and ${\mathcal{G}}_{n}[1,2w_{\mathrm{hh}})$, define also the collections of
random variables

$\displaystyle\widehat{\Psi}_{n}^{{\scriptscriptstyle(\mathrm{cond,unsure})}}$
$\displaystyle:=\big{\\{}\widehat{\varphi}_{u,v}:u\in\widehat{\Xi}_{n}^{\scriptscriptstyle(\mathrm{unsure})}[2w_{\mathrm{hh}},\infty),v\in{\mathcal{S}}(u)\big{\\}},$
(5.26)
$\displaystyle\widehat{\Psi}_{n}^{{\scriptscriptstyle(\mathrm{cond,sure})}}$
$\displaystyle:=\big{\\{}\widehat{\varphi}_{u,v}:u\in\widehat{\Xi}_{n}^{\scriptscriptstyle(\mathrm{sure})}[2w_{\mathrm{hh}},\infty),v\in{\mathcal{S}}(u)\big{\\}},$
(5.27)

so that for different vertices
$u_{1},u_{2}\in\widehat{\Xi}_{n}[2w_{\mathrm{hh}},\infty)$, the collections
$\\{\widehat{\varphi}_{u_{1},v}\\}_{v\in{\mathcal{S}}(u_{1})}$ and
$\\{\widehat{\varphi}_{u_{2},v^{\prime}}\\}_{v^{\prime}\in{\mathcal{S}}(u_{2})}$
are _independent_. The joint distribution of
$\\{\widehat{\varphi}_{u,v}\\}_{v\in{\mathcal{S}}(u)}$ for a single
$u\in\widehat{\Xi}_{n}^{\scriptscriptstyle(\mathrm{unsure})}[2w_{\mathrm{hh}},\infty)$
is as follows: for _any_ sequence
$(z_{u,v})_{v\in{\mathcal{S}}(u)}\in[0,1]^{{\mathcal{S}}(u)}$ of length
$s_{k}$, and with $r_{k}=1-2^{-1/s_{k}}$,

$\displaystyle\mathbb{P}\big{(}\forall
v\in{\mathcal{S}}(u):\widehat{\varphi}_{u,v}\leq z_{u,v}$ $\displaystyle\mid
u\in\widehat{\Xi}_{n}^{\scriptscriptstyle(\mathrm{unsure})}[2w_{\mathrm{hh}},\infty)\big{)}$
(5.28) $\displaystyle:=\mathbb{P}\big{(}\forall
v\in{\mathcal{S}}(u):U_{u,v}\leq z_{u,v}\mid\forall
v\in{\mathcal{S}}(u):U_{u,v}>r_{k}\big{)}.$

Similarly we define the joint distribution of
$\\{\widehat{\varphi}_{u,v}\\}_{v\in{\mathcal{S}}(u)}$ for a single
$u\in\widehat{\Xi}_{n}^{\scriptscriptstyle(\mathrm{sure})}[2w_{\mathrm{hh}},\infty)$
as follows: for any sequence
$(z_{u,v})_{v\in{\mathcal{S}}(u)}\in[0,1]^{{\mathcal{S}}(u)}$ of length
$s_{k}$,

$\displaystyle\mathbb{P}\big{(}\forall
v\in{\mathcal{S}}(u):\widehat{\varphi}_{u,v}\leq z_{u,v}$ $\displaystyle\mid
u\in\widehat{\Xi}_{n}^{\scriptscriptstyle(\mathrm{sure})}[2w_{\mathrm{hh}},\infty)\big{)}$
(5.29) $\displaystyle:=\mathbb{P}\big{(}\forall
v\in{\mathcal{S}}(u):U_{u,v}\leq z_{u,v}\mid\exists
v\in{\mathcal{S}}(u):U_{u,v}\leq r_{k}\big{)}.$

We define $\widehat{\mathcal{G}}_{n}$ as the graph encoded by
$(\widehat{\Xi}_{n},\widehat{\Psi}_{n},\mathrm{p})$, where

$\displaystyle\widehat{\Xi}_{n}$
$\displaystyle:=\Xi_{n}[1,2w_{\mathrm{hh}})\cup\widehat{\Xi}_{n}^{\scriptscriptstyle(\mathrm{unsure})}[2w_{\mathrm{hh}},\infty)\cup\widehat{\Xi}_{n}^{\scriptscriptstyle(\mathrm{sure})}[2w_{\mathrm{hh}},\infty),$
(5.30) $\displaystyle\widehat{\Psi}_{n}$
$\displaystyle:=\Psi_{n}[1,2w_{\mathrm{hh}})\cup\widehat{\Psi}_{n}^{{\scriptscriptstyle(\mathrm{iid})}}\cup\widehat{\Psi}_{n}^{{\scriptscriptstyle(\mathrm{cond,unsure})}}\cup\widehat{\Psi}_{n}^{{\scriptscriptstyle(\mathrm{cond,sure})}}.$

An immediate corollary is the following statement.

######  Corollary 5.7.

Consider an i-KSRG $\widehat{\mathcal{G}}_{n}$ following Definition 5.6 for
some $n,k$. On the event ${\mathcal{A}}_{\mathrm{bb}}(n,k)$, every vertex in
$\widehat{\Xi}_{n}^{\scriptscriptstyle(\mathrm{sure})}[2w_{\mathrm{hh}},\infty)$
is connected by an edge to ${\mathcal{C}}_{\mathrm{bb}}(n,k)$ in
$\widehat{\mathcal{G}}_{n}$.

###### Proof.

The conditioning in (5.29) guarantees that for each
$u\in\widehat{\Xi}_{n}^{\scriptscriptstyle(\mathrm{sure})}[2w_{\mathrm{hh}},\infty)$
at least one $\widehat{\varphi}_{u,v}\leq r_{k}$ occurs among the edge-
variables $\\{\widehat{\varphi}_{u,v}:v\in{\mathcal{S}}(u)\\}$, where
${\mathcal{S}}(u)\subseteq{\mathcal{C}}_{\mathrm{bb}}$ (cf. (5.22)). Then
since $\widehat{\varphi}_{u,v}\leq r_{k}\leq\mathrm{p}(u,v)$ holds by Claim
5.5, this ensures that $\\{u,v\\}$ is in the edge set of
$\widehat{\mathcal{G}}_{n}$ by the graph-encoding in Definition 5.3. ∎

######  Proposition 5.8.

Fix a connectivity function $\mathrm{p}$. The law of the random graph
$\widehat{\mathcal{G}}_{n}$ formed by Definition 5.6 is identical to the law
of the random graph ${\mathcal{G}}_{n}$ formed by Definition 5.3.

###### Proof.

By Corollary 5.4 it is sufficient to show that
$(\widehat{\Xi}_{n},\widehat{\Psi}_{n})$ defined in (5.30) has the same
distribution as $(\Xi_{n},\Psi_{n})$ from Definition 5.6. By standard
properties of PPPs, $\Xi_{n}$ can be written as two independent PPPs defined
on the same space, each having half the intensity measure of $\Xi_{n}$. Thus,
by the construction of $\widehat{\Xi}_{n}[1,2w_{\mathrm{hh}})$ in Definition
5.6, it holds that the PPPs $\Xi_{n}$ and $\widehat{\Xi}_{n}$ have the same
law. Hence, we may couple the vertex sets so that a.s.
$\Xi_{n}=\widehat{\Xi}_{n}$, and only need to show that the collection of
variables encoding the edges, $\Psi_{n}$ and $\widehat{\Psi}_{n}$, share the
same law, conditionally under the given vertex set realization (say)
$\widehat{\Xi}_{n}$. By (5.30) in Definition 5.6, the graph
${\mathcal{G}}_{n,2}$ spanned on
$\Xi_{n}[1,2w_{\mathrm{hh}})\subseteq\widehat{\Xi}_{n}$ is determined by
$\Psi_{n}[1,2w_{\mathrm{hh}})=\\{\varphi_{u,v}:u,v\in\Xi_{n}[1,2w_{\mathrm{hh}})\\}$
in Definition 5.3. Thus ${\mathcal{G}}_{n,2}$ has the same distribution both
in Definition 5.3 and in Definition 5.6.

(i) If now $\Psi_{n}[1,2w_{\mathrm{hh}})$ is such that the graph
${\mathcal{G}}_{n,2}$ does _not_ satisfy the event
${\mathcal{A}}_{\mathrm{bb}}$, by (i) of Definition 5.6, the statement holds
since both $\\{\varphi_{u,v}\\}$ and $\\{U_{u,v}\\}$ are iid uniforms whenever
$u\in\widehat{\Xi}_{n}[2w_{\mathrm{hh}},\infty)$, i.e.,
$\widehat{\Psi}_{n}\setminus\Psi_{n}[1,2w_{\mathrm{hh}})=\Sigma_{n}$ and
$\Psi_{n}\setminus\Psi_{n}[1,2w_{\mathrm{hh}})$ have the same distribution.

(ii) If $\Psi_{n}[1,2w_{\mathrm{hh}})$ is such that the graph
${\mathcal{G}}_{n,2}$ does satisfy the event ${\mathcal{A}}_{\mathrm{bb}}$,
then we work conditionally on a realization of the graph
${\mathcal{G}}_{n,2}=(\Xi_{n}[1,2w_{\mathrm{hh}}),\Psi_{n}[1,2w_{\mathrm{hh}}),\mathrm{p})$,
and also on the coupled realization of the PPPs
$\Xi_{n}[2w_{\mathrm{hh}},\infty)=\widehat{\Xi}_{n}[2w_{\mathrm{hh}},\infty)$.
Let us define the conditional probability measure (of the edges) under the
coupling by

$\mathbb{P}^{\star}(\cdot):=\mathbb{P}(\,\cdot\mid{\mathcal{G}}_{n,2},\Xi_{n}[2w_{\mathrm{hh}},\infty))=\mathbb{P}(\,\cdot\mid{\mathcal{G}}_{n,2},\mbox{unlabeled
}\widehat{\Xi}_{n}[2w_{\mathrm{hh}},\infty)),$ (5.31)

where in the conditioning we do _not_ reveal to which sub-PPP
($\widehat{\Xi}_{n}^{\scriptscriptstyle(\mathrm{sure})}[2w_{\mathrm{hh}},\infty)$
or
$\widehat{\Xi}_{n}^{\scriptscriptstyle(\mathrm{unsure})}[2w_{\mathrm{hh}},\infty)$)
a vertex in $\widehat{\Xi}_{n}[2w_{\mathrm{hh}},\infty)$ belongs to. Using
$\widehat{\Psi}_{n}$ from (5.30) and
$\widehat{\Psi}_{n}^{\scriptscriptstyle(\mathrm{iid})}\subseteq\Sigma_{n}$
from (5.25) (containing _independent_ copies $U_{u,v}$ of $\mathrm{Unif}[0,1]$
random variables, like $\Psi$ in Definition 5.3), we see that variables in
$\Psi_{n}\setminus\Psi_{n}[1,2w_{\mathrm{hh}})$ and
$\widehat{\Psi}_{n}\setminus\Psi_{n}[1,2w_{\mathrm{hh}})$ also share the same
(joint) law of iid $\mathrm{Unif}[0,1]$ whenever $u$ and $v$ are such
$u\in\widehat{\Xi}_{n}[2w_{\mathrm{hh},\infty})$ and that
$v\notin{\mathcal{S}}(u)$. Moreover, in (5.26)-(5.27), the collections
$\\{\widehat{\varphi}_{u,v}\\}_{v\in{\mathcal{S}}(u)}$ are independent across
$u$ for different vertices $u\in\widehat{\Xi}_{n}[2w_{\mathrm{hh}},\infty)$.
So for $\widehat{\mathcal{G}}_{n}\ {\buildrel d\over{=}}\ {\mathcal{G}}_{n}$
it is left to show that for any fixed
$u\in\widehat{\Xi}_{n}[2w_{\mathrm{hh}},\infty)=\Xi_{n}[2w_{\mathrm{hh}},\infty)$,
under the measure $\mathbb{P}^{\star}$,

$\big{\\{}\varphi_{u,v},v\in{\mathcal{S}}(u)\big{\\}}\overset{d}{=}\big{\\{}\widehat{\varphi}_{u,v},v\in{\mathcal{S}}(u)\big{\\}}.$
(5.32)

We first analyze the distribution of the left-hand side, i.e., $\varphi_{u,v}$
being iid from Definition 5.3. Let
$(z_{u,v})_{v\in{\mathcal{S}}(u)}\in[0,1]^{{\mathcal{S}}(u)}$ be any sequence
of length $s_{k}$. By Claim 5.5, and the law of total probability

$\displaystyle\mathbb{P}^{\star}\big{(}\forall
v\in{\mathcal{S}}(u):\varphi_{u,v}\leq z_{u,v}\big{)}$
$\displaystyle=(1/2)\mathbb{P}^{\star}\big{(}\forall
v\in{\mathcal{S}}(u):\varphi_{u,v}\leq z_{u,v}\mid\forall
v\in{\mathcal{S}}(u):\varphi_{u,v}>r_{k}\big{)}$ (5.33) $\displaystyle\hskip
15.0pt+(1/2)\mathbb{P}^{\star}\big{(}\forall
v\in{\mathcal{S}}(u):\varphi_{u,v}\leq z_{u,v}\mid\exists
v\in{\mathcal{S}}(u):\varphi_{u,v}\leq r_{k}\big{)}.$

We now analyze the right-hand side in (5.32). By the construction in (5.24),
$\widehat{\Xi}_{n}[2w_{\mathrm{hh}},\infty)$ is the union of two iid sub-PPPs.
Under $\mathbb{P}^{\star}$ in (5.31) we did not reveal to which sub-PPP
vertices belong to. Hence, for each
$u\in\widehat{\Xi}_{n}[2w_{\mathrm{hh}},\infty)$, independently of each other

$\mathbb{P}^{\star}(u\in\widehat{\Xi}_{n}^{\scriptscriptstyle(\mathrm{unsure})}[2w_{\mathrm{hh}},\infty)\mid
u\in\widehat{\Xi}_{n}[2w_{\mathrm{hh}},\infty))=\mathbb{P}^{\star}(u\in\widehat{\Xi}_{n}^{\scriptscriptstyle(\mathrm{sure})}[2w_{\mathrm{hh}},\infty)\mid
u\in\widehat{\Xi}_{n}[2w_{\mathrm{hh}},\infty))=1/2.$

Thus, by the law of total probability, and using the distributions of
$(\varphi_{u,v})_{v\in{\mathcal{S}}(u)}$ given by (5.28), (5.29),

$\displaystyle\mathbb{P}^{\star}\big{(}\forall
v\in{\mathcal{S}}(u):\widehat{\varphi}_{u,v}\leq z_{u,v}\big{)}$
$\displaystyle=(1/2)\mathbb{P}^{\star}\big{(}\forall
v\in{\mathcal{S}}(u):\widehat{\varphi}_{u,v}\leq z_{u,v}\mid
u\in\widehat{\Xi}_{n}^{\scriptscriptstyle(\mathrm{unsure})}[2w_{\mathrm{hh}},\infty)\big{)}$
(5.34) $\displaystyle\hskip 15.0pt+(1/2)\mathbb{P}^{\star}\big{(}\forall
v\in{\mathcal{S}}(u):\widehat{\varphi}_{u,v}\leq z_{u,v}\mid
u\in\widehat{\Xi}_{n}^{\scriptscriptstyle(\mathrm{sure})}[2w_{\mathrm{hh}},\infty)\big{)}$
$\displaystyle=(1/2)\mathbb{P}^{\star}\big{(}\forall
v\in{\mathcal{S}}(u):U_{u,v}\leq z_{u,v}\mid\forall
v\in{\mathcal{S}}(u):U_{u,v}>r_{k}\big{)}$ $\displaystyle\hskip
15.0pt+(1/2)\mathbb{P}^{\star}\big{(}\forall v\in{\mathcal{S}}(u):U_{u,v}\leq
z_{u,v}\mid\exists v\in{\mathcal{S}}(u):U_{u,v}\leq r_{k}\big{)}.$

Note that $\\{U_{u,v}\\}_{u,v}$ and $\\{\varphi_{u,v}\\}_{u,v}$ are both sets
of independent $\mathrm{Unif}[0,1]$ random variables by Definitions 5.6 and
5.3, respectively. Hence, (5.32) follows by combining (5.33) and (5.34). ∎

For the remainder of this section, we assume that we construct
${\mathcal{G}}_{n}$ following Definition 5.6 and write

$\Xi_{n}[2w_{\mathrm{hh}},\infty):=\Xi_{n}^{\scriptscriptstyle(\mathrm{unsure})}[2_{\mathrm{hh}},\infty)\cup\Xi_{n}^{\scriptscriptstyle(\mathrm{sure})}[2w_{\mathrm{hh}},\infty)$

as the union of two independent PPPs of equal intensity, such that if
${\mathcal{G}}_{n,2}={\mathcal{G}}[1,2w_{\mathrm{hh}})$ satisfies
${\mathcal{A}}_{\mathrm{bb}}$ in (5.8), every vertex in
$\Xi_{n}^{\scriptscriptstyle(\mathrm{sure})}[2w_{\mathrm{hh}},\infty)$
connects by an edge to ${\mathcal{C}}_{\mathrm{bb}}$, by Corollary 5.7.

To finish Step 3, on the event ${\mathcal{A}}_{\mathrm{bb}}$, we define
${\mathcal{G}}_{n,3}:=({\mathcal{V}}_{n,3},\Psi_{n,3},\mathrm{p})$, with

$\displaystyle{\mathcal{V}}_{n,3}:=\Xi_{n}[1,2w_{\mathrm{hh}})\cup\Xi_{n}^{\scriptscriptstyle(\mathrm{unsure})}[2_{\mathrm{hh}},\infty),\quad\Psi_{n,3}:=\Psi_{n}[1,2w_{\mathrm{hh}})\cup\widehat{\Psi}_{n}^{{\scriptscriptstyle(\mathrm{iid},\mathrm{unsure})}}\cup\widehat{\Psi}_{n}^{{\scriptscriptstyle(\mathrm{cond},\mathrm{unsure})}},$
(5.35)

i.e., the graph spanned on ${\mathcal{V}}_{n,3}$. We call the vertices in
$\Xi_{n}^{\scriptscriptstyle(\mathrm{sure})}[2w_{\mathrm{hh}},\infty)$ _sure-
connector_ vertices. If the event $A_{\mathrm{bb}}$ does not hold then we say
that the construction failed and we leave ${\mathcal{G}}_{n,3}$ undefined.

### Step 4. Cover expansion

In this step, we ensure that all components of size at least $k$ of
${\mathcal{G}}_{n,3}$ merge with the giant component of ${\mathcal{G}}_{n}$
via edges towards sure-connector vertices, with error probability
$\mathrm{err}_{n,k}$ from (3.1). The next lemma proves this using the cover-
expansion technique of Section 4. The notion of expandability is from
Definition 4.1, and $s_{k}$ is from (5.7). Let for some $c_{\ref{lemma:hh-
expandable}}^{\prime}>0$

${\mathcal{A}}_{\mathrm{exp}}(c_{\ref{lemma:hh-
expandable}}^{\prime}):={\mathcal{A}}_{\mathrm{exp}}(n,k,c_{\ref{lemma:hh-
expandable}}^{\prime}):=\big{\\{}{\mathcal{V}}_{n,3}\mbox{ is
$(c_{\ref{lemma:hh-expandable}}^{\prime}s_{k})$-expandable}\big{\\}}.$ (5.36)

######  Lemma⋆ 5.9 (Cover-expansion and $\sigma\leq\tau-1$).

Consider an i-KSRG satisfying the conditions of Proposition 5.1. For any
$c_{\ref{lemma:hh-expandable}}^{\prime}>0$

$\mathbb{P}\big{(}\neg{\mathcal{A}}_{\mathrm{exp}}(c_{\ref{lemma:hh-
expandable}}^{\prime})\big{)}\leq C_{\ref{lem:expandable-
ppp}}n\exp(-c_{\ref{lemma:hh-expandable}}^{\prime}s_{k}).$ (5.37)

Moreover, if $\sigma\leq\tau-1$, there exist $k_{2},c_{\ref{lemma:hh-
expandable}},c_{\ref{lemma:hh-expandable}}^{\prime}>0$, such that
conditionally on any realization of ${\mathcal{G}}_{n,3}$ satisfying
${\mathcal{A}}_{\mathrm{bb}}\cap{\mathcal{A}}_{\mathrm{exp}}(c_{\ref{lemma:hh-
expandable}}^{\prime})$, for all $k\geq k_{2}$ and any connected component
${\mathcal{C}}$ of ${\mathcal{G}}_{n,3}$ with $|{\mathcal{C}}|\geq k$,

$\mathbb{P}\big{(}{\mathcal{C}}\not\leftrightarrow\Xi_{n}^{\scriptscriptstyle(\mathrm{sure})}[2w_{\mathrm{hh}},\infty)\mid{\mathcal{G}}_{n,3},{\mathcal{A}}_{\mathrm{bb}}\cap{\mathcal{A}}_{\mathrm{exp}}(c_{\ref{lemma:hh-
expandable}}^{\prime})\big{)}\leq\exp\big{(}-c_{\ref{lemma:hh-
expandable}}s_{k}\big{)}.$ (5.38)

###### Proof.

The statement (5.37) follows directly for any $c_{\ref{lemma:hh-
expandable}}^{\prime}>0$ from its definition in (5.36) and Lemma 4.11. In
Proposition 4.2, for a given mark $\underline{w}$, the function
$s(\underline{w})$ in (4.1) describes the necessary “expandability parameter”,
such that all vertices with mark at least $\underline{w}$ in
${\mathcal{K}}_{n}({\mathcal{L}})$ connect to any
$s(\underline{w})$-expandable set ${\mathcal{L}}$ of vertices with probability
at least $p/2$. We shall take $\underline{w}:=2w_{\mathrm{hh}}$, the lowest
possible mark in $\Xi_{n}^{{\scriptscriptstyle(\mathrm{sure})}}$. By
Definition 4.1 of expandability, if ${\mathcal{V}}_{n,3}$ is
$(c_{\ref{lemma:hh-expandable}}^{\prime}s_{k})$-expandable, it is also
$s(2w_{\mathrm{hh}})$-expandable whenever

$s(2w_{\mathrm{hh}})\geq c_{\ref{lemma:hh-expandable}}^{\prime}s_{k}.$ (5.39)

We compute the left-hand side using the value of $w_{\mathrm{hh}}$ from (5.7)
and the function $s(\underline{w})$ from (4.1):

$s(2w_{\mathrm{hh}})=\big{(}2^{d+1}\beta
C_{1}^{-1/(\tau-1)}\big{)}^{1/(1-1/\alpha)}k^{\gamma_{\mathrm{hh}}/(1-1/\alpha)}.$

By (5.7) and (1.12), it holds that
$s_{k}=\Theta(k^{1-\gamma_{\mathrm{hh}}(\tau-1)})=\Theta(k^{\zeta_{\mathrm{hh}}})$.
Using (1.12), it is elementary to verify that
$\gamma_{\mathrm{hh}}/(1-1/\alpha)\geq\zeta_{\mathrm{hh}}$ if and only if
$\sigma\leq\tau-1$. Thus, if $\sigma\leq\tau-1$, (5.39) holds for some
$c_{\ref{lemma:hh-expandable}}^{\prime}>0$, whenever $k>k_{2}$ for some
sufficiently large $k_{2}\geq 1$, that ensures the condition
$\underline{w}>2^{d}d^{d/2}/\beta$ in Proposition 4.2. On the event
${\mathcal{A}}_{\mathrm{exp}}(c_{\ref{lemma:hh-expandable}}^{\prime})$,
${\mathcal{V}}_{n,3}$ is thus also $s(2w_{\mathrm{hh}})$-expandable (since it
is $c_{\ref{lemma:hh-expandable}}^{\prime}s_{k}$ expandable). Since
expandability carries through for subsets of ${\mathcal{V}}_{n,3}$ (below
Definition 4.1), _any_ subset of ${\mathcal{V}}_{n,3}$ is
$s(2w_{\mathrm{hh}})$-expandable. Hence, Proposition 4.2 is applicable for any
set ${\mathcal{L}}\subseteq{\mathcal{V}}_{n,3}$ and
$\underline{w}:=2w_{\mathrm{hh}}$, and guarantees the existence of a set
${\mathcal{K}}_{n}({\mathcal{L}})\subseteq\Lambda_{n}$ satisfying (4.2) and
(4.3).

