t^{*}=2r^{*}$,
$l^{*}\geq r^{*}$, $s^{*}=t^{*}+2\hat{k}$, and
$\big{(}F^{v}_{(d-1)^{+}}(l^{*},t^{*})+x\big{)}\subseteq
B\big{(}[-l^{*},l^{*}]^{d-2}\times l^{*}\times[0,t^{*}]\big{)}$, it follows
from the construction of $A_{x}^{2}$ that $A_{x}^{2}\subseteq A_{x}$. So, we
have

$\displaystyle\mathbb{P}_{p}\Big{(}A_{x}\Big{)}\geq$ $\displaystyle\hskip
2.84526pt\mathbb{P}_{p}\Big{(}A_{x}^{2}\Big{)}$ (A.33) $\displaystyle\geq$
$\displaystyle\hskip 2.84526pt\mathbb{P}_{p}\Big{(}A_{x}^{2}\mid
A_{x}^{1}\Big{)}\times\mathbb{P}_{p}\Big{(}A_{x}^{1}\Big{)}$
$\displaystyle\geq$ $\displaystyle\hskip
2.84526pt\mathbb{P}_{p}\Big{(}A_{x}^{2}\mid
A_{x}^{1}\Big{)}\times\mathbb{P}_{p}\bigg{(}A_{x}^{1}\hskip
5.69054pt\biggm{|}\hskip
5.69054pt\Big{|}\Big{(}C^{\beta}_{\big{(}B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,t^{*}]\big{)}+x\big{)}}(D^{*}+x)\cap\big{(}F^{v}_{(d-1)^{+}}(l^{*},t^{*})+x\big{)}\Big{)}\Big{|}\geq
n^{*}\bigg{)}$
$\displaystyle\times\mathbb{P}_{p}\bigg{(}\Big{|}\Big{(}C^{\beta}_{\big{(}B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,t^{*}]\big{)}+x\big{)}}(D^{*}+x)\cap\big{(}F^{v}_{(d-1)^{+}}(l^{*},t^{*})+x\big{)}\Big{)}\Big{|}\geq
n^{*}\bigg{)}.$

Define the mapping
$S_{1}:A_{x}^{1}\longrightarrow\mathcal{P}\big{(}\mathbb{H}\big{)}$ such that
for all $\omega\in A_{x}^{1}$,

  1. (i)

$S_{1}(\omega)\subseteq\Big{(}C^{\beta}_{\big{(}B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,t^{*}]\big{)}+x\big{)}}(D^{*}+x)\cap\big{(}F^{v}_{(d-1)^{+}}(l^{*},t^{*})+x\big{)}\Big{)}$,
and

  2. (ii)

$|S_{1}(\omega)|\geq m^{*}$, where $L^{\infty}(y_{1},y_{2})\geq
4r^{*}+8\hat{k}+1$ for all distinct $y_{1},y_{2}\in S_{1}(\omega)$.

By the definition of $m^{*}$, $A_{x}^{1}$, $S_{1}$ and $A_{x}^{2}$, along with
Remark A.1 and the construction of the $\beta$-backbend percolation process,
we have

$\displaystyle\mathbb{P}_{p}\Big{(}A_{x}^{2}\mid A_{x}^{1}\Big{)}$ (A.34)
$\displaystyle\geq$ $\displaystyle\mathbb{P}_{p}\left(\begin{aligned}
&\exists\hskip 2.84526pty\in S_{1}\mbox{ such that }\\\
&\Big{(}B\big{(}[-r^{*},r^{*}]^{d-2}\times[0,2r^{*}]\times[2r^{*}+2\hat{k},2r^{*}+4\hat{k}]\big{)}+y\Big{)}\subseteq
C^{0}_{\big{(}B\big{(}[-r^{*},r^{*}]^{d-2}\times[0,2r^{*}]\times[0,2r^{*}+4\hat{k}]\big{)}+y\big{)}}(\\{y\\})\end{aligned}\hskip
9.95845pt\vrule\hskip 9.95845ptA_{x}^{1}\right)$ $\displaystyle>$
$\displaystyle 1-\frac{\epsilon^{*}}{2}.$

By the definition of $A_{x}^{1}$ and $n^{*}$,

$\mathbb{P}_{p}\bigg{(}A_{x}^{1}\hskip 5.69054pt\biggm{|}\hskip
5.69054pt\Big{|}\Big{(}C^{\beta}_{\big{(}B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,t^{*}]\big{)}+x\big{)}}(D^{*}+x)\cap\big{(}F^{v}_{(d-1)^{+}}(l^{*},t^{*})+x\big{)}\Big{)}\Big{|}\geq
n^{*}\bigg{)}=1,$ (A.35)

and by Lemma A.6 and the construction of the $\beta$-backbend percolation
process,