Consider an arbitrary connected component ${\mathcal{C}}$ of
${\mathcal{G}}_{n,3}$ that satisfies $|{\mathcal{C}}|\geq k$. With
${\mathcal{K}}_{n}({\mathcal{C}})$ from Proposition 4.2, we define the set of
sure-connector vertices inside ${\mathcal{K}}_{n}({\mathcal{C}})$ connected to
${\mathcal{C}}$ as

${\mathcal{H}}_{{\mathcal{C}}}:=\\{v\in{\mathcal{K}}_{n}({\mathcal{C}})\cap\Xi_{n}^{\scriptscriptstyle(\mathrm{sure})}[2w_{\mathrm{hh}},\infty)\big{)}:v\leftrightarrow{\mathcal{C}}\\}.$
(5.40)

Since $\Xi^{\scriptscriptstyle(\mathrm{sure})}[2w_{\mathrm{hh}},\infty)$ is a
Poisson process, the number of its points in
${\mathcal{K}}_{n}({\mathcal{C}})$ follows a Poisson distribution. Since each
of these points connects by an edge _independently_ to ${\mathcal{C}}$ with
probability at least $p/2$ by (4.3), and an independent thinning of a PPP is
another PPP; we obtain using the intensity measure in Definition 5.6 by the
volume bound (4.2) on ${\mathcal{K}}_{n}({\mathcal{C}})$ for
$|{\mathcal{C}}|\geq k$ and
$c:=(p/4)2^{-(4d+1)}2^{-(\tau-1)}d^{-d/2}/{\mathrm{e}}>0$

$\displaystyle\mathbb{P}\big{(}|{\mathcal{H}}_{\mathcal{C}}|=0\mid{\mathcal{G}}_{n,3},{\mathcal{A}}_{\mathrm{bb}}\cap{\mathcal{A}}_{\mathrm{exp}}(c_{\ref{lemma:hh-
expandable}}^{\prime})\big{)}$
$\displaystyle\leq\mathbb{P}\big{(}{\mathrm{Poi}}\big{(}(p/2)\cdot(1/2)\cdot\mathrm{Vol}({\mathcal{K}}_{n}({\mathcal{C}}))\cdot(2w_{\mathrm{hh}})^{-(\tau-1)}\big{)}=0\big{)}$
$\displaystyle\leq\exp\big{(}-(p/4)\mathrm{Vol}({\mathcal{K}}_{n}({\mathcal{C}}))(2w_{\mathrm{hh}})^{-(\tau-1)}\big{)}\big{)}$
$\displaystyle\leq\exp\big{(}-c\cdot
kw_{\mathrm{hh}}^{-(\tau-1)}\big{)}=\exp\big{(}-16c\cdot s_{k}\big{)},$

where we used $s_{k}$ from (5.7) in the last step. Since
$\\{|{\mathcal{H}}_{{\mathcal{C}}}|>0\\}$ in (5.40) implies
$\\{{\mathcal{C}}\leftrightarrow\Xi_{n}^{\scriptscriptstyle(\mathrm{sure})}[2w_{\mathrm{hh}},\infty)\\}$,
this finishes the proof of (5.38) with $c_{\ref{lemma:hh-
expandable}}:=16(p/4)2^{-(4d+1)}2^{-(\tau-1)}d^{-d/2}/{\mathrm{e}}$. ∎

### Combining everything: preventing too large components

###### Proof of Proposition 5.1.

Assume that $k\geq\max\\{k_{1},k_{2}\\}$ defined in Lemmas 5.2 and 5.9,
respectively, and that $n>k^{1+\zeta_{\mathrm{hh}}/\alpha}$, i.e., $n$
satisfies the bound in Lemma 5.2. We construct
${\mathcal{G}}_{n}\supseteq{\mathcal{G}}_{n,3}$ following Definition 5.6,
where ${\mathcal{G}}_{n,3}$ from (5.35) is the subgraph of ${\mathcal{G}}_{n}$
induced on
${\mathcal{V}}_{n,3}=\Xi_{n}[1,2w_{\mathrm{hh}})\cup\Xi_{n}^{\scriptscriptstyle(\mathrm{unsure})}[2w_{\mathrm{hh}},\infty)$.
The events ${\mathcal{A}}_{\mathrm{bb}}$ in (5.8) and
${\mathcal{A}}_{\mathrm{exp}}(c_{\ref{lemma:hh-
expandable}}^{\prime}):={\mathcal{A}}_{\mathrm{exp}}$ in (5.36) are measurable
with respect to ${\mathcal{G}}_{n,3}$. By the law of total probability (taking
expectation over realizations of ${\mathcal{G}}_{n,3}$), we obtain

$\displaystyle\mathbb{P}\big{(}|{\mathcal{C}}_{n}^{\scriptscriptstyle(2)}|\geq
k\big{)}$
$\displaystyle\leq\mathbb{E}\big{[}\mathbbm{1}_{\\{{\mathcal{A}}_{\mathrm{bb}}\cap{\mathcal{A}}_{\mathrm{exp}}\\}}\mathbb{P}\big{(}|{\mathcal{C}}_{n}^{\scriptscriptstyle(2)}|\geq
k\mid{\mathcal{G}}_{n,3},{\mathcal{A}}_{\mathrm{exp}}\cap{\mathcal{A}}_{\mathrm{bb}}\big{)}\big{]}+\mathbb{P}\big{(}\neg{\mathcal{A}}_{\mathrm{bb}}\big{)}+\mathbb{P}\big{(}\neg{\mathcal{A}}_{\mathrm{exp}}\big{)}.$
(5.41)

The not-yet revealed vertices after Step 3 are
${\mathcal{V}}_{n}\setminus{\mathcal{V}}_{n,3}=\Xi^{\scriptscriptstyle(\mathrm{sure})}_{n}[2w_{\mathrm{hh}},\infty)$,
and by Corollary 5.7 each vertex in
$\Xi^{\scriptscriptstyle(\mathrm{sure})}_{n}[2w_{\mathrm{hh}},\infty)$
connects by an edge to ${\mathcal{C}}_{\mathrm{bb}}$. Thus each component
${\mathcal{C}}\nsupseteq{\mathcal{C}}_{\mathrm{bb}}$ of ${\mathcal{G}}_{n,3}$
either remains the same in ${\mathcal{G}}_{n}$ or it merges with the component
containing ${\mathcal{C}}_{\mathrm{bb}}$ via a vertex in
${\mathcal{V}}\setminus{\mathcal{V}}_{n,3}$. Hence, conditionally on
${\mathcal{A}}_{\mathrm{bb}}$ and ${\mathcal{G}}_{n,3}$,

$\big{\\{}|{\mathcal{C}}^{\scriptscriptstyle(2)}_{n}|\geq
k\big{\\}}\subseteq\\{\exists\mbox{ a component }{\mathcal{C}}\mbox{ of
}{\mathcal{G}}_{n,3}\mbox{ with }|{\mathcal{C}}|\geq
k:{\mathcal{C}}\not\leftrightarrow\Xi^{\scriptscriptstyle(\mathrm{sure})}_{n}[2w_{\mathrm{hh}},\infty)\\}.$
(5.42)

By a union bound over the at most $|{\mathcal{V}}_{n,3}|/k$ components of size
at least $k$, (5.38) of Lemma 5.9 yields

$\mathbb{P}\big{(}|{\mathcal{C}}_{n}^{\scriptscriptstyle(2)}|\geq
k\mid{\mathcal{G}}_{n,3},{\mathcal{A}}_{\mathrm{exp}}\cap{\mathcal{A}}_{\mathrm{bb}}\big{)}\leq(|{\mathcal{V}}_{n,3}|/k)\exp\big{(}-c_{\ref{lemma:hh-
expandable}}s_{k}\big{)}.$

Substituting this bound into (5.41), and using Lemmas 5.2 and 5.9 to bound the
last two terms yields

$\mathbb{P}\big{(}|{\mathcal{C}}_{n}^{\scriptscriptstyle(2)}|\geq
k\big{)}\leq(\mathbb{E}\big{[}|{\mathcal{V}}_{n,3}|\big{]}/k)\exp\big{(}-c_{\ref{lemma:hh-
expandable}}s_{k}\big{)}+(3n/k)\exp\big{(}-c_{\ref{lemma:upper-hh-
bb}}s_{k}\big{)}+C_{\ref{lem:expandable-ppp}}n\exp\big{(}-c_{\ref{lemma:hh-
expandable}}^{\prime}s_{k}\big{)}.$

Since ${\mathcal{V}}_{n,3}\subseteq\Xi_{n}$ by construction,
$\mathbb{E}[|\Xi_{n}|]=n$ by (2.2), and
$s_{k}=\Theta(k^{\zeta_{\mathrm{hh}}})$ by (5.7), this finishes the proof of
Proposition 5.1 for $k\geq\max\\{k_{1},k_{2}\\}$ and
$n>k^{1+\zeta_{\mathrm{hh}}/\alpha}$. For $k<\max\\{k_{1},k_{2}\\}$, (5.1) is
trivially satisfied for $c_{\ref{prop:2nd-upper-bound-hh}}>0$ sufficiently
small. ∎

### The backbone: intermediate results

We state two corollaries of the proof of Proposition 5.1, and two propositions
based on the backbone constructions for later use. We defer the detailed
proofs to the appendix and only give a sketch here. We start with a corollary
of the proof of Proposition 5.1.

######  Corollary⋆ 5.10 (Backbone becoming part of the giant).

Consider an i-KSRG satisfying the conditions of Proposition 5.1. Then
conditionally on the graph
${\mathcal{G}}_{n,2}={\mathcal{G}}_{n}[1,2w_{\mathrm{hh}})$ satisfying
${\mathcal{A}}_{\mathrm{bb}}(n,k)$ in (5.8),

$\mathbb{P}\big{(}{\mathcal{C}}_{\mathrm{bb}}(n,k)\nsubseteq{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}\mid{\mathcal{G}}_{n,2},{\mathcal{A}}_{\mathrm{bb}}(n,k)\big{)}\leq(n/c_{\ref{eq:prop-2nd-
upper-bound}})\exp\big{(}-c_{\ref{eq:prop-2nd-upper-
bound}}k^{\zeta_{\mathrm{hh}}}\big{)}.$ (5.43)

###### Proof sketch.

By definition of the backbone in (5.8), the backbone contains at least $k$
vertices. The proof of Proposition 5.1 “merges” each component of size at
least $k$ with the backbone, with error probability given on the right-hand
side of (5.43). Hence, the component containing the backbone is the only
remaining component of size at least $k$. ∎

The next corollary follows from Lemma 5.2. It is not sharp but it yields a
useful temporary estimate.

######  Corollary 5.11 (Lower bound on largest component).

Consider a supercritical interpolating KSRG model as in Definition 2.1 with
parameters $\alpha>1$, $\tau\in(2,2+\sigma)$, $\sigma\geq 0$,
$d\in\mathbb{N}$. For each $\delta>0$, there exists a constant $A>0$ such that
for all $n$ sufficiently large

$\mathbb{P}\big{(}|{\mathcal{C}}_{n}^{(1)}|\leq
n(A\log(n))^{-1/\zeta_{\mathrm{hh}}}\big{)}\leq n^{-\delta}.$

###### Proof.

If ${\mathcal{A}}_{\mathrm{bb}}(n,k_{n})$ holds for some $k_{n}$, then the
largest component ${\mathcal{C}}_{n}^{\scriptscriptstyle(1)}$ must have at
least the size of the backbone ${\mathcal{C}}_{\mathrm{bb}}(n,k_{n})$. Setting
$k=k_{n}=(A\log(n))^{1/\zeta_{\mathrm{hh}}}$ in (5.9), the backbone exists
with probability at least $1-n^{-\delta}$ for $A=A(\delta)$ sufficiently
large, since $s_{k}=s_{k_{n}}=(C_{1}/16)A\log(n)$ by (5.7). Then its size is
at least
$(n^{\prime}/k)s_{k}\geq(n^{\prime}/k)=\Theta(nk_{n}^{-1/\zeta_{\mathrm{hh}}})$
by definition of $n^{\prime}$ in (5.2), finishing the proof. ∎

The next proposition shows that all vertices with a sufficiently high mark
(simultaneously) belong to the largest component
${\mathcal{C}}_{n}^{{\scriptscriptstyle(1)}}$ with polynomially small error
probability.

######  Proposition⋆ 5.12 (Controlling marks of non-giant vertices).

Consider an i-KSRG in the setting of Theorem 2.2(ii). For _all_ $\delta>0$,
there exists $M_{\delta}>0$ such that for
$\overline{w}(n,\delta)=(M_{\delta}\log(n))^{(1-\sigma\gamma_{\mathrm{hh}})/\zeta_{\mathrm{hh}}}$

$\mathbb{P}\big{(}\exists
u\in\Xi_{n}[\overline{w}(n,\delta),\infty):u\notin{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}\big{)}\leq
n^{-\delta}.$ (5.44)

###### Proof sketch.

We give the detailed proof in the Appendix on page A, and here a sketch. We
consider $k$ as a free parameter, so using Lemma 5.2 with
$k=k_{n}=\Theta\big{(}\log^{1/\zeta_{\mathrm{hh}}}(n)\big{)}$, a backbone
${\mathcal{C}}_{\mathrm{bb}}(n,k_{n})$ exists and satisfies
${\mathcal{C}}_{\mathrm{bb}}(n,k_{n})\subseteq{\mathcal{C}}_{n}^{{\scriptscriptstyle(1)}}$,
with probability at least $1-n^{-\delta}$ by Corollary 5.10 and Lemma 5.2 (the
same calculation as the proof of Corollary 5.11). The choice of
$\overline{w}=\overline{w}(n,\delta)$ is so that a vertex $u$ with mark
$w_{u}\geq\overline{w}(n,\delta)$ connects to each backbone-vertex in its own
subbox with probability at least $p$ in (2.3). For $u$ to not be contained in
${\mathcal{C}}_{n}^{{\scriptscriptstyle(1)}}$, these $s_{k_{n}}$ many edges
must be all absent. Then we apply concentration inequalities to obtain the
result. ∎

######  Remark 5.13.

Combined with the proof of the lower bound of Theorem 2.4 below in Section 7,
one may show that Proposition 5.12 is sharp up to a constant factor, i.e.,
there exist constants $\delta,m_{w}>0$ such that for all $n$ sufficiently
large

$\mathbb{P}\big{(}\exists
v\in\Xi_{n}[(m_{w}\log(n))^{(1-\sigma\gamma_{\mathrm{hh}})/\zeta_{\mathrm{hh}}},\infty):v\notin{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}\big{)}\geq
1-n^{-\delta}.$

We state a proposition that 0 is with constant probability in a linear-sized
component in the induced subgraph
${\mathcal{G}}_{n,2}={\mathcal{G}}_{n}[1,2w_{\mathrm{hh}}(k))$ with
$k=k_{n}=\mathrm{polylog}(n)$ and $w_{\mathrm{hh}}(k)$ from (5.7). Let
${\mathcal{C}}_{n,2}(0)$ be the component containing $0$ in
${\mathcal{G}}_{n,2}\subseteq{\mathcal{G}}_{n}$, by setting
${\mathcal{C}}_{n,2}(0)$ to be the empty set if $w_{0}\geq
2w_{\mathrm{hh}}(k)$.

######  Proposition 5.14 (Existence of a large component).

Consider a supercritical i-KSRG model as in Definition 2.1, with parameters
$\alpha>1$, $\tau\in(2,2+\sigma)$, $\sigma\geq 0$, $d\in{\mathbb{N}}$. There
exist constants $\rho,m>0$ such that for all $n$ sufficiently large, when
$k=k_{n}=(m\log(n))^{1/\zeta_{\mathrm{hh}}}$,

$\displaystyle\mathbb{P}^{\scriptscriptstyle
0}\big{(}|{\mathcal{C}}_{n,2}(0)|\geq\rho
n\big{)}\geq\rho,\qquad\mbox{and}\qquad\mathbb{P}^{\scriptscriptstyle
0}\big{(}|{\mathcal{C}}_{\infty}(0)|=\infty\big{)}\geq\rho.$ (5.45)

###### Proof sketch.

For the first inequality in (5.45), we build a connected backbone on vertices
with mark in $[w_{\mathrm{hh}}(k_{n}),2w_{\mathrm{hh}}(k_{n}))$ using Lemma
5.2. Then we use a second-moment method to show that the origin and linearly
many other vertices are connected to this backbone via paths along which the
vertex marks are increasing. The second statement follows similarly, forming
an infinite path along which the marks are increasing. The detailed proof can
be found in Appendix A.1. ∎

##  6\. Upper bound: subexponential decay

In this section we prove Theorem 2.2(ii). We carry out the plan in Section
3.1.2 in detail. Instead of arguing directly for KSRGs with parameters
described in Theorem 2.2(ii), we derive general conditions that ensure that a
bound on the size of the second-largest component as in Proposition 5.1
readily implies subexponential decay. Recall from Definition 2.1 that
$\mathbb{P}^{\scriptscriptstyle x}$ denotes the conditional measure that
${\mathcal{V}}$ contains a vertex at location $x\in\Lambda_{n}$, that has an
unknown mark from distribution $F_{W}$.

######  Proposition 6.1 (Prerequisites for subexponential decay).

Consider a supercritical i-KSRG model as in Definition 2.1, with parameters
$\alpha>1,\tau>2$ and $d\in{\mathbb{N}}$. Assume that there exist
$\zeta,\eta,c,c^{\prime},\eta,M_{c}>0$, and a function
$n_{0}(k)=O(k^{1+c^{\prime}})$, such that for all $k$ sufficiently large
constant and whenever $n\in[n_{0}(k),\infty)$, with
$\overline{w}(n,c):=M_{c}\log^{\eta}(n)$ it holds that for any
$x\in\Lambda_{n}$

$\displaystyle\mathbb{P}^{\scriptscriptstyle
x}\big{(}|{\mathcal{C}}_{n}^{\scriptscriptstyle(2)}|\geq k\big{)}$
$\displaystyle\leq n^{c^{\prime}}\exp\big{(}-ck^{\zeta}\big{)},$ (6.1)
$\displaystyle\mathbb{P}^{\scriptscriptstyle
x}\big{(}|{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}|\leq n^{c}\big{)}$
$\displaystyle\leq n^{-1-c},$ (6.2)
$\displaystyle\mathbb{P}^{\scriptscriptstyle x}\big{(}\exists
v\in\Xi_{n}[\overline{w}(n,c),\infty):v\notin{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}\big{)}$
$\displaystyle\leq n^{-c}.$ (6.3)

Then there exists a constant $A>0$ such that for all $k$ sufficiently large
constant and $n$ satisfying $n\in[n_{0}(k),\infty]$,

$\mathbb{P}^{\scriptscriptstyle 0}\big{(}|{\mathcal{C}}_{n}(0)|\geq
k,0\notin{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}\big{)}\leq
A\exp\big{(}-(1/A)k^{\zeta}\big{)},$ (6.4)

and

$\frac{|{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}|}{n}\overset{\mathbb{P}}{\longrightarrow}\mathbb{P}\big{(}|{\mathcal{C}}_{\infty}(0)|=\infty\big{)},\qquad\mbox{as
$n\to\infty$.}$ (6.5)

Observe that (6.4) does not follow from a naive application of (6.1), since we
sharpened the polynomial prefactor on the right-hand side of (6.1) to a
universal constant $A$ in (6.4), and $n=\infty$ is also allowed in (6.4). The
inequalities (6.1)–(6.3) are satisfied when $\tau\in(2,2+\sigma)$ and
$\sigma\leq\tau-1$ by Propositions 5.1 and 5.12 and Corollary 5.11 (we leave
it to the reader to verify that the results hold also for the Palm-version
$\mathbb{P}^{\scriptscriptstyle x}$ of $\mathbb{P}$). Thus, Theorem 2.2(ii)
follows immediately after we prove Proposition 6.1. We state and prove an
intermediate claim that we need for Proposition 6.1. We write
${\mathcal{C}}_{{\mathcal{Q}}}^{\scriptscriptstyle(1)}$ for the largest
component in the graph induced on vertices in
${\mathcal{Q}}\subseteq{\mathbb{R}}^{d}$.

######  Claim 6.2 (Leaving the giant).

Consider a supercritical i-KSRG model as in Definition 2.1, under the same
setting as Proposition 6.1. Assume that (6.1)–(6.3) hold, $n$ is sufficiently
large and let $N\in[n,\infty]$. Then there exists $\delta>0$ such that for any
box ${\mathcal{Q}}_{n}\subseteq\Lambda_{N}$ with
$\mathrm{Vol}({\mathcal{Q}}_{n})=n$, any $x\in{\mathcal{Q}}_{n}$, $n$
sufficiently large, and all $N\geq n$, it holds for $u:=(x,w_{u})$ that

$\mathbb{P}^{\scriptscriptstyle
x}\big{(}u\in{\mathcal{C}}_{{\mathcal{Q}}_{n}}^{\scriptscriptstyle(1)},u\notin{\mathcal{C}}_{N}^{\scriptscriptstyle(1)}\big{)}\leq
n^{-\delta}.$ (6.6)

###### Proof.

We will first prove the following bound that holds generally for a sequence of
increasing graphs $G_{n}\subseteq G_{n+1}\subseteq\dots$, whose largest and
second-largest components we denote by $C_{n}^{\scriptscriptstyle(1)}$ and
$C_{n}^{\scriptscriptstyle(2)}$, respectively. Let $(k_{n})_{n\geq 0}$ and
$(K_{n})_{n\geq 0}$ be two non-negative sequences such that $k_{n+1}<K_{n}$
for all $n\geq 0$. Then, for all $0<n\leq N\leq\infty$,

$\mathbb{P}\big{(}u\in C_{n}^{\scriptscriptstyle(1)},u\notin
C_{N}^{\scriptscriptstyle(1)}\big{)}\leq\sum_{\tilde{n}=n}^{N}\Big{(}\mathbb{P}\big{(}|C_{\tilde{n}}^{\scriptscriptstyle(1)}|<K_{\tilde{n}}\big{)}+\mathbb{P}\big{(}|C_{\tilde{n}}^{\scriptscriptstyle(2)}|>k_{\tilde{n}}\big{)}\Big{)}.$
(6.7)

We verify the bound using an inductive argument. We define for $\tilde{n}\geq
n$ the events

${\mathcal{A}}(\tilde{n}):=\big{\\{}|C_{\tilde{n}}^{\scriptscriptstyle(1)}|\geq
K_{\tilde{n}}\big{\\}}\cap\big{\\{}|C_{\tilde{n}+1}^{\scriptscriptstyle(2)}|\leq
k_{\tilde{n}+1}\big{\\}}.$

Since by assumption $k_{\tilde{n}+1}<K_{\tilde{n}}$, the event
${\mathcal{A}}(\tilde{n})$ ensures that
$|C_{\tilde{n}}^{\scriptscriptstyle(1)}|$ is already larger than
$|C_{\tilde{n}+1}^{\scriptscriptstyle(2)}|$, hence ${\mathcal{A}}(\tilde{n})$
implies that $C_{\tilde{n}}^{\scriptscriptstyle(1)}\subseteq
C_{\tilde{n}+1}^{\scriptscriptstyle(1)}$. Iteratively applying this argument
yields that $\cap_{\tilde{n}\in[n,N]}{\mathcal{A}}(\tilde{n})$ implies that
$\\{C_{n}^{\scriptscriptstyle(1)}\subseteq C_{N}^{\scriptscriptstyle(1)}\\}$.
We combine this with the observation that given that $u\in V_{n}$, it holds
that $\\{u\in C_{n}^{\scriptscriptstyle(1)},u\notin
C_{N}^{\scriptscriptstyle(1)}\\}\subseteq\\{C_{n}^{\scriptscriptstyle(1)}\nsubseteq
C_{N}^{\scriptscriptstyle(1)}\\}$. This yields

$\displaystyle\mathbb{P}\big{(}u\in C_{n}^{\scriptscriptstyle(1)},u\notin
C_{N}^{\scriptscriptstyle(1)}\big{)}\leq\mathbb{P}\big{(}C_{n}^{\scriptscriptstyle(1)}\nsubseteq
C_{N}^{\scriptscriptstyle(1)}\big{)}$
$\displaystyle\leq\mathbb{P}\bigg{(}\\{C_{n}^{\scriptscriptstyle(1)}\nsubseteq
C_{N}^{\scriptscriptstyle(1)}\\}\cap\bigcap_{\tilde{n}=n}^{N-1}{\mathcal{A}}(\tilde{n})\bigg{)}+\sum_{\tilde{n}=n}^{N-1}\mathbb{P}\big{(}\neg{\mathcal{A}}(\tilde{n})\big{)}$
(6.8) $\displaystyle\leq
0+\sum_{\tilde{n}=n}^{N}\Big{(}\mathbb{P}\big{(}|C_{\tilde{n}}^{\scriptscriptstyle(1)}|<K_{\tilde{n}}\big{)}+\mathbb{P}\big{(}|C_{\tilde{n}}^{\scriptscriptstyle(2)}|>k_{\tilde{n}}\big{)}\Big{)},$

showing (6.7). We move on to (6.6) for which we have to define the increasing
sequence of graphs. Consider any sequence of boxes
$({\mathcal{Q}}_{\tilde{n}})_{\tilde{n}\geq n}$ such that
${\mathcal{Q}}_{n}\subseteq{\mathcal{Q}}_{n+1}\subseteq\dots\subseteq{\mathcal{Q}}_{N}:=\Lambda_{N}$
and $\mathrm{Vol}({\mathcal{Q}}_{\tilde{n}})=\tilde{n}$, and let
$G_{\tilde{n}}$ denote the induced subgraph of ${\mathcal{G}}$ on
${\mathcal{Q}}_{\tilde{n}}$ for $\tilde{n}\in[n,N]$, that is conditioned to
contain a vertex at location $x\in{\mathcal{Q}}_{n}$. Using the assumed lower
bound on $|\mathcal{C}_{\tilde{n}}^{\scriptscriptstyle(1)}|$ in (6.2), we set
$K_{\tilde{n}}:=\tilde{n}^{c}$, and using the assumed upper bound on
$|{\mathcal{C}}_{\tilde{n}}^{\scriptscriptstyle(2)}|$ in (6.1), there exists a
large constant $A$ so that whenever $n$ is sufficiently large, and
$\tilde{n}\geq n$, by setting
$k_{\tilde{n}}:=(A\log(\tilde{n}))^{1/\zeta_{\mathrm{hh}}}$, it holds that

$\mathbb{P}^{\scriptscriptstyle
x}\big{(}|{\mathcal{C}}_{{\mathcal{Q}}_{\tilde{n}}}^{\scriptscriptstyle(1)}|<K_{\tilde{n}}\big{)}\leq\tilde{n}^{-c-1},\qquad\mathbb{P}^{\scriptscriptstyle
x}\big{(}|{\mathcal{C}}_{{\mathcal{Q}}_{\tilde{n}}}^{\scriptscriptstyle(2)}|>k_{\tilde{n}}\big{)}\leq\tilde{n}^{-c-1}.$

Clearly $k_{\tilde{n}+1}<K_{\tilde{n}}$ for $\tilde{n}\geq n$, so that
substituting the bounds into (6.8) and summing over $\tilde{n}\geq n$ yields
the assertion (6.6) for any $\delta<c$ and $n$ sufficiently large. ∎

We continue to prove Proposition 6.1, starting with some notation. For some
$\varepsilon\in(0,\min\\{\alpha-1,\tau-2\\})$, and using
$c,c^{\prime},\eta,\zeta$, and $\overline{w}(n,c)$ from the statement of
Proposition 6.1

$\displaystyle N_{k}$ $\displaystyle:=\exp(k^{\zeta}c/(2c^{\prime})),$
$\displaystyle n_{k}$ $\displaystyle:=N_{k}^{\varepsilon},$
$\displaystyle\overline{w}_{N_{k}}$
$\displaystyle:=\overline{w}(N_{k},c)=M_{c}(k^{\zeta})^{\eta}(\tfrac{c}{2c^{\prime}})^{\eta},$
$\displaystyle t_{k}$ $\displaystyle:=n_{k}^{1/d}/(2k).$ (6.9)

Note that $N_{k},n_{k}=\exp(\Theta(k^{\zeta}))$. For $n\leq N_{k}$, the
statement (6.4) follows directly from (6.1), since when $n=N_{k}$, the right-
hand side of (6.1) becomes $\exp(-k^{\zeta}c/2)$, so we may set any $A$ such
that $1/A\leq c/2$ in (6.4). So we may assume in the remainder of the section
that $n>N_{k}$. We write for ${\mathcal{C}}_{\Lambda(x,n)}(u)$ the component
of vertex $u:=(x,w_{u})\in{\mathcal{V}}$ in the graph ${\mathcal{G}}_{\infty}$
restricted to $\Lambda(x,n)$. Define for $x\in{\mathbb{R}}^{d}$ the two events

$\displaystyle{\mathcal{A}}_{\mathrm{low-
edge}}(x,n_{k},N_{k},\overline{w}_{N_{k}})$
$\displaystyle:=\left\\{\begin{aligned}
&|{\mathcal{C}}_{\Lambda(x,n_{k})}(u)|<k,\exists
v_{1},v_{2}\in\Xi_{\Lambda(x,N_{k})}[1,\overline{w}_{N_{k}})\mbox{ such that
}\\\ &v_{1}\in{\mathcal{C}}_{\Lambda(x,n_{k})}(u),\|x_{v_{1}}-x_{v_{2}}\|\geq
t_{k},\mbox{ and }v_{1}\leftrightarrow v_{2}\end{aligned}\right\\},$ (6.10)
$\displaystyle{\mathcal{A}}_{\mathrm{long-
edge}}(x,n_{k},N_{k},\overline{w}_{N_{k}})$ $\displaystyle:=\big{\\{}\exists
v_{1}\in\Xi_{\Lambda(x,n_{k})}[1,\overline{w}_{N_{k}}),\exists
v_{2}\in\Xi\setminus\Xi_{\Lambda(x,N_{k})},v_{1}\leftrightarrow
v_{2}\big{\\}}.$ (6.11)

The next lemma relates the probability of the event
$\\{|{\mathcal{C}}_{n}(u)|\geq
k,u\notin{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}\\}$ to the events
${\mathcal{A}}_{\mathrm{low-edge}}$ and ${\mathcal{A}}_{\mathrm{long-edge}}$
using the assumed bounds in Proposition 6.1.

######  Lemma 6.3 (Extending the box-sizes).

Consider a supercritical i-KSRG model as in Definition 2.1, under the same
setting as Proposition 6.1. Assume that (6.1)–(6.3) hold. Then there exists
$A^{\prime}>0$ such that for all $n$ with $n\in[N_{k},\infty]$, whenever
$u=(x,w_{u})\in{\mathcal{V}}$ is a vertex with $\|x-\partial\Lambda_{n}\|\geq
N_{k}^{1/d}/2$,

$\displaystyle\mathbb{P}^{\scriptscriptstyle x}\big{(}|{\mathcal{C}}_{n}(u)|$
$\displaystyle\geq k,u\notin{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}\big{)}$
(6.12) $\displaystyle\leq
A^{\prime}\exp\big{(}-(1/A^{\prime})k^{\zeta}\big{)}+\mathbb{P}^{\scriptscriptstyle
0}\big{(}{\mathcal{A}}_{\mathrm{low-
edge}}(0,n_{k},N_{k},\overline{w}_{N_{k}})\big{)}+\mathbb{P}^{\scriptscriptstyle
0}\big{(}{\mathcal{A}}_{\mathrm{long-
edge}}(0,n_{k},N_{k},\overline{w}_{N_{k}})\big{)}.$

###### Proof.

Let $\tilde{n}\geq 1$, and denote by
${\mathcal{C}}_{\Lambda(x,\tilde{n})}^{\scriptscriptstyle(1)}$ the largest
connected component in the induced subgraph of ${\mathcal{G}}_{\infty}$ inside
the box $\Lambda(x,\tilde{n})\subseteq\Lambda_{n}$. For a vertex
$u=(x,w_{u})\in{\mathcal{V}}$ define

$\displaystyle{\mathcal{A}}_{\mathrm{leave-giant}}(x,\tilde{n})$
$\displaystyle:=\\{u\in{\mathcal{C}}_{\Lambda(x,\tilde{n})}^{\scriptscriptstyle(1)},u\notin{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}\\},$
(6.13) $\displaystyle{\mathcal{A}}_{\mathrm{mark-
giant}}(x,N_{k},\overline{w}_{N_{k}})$ $\displaystyle:=\big{\\{}\forall
v\in\Xi_{\Lambda(x,N_{k})}[\overline{w}_{N_{k}},\infty):v\in{\mathcal{C}}_{\Lambda(x,N_{k})}^{\scriptscriptstyle(1)}\big{\\}}.$
(6.14)

The first events relates to (6.6) in Claim 6.2, while the second one to (6.3)
of Proposition 6.1. The values of $n_{k}<N_{k}\leq n$ from (6.9) and that we
assumed $\|x-\partial\Lambda_{n}\|\geq N_{k}^{1/d}/2$ ensure that
$\Lambda(x,n_{k})\\!\subseteq\\!\Lambda(x,N_{k})\\!\subseteq\\!\Lambda_{n}$.
Then we bound

$\displaystyle\big{\\{}|{\mathcal{C}}_{n}(u)|\geq k$
$\displaystyle,u\notin{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}\big{\\}}$
$\displaystyle\subseteq\big{\\{}|{\mathcal{C}}_{n}(u)|\geq
k,u\notin{\mathcal{C}}_{n}^{\scriptscriptstyle(1)},u\notin{\mathcal{C}}_{\Lambda(x,n_{k})}^{\scriptscriptstyle(1)},|{\mathcal{C}}_{\Lambda(x,n_{k})}^{\scriptscriptstyle(2)}|\geq
k\big{\\}}\,\cup\,\big{\\{}u\in{\mathcal{C}}_{\Lambda(x,n_{k})}^{\scriptscriptstyle(1)},u\notin{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}\big{\\}}$
$\displaystyle\hskip 15.0pt\cup\big{\\{}|{\mathcal{C}}_{n}(u)|\geq
k,u\notin{\mathcal{C}}_{n}^{\scriptscriptstyle(1)},u\notin{\mathcal{C}}_{\Lambda(x,n_{k})}^{\scriptscriptstyle(1)},|{\mathcal{C}}_{\Lambda(x,n_{k})}^{\scriptscriptstyle(2)}|<k\big{\\}}$
$\displaystyle\begin{aligned}
&\subseteq\big{\\{}|{\mathcal{C}}_{\Lambda(x,n_{k})}^{\scriptscriptstyle(2)}|\geq
k\big{\\}}\,\cup\,{\mathcal{A}}_{\mathrm{leave-giant}}(x,n_{k})\\\ &\hskip
15.0pt\cup\,\big{\\{}|{\mathcal{C}}_{n}(u)|\geq
k,u\notin{\mathcal{C}}_{n}^{\scriptscriptstyle(1)},u\notin{\mathcal{C}}_{\Lambda(x,n_{k})}^{\scriptscriptstyle(1)},|{\mathcal{C}}_{\Lambda(x,n_{k})}(u)|<k\big{\\}}.\end{aligned}$
(6.15)

Applying probabilities on both sides we obtain the inequality stated in (3.6)
for $x\\!=\\!0$. We introduce a shorthand notation for the third event on the
right-hand side of (6.15), i.e.,

${\mathcal{A}}_{\mathrm{goal}}:=\big{\\{}|{\mathcal{C}}_{n}(u)|\geq
k,u\notin{\mathcal{C}}_{n}^{\scriptscriptstyle(1)},u\notin{\mathcal{C}}_{\Lambda(x,n_{k})}^{\scriptscriptstyle(1)},|{\mathcal{C}}_{\Lambda(x,n_{k})}(u)|<k\big{\\}}$

Define the auxiliary events

$\displaystyle{\mathcal{A}}_{\mathrm{becomes-large}}$
$\displaystyle:=\\{|{\mathcal{C}}_{\Lambda(x,n_{k})}(u)|<k,|{\mathcal{C}}_{n}(u)|\geq
k\\},$ (6.16) $\displaystyle{\mathcal{A}}_{\mathrm{outof-giant}}(n_{k},n)$
$\displaystyle:=\\{u\notin{\mathcal{C}}_{n}^{\scriptscriptstyle(1)},u\notin{\mathcal{C}}_{\Lambda(x,n_{k})}^{\scriptscriptstyle(1)}\\},$

and observe that
${\mathcal{A}}_{\mathrm{goal}}={\mathcal{A}}_{\mathrm{becomes-
large}}\cap{\mathcal{A}}_{\mathrm{outof-giant}}(n_{k},n).$ In order to bound
$\mathbb{P}({\mathcal{A}}_{\mathrm{goal}})$, we distinguish whether $u$ enters
the giant at the intermediate box of size $N_{k}\in(n_{k},n)$ or not:

$\displaystyle{\mathcal{A}}_{\mathrm{goal}}$
$\displaystyle\subseteq\\{u\in{\mathcal{C}}_{\Lambda(x,N_{k})}^{\scriptscriptstyle(1)},u\notin{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}\\}\cup\\{|{\mathcal{C}}_{n}(u)|\geq
k,u\notin{\mathcal{C}}_{\Lambda(x,N_{k})}^{\scriptscriptstyle(1)},u\notin{\mathcal{C}}_{\Lambda(x,n_{k})}^{\scriptscriptstyle(1)},|{\mathcal{C}}_{\Lambda(x,n_{k})}(u)|<k\\}$
$\displaystyle={\mathcal{A}}_{\mathrm{leave-
giant}}(x,N_{k})\cup\big{(}{\mathcal{A}}_{\mathrm{becomes-
large}}\cap{\mathcal{A}}_{\mathrm{outof-giant}}(n_{k},N_{k})\big{)},$ (6.17)

with ${\mathcal{A}}_{\mathrm{leave-giant}}(x,\tilde{n})$ defined in (6.14). We
observe that ${\mathcal{A}}_{\mathrm{becomes-large}}$ in (6.16) implies that
at least one of the at most $k-1$ vertices in
${\mathcal{C}}_{\Lambda(x,n_{k})}(u)$ has an incident edge crossing the
boundary of $\Lambda(x,n_{k})$, and so there must exist a “fairly” long edge
either inside the cluster ${\mathcal{C}}_{\Lambda(x,n_{k})}(u)$ or between a
vertex in ${\mathcal{C}}_{\Lambda(x,n_{k})}(u)$ and a vertex in
${\mathcal{C}}_{n}(u)\setminus{\mathcal{C}}_{\Lambda(x,n_{k})}(u)$. More
precisely, recalling $t_{k}=n_{k}^{1/d}/(2k)$, define

${\mathcal{A}}_{\mathrm{edge}}:=\big{\\{}|{\mathcal{C}}_{\Lambda(x,n_{k})}(u)|<k,\exists
v_{1}\in{\mathcal{C}}_{\Lambda(x,n_{k})}(u),v_{2}\in\Xi:v_{1}\leftrightarrow
v_{2},\|v_{1}-v_{2}\|\geq t_{k}\big{\\}}.$ (6.18)

We argue that ${\mathcal{A}}_{\mathrm{becomes-
large}}\subseteq{\mathcal{A}}_{\mathrm{edge}}$. By the pigeon-hole principle,
if all edges adjacent to all vertices in ${\mathcal{C}}_{\Lambda(x,n_{k})}(u)$
were shorter than $t_{k}$, the furthest point that could be reached from $x$
with at most $k-1$ edges has Euclidean norm at most
$(k-1)t_{k}<n_{k}^{1/d}/2$, and thus its location would be inside
$\Lambda(x,n_{k})$, contradicting the definition of
${\mathcal{A}}_{\mathrm{becomes-large}}$ in (6.16). Returning to (6.17), we
obtain that

$\displaystyle{\mathcal{A}}_{\mathrm{goal}}$
$\displaystyle\subseteq{\mathcal{A}}_{\mathrm{leave-
giant}}(x,N_{k})\cup\big{(}{\mathcal{A}}_{\mathrm{edge}}\cap{\mathcal{A}}_{\mathrm{outof-
giant}}(n_{k},N_{k})\big{)}$
$\displaystyle\subseteq{\mathcal{A}}_{\mathrm{leave-
giant}}(x,N_{k})\cup\big{(}{\mathcal{A}}_{\mathrm{edge}}\cap\\{u\notin{\mathcal{C}}^{\scriptscriptstyle(1)}_{\Lambda(x,N_{k})}\\}\big{)}.$

In order to bound the existence of long edges, we put restrictions on the
marks: we distinguish whether all vertices in
$\Xi_{\Lambda(x,N_{k})}\setminus{\mathcal{C}}_{\Lambda(x,N_{k})}^{\scriptscriptstyle(1)}$
have mark at most $\overline{w}_{N_{k}}$ – this is the event
${\mathcal{A}}_{\mathrm{mark-giant}}(x,N_{k},\overline{w}_{N_{k}})$ in (6.14)
– or not. We obtain

$\displaystyle{\mathcal{A}}_{\mathrm{goal}}$
$\displaystyle\subseteq{\mathcal{A}}_{\mathrm{leave-
giant}}(x,N_{k})\cup\big{(}\neg{\mathcal{A}}_{\mathrm{mark-
giant}}(x,N_{k},\overline{w})\big{)}$ (6.19) $\displaystyle\hskip
10.0pt\cup\big{(}{\mathcal{A}}_{\mathrm{edge}}\cap\\{u\notin{\mathcal{C}}^{\scriptscriptstyle(1)}_{\Lambda(x,N_{k})}\\}\cap{\mathcal{A}}_{\mathrm{mark-
giant}}(x,N_{k},\overline{w})\big{)}.$

The intersection with
$\\{u\notin{\mathcal{C}}^{\scriptscriptstyle(1)}_{\Lambda(x,N_{k})}\\}\cap{\mathcal{A}}_{\mathrm{mark-
giant}}(x,N_{k},\overline{w}_{N_{k}})$ in the last event ensures that _all_
vertices in the cluster of $u$ with location in
$\Lambda(x,N_{k})\supseteq\Lambda(x,n_{k})$ have mark at most
$\overline{w}_{N_{k}}$. We make another case distinction, with respect to the
locations of the vertices the edge $e_{\geq t_{k}}$ of length at least $t_{k}$
that exists on the event ${\mathcal{A}}_{\mathrm{edge}}$ in (6.18). Namely,
$e_{\geq t_{k}}$ either has both endpoints in $\Lambda_{N_{k}}$ or it has one
endpoint inside $\Lambda_{n_{k}}$ and the other one outside $\Lambda_{N_{k}}$.
For the first event, we obtain the event ${\mathcal{A}}_{\mathrm{low-
edge}}(x,n_{k},N_{k},\overline{w}_{k})$, and for the latter
${\mathcal{A}}_{\mathrm{long-edge}}(x,n_{k},N_{k},\overline{w}_{k})$,
respectively (defined in (6.10)—(6.11)). Hence,