$\mathbb{P}_{p}\bigg{(}\Big{|}\Big{(}C^{\beta}_{\big{(}B\big{(}[-l^{*},l^{*}]^{d-1}\times[0,t^{*}]\big{)}+x\big{)}}(D^{*}+x)\cap\big{(}F^{v}_{(d-1)^{+}}(l^{*},t^{*})+x\big{)}\Big{)}\Big{|}\geq
n^{*}\bigg{)}>1-\frac{\epsilon^{*}}{2}.$

This, together with (A.33), (A.34), (A.35), yields

$\mathbb{P}_{p}\Big{(}A_{x}\Big{)}>1-\epsilon^{*}.$ (A.36)

Note that the event $A_{x}$ depends only on the edges in the set
$\Big{(}B\big{(}[-2l^{*},2l^{*}]^{d-2}\times[-l^{*},3l^{*}]\times[0,2s^{*}]\big{)}\Big{)}^{2}$.
Since this set is finite, the proof of Lemma A.7 follows from (A.36) by using
an argument similar to the one which we use to complete the proof of Lemma A.2
from (A.21). ∎

##  Appendix B Proof of Proposition 3.2

Fix an arbitrary $l\in\mathbb{N}$. Using a similar argument as we have
suggested for the proof of the “if” part of Theorem 3.4, we have
$p^{\tilde{\beta}}_{c}(\mathbb{Q}_{l}^{2})\leq
p^{\beta}_{c}(\mathbb{Q}_{l}^{2})$. We proceed to show that
$p^{\beta}_{c}(\mathbb{Q}_{2l}^{2})\leq
p^{\tilde{\beta}}_{c}(\mathbb{Q}_{l}^{2})$. In order to prove it, we show that
for all $p\in[0,1]$, $\theta^{\tilde{\beta}}_{\mathbb{Q}_{l}^{2}}(p)>0$
implies $\theta^{\beta}_{\mathbb{Q}_{2l}^{2}}(p)>0$. If $p=1$, then there is
nothing to prove. Fix an arbitrary $p\in[0,1)$ such that
$\theta^{\tilde{\beta}}_{\mathbb{Q}_{l}^{2}}(p)>0$.

Since $\underset{n\rightarrow\infty}{\lim}\hskip
2.84526pt\tilde{\beta}_{kn+i}=\beta_{i}$ for all $i\in\\{0,\ldots,k-1\\}$,
there exists $n^{*}\in\mathbb{N}$ such that $\tilde{\beta}_{kn+i}=\beta_{i}$
for all $n\geq n^{*}$ and all $i\in\\{0,\ldots,k-1\\}$. Let $t^{*}\in
k\mathbb{N}$ be such that $t^{*}\geq\max\\{\beta_{0},\ldots,\beta_{k-1}\\}$.
Let us define the following events:

$\displaystyle A:=\left\\{\omega\in\Omega:\begin{aligned} &\hskip
2.84526pt\mbox{there is an infinite open $\tilde{\beta}$-backbend path
}(x^{0},x^{1},\ldots)\mbox{ in }\mathbb{Q}_{l}^{2}\\\ &\hskip
2.84526pt\mbox{from the origin such that }x^{j}_{d}=kn^{*}+t^{*}\mbox{ for
some }j\end{aligned}\right\\},\mbox{ and}$ $\displaystyle
B:=\left\\{\omega\in\Omega:\begin{aligned} &\hskip 2.84526pt\mbox{there is an
infinite open $\tilde{\beta}$-backbend path }(x^{0},x^{1},\ldots)\mbox{ in
}\mathbb{Q}_{l}^{2}\\\ &\hskip 2.84526pt\mbox{from the origin such that
}x^{j}_{d}<kn^{*}+t^{*}\mbox{ for all }j\end{aligned}\right\\}.$

By the constructions of $A$ and $B$, we have $\mathbb{P}_{p}(A\cup
B)=\theta^{\tilde{\beta}}_{\mathbb{Q}_{l}^{2}}(p)$. Furthermore, since
$B\big{(}[-l,l]^{d-2}\times\mathbb{Z}\times[0,kn^{*}+t^{*}]\big{)}$ is a one-
dimensional cylinder, the fact $p<1$ implies that $\mathbb{P}_{p}(B)=0$.
Combining the facts that $\theta^{\tilde{\beta}}_{\mathbb{Q}_{l}^{2}}(p)>0$,
$\mathbb{P}_{p}(A\cup B)=\theta^{\tilde{\beta}}_{\mathbb{Q}_{l}^{2}}(p)$, and
$\mathbb{P}_{p}(B)=0$, we have $\mathbb{P}_{p}(A)>0$. For all
$s\in\mathbb{N}$, let us define the following event:

$A_{s}:=\left\\{\omega\in\Omega:\begin{aligned} &\hskip 2.84526pt\mbox{there
is an infinite open $\tilde{\beta}$-backbend path
}\pi=(x^{0},x^{1},\ldots)\mbox{ in }\mathbb{Q}_{l}^{2}\\\ &\hskip
2.84526pt\mbox{from the origin such that }|x^{j}_{d-1}|<s\mbox{ for all
}j\mbox{ with }h^{j}(\pi)<kn^{*}+t^{*}\end{aligned}\right\\}.$

By the constructions of $A$ and $A_{s}$, it follows that
$A_{s}\underset{s\rightarrow\infty}{\uparrow}A$. Since $\mathbb{P}_{p}(A)>0$,
this implies that there exists $s^{*}\in\mathbb{N}$ such that

$\mathbb{P}_{p}(A_{s^{*}})>0.$ (B.1)

######  Claim B.1.

$A_{s^{*}}\subseteq\bigg{\\{}\omega\in\Omega:\Big{|}C^{\beta}_{B\big{(}[-l,l]^{d-2}\times\mathbb{Z}\times[kn^{*},\infty]\big{)}}\Big{(}B\big{(}[-l,l]^{d-2}\times[-s^{*},s^{*}]\times
kn^{*}\big{)}\Big{)}\Big{|}=\infty\bigg{\\}}$.

###### Proof of Claim B.1.

Fix an arbitrary configuration $\omega^{*}\in A_{s^{*}}$. Let
$\pi=(x^{0},x^{1},\ldots)$ be an infinite open $\tilde{\beta}$-backbend path
in $\mathbb{Q}_{l}^{2}$ from the origin such that $|x^{j}_{d-1}|<s^{*}$ for
all $j$ with $h^{j}(\pi)<kn^{*}+t^{*}$. Since
$B\big{(}[-l,l]^{d-2}\times[-s^{*},s^{*}]\times[0,kn^{*}+t^{*}]\big{)}$ is a
finite set of vertices, it follows from the assumptions on $\pi$ that there
exists a finite sub-path $\pi_{1}=(x^{0},\ldots,x^{m^{*}})$ of $\pi$ from the
origin such that $\pi_{1}$ is in
$B\big{(}[-l,l]^{d-2}\times[-s^{*},s^{*}]\times[0,kn^{*}+t^{*}]\big{)}$ and
$x^{m^{*}}_{d}=kn^{*}+t^{*}$. Let $z^{*}$ be the last vertex of $\pi_{1}$ such
that $z^{*}_{d}=kn^{*}$. Note that by construction, $z^{*}\in
B\big{(}[-l,l]^{d-2}\times[-s^{*},s^{*}]\times kn^{*}\big{)}$. Consider the
sub-path $\pi^{*}$ of $\pi$ from $z^{*}$. Clearly, $\pi^{*}$ is an infinite
open path.

First, we show that $\pi^{*}$ is in
$B\big{(}[-l,l]^{d-2}\times\mathbb{Z}\times[kn^{*},\infty]\big{)}$. We divide
$\pi^{*}$ into two parts. The first part $\pi^{*}_{1}$ is the sub-path of
$\pi^{*}$ from $z^{*}$ to $x^{m^{*}}$, and the second part $\pi^{*}_{2}$ is
the sub-path of $\pi^{*}$ from $x^{m^{*}}$. By construction, $\pi^{*}_{1}$ is
in
$B\big{(}[-l,l]^{d-2}\times[-s^{*},s^{*}]\times[kn^{*},kn^{*}+t^{*}]\big{)}$.
Furthermore, because of the facts that $\pi$ is a $\tilde{\beta}$-backbend
path in $\mathbb{Q}_{l}^{2}$, $x^{m^{*}}_{d}=kn^{*}+t^{*}$,
$\tilde{\beta}_{i}=\beta_{i}$ for all $i\geq kn^{*}$, and
$t^{*}\geq\max\\{\beta_{0},\ldots,\beta_{k-1}\\}$, it follows by construction
that $\pi^{*}_{2}$ is in
$B\big{(}[-l,l]^{d-2}\times\mathbb{Z}\times[kn^{*},\infty]\big{)}$. Since
$\pi^{*}_{1}$ is in
$B\big{(}[-l,l]^{d-2}\times[-s^{*},s^{*}]\times[kn^{*},kn^{*}+t^{*}]\big{)}$,
and $\pi^{*}_{2}$ is in
$B\big{(}[-l,l]^{d-2}\times\mathbb{Z}\times[kn^{*},\infty]\big{)}$, it follows
that $\pi^{*}$ is in
$B\big{(}[-l,l]^{d-2}\times\mathbb{Z}\times[kn^{*},\infty]\big{)}$.