${\mathcal{A}}_{\mathrm{edge}}\cap\\{u\notin{\mathcal{C}}^{\scriptscriptstyle(1)}_{\Lambda(x,N_{k})}\\}\cap{\mathcal{A}}_{\mathrm{mark-
giant}}(x,N_{k},\overline{w}_{N_{k}})\subseteq{\mathcal{A}}_{\mathrm{low-
edge}}(x,n_{k},N_{k},\overline{w}_{k})\cup{\mathcal{A}}_{\mathrm{long-
edge}}(x,n_{k},N_{k},\overline{w}_{k}).$

Using this in (6.19), then substituting (6.19) back into (6.15), and then
taking probabilities yields

$\displaystyle\mathbb{P}^{\scriptscriptstyle
x}\big{(}|{\mathcal{C}}_{n}(u)|\geq k$
$\displaystyle,u\notin{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}\big{)}$
$\displaystyle\leq\mathbb{P}^{\scriptscriptstyle
x}\big{(}|{\mathcal{C}}_{\Lambda(x,n_{k})}^{\scriptscriptstyle(2)}|\geq
k\big{)}+\mathbb{P}^{\scriptscriptstyle
x}\big{(}\neg{\mathcal{A}}_{\mathrm{mark-
giant}}(x,N_{k},\overline{w})\big{)}+\sum_{\tilde{n}\in\\{n_{k},N_{k}\\}}\mathbb{P}^{\scriptscriptstyle
x}\big{(}{\mathcal{A}}_{\mathrm{leave-giant}}(x,\tilde{n})\big{)}$
$\displaystyle\hskip 15.0pt+\mathbb{P}^{\scriptscriptstyle
x}\big{(}{\mathcal{A}}_{\mathrm{low-
edge}}(x,n_{k},N_{k},\overline{w})\big{)}+\mathbb{P}^{\scriptscriptstyle
x}\big{(}{\mathcal{A}}_{\mathrm{long-
edge}}(x,n_{k},N_{k},\overline{w})\big{)}.$

The event ${\mathcal{A}}_{\mathrm{leave-
giant}}(x,\tilde{n})\\!=\\!\\{u\\!\in\\!{\mathcal{C}}_{\Lambda(x,\tilde{n})}^{\scriptscriptstyle(1)},u\\!\notin\\!{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}\\}$
considers the graph in the box $\Lambda_{n}$ (which is centered at the origin)
and therefore does not necessarily have the same probability for all
$x\in\Lambda_{n}$. The four other events consider the graph in boxes centered
at $x$. Hence, we translate those events (and the Palm measure
$\mathbb{P}^{\scriptscriptstyle x}$) by $-x$ to obtain

$\displaystyle\mathbb{P}^{\scriptscriptstyle
x}\big{(}|{\mathcal{C}}_{n}(u)|\geq k$
$\displaystyle,u\notin{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}\big{)}$
$\displaystyle\leq\mathbb{P}^{\scriptscriptstyle
0}\big{(}|{\mathcal{C}}_{n_{k}}^{\scriptscriptstyle(2)}|\geq
k\big{)}+\mathbb{P}^{\scriptscriptstyle
0}\big{(}\neg{\mathcal{A}}_{\mathrm{mark-
giant}}(0,N_{k},\overline{w})\big{)}+\sum_{\tilde{n}\in\\{n_{k},N_{k}\\}}\mathbb{P}^{\scriptscriptstyle
x}\big{(}u\in{\mathcal{C}}_{\Lambda(x,\tilde{n})}^{\scriptscriptstyle(1)},u\notin{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}\big{)}$
$\displaystyle\hskip 15.0pt+\mathbb{P}^{\scriptscriptstyle
0}\big{(}{\mathcal{A}}_{\mathrm{low-
edge}}(0,n_{k},N_{k},\overline{w})\big{)}+\mathbb{P}^{\scriptscriptstyle
0}\big{(}{\mathcal{A}}_{\mathrm{long-
edge}}(0,n_{k},N_{k},\overline{w})\big{)},$

The first two terms can be bounded by substituting the definitions
$n_{k},N_{k}=\exp\big{(}\Theta(k^{\zeta})\big{)}$ in (6.9) into the assumed
bounds on the probabilities in Proposition 6.1. The terms in the sum are
bounded from above by $n_{k}^{-\delta}=\exp\big{(}-\Theta(k^{\zeta})\big{)}$
by Claim 6.2. This finishes the proof of (6.12). ∎

We move on to bounding $\mathbb{P}^{\scriptscriptstyle
0}({\mathcal{A}}_{\mathrm{low-edge}})$ on the right-hand side of (6.12) in
Proposition 6.3, with ${\mathcal{A}}_{\mathrm{low-edge}}$ from (6.10). To do
so, we need an auxiliary claim that controls the probability that for every
point in $\Xi_{N_{k}}[1,\overline{w}_{N_{k}})$ there are not “too many” points
at distance at least $t_{k}$, with $t_{k}=n_{k}^{1/d}/(2k)$. We define first
for $i\geq 1$, and $u=(x_{u},w_{u})\in\Xi_{n_{k}}[0,\overline{w})$ the annuli

${\mathcal{R}}_{i}(x_{u}):=\big{(}\Lambda\big{(}x_{u},(2^{i}t_{k})^{d}\big{)}\setminus\Lambda\big{(}x_{u},(2^{i-1}t_{k})^{d}\big{)}\big{)}\times[1,\infty).$
(6.20)

With the measure $\mu_{\tau}$ from (2.2), we then define the _bad_ events

$\displaystyle{\mathcal{A}}_{\mathrm{dense}}$ $\displaystyle:=\big{\\{}\exists
i\geq
1,u\in\Xi_{n_{k}}[1,\overline{w}_{N_{k}}):|\Xi_{N_{k}}\cap{\mathcal{R}}_{i}(x_{u})|>2\cdot\mu_{\tau}\big{(}{\mathcal{R}}_{i}(x_{u})\big{)}\big{\\}}.$
(6.21)

In the following auxiliary claim we give an upper bound on
$\mathbb{P}\big{(}{\mathcal{A}}_{\mathrm{dense}}\big{)}$. Its proof is
standard, based on Palm theory and Chernoff bounds, see page B of Appendix B.

######  Claim 6.4.

For all $c,\delta>0$, there exists $n_{0}$ such that for all
$n_{k}\geq\max\\{n_{0},k^{d+\delta}\\}$, $\mathbb{P}^{\scriptscriptstyle
0}\big{(}{\mathcal{A}}_{\mathrm{dense}}\big{)}\leq n_{k}^{-c}$.

We can now analyze $\mathbb{P}^{\scriptscriptstyle
0}({\mathcal{A}}_{\mathrm{low-edge}})$ in Lemma 6.3. The next claim holds for
the specific choices of $n_{k},N_{k},\overline{w}_{N_{k}}$ and $t_{k}$ in
(6.9) for all $k$ is sufficiently large, but extends to more general settings,
e.g. KSRGs defined on any vertex set that satisfies
$\neg{\mathcal{A}}_{\mathrm{dense}}$ in (6.21).

######  Claim 6.5 (No _low_ -mark edge from a small component).

Consider an i-KSRG that satisfies the conditions in Proposition 6.1. For any
$\varepsilon\in(0,\min\\{\alpha-1,\tau-2\\})$ in (6.9), there exists a
constant $A^{\prime}>0$, such that for $k$ sufficiently large

$\mathbb{P}^{\scriptscriptstyle 0}\big{(}{\mathcal{A}}_{\mathrm{low-
edge}}(0,n_{k},N_{k},\overline{w})\mid\neg{\mathcal{A}}_{\mathrm{dense}}\big{)}\leq\begin{dcases}A^{\prime}\exp(-(1/A^{\prime})k^{\zeta}),&\text{if
}\alpha<\infty,\\\ 0,&\text{if }\alpha=\infty.\end{dcases}$ (6.22)

###### Proof of Claim 6.5.

Assume first $\alpha=\infty$. The event ${\mathcal{A}}_{\mathrm{low-
edge}}(0,n_{k},N_{k},\overline{w}_{N_{k}})$ is by definition in (6.10)
restricted to vertices of mark at most
$\overline{w}_{N_{k}}=\overline{w}(N_{k},c)$ in (6.9). We write $t_{k}$ in
(6.9) as $t_{k}=\exp((\varepsilon/d)k^{\zeta})/(2k)$, which is larger than
$\beta\overline{w}_{N_{k}}^{1+\sigma}=\beta
M_{c}^{1+\sigma}((c/(2c^{\prime}))k^{\zeta})^{\eta(\sigma+1)}$ for $k$
sufficiently large. Hence, the indicator in $\mathrm{p}(u,v)$ is then $0$ by
(2.3), so a connection between $u,v$ can not occur.

Assume $\alpha<\infty$. To obtain an upper bound on the left-hand side of
(6.22), we condition on the full realization $\Xi$ satisfying the event
$\neg{\mathcal{A}}_{\mathrm{dense}}$ containing $0$:

$\mathbb{P}^{\scriptscriptstyle 0}\big{(}{\mathcal{A}}_{\mathrm{low-
edge}}(0,n_{k},N_{k},\overline{w})\mid\neg{\mathcal{A}}_{\mathrm{dense}}\big{)}=\mathbb{E}^{\scriptscriptstyle
0}\big{[}\mathbb{P}^{\scriptscriptstyle 0}\big{(}{\mathcal{A}}_{\mathrm{low-
edge}}(0,n_{k},N_{k},\overline{w}_{N_{k}})\mid\Xi,\neg{\mathcal{A}}_{\mathrm{dense}}\big{)}\big{]}.$
(6.23)

Let us denote the subgraph of ${\mathcal{G}}_{n_{k}}$ with edges of length at
most $t_{k}=n_{k}^{1/d}/(2k)$ by ${\mathcal{G}}_{n_{k}}(\leq t_{k})$ and write
${\mathcal{C}}_{n_{k}}(0,\leq t_{k})$ for the component in this graph
containing the origin. Clearly,

$\left\\{\begin{aligned} &|{\mathcal{C}}_{n_{k}}(0)|<k,\exists
v_{1},v_{2}\in\Xi_{N_{k}}[1,\overline{w}_{N_{k}})\mbox{ s.t. }\\\
&v_{1}\in{\mathcal{C}}_{n_{k}}(0),\|x_{v_{1}}-x_{v_{2}}\|\geq t_{k},\mbox{ and
}v_{1}\leftrightarrow
v_{2}\end{aligned}\right\\}\subseteq\left\\{\begin{aligned}
&|{\mathcal{C}}_{n_{k}}(0,\leq t_{k})|<k,\exists
v_{1},v_{2}\in\Xi_{N_{k}}[1,\overline{w}_{N_{k}})\mbox{ s.t. }\\\
&v_{1}\in{\mathcal{C}}_{n_{k}}(0,\leq t_{k}),\|x_{v_{1}}-x_{v_{2}}\|\geq
t_{k},\mbox{ and }v_{1}\leftrightarrow v_{2}\end{aligned}\right\\},$

where the left-hand side is the definition of ${\mathcal{A}}_{\mathrm{low-
edge}}$ in (6.10). Conditionally on $\Xi$, all edges of length at least
$t_{k}$ are present independently of edges shorter than $t_{k}$. We obtain by
a union bound over all vertices in
$\Xi_{n_{k}}[1,\overline{w}_{N_{k}})\subseteq\Xi$,

$\displaystyle\mathbb{P}^{\scriptscriptstyle 0}$
$\displaystyle\big{(}\neg{\mathcal{A}}_{\mathrm{low-
edge}}(0,n_{k},N_{k},\overline{w}_{N_{k}})\mid\Xi,\neg{\mathcal{A}}_{\mathrm{dense}}\big{)}$
$\displaystyle\leq\sum_{v_{1}\in\Xi_{n_{k}}[1,\overline{w}_{N_{k}})}\mathbb{P}^{\scriptscriptstyle
0}\big{(}v_{1}\in{\mathcal{C}}_{n_{k}}(0,\leq
t_{k}),|{\mathcal{C}}_{n_{k}}(0,\leq
t_{k})|<k\mid\Xi,\neg{\mathcal{A}}_{\mathrm{dense}}\big{)}\underbrace{\sum_{\begin{subarray}{c}v_{2}\in\Xi_{N_{k}}[1,\overline{w}_{N_{k}}):\\\
\|x_{v_{1}}-x_{v_{2}}\|\geq
t_{k}\end{subarray}}\mathrm{p}(v_{1},v_{2})}_{:=T(v_{1})}.$ (6.24)

Using the definitions of $\mathrm{p}$ in (2.3), $\kappa_{1,\sigma}$ from
(1.10), the upper bound on $|{\mathcal{R}}_{i}(u)|$ in
${\mathcal{A}}_{\mathrm{dense}}$ in (6.21), mark bounds
$w_{v_{1}},w_{v_{2}}\leq\overline{w}_{N_{k}}$, and the distance bound
$\|x_{v_{1}}-x_{v_{2}}\|\geq 2^{(i-1)}t_{k}$ when
$x_{v_{2}}\in{\mathcal{R}}_{i}(x_{u})$ in (6.20), the following bound holds
uniformly for all $v_{1}\in\Xi_{N_{k}}[1,\overline{w}_{k})$:

$\displaystyle T(v_{1})$ $\displaystyle\leq\sum_{i\geq
1}\sum_{\begin{subarray}{c}v_{2}\in\Xi_{N_{k}}[1,\overline{w}_{k})\\\
v_{2}\in{\mathcal{R}}_{i}(v_{1})\end{subarray}}\mathrm{p}(v_{1},v_{2})\leq\sum_{i\geq
1}\sum_{\begin{subarray}{c}v_{2}\in\Xi_{N_{k}}[1,\overline{w}_{k})\\\
v_{2}\in{\mathcal{R}}_{i}(v_{1})\end{subarray}}p\beta^{\alpha}\kappa_{1,\sigma}^{\alpha}(\overline{w}_{N_{k}},\overline{w}_{N_{k}})2^{-(i-1)\alpha
d}t_{k}^{-\alpha d}$ $\displaystyle\leq 2\sum_{i\geq
1}t_{k}^{d}(2^{id}-2^{(i-1)d})p\beta^{\alpha}\kappa_{1,\sigma}^{\alpha}(\overline{w}_{N_{k}},\overline{w}_{N_{k}})2^{-(i-1)\alpha
d}t_{k}^{-\alpha d}$
$\displaystyle=2(2^{d}-1)p\beta^{\alpha}\overline{w}_{N_{k}}^{\alpha(\sigma+1)}t_{k}^{(1-\alpha)d}\sum_{i\geq
1}2^{-(\alpha-1)(i-1)d}.$ (6.25)

Since $\alpha>1$ by assumption in Theorem 2.2, the sum on the right-hand side
is finite. This gives a bound on $T(v_{1})$ in (6.24) that does not depend on
$v_{1}$. Hence, returning to (6.24),

$\displaystyle\sum_{v_{1}\in\Xi_{n_{k}}[1,\overline{w}_{N_{k}})}\mathbb{P}^{\scriptscriptstyle
0}\big{(}v_{1}$ $\displaystyle\in{\mathcal{C}}_{n_{k}}(0,\leq
t_{k}),|{\mathcal{C}}_{n_{k}}(0,\leq
t_{k})|<k\mid\Xi,\neg{\mathcal{A}}_{\mathrm{dense}}\big{)}$
$\displaystyle=\mathbb{E}^{\scriptscriptstyle
0}\Bigg{[}\mathbbm{1}_{\\{|{\mathcal{C}}_{n_{k}}(0,\leq
t_{k})|<k\\}}\sum_{v_{1}\in\Xi_{n_{k}}[1,\overline{w}_{N_{k}})}\mathbbm{1}_{\\{v_{1}\in{\mathcal{C}}_{n_{k}}(0,\leq
t_{k})\\}}\mid\Xi,\neg{\mathcal{A}}_{\mathrm{dense}}\Bigg{]}<k,$

since on realizations of the graph satisfying $\\{{\mathcal{C}}_{n_{k}}(0,\leq
t_{k})\leq k\\}$, the sum that follows is at most $k-1$. Substituting this
with (6.25) into (6.24) yields for (6.23), that for some constant $C>0$

$\mathbb{P}^{\scriptscriptstyle 0}\big{(}{\mathcal{A}}_{\mathrm{low-
edge}}(0,n_{k},N_{k},\overline{w}_{N_{k}})\mid\neg{\mathcal{A}}_{\mathrm{dense}}\big{)}\leq
C\,k\,\overline{w}_{N_{k}}^{\alpha(\sigma+1)}\,t_{k}^{(1-\alpha)d}.$

Substituting on the right-hand side the choices of $t_{k}=n_{k}^{1/d}/(2k)$,
and $n_{k}=\exp(\varepsilon(c/2)k^{\zeta})$ from (6.9), yield (6.22) for any
$\varepsilon>0$ (using $\overline{w}_{N_{k}}$ is polynomial in $k$). ∎

The last claim bounds the last term in Lemma 6.3. Recall
${\mathcal{A}}_{\mathrm{long-edge}}$ from (6.11).

######  Claim 6.6 (No _long_ edge from a small component).

Consider a supercritical i-KSRG model in Definition 2.1 with parameters
$\alpha>1,\tau>2$ and $d\in{\mathbb{N}}$. Assume $N\geq n\geq 1$, and
$\overline{w}\geq 1$ such that

$(N^{1/d}-n^{1/d})/2\geq\max\big{\\{}\sqrt{d}n^{1/d},(\beta\overline{w}^{1+\sigma})^{1/d},N^{1/d}/4\big{\\}}.$
(6.26)

There exists a constant $C_{\ref{claim:edge-long}}>0$ such that

$\mathbb{P}^{\scriptscriptstyle 0}\big{(}\exists
v_{1}\in\Xi_{n}[1,\overline{w}),\exists
v_{2}\in\Xi\setminus\Xi_{N}:v_{1}\leftrightarrow v_{2}\big{)}\leq
C_{\ref{claim:edge-long}}\overline{w}^{c_{\ref{claim:edge-
long}}}nN^{-\min\\{\alpha-1,\tau-2\\}}(1+\mathbbm{1}_{\\{\alpha=\tau-1\\}}\log(N)).$
(6.27)

In particular, for $n_{k}$, $N_{k}$, $\overline{w}_{N_{k}}$ as in (6.9), if
$\varepsilon=\varepsilon(M_{c},\eta)\in(0,\min\\{\alpha-1,\tau-2\\})$ is
sufficiently small, then for $k$ sufficiently large, with
${\mathcal{A}}_{\mathrm{long-edge}}$ defined in (6.11),

$\mathbb{P}^{\scriptscriptstyle 0}\big{(}{\mathcal{A}}_{\mathrm{long-
edge}}(0,n_{k},N_{k},\overline{w}_{N_{k}})\big{)}\leq\exp\big{(}-\varepsilon
k^{\zeta_{\mathrm{hh}}}\big{)}.$ (6.28)

###### Proof.

We defer the proof of (6.27) (based on a first-moment method) to Appendix B on
page B. The bound (6.28) follows directly from (6.27) by substituting $n_{k}$,
$N_{k}$ and $\overline{w}_{N_{k}}$ from (6.9) to (6.27), then using that
$\overline{w}_{N_{k}}$ and $\log(N_{k})$ are polynomial in $k$ and of much
smaller order than $n_{k}$ and $N_{k}$. ∎

Having bounded all terms on the right-hand side in (6.12), we finish the
section:

###### Proof of Proposition 6.1.

For $n\leq N_{k}$, using that $N_{k}=\exp\big{(}(c/2)k^{\zeta}\big{)}$,
Proposition 6.1 follows directly from (6.1), since

$\displaystyle\mathbb{P}^{\scriptscriptstyle
0}\big{(}|{\mathcal{C}}_{n}(0)|\geq
k,0\notin{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}\big{)}\leq\mathbb{P}^{\scriptscriptstyle
0}\big{(}|{\mathcal{C}}_{n}^{\scriptscriptstyle(2)}|\geq k\big{)}$
$\displaystyle\leq
N_{k}\exp\big{(}-ck^{\zeta}\big{)}=\exp\big{(}-(c/2)k^{\zeta}\big{)}.$

We now consider $n>N_{k}$. Recall the values of $n_{k},\overline{w}_{N_{k}}$,
and $t_{k}$ from (6.9). Lemma 6.3 and Claims 6.4–6.6 directly imply (6.4) in
Proposition 6.1. We will now prove the law of large numbers (6.5). In [35] it
is shown that finite i-KSRGs
${\mathcal{G}}_{n}=({\mathcal{V}}_{n},{\mathcal{E}}_{n})$ rooted at a vertex
at the origin (see Definition 2.1) converge locally to their infinite rooted
version $({\mathcal{G}}_{\infty},0)$ as $n\to\infty$. We refer to [34] and its
references for an introduction to local limits. We use the concept of local
limits as a black box and verify a necessary and sufficient condition for the
law of large numbers for the size of the giant component for graphs that have
a local limit by Van der Hofstad [33, Theorem 2.2]. We state the condition:
let $({\mathcal{G}}_{n})_{n\geq 1}$ be a sequence of graphs with
$|{\mathcal{V}}_{n}|=n$ that converges locally in probability to
$({\mathcal{G}}_{\infty},\varnothing)$ (here $\varnothing$ denotes the root of
the graph). Assume that

$\lim_{k\to\infty}\limsup_{n\to\infty}\frac{1}{n^{2}}\mathbb{E}\Big{[}\sum_{u,v\in{\mathcal{V}}_{n}}\mathbbm{1}_{\\{|{\mathcal{C}}_{n}(u)|\geq
k,\,|{\mathcal{C}}_{n}(v)|\geq
k,\,{\mathcal{C}}(u)\neq{\mathcal{C}}(v)\\}}\Big{]}=0,$ (6.29)

then, as $n\to\infty$,

$\frac{|{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}|}{n}\overset{\mathbb{P}}{\longrightarrow}\mathbb{P}^{\scriptscriptstyle
0}\big{(}|{\mathcal{C}}_{\infty}(\varnothing)|=\infty\big{)}.$