In order to complete the proof of Claim B.1, it remains to show that $\pi^{*}$
is a $\beta$-backbend path. Assume for contradiction that $\pi^{*}$ is not a
$\beta$-backbend path. Since $\pi^{*}$ is not a $\beta$-backbend path, there
exists a vertex $x^{*}$ in $\pi^{*}$ such that
$x^{*}_{d}<h^{*}-\beta_{h^{*}}$, where $h^{*}$ is the record level attained by
the path $\pi^{*}$ till $x^{*}$. Let $\hat{h}$ be the record level attained by
the path $\pi$ till $x^{*}$. Clearly, $\hat{h}\geq h^{*}\geq kn^{*}$. Because
$\pi$ is a $\tilde{\beta}$-backbend path, it must be that
$x^{*}_{d}\geq\hat{h}-\tilde{\beta}_{\hat{h}}$. Since $\hat{h}\geq kn^{*}$ and
$\tilde{\beta}_{i}=\beta_{i}$ for all $i\geq kn^{*}$, this implies
$x^{*}_{d}\geq\hat{h}-\beta_{\hat{h}}$. Combining the facts that
$x^{*}_{d}<h^{*}-\beta_{h^{*}}$ and $x^{*}_{d}\geq\hat{h}-\beta_{\hat{h}}$, we
have

$h^{*}-\beta_{h^{*}}>\hat{h}-\beta_{\hat{h}}.$ (B.2)

The assumptions on $\beta$ imply that $\beta_{l}-\beta_{m}\leq l-m$ for all
$l\geq m$. Since $\hat{h}\geq h^{*}$, this yields
$\beta_{\hat{h}}-\beta_{h^{*}}\leq\hat{h}-h^{*}$, a contradiction to (B.2).
So, it must be that $\pi^{*}$ is a $\beta$-backbend path. This completes the
proof of Claim B.1. ∎

Now, we complete the proof of Proposition 3.2. Because
$B\big{(}[-l,l]^{d-2}\times[-s^{*},s^{*}]\times kn^{*}\big{)}$ is a finite
set, (B.1) and Claim B.1 together imply that there exists some $x\in
B\big{(}[-l,l]^{d-2}\times[-s^{*},s^{*}]\times kn^{*}\big{)}$ such that
$\mathbb{P}_{p}\Big{(}\big{|}C^{\beta}_{(x+\mathbb{Q}_{2l}^{2})}(\\{x\\})\big{|}=\infty\Big{)}>0$.
Furthermore, since $\beta$ is $k$-cyclic and $x_{d}=kn^{*}$, by the
construction of the $\beta$-backbend percolation process, we have
$\mathbb{P}_{p}\Big{(}\big{|}C^{\beta}_{\mathbb{Q}_{2l}^{2}}\big{|}=\infty\Big{)}=\mathbb{P}_{p}\Big{(}\big{|}C^{\beta}_{(x+\mathbb{Q}_{2l}^{2})}(\\{x\\})\big{|}=\infty\Big{)}$,
and hence
$\mathbb{P}_{p}\Big{(}\big{|}C^{\beta}_{\mathbb{Q}_{2l}^{2}}\big{|}=\infty\Big{)}>0$.
This completes the proof of Proposition 3.2. ∎

## References

  * Balister et al. (1993) Paul Balister, Bela Bollobas, and Alan Stacey.  Upper bounds for the critical probability of oriented percolation in two dimensions.  _Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences_ , 440(1908):201–220, 1993\.
  * Bezuidenhout et al. (1990) Carol Bezuidenhout, Geoffrey Grimmett, et al.  The critical contact process dies out.  _The Annals of Probability_ , 18(4):1462–1482, 1990.
  * Durrett (1984) Richard Durrett.  Oriented percolation in two dimensions.  _The Annals of Probability_ , pages 999–1040, 1984.
  * Durrett (1988) Richard Durrett.  _Lecture notes on particle systems and percolation_.  Brooks/Cole Pub Co, 1988.
  * Gliozzi et al. (2005) Ferdinando Gliozzi, S Lottini, Marco Panero, and A Rago.  Random percolation as a gauge theory.  _Nuclear Physics B_ , 719(3):255–274, 2005.
  * Grimmett (2013) Geoffrey R Grimmett.  _Percolation_ , volume 321.  Springer Science & Business Media, 2013.
  * Liggett (2012) Thomas Milton Liggett.  _Interacting particle systems_ , volume 276.  Springer Science & Business Media, 2012.
  * Perlsman and Havlin (2002) E Perlsman and S Havlin.  Method to estimate critical exponents using numerical studies.  _EPL (Europhysics Letters)_ , 58(2):176, 2002\.
  * Roy et al. (1998) Rahul Roy, Anish Sarkar, and Damien G White.  Backbends in directed percolation.  _Journal of statistical physics_ , 91(5-6):889–908, 1998.