To verify condition (6.29), we have to consider a model satisfying
$|{\mathcal{V}}_{n}|=n$. Therefore, we consider i-KSRGs on $\Lambda_{n}$ that
are conditioned to have $|{\mathcal{V}}_{n}|=n$. All our previous results
extend to this model, as remarked in Section 3.3. We analyze the indicator
random variables in (6.29) for a fixed pair $u,v\in{\mathcal{V}}_{n}$. We
distinguish whether one of them is part of the largest component to obtain

$\\{|{\mathcal{C}}_{n}(u)|\geq k,|{\mathcal{C}}_{n}(v)|\geq
k,{\mathcal{C}}_{n}(u)\neq{\mathcal{C}}_{n}(v)\\}\subseteq\\{|{\mathcal{C}}_{n}(u)|\geq
k,u\notin{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}\\}\cup\\{|{\mathcal{C}}_{n}(v)|\geq
k,v\notin{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}\\}.$

Using linearity of expectation over the $n^{2}$ vertex pairs in the sum of
(6.29), and that all $n$ vertices are identically distributed, we obtain that

$\lim_{k\to\infty}\limsup_{n\to\infty}\frac{1}{n^{2}}\mathbb{E}\Big{[}\sum_{u,v\in{\mathcal{V}}_{n}}\mathbbm{1}_{\\{|{\mathcal{C}}_{n}(u)|\geq
k,\,|{\mathcal{C}}_{n}(v)|\geq
k,\,{\mathcal{C}}(u)\neq{\mathcal{C}}(v)\\}}\Big{]}\leq\lim_{k\to\infty}\limsup_{n\to\infty}2\,\mathbb{P}\big{(}|{\mathcal{C}}_{n}(U)|\geq
k,U\notin{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}\big{)},$

where $U$ is a vertex with a uniform location $X\in\Lambda_{n}$ and mark
$w_{u}$ following distribution $F_{W}$. We condition on the location $X=x$, to
obtain that

$\mathbb{P}\big{(}|{\mathcal{C}}_{n}(U)|\geq
k,U\notin{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}\big{)}\leq\frac{1}{n}\int_{x\in\Lambda_{n}:\|x-\partial\Lambda_{n}\|\geq
N_{k}^{1/d}/2}\\!\\!\\!\\!\mathbb{P}^{x}\big{(}|{\mathcal{C}}_{n}(u)|\\!\geq\\!k,u\notin{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}\big{)}\mathrm{d}x+\mathbb{P}\big{(}\|X-\partial\Lambda_{n}\|\\!<\\!N_{k}^{1/d}/2\big{)}.$

The second term tends to zero as $n\to\infty$, while the first term is of
order $\exp\big{(}-\Omega(k^{\zeta})\big{)}$ by Lemma 6.3 and Claims 6.4–6.6
that apply for all $n\geq N_{k}$ and $x$ such that
$\|x-\partial\Lambda_{n}\|\geq N_{k}^{1/d}/2$. Thus, the first term tends to
zero as $k\to\infty$. This finishes the proof of the condition (6.29). The law
of large numbers (6.5) follows. ∎

##  7\. Lower bounds

In this section we prove Theorem 2.2(i). We first show that certain
prerequisite inequalities imply the lower bound on the cluster-size decay, and
afterwards we verify these inequalities for $\tau\in(2,2+\sigma)$. Eventually,
we prove Theorem 2.5. We define for some constant $\eta>0$ the function and
set

$\displaystyle w_{\eta}(\ell)$
$\displaystyle:=(\log(\ell))^{\eta},\qquad\Lambda_{\ell,\eta}:=\Lambda_{\ell}\times[1,w_{\eta}(\ell)),$
(7.1) $\displaystyle{\mathcal{C}}_{\ell,\eta}(0)$ $\displaystyle:=\mbox{the
component containing $(0,w_{0})$ in }{\mathcal{G}}_{\ell}[1,w_{\eta}(\ell)),$

i.e., ${\mathcal{C}}_{\ell,\eta}(0)$ is the component of $(0,w_{0})$ in the
graph induced on vertices in $\Lambda_{\ell,\eta}$ – and let
${\mathcal{C}}_{\ell,\eta}(0):=\emptyset$ if $w_{0}\geq w_{\eta}(\ell)$. We
recall $\mathfrak{m}({\mathcal{Z}})$, the multiplicity of the maximum of
${\mathcal{Z}}$, from (2.5); and the exponents $\zeta_{\mathrm{ll}}$,
$\zeta_{\mathrm{lh}}$, $\zeta_{\mathrm{hh}}$, $\zeta_{\mathrm{nn}}$ from
(1.7), (1.8), (1.12), and (1.9). Define for some small constant
$\varepsilon>0$ to be defined later, and
${\mathcal{Z}}=\\{\zeta_{\mathrm{ll}},\zeta_{\mathrm{lh}},\zeta_{\mathrm{hh}},\zeta_{\mathrm{nn}}\\}$

$\underline{k}_{n,\varepsilon}:=\begin{dcases}\big{(}\varepsilon\log(n)/(\log\log(n))^{\mathfrak{m}_{\mathcal{Z}}-1}\big{)}^{1/\max({\mathcal{Z}})},&\text{if
}\max({\mathcal{Z}})>0,\\\
\exp\big{(}(\varepsilon\log(n))^{1/(\mathfrak{m}_{\mathcal{Z}}-1)}\big{)},&\text{if
}\max({\mathcal{Z}})=0,\mbox{ and }\mathfrak{m}_{\mathcal{Z}}>1.\end{dcases}$
(7.2)

Note that $\zeta_{\mathrm{nn}}=\tfrac{d-1}{d}$, which is equal to 0 only if
$d=1$, and positive otherwise. Thus, only in dimension $d=1$ one can have
$\underline{k}_{n,\varepsilon}$ different from $\Theta(\mathrm{polylog}(n))$.
More precisely, for $d=1$,
$\underline{k}_{n,\varepsilon}=\Theta(\mathrm{poly}(n))$ if only one out of
$\\{\zeta_{\mathrm{ll}},\zeta_{\mathrm{lh}},\zeta_{\mathrm{hh}}\\}$ is zero,
and the others are negative; and it is stretched exponential in the logarithm
if at least two elements out of
$\\{\zeta_{\mathrm{ll}},\zeta_{\mathrm{lh}},\zeta_{\mathrm{hh}}\\}$ are zero,
and none of them is positive. If the three values
$\zeta_{\mathrm{ll}},\zeta_{\mathrm{lh}},\zeta_{\mathrm{hh}}$ are all
negative, then the model is always subcritical when $d=1$ as shown by Gracar,
Lüchtrath, and Mönch [27].

By an “i-KSRG on ${\mathbb{Z}}^{d}$” we mean the model mentioned in Section
3.3, i.e., when $\Xi$ is replaced by vertices in ${\mathbb{Z}}^{d}$ having iid
marks from distribution $F_{W}$ in (2.2) in Definition 2.1.

######  Proposition 7.1 (Lower bound holds when linear-sized giant on
truncated weights exists).

Consider a supercritical i-KSRG model in Definition 2.1 with parameters
$\alpha\in(1,\infty]$, $\tau\in(2,\infty]$, $\sigma\geq 0$,
$d\in{\mathbb{N}}$, or a supercritical i-KSRG on ${\mathbb{Z}}^{d}$ with
additionally $\min\\{p,p\beta^{\alpha}\\}<1$ in (2.3). Assume that there exist
constants $\eta,\rho>0$ such that for all $\ell$ sufficiently large, and with
${\mathcal{C}}_{\ell,\eta}(0)$ from (7.1),

$\displaystyle\mathbb{P}^{\scriptscriptstyle
0}\big{(}|{\mathcal{C}}_{\ell,\eta}(0)|\geq\rho\ell\big{)}\geq\rho.$ (7.3)

Then there exists $A>0$ such that for all $n\in[Ak,\infty]$, with
${\mathcal{Z}}=\\{\zeta_{\mathrm{ll}},\zeta_{\mathrm{lh}},\zeta_{\mathrm{hh}},\zeta_{\mathrm{nn}}\\}$,

$\mathbb{P}^{\scriptscriptstyle 0}\big{(}|{\mathcal{C}}_{n}(0)|\geq
k,0\notin{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}\big{)}\geq\exp\big{(}-Ak^{\max({\mathcal{Z}})}\log^{\mathfrak{m}_{\mathcal{Z}}-1}(k)\big{)}.$
(7.4)

Moreover, there exists $\delta,\varepsilon>0$, such that for all $n$
sufficiently large, with $k_{n,\varepsilon}$ from (7.2),

$\mathbb{P}\big{(}|{\mathcal{C}}_{n}^{\scriptscriptstyle(2)}|>\underline{k}_{n,\varepsilon}\big{)}\geq
1-n^{-\delta}.$ (7.5)

By Proposition 5.14, the condition (7.3) is satisfied when
$\tau\in(2,2+\sigma)$, implying Theorem 2.2(i). In [43], we show that (7.3)
holds whenever
$\max\\{\zeta_{\mathrm{ll}},\zeta_{\mathrm{lh}},\zeta_{\mathrm{hh}}\\}>0$ and
the model is supercritical. We prove Proposition 7.1 by formalizing the
reasoning at the beginning of Section 1.2. First, we assume that the vertex
set is given by the PPP $\Xi$ in Definition 2.1. Then we explain how to extend
to i-KSRGs on ${\mathbb{Z}}^{d}$.

###  7.1. Finding a localized component

The goal of this section is to prove Proposition 7.1. First we aim to bound
the probability of $\\{{\mathcal{C}}_{n}(0)\geq
k,0\notin{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}\\}$ in (7.4) from below. To
do so, we write it as the intersection of “almost independent” events, for
which we introduce some notation now:

#### Two large components.

We write $\Lambda(x,s)=\Lambda_{s}(x)$ for a box of volume $s$ centered at
$x$, see (2.1). We define for a constant $M_{\mathrm{in}}>0$ the radius and
Euclidean ball

$r_{k}:=(kM_{\mathrm{in}})^{1/d},\qquad{\mathcal{B}}_{kM_{\mathrm{in}}}:=\\{x\in{\mathbb{R}}^{d}:\|x\|\leq
r_{k}\\},$ (7.6)

where $M_{\mathrm{in}}>1/\rho$ is implicitly defined given $\rho$ from
Proposition 7.1, such that
$\|\partial\Lambda_{k/\rho}-\partial{\mathcal{B}}_{kM_{\mathrm{in}}}\|=r_{k}/2$,
implying also that
$\Lambda_{k/\rho}\subseteq{\mathcal{B}}_{kM_{\mathrm{in}}}$. We would like to
constrain ${\mathcal{C}}_{n}(0)$ to the ball
${\mathcal{B}}_{kM_{\mathrm{in}}}$, and we would like to find a component
outside ${\mathcal{B}}_{kM_{\mathrm{in}}}$ that is larger than
$|{\mathcal{C}}_{n}(0)|$. We ‘construct’ these two components, with the
constant $M_{\mathrm{out}}:=2^{d+2}M_{\mathrm{in}}$, on vertices in the
(hyper)rectangles (see also (7.1))

$\displaystyle{\mathcal{R}}_{\mathrm{in}}$
$\displaystyle:=\Lambda_{k/\rho,\eta}=\Lambda(0,k/\rho)\times\big{[}1,w_{\eta}(k/\rho)\big{)},\quad$
$\displaystyle{\mathcal{R}}_{\mathrm{out}}:=\Lambda(x_{\mathrm{out}},kM_{\mathrm{out}}/\rho)\times\big{[}1,w_{\eta}(kM_{\mathrm{out}}/\rho)\big{)},$
(7.7) $\displaystyle\Lambda_{\mathrm{in}}$
$\displaystyle:=\Lambda(0,k/\rho),\qquad\qquad$
$\displaystyle\Lambda_{\mathrm{out}}:=\Lambda(x_{\mathrm{out}},(kM_{\mathrm{out}}/\rho)),$

where $x_{\mathrm{out}}:=(x^{\mathrm{out}}_{1},0,\dots,0)\in{\mathbb{R}}^{d}$
is defined implicitly given $\rho$ such that
$\|\partial\Lambda(x_{\mathrm{out}},kM_{\mathrm{out}}/\rho)-\partial{\mathcal{B}}_{kM_{\mathrm{in}}}\|:=r_{k}/2$.
We assume that $n$ is sufficiently large so that
$\Lambda_{\mathrm{in}}\cup\Lambda_{\mathrm{out}}\subseteq\Lambda_{n}$. Let
${\mathcal{C}}_{\mathrm{in}}(0)$ be the component of $(0,w_{0})$ in the
subgraph of ${\mathcal{G}}_{n}$ induced on vertices in
${\mathcal{R}}_{\mathrm{in}}$, where
${\mathcal{C}}_{\mathrm{in}}(0)=\emptyset$ if $w_{0}\geq w_{\eta}(k/\rho)$.
Since ${\mathcal{R}}_{\mathrm{in}}\subseteq{\mathcal{B}}_{kM_{\mathrm{in}}}$,
it is immediate that
${\mathcal{C}}_{\mathrm{in}}(0)\subseteq\Xi_{{\mathcal{B}}_{kM_{\mathrm{in}}}}$.
Let ${\mathcal{C}}_{\mathrm{out}}^{\scriptscriptstyle(1)}$ be the largest
component in the subgraph of ${\mathcal{G}}_{n}$ induced on vertices in
${\mathcal{R}}_{\mathrm{out}}$. Define the events

$\displaystyle{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{ingiant}}$
$\displaystyle:=\\{|{\mathcal{C}}_{\mathrm{in}}(0)|\geq k\\},\qquad$
$\displaystyle{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{outgiant}}$
$\displaystyle:=\\{|{\mathcal{C}}_{\mathrm{out}}^{\scriptscriptstyle(1)}|\geq
kM_{\mathrm{out}}\\},$ (7.8)
$\displaystyle{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{insmall}}$
$\displaystyle:=\\{|\Xi_{{\mathcal{B}}_{kM_{\mathrm{in}}}}|\leq
kM_{\mathrm{out}}/2\\},\qquad$
$\displaystyle{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{components}}$
$\displaystyle:={\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{ingiant}}\cap{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{outgiant}}.$

Figure 3. A visualization of the $\gamma$-suppressed profile
${\mathcal{M}}_{\gamma}$. The horizontal axis represents space, the vertical
axis represents marks. $\\{\Xi\leq{\mathcal{M}}_{\gamma}\\}$ demands no
vertices in the yellow region. ${\mathcal{A}}_{\mathrm{noedge}}(\gamma)$
demands that there is no edge between vertices in the inner and vertices in
the outer blue region, ${\mathcal{A}}_{\mathrm{components}}$ requires that the
two red areas contain large components;
${\mathcal{A}}_{\mathrm{regular}}(\eta)$ ensures that $\Xi$ is ‘close to being
typical’ in the red areas ${\mathcal{R}}_{\mathrm{in}}$ and
${\mathcal{R}}_{\mathrm{out}}$.

#### Isolation

On ${\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{components}}$,
${\mathcal{C}}_{n}(0)\supseteq{\mathcal{C}}_{\mathrm{in}}(0)$ could still
connect to the giant/infinite component. To prevent this, we will ban edges
that cross the boundary of ${\mathcal{B}}_{kM_{\mathrm{in}}}$ as follows. In
Section 3.2 we informally described an “optimally-suppressed mark-profile”. We
first define a suppressed mark-profile, then optimize its shape. Let
$\gamma\in(0,1/(\sigma+1)]$ be a constant. We define, for
$C_{\beta}:=(2\beta)^{1/d}$, and for $x\in{\mathbb{R}}^{d}$ with
$\|x-\partial{\mathcal{B}}_{kM_{\mathrm{in}}}\|=|\|x\|-r_{k}|=:z$, the
$\gamma$-suppressed profile by

$\displaystyle f_{\gamma}(z)$ $\displaystyle:=\begin{dcases}1&\text{if }z\leq
C_{\beta},\\\ (z/C_{\beta})^{\gamma d}&\text{if }z\in(C_{\beta},r_{k}],\\\
(z/C_{\beta})^{d}(r_{k}/C_{\beta})^{-d(1-\gamma)}&\text{if
}z>r_{k},\end{dcases}$ (7.9) $\displaystyle{\mathcal{M}}_{\gamma}$
$\displaystyle:=\\{(x_{v},f_{\gamma}(|\|x_{v}\|-r_{k}|)):x_{v}\in{\mathbb{R}}^{d}\\}.$
(7.10)

We say that $v$ is _below, on, or above_ ${\mathcal{M}}_{\gamma}$ if $w_{v}$
is at most, equal to, or strictly larger than $f_{\gamma}(|\|x_{v}\|-r_{k}|)$,
respectively. We say $\Xi\leq{\mathcal{M}}_{\gamma}$ if all points in $\Xi$
are below ${\mathcal{M}}_{\gamma}$, see Figure 3. We split the PPP $\Xi$ into
four independent PPPs, whether points fall below or above
${\mathcal{M}}_{\gamma}$, and inside or outside
${\mathcal{B}}_{kM_{\mathrm{in}}}$:

$\displaystyle\Xi_{\leq{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{in}}$
$\displaystyle:=\\{(x_{u},w_{u})\in\Xi:x_{u}\in{\mathcal{B}}_{kM_{\mathrm{in}}},w_{u}\leq
f_{\gamma}(|\|x_{u}\|-r_{k}|)\\},$ (7.11)
$\displaystyle\Xi_{\leq{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{out}}$
$\displaystyle:=\\{(x_{v},w_{v})\in\Xi:x_{v}\notin{\mathcal{B}}_{kM_{\mathrm{in}}},w_{v}\leq
f_{\gamma}(|\|x_{v}\|-r_{k}|)\\},$
$\displaystyle\Xi_{>{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{in}}$
$\displaystyle:=\\{(x_{u},w_{u})\in\Xi:x_{u}\in{\mathcal{B}}_{kM_{\mathrm{in}}},w_{u}>f_{\gamma}(|\|x_{u}\|-r_{k}|)\\},$
$\displaystyle\Xi_{>{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{out}}$
$\displaystyle:=\\{(x_{v},w_{v})\in\Xi:x_{v}\notin{\mathcal{B}}_{kM_{\mathrm{in}}},w_{v}>f_{\gamma}(|\|x_{v}\|-r_{k}|)\\}.$

For $A,B\subseteq\Xi$ we denote by $|{\mathcal{E}}(A,B)|$ the number of edges
between vertices in $A$ and $B$. Define

$\\{\Xi\leq{\mathcal{M}}_{\gamma}\\}=\\{|\Xi_{>{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{in}}\cup\Xi_{>{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{out}}|=0\\},\quad{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{noedge}}(\gamma):=\\{|{\mathcal{E}}(\Xi_{\leq{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{in}},\Xi_{\leq{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{out}})|=0\\}.$
(7.12)

Under
$\\{\Xi\leq{\mathcal{M}}_{\gamma}\\}\cap{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{noedge}}(\gamma)$,
the vertices in ${\mathcal{B}}_{kM_{\mathrm{in}}}$ are not connected to the
unique infinite component when $n=\infty$ and are isolated from the rest of
${\mathcal{G}}_{n}$ when $n<\infty$. We comment on the profile function
$f_{\gamma}$ in (7.9). The event $\\{\Xi\leq{\mathcal{M}}_{\gamma}\\}$ demands
no vertices within distance $C_{\beta}$ from
$\partial{\mathcal{B}}_{M_{\mathrm{in}}}$, since $f_{\gamma}(|\|x\|-r_{k}|)=1$
for $\|x-\partial{\mathcal{B}}_{kM_{\mathrm{in}}}\|\leq C_{\beta}$, and vertex
marks are above $1$ a.s. The function $f_{\gamma}$ is continuous and
increasing in $z$: the closer a point $x$ is to the boundary of
${\mathcal{B}}_{kM_{\mathrm{in}}}$, the stronger the restriction on its mark
in $\Xi\leq{\mathcal{M}}_{\gamma}$. This is natural since vertices with higher
mark close to $\partial{\mathcal{B}}_{kM_{\mathrm{in}}}$ are more likely to
have an edge crossing this boundary, which we want to prevent. While the event
$\\{\Xi\leq{\mathcal{M}}_{\gamma}\\}$ becomes less likely when $\gamma$ is
small, ${\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{noedge}}(\gamma)$
becomes more likely. This naturally leads to an optimization problem, that we
set up next. Let
${\mathcal{Y}}=\\{0,\alpha-(\tau-1),(\sigma+1)\alpha-2(\tau-1)\\}$, and define

$\displaystyle\begin{aligned} \gamma_{\mathrm{\neg
nn}}&:=\begin{dcases}\max_{0<\gamma\leq
1/(\sigma+1)}\big{\\{}\gamma:1-\gamma(\tau-1)\overset{(\star)}{\geq}2-\alpha+\gamma\max({\mathcal{Y}})\\},&\text{if
}\alpha<\infty,\\\ 1/(\sigma+1),&\text{if }\alpha=\infty,\end{dcases}\\\
\qquad\zeta_{\mathrm{\neg nn}}&:=1-\gamma_{\mathrm{\neg
nn}}(\tau-1),\end{aligned}$ (7.13)

The inequality $(\star)$ holds for $\gamma=0$ since $\alpha>1$ by assumption,
therefore $\gamma_{\mathrm{\neg nn}}>0$ and is well-defined. We call
$f_{\star}:=f_{\gamma_{\mathrm{\neg nn}}}$ the _optimally-suppressed mark-
profile_ (and show below that it is indeed “optimal”) and write
${\mathcal{M}}_{\star}:={\mathcal{M}}_{\gamma_{\mathrm{\neg nn}}}$. We define
the isolation event using the events in (7.12), see also Figure 3:

${\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{isolation}}:=\\{\Xi\leq{\mathcal{M}}_{\star}\\}\cap{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{noedge}}(\gamma_{\mathrm{\neg
nn}}).$ (7.14)

#### Ensuring almost independence

The events ${\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{isolation}}$ and
$\\{|{\mathcal{C}}_{\mathrm{in}}(0)|\geq k\\}$ in (7.8) are correlated, since
$\\{|{\mathcal{C}}_{\mathrm{in}}(0)|\geq k\\}$ may push up the number of high-
mark vertices in ${\mathcal{R}}_{\mathrm{in}}$, making
${\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{isolation}}$ less likely. To
overcome the dependence, we introduce two auxiliary events that ensure
regularity of $\Xi$ in the hyperrectangles
${\mathcal{R}}_{\mathrm{in}},{\mathcal{R}}_{\mathrm{out}}$ from (7.7). Using
$w_{\eta}$ from (7.1), define, using
$c_{\mathrm{in}}:=1/\rho,c_{\mathrm{out}}:=M_{\mathrm{out}}/\rho$:

$\displaystyle
j^{\star}_{\mathrm{in}}:=\min\\{j\in{\mathbb{N}}:w_{\eta}(kc_{\mathrm{in}})/2^{j}<1\\},\quad
j^{\star}_{\mathrm{out}}:=\min\\{j\in{\mathbb{N}}:w_{\eta}(kc_{\mathrm{out}})/2^{j}<1)\\}.$
(7.15)

and writing $\mathrm{loc}\in\\{\mathrm{in},\mathrm{out}\\}$, define

$I_{j}^{{\scriptscriptstyle\mathrm{loc}}}:=[w_{\eta}(kc_{\mathrm{loc}})/2^{j},w_{\eta}(kc_{\mathrm{loc}})/2^{j-1}),\qquad
1\leq j<j^{\star}_{\mathrm{loc}}.$ (7.16)

Using $\Lambda_{\mathrm{in}},\Lambda_{\mathrm{out}}$ in (7.7), the intensity
measure $\mu_{\tau}$ of $\Xi$ in (2.2), and
$\Xi_{\mathrm{loc}}(I_{j}^{{\scriptscriptstyle\mathrm{loc}}})$ for the
vertices in $\Xi\cap(\Lambda_{\mathrm{loc}}\times
I_{j}^{{\scriptscriptstyle\mathrm{loc}}})$, consider the following events (see
also Figure 3) for $\mathrm{loc}\in\\{\mathrm{in},\mathrm{out}\\}$:

$\displaystyle{\mathcal{A}}_{\mathrm{regular}}^{\scriptscriptstyle(k,\mathrm{loc})}(\eta)$
$\displaystyle:=\\{\forall j\leq
j^{\star}_{\mathrm{loc}}:|\Xi_{\mathrm{loc}}(I_{j}^{{\scriptscriptstyle\mathrm{loc}}})|\leq
2\mu_{\tau}(\Lambda_{\mathrm{loc}}\times
I_{j}^{{\scriptscriptstyle\mathrm{loc}}})\\},$ (7.17)
$\displaystyle{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{regular}}({\eta})$
$\displaystyle:={\mathcal{A}}_{\mathrm{regular}}^{\scriptscriptstyle(k,\mathrm{in})}({\eta})\cap{\mathcal{A}}_{\mathrm{regular}}^{\scriptscriptstyle(k,\mathrm{out})}({\eta}).$

Finally, fix a realization of the induced subgraphs
${\mathcal{G}}_{{\mathcal{R}}_{\mathrm{in}}}\cup{\mathcal{G}}_{{\mathcal{R}}_{\mathrm{out}}}=(\Xi_{{\mathcal{R}}_{\mathrm{in}}},{\mathcal{E}}({\mathcal{G}}_{{\mathcal{R}}_{\mathrm{in}}}))\cup(\Xi_{{\mathcal{R}}_{\mathrm{out}}},{\mathcal{E}}({\mathcal{G}}_{{\mathcal{R}}_{\mathrm{out}}}))$
so that the vertex set
$\Xi_{{\mathcal{R}}_{\mathrm{in}}}\cup\Xi_{{\mathcal{R}}_{\mathrm{out}}}$
satisfies the event
${\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{regular}}(\eta)$ for some
$\eta>0$, and define the conditional probability measure and expectation by

$\widetilde{\mathbb{P}}_{\mathrm{io}}\big{(}\,\cdot\,\big{)}:=\mathbb{P}\big{(}\,\cdot\mid{\mathcal{G}}_{{\mathcal{R}}_{\mathrm{in}}}\cup{\mathcal{G}}_{{\mathcal{R}}_{\mathrm{out}}},{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{regular}}(\eta)\big{)},\qquad\widetilde{\mathbb{E}}_{\mathrm{io}}[\,\cdot\,]:=\widetilde{\mathbb{E}}_{\mathrm{io}}\big{[}\,\cdot\mid{\mathcal{G}}_{{\mathcal{R}}_{\mathrm{in}}}\cup{\mathcal{G}}_{{\mathcal{R}}_{\mathrm{out}}},{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{regular}}(\eta)\big{]}$
(7.18)

In the conditioning we reveal both the vertex and edge sets in the disjoint
boxes ${\mathcal{R}}_{\mathrm{in}},{\mathcal{R}}_{\mathrm{out}}.$ The event
${\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{regular}}(\eta)$ checks the
number of vertices in hyperrectangles inside
${\mathcal{R}}_{\mathrm{in}},{\mathcal{R}}_{\mathrm{out}}$, hence
${\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{regular}}(\eta)$ is measurable
with respect to
${\mathcal{G}}_{{\mathcal{R}}_{\mathrm{in}}},{\mathcal{G}}_{{\mathcal{R}}_{\mathrm{out}}}$
and in principle can be left out of the conditioning in (7.18). We state a
lemma that will imply Proposition 7.1.

######  Lemma 7.2 (Lower bound for isolation).

Consider a supercritical i-KSRG model in Definition 2.1 with parameters
$\alpha\in(1,\infty]$, $\tau\in(2,\infty]$, $\sigma\geq 0$, and
$d\in{\mathbb{N}}$, or a supercritical i-KSRG on ${\mathbb{Z}}^{d}$ with
additionally $p<1$ in (2.3). For any constant $\eta>0$ in (7.1), there exists
$A>0$, such that for any realization of
${\mathcal{G}}_{{\mathcal{R}}_{\mathrm{in}}}\cup{\mathcal{G}}_{{\mathcal{R}}_{\mathrm{out}}}$
that satisfies
${\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{regular}}(\eta)$, with
${\mathcal{Z}}=\\{\zeta_{\mathrm{ll}},\zeta_{\mathrm{lh}},\zeta_{\mathrm{hh}},\zeta_{\mathrm{nn}}\\}$,
with $\widetilde{\mathbb{P}}_{\mathrm{io}}$ from (7.18)

$\displaystyle\widetilde{\mathbb{P}}_{\mathrm{io}}\big{(}{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{isolation}}\big{)}\geq\exp\big{(}-Ar_{k}^{d\max({\mathcal{Z}})}\log^{\mathfrak{m}_{\mathcal{Z}}-1}(r_{k})\big{)}.$
(7.19)

The same bound holds for the Palm-version
$\widetilde{\mathbb{P}}_{\mathrm{io}}^{\scriptscriptstyle 0}$ of
$\widetilde{\mathbb{P}}_{\mathrm{io}}$.

We now show that Proposition 7.1 follows directly from Lemma 7.2.

###### Proof of Proposition 7.1, assuming Lemma 7.2..

We first show (7.4). Recall
${\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{components}},{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{insmall}}$
from (7.8), and
${\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{isolation}},{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{regular}}({\eta})$
from (7.14), (7.17) respectively. The intersection of these events implies
$\\{|{\mathcal{C}}_{n}(0)|\geq
k,0\notin{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}\\}$, since
$|{\mathcal{C}}_{n}(0)|\geq|{\mathcal{C}}_{\mathrm{in}}(0)|\geq k$, and
${\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{isolation}}$ ensures that
${\mathcal{C}}_{n}(0)$ is fully contained in
${\mathcal{B}}_{kM_{\mathrm{in}}}$. Hence,
$|{\mathcal{C}}_{n}(0)|\leq|\Xi_{{\mathcal{B}}_{kM_{\mathrm{in}}}}|\leq
kM_{\mathrm{out}}/2$ and $|{\mathcal{C}}_{\mathrm{out}}|\geq
kM_{\mathrm{out}}$ ensure that ${\mathcal{C}}_{n}(0)$ is not the largest
component of ${\mathcal{G}}_{n}$. So, by the law of total probability

$\displaystyle\mathbb{P}^{\scriptscriptstyle 0}\big{(}|{\mathcal{C}}_{n}$
$\displaystyle(0)|\geq
k,0\notin{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}\big{)}$ (7.20)
$\displaystyle\geq\mathbb{P}^{\scriptscriptstyle
0}\big{(}{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{components}}\cap{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{isolation}}\cap{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{insmall}}\cap{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{regular}}(\eta)\big{)}$
$\displaystyle\geq\mathbb{P}^{\scriptscriptstyle
0}\big{(}{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{components}}\cap{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{isolation}}\cap{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{regular}}(\eta)\big{)}-\mathbb{P}^{\scriptscriptstyle
0}\big{(}\neg{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{insmall}}\big{)}$
$\displaystyle=\mathbb{P}^{\scriptscriptstyle
0}\big{(}{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{components}}\cap{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{regular}}(\eta)\big{)}\mathbb{P}^{\scriptscriptstyle
0}\big{(}{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{isolation}}\mid{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{components}}\cap{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{regular}}(\eta)\big{)}-\mathbb{P}^{\scriptscriptstyle
0}\big{(}\neg{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{insmall}}\big{)}.$

Recall
${\mathcal{A}}_{\mathrm{insmall}}^{{\scriptscriptstyle(k)}}=\\{|\Xi_{{\mathcal{B}}_{kM_{\mathrm{in}}}}|\leq
kM_{\mathrm{out}}/2\\}$ from (7.8), and
$M_{\mathrm{out}}=2^{d+2}M_{\mathrm{in}}$ above (7.7). The box with side-
length $2r_{k}=2(kM_{\mathrm{in}})^{1/d}$ (by definition in (7.6)) centered at
the origin is the smallest box that contains
${\mathcal{B}}_{kM_{\mathrm{in}}}$. Using the intensity measure $\mu_{\tau}$
from (2.2), and writing
${\mathcal{B}}_{kM_{\mathrm{in}}}^{\leq}:=\\{(x,w_{x})\in{\mathbb{R}}^{d+1}:x\in{\mathcal{B}}_{kM_{\mathrm{in}}};(x,w_{v})\leq{\mathcal{M}}_{\gamma}\\}$,
we have

$\displaystyle\mu_{\tau}({\mathcal{B}}_{kM_{\mathrm{in}}}^{\leq})\leq
2^{d}kM_{\mathrm{in}}=2^{d+2}kM_{\mathrm{in}}/4=kM_{\mathrm{out}}/4.$

By a standard concentration inequality for Poisson random variables (see Lemma
C.1 for $x=2$), there exist $c^{\prime},c>0$ such that, since
$r_{k}=\Theta(k^{1/d})$,

$\displaystyle\mathbb{P}^{\scriptscriptstyle
0}\big{(}\neg{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{insmall}}\big{)}\leq\exp(-c^{\prime}r_{k}^{d})=\exp(-ck).$
(7.21)

Returning to (7.20), the event
${\mathcal{A}}_{\mathrm{regular}}^{{\scriptscriptstyle(k)}}(\eta)={\mathcal{A}}_{\mathrm{regular}}^{\scriptscriptstyle(k,\mathrm{in})}(\eta)\cap{\mathcal{A}}_{\mathrm{regular}}^{\scriptscriptstyle(k,\mathrm{out})}(\eta)$,
defined in (7.17), holds with probability tending to $1$ as $k\to\infty$,
again by concentration inequalities for Poisson random variables (see Lemma
C.1 for $x=2$). Hence,

$\displaystyle\mathbb{P}^{\scriptscriptstyle
0}\big{(}{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{components}}\cap{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{regular}}(\eta)\big{)}$
$\displaystyle\geq\mathbb{P}^{\scriptscriptstyle
0}\big{(}{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{components}}\big{)}-\mathbb{P}^{\scriptscriptstyle
0}\big{(}\neg({\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{regular}}(\eta)\big{)}$
(7.22) $\displaystyle=\mathbb{P}^{\scriptscriptstyle
0}\big{(}{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{components}}\big{)}-o_{k}(1).$

We recall from (7.8) that
${\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{components}}=\\{|{\mathcal{C}}_{\mathrm{in}}(0)|\geq
k\\}\cap\\{|{\mathcal{C}}_{\mathrm{out}}^{\scriptscriptstyle(1)}|\geq
kM_{\mathrm{out}}\\}$. Translate the hyperrectangle
${\mathcal{R}}_{\mathrm{out}}$ in (7.7) containing
${\mathcal{C}}_{\mathrm{out}}^{\scriptscriptstyle(1)}$ to the origin of
${\mathbb{R}}^{d}$ (using $w_{\eta}(\ell))$ in (7.1)):

${\mathcal{R}}_{\mathrm{out}}^{\prime}:=\Lambda(0,kM_{\mathrm{out}}/\rho)\times[1,w_{\eta}(kM_{\mathrm{out}}/\rho))=\Lambda_{kM_{\mathrm{out}}/\rho,\eta},$
(7.23)

and write
${\mathcal{C}}_{\mathrm{out^{\prime}}}^{\scriptscriptstyle(1)},{\mathcal{C}}_{\mathrm{out^{\prime}}}(0)$
for the largest component and for the component containing $(0,w_{0})$ in the
subgraph of ${\mathcal{G}}_{n}$ induced by vertices in
${\mathcal{R}}^{\prime}_{\mathrm{out}}$. As before, we ignore the conditioning
$(0,w_{0})\in\Xi$ in Definition 2.1 in our computations. By the translation
invariance of the annealed measure and that the events
$\\{|{\mathcal{C}}_{\mathrm{in}}(0)|\geq k\\}$ and
$\\{|{\mathcal{C}}_{\mathrm{out}}^{\scriptscriptstyle(1)}|\geq
M_{\mathrm{out}}k\\}$ are independent because they are induced subgraphs of
the disjoint hyperrectangles ${\mathcal{R}}_{\mathrm{in}}$ and
${\mathcal{R}}_{\mathrm{out}}$ in (7.7),

$\mathbb{P}^{\scriptscriptstyle
0}\big{(}{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{components}}\big{)}=\mathbb{P}^{\scriptscriptstyle
0}\big{(}|{\mathcal{C}}_{\mathrm{in}}(0)|\geq
k\big{)}\mathbb{P}\big{(}|{\mathcal{C}}_{\mathrm{out}^{\prime}}^{\scriptscriptstyle(1)}|\geq
M_{\mathrm{out}}k\big{)}\geq\mathbb{P}^{\scriptscriptstyle
0}\big{(}|{\mathcal{C}}_{\mathrm{in}}(0)|\geq
k\big{)}\mathbb{P}^{\scriptscriptstyle
0}\big{(}|{\mathcal{C}}_{\mathrm{out}^{\prime}}(0)|\geq
M_{\mathrm{out}}k\big{)}.$

The bound $\mathbb{P}^{\scriptscriptstyle
0}\big{(}|{\mathcal{C}}_{\ell,\eta}(0)|\geq\rho\ell\big{)}\geq\rho$ in (7.3)
in Proposition 7.1 holds for all $\ell$ sufficiently large. In particular,
since ${\mathcal{C}}_{\mathrm{in}}(0)={\mathcal{C}}_{k/\rho,\eta}(0)$ by
definition of ${\mathcal{R}}_{\mathrm{in}}$ in (7.7), and
${\mathcal{C}}_{\mathrm{out^{\prime}}}(0)={\mathcal{C}}_{kM_{\mathrm{out}}/\rho,\eta}(0)$
by definition of ${\mathcal{R}}_{\mathrm{out}}^{\prime}$ in (7.23), we obtain
for $k$ sufficiently large

$\mathbb{P}^{\scriptscriptstyle
0}\big{(}{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{components}}\big{)}\geq\mathbb{P}^{{\scriptscriptstyle
0}}\big{(}|{\mathcal{C}}_{k/\rho,\eta}(0)|\geq
k\big{)}\mathbb{P}^{\scriptscriptstyle
0}\big{(}|{\mathcal{C}}_{kM_{\mathrm{out}}/\rho,\eta}(0)|\geq
M_{\mathrm{out}}k\big{)}\geq\rho^{2},$ (7.24)

implying that $\mathbb{P}^{\scriptscriptstyle
0}\big{(}{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{components}}\cap{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{regular}}(\eta)\big{)}\geq\rho^{2}-o_{k}(1)>3\rho^{2}/4$
in (7.22). Since the event
${\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{components}}\cap{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{regular}}(\eta)$
is measurable with respect to the $\sigma$-algebra generated by the subgraph
${\mathcal{G}}_{{\mathcal{R}}_{\mathrm{in}}}\cup{\mathcal{G}}_{{\mathcal{R}}_{\mathrm{out}}}$,
we take expectation over all possible realizations of the latter, and
recalling the measure $\widetilde{\mathbb{P}}_{\mathrm{io}}$ from (7.18),

$\displaystyle\mathbb{P}^{\scriptscriptstyle
0}\big{(}{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{isolation}}\mid{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{components}}\cap{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{regular}}(\eta)\big{)}$
$\displaystyle=\mathbb{E}^{\scriptscriptstyle
0}\big{[}\mathbb{P}^{\scriptscriptstyle
0}\big{(}{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{isolation}}\mid{\mathcal{G}}_{{\mathcal{R}}_{\mathrm{in}}}\cup{\mathcal{G}}_{{\mathcal{R}}_{\mathrm{out}}},{\mathcal{A}}_{\mathrm{regular}}^{\scriptscriptstyle(k)}(\eta)\big{)}\big{]}$
$\displaystyle=\mathbb{E}^{\scriptscriptstyle
0}\big{[}\widetilde{\mathbb{P}}^{\scriptscriptstyle
0}_{\mathrm{io}}({\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{isolation}})\big{]}.$

Apply now Lemma 7.2 on the right-hand side, and substitute the bound
$3\rho^{2}/4$ below (7.24) into (7.22) and then in turn into (7.20) and
(7.21), to obtain for $k$ sufficiently large

$\displaystyle\mathbb{P}^{\scriptscriptstyle
0}\big{(}{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{components}}\cap{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{isolation}}\cap{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{insmall}}\cap{\mathcal{A}}^{\scriptscriptstyle(k)}_{\mathrm{regular}}(\eta)\big{)}$
$\displaystyle\geq(\rho^{2}/2)\exp\big{(}-Ar_{k}^{d\max({\mathcal{Z}})}\log^{\mathfrak{m}_{\mathcal{Z}}-1}(r_{k})\big{)}$
(7.25)
$\displaystyle\geq(\rho^{2}/2)\exp\big{(}-A^{\prime}k^{\max({\mathcal{Z}})}\log^{\mathfrak{m}_{\mathcal{Z}}-1}(k)\big{)}.$

We obtained the second row by substituting $r_{k}=(kM_{\mathrm{in}})^{1/d}$ in
(7.6) and setting $A^{\prime}:=AM_{\mathrm{in}}^{\max({\mathcal{Z}})}/2$ that
also compensates for the constants from the $\log$-correction term. Observe
that $\max({\mathcal{Z}})<1$, so that the right-hand side in (7.21) is of
smaller order than the right-hand side in (7.19). By (7.20) this finishes the
proof of (7.4). We turn to the proof of (7.5). _Lower bound on second-largest
component._ We generalize an argument from [46]. We have to bound
$\mathbb{P}\big{(}|{\mathcal{C}}_{n}^{\scriptscriptstyle(2)}|\leq\underline{k}_{n,\varepsilon}\big{)}$
from above for a suitably chosen $\varepsilon$ in the definition of
$\underline{k}_{n,\varepsilon}$ in (7.2). To do so, we fix $\vartheta>0$ to be
specified later, and assume for simplicity that
$n^{(1-\vartheta)/d}\in{\mathbb{N}}$. We then partition $\Lambda_{n}$ into
$m_{n}:=n^{1-\vartheta}$ many subboxes
$\Lambda_{\tilde{n}}^{\scriptscriptstyle(1)},\dots,\Lambda_{\tilde{n}}^{\scriptscriptstyle(m_{n})}$,
centered respectively at $x_{1},\dots,x_{m_{n}}$, each of volume
$\tilde{n}:=n^{\vartheta}$. By disjointness, the induced subgraphs
${\mathcal{G}}_{\tilde{n}}^{\scriptscriptstyle(1)},\dots,{\mathcal{G}}_{\tilde{n}}^{\scriptscriptstyle(m_{n})}$
in these boxes are independent realizations of ${\mathcal{G}}_{\tilde{n}}$,
translated to $x_{1},\dots,x_{m_{n}}$. We write
$\widetilde{\mathcal{G}}_{\tilde{n}}^{\scriptscriptstyle(1)},\dots,\widetilde{\mathcal{G}}_{\tilde{n}}^{\scriptscriptstyle(m_{n})}$
for the induced subgraphs, translated back to the origin;
$\widetilde{\Xi}_{\leq{\mathcal{M}}_{\star}}^{\scriptscriptstyle(\mathrm{in},i)}$
for the vertex set in
$\widetilde{\mathcal{G}}_{\tilde{n}}^{\scriptscriptstyle(i)}$ that is below
${\mathcal{M}}_{\star}$ and inside ${\mathcal{B}}_{kM_{\mathrm{in}}}$ after
the translation, see (7.11); and write
$\Xi_{\leq{\mathcal{M}}_{\star}}^{\scriptscriptstyle(\mathrm{in},i)}$ for the
same vertex set before the translation. For the translated subgraphs
$\big{(}\widetilde{\mathcal{G}}_{\tilde{n}}^{\scriptscriptstyle(i)})_{i\leq
m_{n}}$, we define for $k=\underline{k}_{n,\varepsilon}$ the same events as in
(7.8), (7.14), (7.17),

${\mathcal{A}}^{\scriptscriptstyle(i)}_{\mathrm{good}}:={\mathcal{A}}^{\scriptscriptstyle(\underline{k}_{n,\varepsilon}),i}_{\mathrm{components}}\cap{\mathcal{A}}^{\scriptscriptstyle(\underline{k}_{n,\varepsilon}),i}_{\mathrm{isolation}}\cap{\mathcal{A}}^{\scriptscriptstyle(\underline{k}_{n,\varepsilon}),i}_{\mathrm{insmall}}\cap{\mathcal{A}}^{\scriptscriptstyle(\underline{k}_{n,\varepsilon}),i}_{\mathrm{regular}}(\eta),$

where now in the definition of these events we replace
${\mathcal{C}}_{\Box}(0)$ with the component containing the point of $\Xi$
closest to the origin $0\in{\mathbb{R}}^{d}$ for
$\Box\in\\{\mathrm{in},\widetilde{n}\\}$. We also assume that
$\widetilde{n}=n^{\vartheta}$ is sufficiently large compared to
$\underline{k}_{n,\varepsilon}$ in (7.2) so that the spatial projection of the
box ${\mathcal{R}}_{\mathrm{out}}$ still fits within $\Lambda_{\tilde{n}}$.
This can be ensured even if $k_{n,\varepsilon}=\Theta(n^{\varepsilon})$ is
maximal in (7.2) by choosing $\varepsilon<\vartheta$. If
${\mathcal{A}}^{\scriptscriptstyle(i)}_{\mathrm{good}}$ holds for some $i\leq
m_{n}$, then the induced graph
${\mathcal{G}}_{\tilde{n}}^{\scriptscriptstyle(i)}$ in subbox
$\Lambda_{\tilde{n}}^{\scriptscriptstyle(i)}$ contains a component
${\mathcal{C}}_{\tilde{n}}^{{\scriptscriptstyle(i)}}$ in
$\Xi_{\leq{\mathcal{M}}_{\star}}^{\scriptscriptstyle(\mathrm{in},i)}$ (which
we call a ‘candidate’ second-largest component of ${\mathcal{G}}_{n}$) with
size at least $\underline{k}_{n,\varepsilon}$ that is not the largest
component in its own box, and all vertices in
$\Lambda_{\tilde{n}}^{\scriptscriptstyle(i)}$ are below
${\mathcal{M}}_{\star}(x_{i})$, i.e., ${\mathcal{M}}_{\star}$ shifted to
$x_{i}$. If additionally for all $i$ there is no edge between
$\Xi_{\leq{\mathcal{M}}_{\star}}^{\scriptscriptstyle(\mathrm{in},i)}$ to a
vertex outside $\Lambda_{\tilde{n}}^{\scriptscriptstyle(i)}$, then the
component ${\mathcal{C}}_{\tilde{n}}^{\scriptscriptstyle(i)}$ is not the giant
and is larger than $k_{n,\varepsilon}$. Taking complements we obtain that

$\big{\\{}|{\mathcal{C}}_{n}^{\scriptscriptstyle(2)}|\leq\underline{k}_{n,\varepsilon}\\}\subseteq\\{\exists
i\leq
m_{n}:\Xi_{\leq{\mathcal{M}}_{\star}}^{\scriptscriptstyle(\mathrm{in},i)}\leftrightarrow\Xi_{n}\setminus\Xi_{\tilde{n}}^{\scriptscriptstyle(i)}\\}\cup\Bigg{(}\bigcap_{i\leq
m_{n}}\big{(}\neg{\mathcal{A}}_{\mathrm{good}}^{\scriptscriptstyle(i)}\big{)}\Bigg{)}.$

By translation invariance, a union bound, and the independence of
$({\mathcal{G}}_{\tilde{n}}^{\scriptscriptstyle(i)})_{i\leq m_{n}}$,

$\displaystyle\mathbb{P}\big{(}|{\mathcal{C}}_{n}^{\scriptscriptstyle(2)}|\leq\underline{k}_{n,\varepsilon}\big{)}$
$\displaystyle\leq
m_{n}\mathbb{P}\big{(}\Xi_{\leq{\mathcal{M}}_{\star}}^{\scriptscriptstyle\mathrm{in}}\leftrightarrow\Xi_{n}\setminus\Xi_{\tilde{n}}\\}\big{)}+\big{(}1-\mathbb{P}\big{(}{\mathcal{A}}_{\mathrm{good}}^{\scriptscriptstyle(1)}\big{)}\big{)}^{m_{n}}=:T_{1}+T_{2}.$
(7.26)

By the definitions in (7.11) and (7.6), each
$u\in\Xi_{\leq{\mathcal{M}}_{\star}}^{\scriptscriptstyle\mathrm{in}}$ has
$\|x_{u}\|\leq r_{k}$ with $k=k_{n,\varepsilon}$, and mark $w_{u}\leq
f_{\star}(r_{\underline{k}_{n,\varepsilon}})$,
($f_{\star}=f_{\gamma_{\mathrm{\neg nn}}}$ is from below (7.13)), hence
$\Xi_{\leq{\mathcal{M}}_{\star}}^{\scriptscriptstyle\mathrm{in}}\subseteq\Xi_{(2r_{\underline{k}_{n,\varepsilon}})^{d}}[1,f_{\star}(r_{\underline{k}_{n,\varepsilon}}))\subseteq\Xi_{\tilde{n}}$,
whenever
$(2r_{\underline{k}_{n,\varepsilon}})^{d}=k_{n,\varepsilon}M_{\mathrm{in}}2^{d}<n^{\vartheta}$,
which holds whenever $\varepsilon<\vartheta$ by (7.2). Hence we can bound
$T_{1}$ as

$T_{1}\leq
m_{n}\mathbb{P}\big{(}\Xi_{(2r_{\underline{k}_{n,\varepsilon}})^{d}}[1,f_{\star}(r_{\underline{k}_{n,\varepsilon}}))\leftrightarrow\Xi_{n}\setminus\Xi_{\tilde{n}}\\}\big{)}.$

We can directly apply Claim 6.6 on the right-hand side, i.e., setting there
$N:=\tilde{n}=n^{\vartheta}$ and
$n:=(2r_{\underline{k}_{n,\varepsilon}})^{d}=k_{n,\varepsilon}M_{\mathrm{in}}2^{d}$
(by (7.6)) and $\overline{w}:=f_{\star}(r_{\underline{k}_{n,\varepsilon}})$.
The profile $f_{\star}=f_{\gamma_{\mathrm{\neg nn}}}$ is defined below (7.13),
using (7.9) with exponent $\gamma_{\mathrm{\neg nn}}$ and
$r_{k}=(M_{\mathrm{in}}k)^{1/d}$ in (7.6), and finally $k_{n,\epsilon}$ from
(7.2) we obtain

$\overline{w}:=f_{\star}(r_{\underline{k}_{n,\varepsilon}})=C_{\beta}^{-\gamma_{\mathrm{\neg
nn}}d}r_{\underline{k}_{n,\varepsilon}}^{\gamma_{\mathrm{\neg
nn}}d}=C_{\beta}^{-\gamma_{\mathrm{\neg
nn}}d}M_{\mathrm{in}}^{\gamma_{\mathrm{\neg
nn}}}\underline{k}_{n,\varepsilon}^{\gamma_{\mathrm{\neg nn}}}$

Condition (6.26) holds since $N=n^{\vartheta}\gg
k_{n,\varepsilon}\gg\overline{w}$. Then Claim 6.6 yields for some $C>0$

$\displaystyle T_{1}$ $\displaystyle\leq n^{1-\vartheta}C_{\ref{claim:edge-
long}}f_{\star}(r_{\underline{k}_{n,\varepsilon}})^{c_{\ref{claim:edge-
long}}}(2r_{\underline{k}_{n,\varepsilon}})^{d}n^{-\vartheta\min\\{\alpha-1,\tau-2\\}}\big{(}1+\mathbbm{1}_{\alpha=\tau-1}\log(n^{\vartheta})\big{)}$
$\displaystyle\leq
C\log(n)\cdot\underline{k}_{n,\varepsilon}^{1+\gamma_{\mathrm{\neg
nn}}c_{\ref{claim:edge-long}}}\cdot n^{1-\vartheta\min\\{\alpha,\tau-1\\}}.$

Since $\underline{k}_{n,\varepsilon}$ in (7.2) is at most $n^{\varepsilon}$,
as long as $1-\vartheta\min(\alpha,\tau-1)<0$, we can choose $\varepsilon>0$
in (7.2) sufficiently small such that for any
$\delta\in(0,\vartheta\min\\{\alpha,\tau-1\\}-1)$, for all $n$ sufficiently
large,

$T_{1}\leq n^{-\delta}.$ (7.27)

We turn to bound $T_{2}$ in (7.26) using $(1-x)^{m_{n}}\leq\exp(-m_{n}x)$,
where we apply (7.25) on
$x=\mathbb{P}({\mathcal{A}}_{\mathrm{good}}^{\scriptscriptstyle(1)})$ to
obtain a lower bound on the exponent

$\displaystyle
m_{n}\mathbb{P}\big{(}{\mathcal{A}}_{\mathrm{good}}^{\scriptscriptstyle(1)}\big{)}$
$\displaystyle\geq(\rho^{2}/2)n^{1-\vartheta}\exp\big{(}-A^{\prime}\underline{k}_{n,\varepsilon}^{\max({\mathcal{Z}})}\log^{\mathfrak{m}_{\mathcal{Z}}-1}(\underline{k}_{n,\varepsilon})\big{)}$
$\displaystyle=(\rho^{2}/2)\exp\big{(}(1-\vartheta)\log(n)-A^{\prime}\underline{k}_{n,\varepsilon}^{\max({\mathcal{Z}})}\log^{\mathfrak{m}_{\mathcal{Z}}-1}(\underline{k}_{n,\varepsilon})\big{)}.$

In order to show $T_{2}\leq n^{-\delta}$ in (7.26), it is much stronger to
show that with $\mathfrak{m}({\mathcal{Z}})-1=\mathfrak{m}^{\prime}$

$\mbox{$\forall\varepsilon^{\prime}>0$, there exists $\varepsilon_{1}>0$ such
that for all $\varepsilon<\varepsilon_{1}$:
}A^{\prime}\underline{k}_{n,\varepsilon}^{\max({\mathcal{Z}})}\log^{\mathfrak{m}^{\prime}}(\underline{k}_{n,\varepsilon})<\varepsilon^{\prime}\log(n).$
(7.28)

We recall the definition of $\underline{k}_{n,\varepsilon}$ in (7.2). While
$k_{n,\varepsilon}$ may be as large as $n^{\varepsilon}$, only when
$\max({\mathcal{Z}})=0$, so effectively the expression in (7.28) is still
small. We formally check:

_Case 1. $\max({\mathcal{Z}})>0$._ We substitute
$\underline{k}_{n,\varepsilon}$ in the first row of (7.2) to (7.28)

$\displaystyle
A^{\prime}\underline{k}_{n,\varepsilon}^{\max({\mathcal{Z}})}\log^{\mathfrak{m}^{\prime}}(\underline{k}_{n,\varepsilon})=A^{\prime}\frac{\varepsilon\log(n)}{\log\log(n))^{\mathfrak{m}^{\prime}}}\log^{\mathfrak{m}^{\prime}}\Big{(}\big{(}\frac{\varepsilon\log(n)}{(\log\log(n))^{\mathfrak{m}^{\prime}}}\big{)}^{1/\max({\mathcal{Z}})}\Big{)}.$

The last factor is at most
$(\max{{\mathcal{Z}}})^{-\mathfrak{m}^{\prime}}(\log\log(n))^{\mathfrak{m}^{\prime}}$,
and (7.28) follows whenever
$\varepsilon<\varepsilon^{\prime}(\max{\mathcal{Z}})^{\mathfrak{m}^{\prime}}/A^{\prime}.$

_Case 2. $\max({\mathcal{Z}})=0$._ We substitute
$\underline{k}_{n,\varepsilon}$ in the second row of (7.2) to (7.28)

$\displaystyle
A^{\prime}\underline{k}_{n,\varepsilon}^{\max({\mathcal{Z}})}\log^{\mathfrak{m}^{\prime}}(\underline{k}_{n,\varepsilon})=A^{\prime}\big{(}\log\big{(}\exp\big{[}(\varepsilon\log(n))^{1/\mathfrak{m}^{\prime}}\big{]}\big{)}\big{)}^{\mathfrak{m}^{\prime}}=A^{\prime}\varepsilon\log(n),$

and (7.28) again follows. Choose now any
$\vartheta<1/\min\\{\alpha,\tau-1\\}$, and then combine (7.27) with $T_{2}\leq
n^{-\delta}$ to bound (7.26). This finishes the proof of (7.5) and hence
Proposition 7.1 subject to Lemma 7.2. ∎

###  7.2. Isolation

We aim to prove Lemma 7.2. We suppress the superscripts $(k)$ of events, and
leave it to the reader to verify that the results extend to the Palm-version
$\widetilde{\mathbb{P}}^{\scriptscriptstyle 0}_{\mathrm{io}}$ of
$\widetilde{\mathbb{P}}_{\mathrm{io}}$, which is defined in (7.18). We work
towards bounding the event in (7.19) via a few lemmas/claims, also related to
the back-of-the-envelope reasoning of Section 1.2. Recall
$f_{\gamma},M_{\gamma}$ from (7.9), (7.10), the PPPs in (7.11), and
$\zeta_{\mathrm{nn}}\\!=\\!\tfrac{d-1}{d}$ from (1.9).

######  Lemma 7.3 (Vertices above ${\mathcal{M}}_{\gamma}$).

Consider a supercritical i-KSRG under the conditions of Lemma 7.2. For each
$\gamma\in(0,1/(\sigma+1)]$, there exists a constant $C_{\ref{lemma:lower-
vertices-above}}>0$ such that

$\widetilde{\mathbb{E}}_{\mathrm{io}}\big{[}|\Xi_{>{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{in}}\cup\Xi_{>{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{out}}|\big{]}\leq\begin{dcases}C_{\ref{lemma:lower-
vertices-above}}r_{k}^{d(1-\gamma(\tau-1))},&\text{if
}1-\gamma(\tau-1)>\zeta_{\mathrm{nn}},\\\ C_{\ref{lemma:lower-vertices-
above}}r_{k}^{d\zeta_{\mathrm{nn}}}\log(r_{k}),&\text{if
}1-\gamma(\tau-1)=\zeta_{\mathrm{nn}},\\\ C_{\ref{lemma:lower-vertices-
above}}r_{k}^{d\zeta_{\mathrm{nn}}},&\text{if
}1-\gamma(\tau-1)<\zeta_{\mathrm{nn}}.\end{dcases}$ (7.29)

Recall the multiplicity $\mathfrak{m}$ of the maximum from (2.5).

######  Lemma 7.4 (Edges between vertices below ${\mathcal{M}}_{\gamma}$).

Consider a supercritical i-KSRG under the conditions of Lemma 7.2 with
$\alpha<\infty$. Let
${\mathcal{Y}}=\\{0,\alpha-(\tau-1),(\sigma+1)\alpha-2(\tau-1)\\}$. For each
$\gamma\in(0,1/(\sigma+1)]$ there exists a constant $C_{\ref{lemma:lower-
edges-below}}=C_{\ref{lemma:lower-edges-below}}(\rho)>0$ such that for any
realization of
$\Xi_{{\mathcal{R}}_{\mathrm{in}}}\cup\Xi_{{\mathcal{R}}_{\mathrm{out}}}$ that
satisfies ${\mathcal{A}}_{\mathrm{regular}}(\eta)$ in (7.17) for some
$\eta>0$,

$\widetilde{\mathbb{E}}_{\mathrm{io}}\big{[}|{\mathcal{E}}\big{(}\Xi_{\leq{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{in}},\Xi_{\leq{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{out}}\big{)}|\big{]}\leq\begin{dcases}C_{\ref{lemma:lower-
edges-
below}}r_{k}^{d(2-\alpha+\gamma\max({\mathcal{Y}}))}\log^{\mathfrak{m}({\mathcal{Y}})-1}(r_{k}),&\text{if
}2-\alpha+\gamma\max({\mathcal{Y}})>\zeta_{\mathrm{nn}},\\\
C_{\ref{lemma:lower-edges-
below}}r_{k}^{d\zeta_{\mathrm{nn}}}\log^{\mathfrak{m}({\mathcal{Y}})}(r_{k}),&\text{if
}2-\alpha+\gamma\max({\mathcal{Y}})=\zeta_{\mathrm{nn}},\\\
C_{\ref{lemma:lower-edges-below}}r_{k}^{d\zeta_{\mathrm{nn}}},&\text{if
}2-\alpha+\gamma\max({\mathcal{Y}})<\zeta_{\mathrm{nn}}.\end{dcases}$ (7.30)

###### Proof sketch of Lemmas 7.3 and 7.4.

We defer the lengthy integrals to the appendix on page B but we give some
intuition. The expectation $\widetilde{\mathbb{E}}_{\mathrm{io}}$ in (7.18) is
conditional on
$\Xi\cap({\mathcal{R}}_{\mathrm{in}}\cup{\mathcal{R}}_{\mathrm{out}})$ where
${\mathcal{R}}_{\mathrm{in}},{\mathcal{R}}_{\mathrm{out}}$ in (7.7) are
hyperrectangles below ${\mathcal{M}}_{\gamma}$ for all $\eta$, since
$w_{\eta}(\ell)$ is poly-logarithmic in $\ell$ in (7.1). Hence
$\Xi_{>{\mathcal{M}}_{\gamma}}$ is independent of the conditioning in
$\widetilde{\mathbb{E}}_{\mathrm{io}}$, so

$\widetilde{\mathbb{E}}_{\mathrm{io}}\big{[}|\Xi_{>{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{in}}\cup\Xi_{>{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{out}}|\big{]}=\mathbb{E}\big{[}|\Xi_{>{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{in}}\cup\Xi_{>{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{out}}|\big{]}=\mu_{\tau}\big{(}{\mathbb{R}}^{d+1}_{>{\mathcal{M}}_{\gamma}}\big{)},$
(7.31)

with the intensity $\mu_{\tau}$ in (2.2), and where we denote
${\mathbb{R}}^{d+1}_{>{\mathcal{M}}_{\gamma}}$ the points of
${\mathbb{R}}^{d+1}$ above the $d$-dimensional surface
${\mathcal{M}}_{\gamma}$, see (7.10). Define the hyperrectangle
${\mathcal{R}}^{\uparrow}:=[-2r_{k},2r_{k}]^{d}\times[r_{k}^{\gamma},\infty)$
and
$\mathrm{A}_{\beta}:=\\{x\in{\mathbb{R}}^{d},\|x\|\in[r_{k}-C_{\beta},r_{k}+C_{\beta}]\\}\times[1,\infty)$,
an annulus in ${\mathbb{R}}^{d}$ times all mark-coordinates. Then by
definition of $f_{\gamma}$ in (7.9),
$({\mathcal{R}}^{\uparrow}\cup\mathrm{A}_{\beta})\subseteq{\mathbb{R}}^{d+1}_{>{\mathcal{M}}_{\gamma}}$,
and
$\mu_{\tau}({\mathcal{R}}^{\uparrow}\cup\mathrm{A}_{\beta})=\Theta(r_{k}^{d(1-\gamma(\tau-1))}+r_{k}^{d-1})$.
Careful integration shows that the right-hand side of (7.31) is of the same
order, unless $1-\gamma(\tau-1)=(d-1)/d=\zeta_{\mathrm{nn}}$, when we get an
extra $\log(r_{k})$ factor.

For Lemma 7.4, we use independence of $\Xi$ in disjoint sets: inside and
outside of ${\mathcal{R}}_{\mathrm{in}},{\mathcal{R}}_{\mathrm{out}}$. When
counting edges with at least one point in
${\mathcal{R}}_{\mathrm{in}}\cup{\mathcal{R}}_{\mathrm{out}}$, we use that
$\Xi$ in these sets is regular, i.e.,
${\mathcal{A}}_{\mathrm{regular}}^{{\scriptscriptstyle(k)}}$ holds, and
outside these sets we integrate using the intensity $\mu_{\tau}$ in (7.29). We
explain now how the exponents of $r_{k}$ in (7.30) arise: the expected number
of edges between vertices of constant mark within constant distance of
$\partial{\mathcal{B}}_{kM_{\mathrm{in}}}$ is
$\Theta(r_{k}^{d-1})=\Theta(r_{k}^{d\zeta_{\mathrm{nn}}})$, which yields lower
bounds for all cases in (7.30). Let $0\leq\gamma_{1}\leq\gamma_{2}\leq\gamma$
such that $\sigma\gamma_{1}+\gamma_{2}\leq\sigma+1$. Using $\mu_{\tau}$ in
(2.2), the expected number of pairs
$(u,v)\in\Xi_{\leq{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{in}}\times\Xi_{\leq{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{out}}$
within distance $\Theta(r_{k})$ from
$\partial{\mathcal{B}}_{kM_{\mathrm{in}}}$, and marks
$w_{u}=\Theta(k^{\gamma_{1}}),w_{v}=\Theta(k^{\gamma_{2}})$ is

$\mathbb{E}[\mathrm{Pairs}(\gamma_{1},\gamma_{2})]:=\Theta\big{(}r_{k}^{d(1-\gamma_{1}(\tau-1))}\cdot
r_{k}^{d(1-\gamma_{2}(\tau-1))}\big{)}=\Theta\big{(}r_{k}^{d(2-(\gamma_{1}+\gamma_{2})(\tau-1))}\big{)}.$

The typical Euclidean distance between such vertices is $\Theta(r_{k})$.
Therefore, by the connection probability $\mathrm{p}$ in (2.3), a pair of such
vertices are connected with probability roughly
$\Theta(r_{k}^{d\alpha(\sigma\gamma_{1}+\gamma_{2}-1)})$, and hence there are

$\mathbb{E}[\mathrm{Edges}(\gamma_{1},\gamma_{2})]:=\mathbb{E}[\mathrm{Pairs}(\gamma_{1},\gamma_{2})]\cdot\Theta(r_{k}^{d\alpha(\sigma\gamma_{1}+\gamma_{2}-1)})=\Theta\big{(}r_{k}^{d(2-\alpha+\gamma_{1}(\sigma\alpha-(\tau-1))+\gamma_{2}(\alpha-(\tau-1)))}\big{)}.$
(7.32)

such edges in expectation. The proof below on page B reveals that the
expectation of
$|{\mathcal{E}}(\Xi_{\leq{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{in}},\Xi_{\leq{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{out}}\big{)}|$
can be bounded by maximizing (7.32) with respect to $\gamma_{1},\gamma_{2}$,
i.e., edges between vertices of mark $\Theta(k^{\gamma_{1}^{\star}})$ and
$\Theta(k^{\gamma_{2}^{\star}})$, with the optimal pair
$(\gamma_{1}^{\star},\gamma_{2}^{\star})$ dominate the whole expectation.
Depending on $(\sigma,\alpha,\tau)$, the optimal
$(\gamma_{1}^{\star},\gamma_{2}^{\star})$ is either

  * (ll)

$(0,0)$, pairs of low-mark vertices, which, when substituted in (7.32) yields
$0\in{\mathcal{Y}}$ in (7.30);

  * (lh)

$(0,\gamma)$, pairs of one low-mark vertex and one high-mark vertex, which,
when substituted in (7.32) yields $\alpha-(\tau-1)\in{\mathcal{Y}}$ in (7.30);

  * (hh)

$(\gamma,\gamma)$, pairs of high-mark vertices, which, when substituted in
(7.32) yields $(\sigma+1)\alpha-2(\tau-1)\in{\mathcal{Y}}$ in (7.30);

or the convex combinations of these if the maximum in ${\mathcal{Y}}$ is non-
unique, leading to the poly-logarithmic correction terms in (7.30). These
three options of $(\gamma_{1}^{\star},\gamma_{2}^{\star})$, combined with the
$\Theta(r_{k}^{d-1})$ many low-mark edges close to the boundary explain what
we call the _dominant types of connectivity_ in Section 1.2. We remark that
without the assumption that $\gamma\leq 1/(\sigma+1)$, the optimal solution in
(hh) would change to $(\gamma\wedge 1/(\sigma+1),\gamma\wedge 1/(\sigma+1))$.
∎

Lemmas 7.3 and 7.4 hold for any $\gamma\leq 1/(\sigma+1)$. Now we show that
setting $\gamma=\gamma_{\mathrm{\neg nn}}$ from (7.13) minimizes the sum of
(7.29) (increasing in $\gamma$) and (7.30) (decreasing in $\gamma$).

######  Claim 7.5 (The optimally-suppressed mark-profile).

Consider a supercritical i-KSRG under the conditions of Lemma 7.2. Let
${\mathcal{Z}}=\\{\zeta_{\mathrm{ll}},\zeta_{\mathrm{lh}},\zeta_{\mathrm{hh}},\zeta_{\mathrm{nn}}\\}$.
There exists a constant $C_{\ref{lemma:zeta-opt-other}}>0$ such that when
$\alpha<\infty$

$\widetilde{\mathbb{E}}_{\mathrm{io}}\big{[}|\Xi_{>{\mathcal{M}}_{\star}}^{\scriptscriptstyle\mathrm{in}}\cup\Xi_{>{\mathcal{M}}_{\star}}^{\scriptscriptstyle\mathrm{out}}|\big{]}\leq
C_{\ref{lemma:zeta-opt-
other}}\widetilde{\mathbb{E}}_{\mathrm{io}}\big{[}|{\mathcal{E}}\big{(}\Xi_{\leq{\mathcal{M}}_{\star}}^{\scriptscriptstyle\mathrm{in}},\Xi_{\leq{\mathcal{M}}_{\star}}^{\scriptscriptstyle\mathrm{out}}\big{)}|\big{]}\leq
C_{\ref{lemma:zeta-opt-other}}C_{\ref{lemma:lower-edges-
below}}r_{k}^{d\max({\mathcal{Z}})}\log^{\mathfrak{m}_{\mathcal{Z}}-1}(r_{k}),$
(7.33)

and when $\alpha=\infty$

$\widetilde{\mathbb{E}}_{\mathrm{io}}\big{[}|\Xi_{>{\mathcal{M}}_{\star}}^{\scriptscriptstyle\mathrm{in}}\cup\Xi_{>{\mathcal{M}}_{\star}}^{\scriptscriptstyle\mathrm{out}}|\big{]}\leq
C_{\ref{lemma:zeta-opt-
other}}r_{k}^{d\max({\mathcal{Z}})}\log^{\mathfrak{m}_{\mathcal{Z}}-1}(r_{k}).$
(7.34)

We postpone the proof, based on elementary rearrangements of the formulas of
$\zeta_{\mathrm{ll}},\zeta_{\mathrm{lh}},\zeta_{\mathrm{hh}},$ and
$\zeta_{\mathrm{hh}}$, to the appendix on page C. We proceed with a claim
towards bounding the event
${\mathcal{A}}_{\mathrm{noedge}}^{\scriptscriptstyle(k)}$ in (7.12), its proof
explaining the restriction $\gamma\leq 1/(\sigma+1)$ in $\gamma_{\mathrm{\neg
nn}}$ in (7.13). We recall $\mathrm{p}(u,v)$ defined in (2.3).

######  Claim 7.6 (Cross-edge probability bounds).

Consider a supercritical i-KSRG under the conditions of Lemma 7.2, and assume
$\gamma\leq 1/(\sigma+1)$. For any
$(u,v)\in\Xi_{\leq{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{in}}\times\Xi_{\leq{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{out}}$

$\mathrm{p}(u,v)\leq\begin{dcases}2^{-\alpha},&\text{if }\alpha<\infty,\\\
0,&\text{if }\alpha=\infty.\end{dcases}$ (7.35)

###### Proof.

We show that whenever $u,v$ are below ${\mathcal{M}}_{\gamma}$ and on
different sides of $\partial{\mathcal{B}}_{kM_{\mathrm{in}}}$, then
$\beta\kappa_{1,\sigma}(w_{u},w_{v})/$ $\|x_{u}-x_{v}\|^{d}\leq 1/2$. By
definition of $\mathrm{p}$ in (2.3), this directly implies (7.35). To see this
bound, for $\sigma\geq 0$ the connection probability $\mathrm{p}$ is
increasing in the marks. Therefore, without loss of generality we will assume
that $u\in{\mathcal{B}}_{kM_{\mathrm{in}}}$ and
$v\notin{\mathcal{B}}_{kM_{\mathrm{in}}}$ fall exactly on
${\mathcal{M}}_{\gamma}$, and that $k$ is large enough that $r_{k}\geq
C_{\beta}$. Since $f_{\gamma}$ in (7.9) switches formula at $r_{k}$ outside
$\partial{\mathcal{B}}_{kM_{\mathrm{in}}}$, we distinguish two cases.

_Case 1. Assume $|\|x_{v}\|-r_{k}|\leq r_{k}$._ Since $f_{\gamma}$ in (7.9)
and the explanation below (7.12) both $u,v$ are at least $C_{\beta}$ distance
from $\partial{\mathcal{B}}_{kM_{\mathrm{in}}}$. Thus, for any $(u,v)$, there
exist $t\geq 2C_{\beta}$, $\nu\in(0,1)$ such that
$\|x_{u}-\partial{\mathcal{B}}_{kM_{\mathrm{in}}}\|=r_{k}-\|x_{u}\|=(1-\nu)t$,
and $\|x_{v}-\partial{\mathcal{B}}_{kM_{\mathrm{in}}}\|=\|x_{v}\|-r_{k}=\nu
t$. By the triangle inequality $\|x_{u}-x_{v}\|\geq t$. Hence,
$w_{v}=C_{\beta}^{-\gamma d}(\nu t)^{\gamma d},w_{u}=C_{\beta}^{-\gamma
d}((1-\nu)t)^{\gamma d}$ by assuming $u,v$ being on ${\mathcal{M}}_{\gamma}$
and $f_{\gamma}$ in (7.9). Using $\kappa_{1,\sigma}$ in (1.10), and
$\nu\in(0,1)$

$\displaystyle\beta\frac{\kappa_{1,\sigma}(w_{u},w_{v})}{\|x_{u}-x_{v}\|^{d}}$
$\displaystyle\leq\beta C_{\beta}^{-\gamma
d(\sigma+1)}\frac{\max\\{(1-\nu)t,\nu t\\}^{\gamma d}\min\\{(1-\nu)t,\nu
t\\}^{\sigma\gamma d}}{t^{d}}\leq\beta C_{\beta}^{-\gamma
d(\sigma+1)}t^{d(\gamma(\sigma+1)-1)}.$

The right-hand side is non-increasing in $t$ _whenever_
$\gamma\leq\frac{1}{\sigma+1}$. Since $t\geq 2C_{\beta}\geq C_{\beta}$, and
$C_{\beta}=(2\beta)^{1/d}$

$\beta\frac{\kappa_{1,\sigma}(w_{u},w_{v})}{\|x_{u}-x_{v}\|^{d}}\leq\beta
C_{\beta}^{-d\gamma(\sigma+1)}C_{\beta}^{d(\gamma(\sigma+1)-1)}=\beta
C_{\beta}^{-d}=1/2.$

_Case 2. Assume $|\|x_{v}\|-r_{k}|>r_{k}$._ In this case
$\|x_{v}-\partial{\mathcal{B}}_{kM_{\mathrm{in}}}\|\geq\|x_{u}-\partial{\mathcal{B}}_{kM_{\mathrm{in}}}\|$,
and
$\|x_{v}-x_{u}\|\geq\|x_{v}-\partial{\mathcal{B}}_{kM_{\mathrm{in}}}\|=|\|x_{v}\|-r_{k}|$.
Since $f_{\gamma}(z)$ is increasing, $w_{v}=f_{\gamma}(|\|x_{v}\|-r_{k}|)\geq
f_{\gamma}(r_{k})\geq f_{\gamma}(|\|x_{u}\|-r_{k}|)=w_{u}$. We obtain by
definition of $\kappa_{1,\sigma}$ in (1.10) and (7.9)

$\displaystyle\beta\frac{\kappa_{1,\sigma}(w_{u},w_{v})}{\|x_{u}-x_{v}\|^{d}}\\!\leq\\!\beta\frac{f_{\gamma}(|\|x_{v}\|-r_{k}|)f_{\gamma}(|\|x_{u}\|-r_{k}|)^{\sigma}}{|\|x_{v}\|-r_{k}|^{d}}$
$\displaystyle\\!\leq\\!\beta C_{\beta}^{-\gamma
d(\sigma+1)}|\|x_{v}\|-r_{k}|^{d}r_{k}^{-d(1-\gamma)}r_{k}^{\sigma\gamma
d}|\|x_{v}\|-r_{k}|^{-d}$ $\displaystyle=\beta C_{\beta}^{-\gamma
d(\sigma+1)}r_{k}^{-d(1-\gamma(\sigma+1))}\leq 1/2.$

where to obtain the second inequality we used that
$f_{\gamma}(|\|x_{u}\|-r_{k}|)^{\sigma}\leq r_{k}^{\sigma\gamma d}$, and to
obtain the second row the power of $|\|x_{v}\|-r_{k}|$ canceled. The bound
$1/2$ follows because $r_{k}\geq C_{\beta}$, $\gamma\leq 1/(\sigma+1)$ and
$C_{\beta}=(2\beta)^{1/d}$. ∎

We are ready to prove Lemma 7.2.

###### Proof of Lemma 7.2.

We will obtain a lower bound on
$\widetilde{\mathbb{P}}_{\mathrm{io}}\big{(}\\{\Xi\\!\leq\\!{\mathcal{M}}_{\gamma}\\}\cap{\mathcal{A}}_{\mathrm{noedge}}^{\scriptscriptstyle(k)}(\gamma)\big{)}$
for _arbitrary_ $\gamma\in(0,1/(\sigma+1)]$. Then we maximize this bound by
setting $\gamma=\gamma_{\mathrm{\neg nn}}$, yielding the event
${\mathcal{A}}_{\mathrm{isolation}}^{\scriptscriptstyle(k)}$ in (7.14) and the
bound in Lemma 7.2. The events $\\{\Xi\\!\leq\\!{\mathcal{M}}_{\gamma}\\}$ and
${\mathcal{A}}_{\mathrm{noedge}}^{\scriptscriptstyle(k)}$ (defined in (7.12))
are independent of each other under $\widetilde{\mathbb{P}}_{\mathrm{io}}$ in
(7.18), since having no points above ${\mathcal{M}}_{\gamma}$ is independent
of the conditioning in $\widetilde{\mathbb{P}}_{\mathrm{io}}$ (since each
point in ${\mathcal{R}}_{\mathrm{in}}\cup{\mathcal{R}}_{\mathrm{out}}$ is
below ${\mathcal{M}}_{\gamma}$ if $k$ is sufficiently large), and
${\mathcal{A}}_{\mathrm{noedge}}^{\scriptscriptstyle(k)}$ only depends on
points of $\Xi$ below ${\mathcal{M}}_{\gamma}$. Hence

$\displaystyle\widetilde{\mathbb{P}}_{\mathrm{io}}\big{(}\\{\Xi\leq{\mathcal{M}}_{\gamma}\\}\cap{\mathcal{A}}_{\mathrm{noedge}}^{\scriptscriptstyle(k)}(\gamma)\big{)}=\widetilde{\mathbb{P}}_{\mathrm{io}}\big{(}\Xi\leq{\mathcal{M}}_{\gamma}\big{)}\widetilde{\mathbb{P}}_{\mathrm{io}}\big{(}{\mathcal{A}}_{\mathrm{noedge}}^{\scriptscriptstyle(k)}(\gamma)\big{)}.$
(7.36)

We analyze the two probabilities separately. For the first term, by the above
independence,

$\displaystyle\widetilde{\mathbb{P}}_{\mathrm{io}}\big{(}\Xi\leq{\mathcal{M}}_{\gamma}\big{)}=\widetilde{\mathbb{P}}_{\mathrm{io}}\big{(}|\Xi_{>{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{in}}\cup\Xi_{>{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{out}}|=0\big{)}$
$\displaystyle=\exp\big{(}-\widetilde{\mathbb{E}}_{\mathrm{io}}\big{[}|\Xi_{>{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{in}}\cup\Xi_{>{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{out}}|\big{]}\big{)},$
(7.37)

for which we will use Lemma 7.3 shortly. We now turn to the second factor in
(7.36). By definition of
${\mathcal{A}}_{\mathrm{noedge}}^{\scriptscriptstyle(k)}$ in (7.12), and using
the conditional independence of edges,

$\widetilde{\mathbb{P}}_{\mathrm{io}}\big{(}{\mathcal{A}}_{\mathrm{noedge}}^{\scriptscriptstyle(k)}(\gamma)\big{)}=\widetilde{\mathbb{E}}_{\mathrm{io}}\Bigg{[}\prod_{u\in\Xi_{\leq{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{in}},v\in\Xi_{\leq{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{out}}}(1-\mathrm{p}(u,v))\Bigg{]}.$
(7.38)

We distinguish now the two cases $\alpha=\infty$ and $\alpha<\infty$.

_Case $\alpha=\infty$_. By Claim 7.6, since $\gamma\leq 1/(\sigma+1)$ by
assumption, $\mathrm{p}(u,v)=0$ for each factor, hence
$\widetilde{\mathbb{P}}_{\mathrm{io}}\big{(}{\mathcal{A}}_{\mathrm{noedge}}^{\scriptscriptstyle(k)}(\gamma)\big{)}=1$.
Since the right-hand side of (7.37) is increasing in $\gamma$, we set
$\gamma=1/(\sigma+1)=\gamma_{\mathrm{\neg nn}}$ when $\alpha=\infty$ by
(7.13). Combining then (7.36) with (7.37) and Claim 7.5, it follows that

$\displaystyle\widetilde{\mathbb{P}}_{\mathrm{io}}\big{(}\\{\Xi\leq{\mathcal{M}}_{\star}\\}\cap{\mathcal{A}}_{\mathrm{noedge}}^{\scriptscriptstyle(k)}(\gamma_{\neg\mathrm{nn}})\big{)}$
$\displaystyle\geq\exp\big{(}-C_{\ref{lemma:lower-vertices-
above}}r_{k}^{d\max({\mathcal{Z}})}\log^{\mathfrak{m}_{\mathcal{Z}}-1}(r_{k})\big{)},$

finishing the proof of the lemma for $\alpha=\infty$.

_Case $\alpha<\infty$_. By Claim 7.6, for all
$(u,v)\in\Xi_{\leq{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{in}}\times\Xi_{\leq{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{out}}$
it holds that $1-\mathrm{p}(u,v)\geq 1-2^{-\alpha}$ since $\gamma\leq
1/(\sigma+1)$ by assumption. Hence, there exists a constant $c>0$ such that
for all such $(u,v)$ we have
$1-\mathrm{p}(u,v)\geq\exp(-c\cdot\mathrm{p}(u,v))$. Using this in (7.38) and
that $s\mapsto\exp(-s)$ is a convex function, Jensen’s inequality gives a
lower bound in terms of the expected number of edges between vertices below
${\mathcal{M}}_{\gamma}$, i.e.,

$\displaystyle\widetilde{\mathbb{P}}_{\mathrm{io}}\big{(}{\mathcal{A}}_{\mathrm{noedge}}^{\scriptscriptstyle(k)}(\gamma)\big{)}\geq\widetilde{\mathbb{E}}_{\mathrm{io}}\Bigg{[}\exp\Bigg{(}-c\sum_{\begin{subarray}{c}u\in\Xi_{\leq{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{in}},\\\
v\in\Xi_{\leq{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{out}}\end{subarray}}\mathrm{p}(u,v)\Bigg{)}\Bigg{]}$
$\displaystyle\geq\exp\Bigg{(}-c\,\widetilde{\mathbb{E}}_{\mathrm{io}}\Bigg{[}\sum_{\begin{subarray}{c}u\in\Xi_{\leq{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{in}},\\\
v\in\Xi_{\leq{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{out}}\end{subarray}}\mathrm{p}(u,v)\Bigg{]}\Bigg{)}$
$\displaystyle=\exp\big{(}-c\,\widetilde{\mathbb{E}}_{\mathrm{io}}\big{[}|{\mathcal{E}}\big{(}\Xi_{\leq{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{in}},\Xi_{\leq{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{out}}\big{)}|\big{]}\big{)}.$
(7.39)

Together with (7.36) and (7.37), we obtain that

$\displaystyle\widetilde{\mathbb{P}}_{\mathrm{io}}\big{(}\\{\Xi\leq{\mathcal{M}}_{\gamma}\\}\cap{\mathcal{A}}_{\mathrm{noedge}}^{\scriptscriptstyle(k)}(\gamma)\big{)}$
$\displaystyle\geq\exp\Big{(}-\big{(}\widetilde{\mathbb{E}}_{\mathrm{io}}\big{[}|\Xi_{>{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{in}}\cup\Xi_{>{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{out}}|\big{]}+c\,\widetilde{\mathbb{E}}_{\mathrm{io}}\big{[}|{\mathcal{E}}\big{(}\Xi_{\leq{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{in}},\Xi_{\leq{\mathcal{M}}_{\gamma}}^{\scriptscriptstyle\mathrm{out}}\big{)}|\big{]}\big{)}\Big{)}.$
(7.40)

By Lemmas 7.3 and 7.4, the first expectation is decreasing, while the second
one is increasing in $\gamma$ on the right-hand side. By (7.33) in Lemma 7.5,
when we set $\gamma=\gamma_{\mathrm{\neg nn}}$ in (7.13), we obtain that
(7.40) turns into (7.19), finishing the proof of Lemma 7.2 when
${\mathcal{V}}=\Xi$. ∎

###### Proof of Lemma 7.2 for i-KSRGs on ${\mathbb{Z}}^{d}$.

We explain how to adjust the proof to i-KSRGs on ${\mathbb{Z}}^{d}$ using the
assumption $\min\\{p,p\beta^{\alpha}\\}<1$ in Lemma 7.2. Since the vertex set
${\mathcal{V}}={\mathbb{Z}}^{d}$, the event $\Xi\leq{\mathcal{M}}_{\gamma}$ as
defined in (7.10) would never hold, since ${\mathbb{Z}}^{d}$ does have points
within distance $C_{\beta}$ from $\partial{\mathcal{B}}_{kM_{\mathrm{in}}}$
(see $f_{\gamma}$ in (7.9) and the reasoning below (7.12)). Hence, we must
adjust the definition of $f_{\gamma}$ within distance $C_{\beta}$ of
$\partial{\mathcal{B}}_{kM_{\mathrm{in}}}$ to be any constant $c>1$. Then the
upper bound $2^{-\alpha}$ on $\mathrm{p}(u,v)$ in Claim 7.6 can be replaced by
$p<1$ (the edge-retention probability for vertices at distance 1 in (2.3)),
that only affects constant prefactors in (7.39) and (7.40).

The proofs of the preliminary Lemmas 7.3 and 7.4 remain valid by replacing
concentration bounds for Poisson random variables by concentration bounds for
Binomial random variables, and replacing integrals over ${\mathbb{R}}^{d}$ by
summations over ${\mathbb{Z}}^{d}$. The proof of Claim 7.5 remains verbatim
valid. ∎

###### Proofs of Theorems 2.2, 2.4, and Corollary 2.3.

Proposition 7.1 yields the lower bounds for both theorems, since its condition
(7.3) on having a large enough component on restricted marks occurs with
positive probability by Proposition 5.14, under the assumption
$\tau<2+\sigma$. The assumption $\tau>2$ is necessary to have a locally finite
graph. Propositions 5.1 and 6.1 imply the upper bounds (i.e., part (ii)) in
Theorems 2.4, 2.2, and Corollary 2.3, respectively. The conditions in
Proposition 6.1 are satisfied by Propositions 5.1, 5.12 and Corollary 5.11
that all hold for the Palm-version $\mathbb{P}^{\scriptscriptstyle x}$ of
$\mathbb{P}$ as well. The condition $\tau<2+\sigma$ is needed for Claim 5.5,
which yields that a constant proportion of high-mark vertices, i.e., vertices
having mark $\Omega(k^{\gamma_{\mathrm{hh}}})$, is connected by an edge to
another high-mark vertex. This allows to construct a backbone of high-mark
vertices (Lemma 5.2), and to merge components of size at least $k$ with the
backbone via a high-mark vertex in (5.42). The additional condition
$\sigma\leq\tau-1$ in Theorems 2.2 and 2.4 is needed in the cover-expansion
step (Lemma 5.9) in the proof of Proposition 5.1. ∎

It remains to prove Theorem 2.5, whose proof is based on Lemma 7.2.

###### Proof of Theorem 2.5.

For $\rho\geq 1$ the statement is trivial. There exists a constant $C>0$ such
that for any $\rho\in(0,1)$ and $n\geq 1$ a box of volume $n$ is contained in
the union of $\lceil C/\rho\rceil$ (partially overlapping) balls of volume
$n\rho/2$. Fix $\rho\in(0,1)$, and write $\Xi^{\scriptscriptstyle(i)}$ for the
vertices in the $i$-th ball of such a cover of balls of volume $n\rho/2$.
Recall that $|{\mathcal{E}}(A,B)|$ denotes the number of edges between the
sets $A,B$. Then

$\\{|{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}|<\rho
n\\}\supseteq\bigcap_{i\leq\lceil
C/\rho\rceil}\\{|{\mathcal{E}}(\Xi^{\scriptscriptstyle(i)},\Xi\setminus\Xi^{\scriptscriptstyle(i)})|=0\\}\cap\\{|\Xi^{\scriptscriptstyle(i)}|<\rho
n\\}$ (7.41)

Indeed, on the event on the right-hand side, each connected component of
${\mathcal{G}}_{n}$ is fully contained in some ball (or the intersection of
some balls) with at most $\rho n$ vertices. We apply an FKG inequality to
bound the probability of intersection from below.

We give a (natural) definition of increasing events, using the collection
$\Psi$ from Definition 5.3 that encodes the presence of edges using a set of
uniform random variables $\Psi_{\Xi}=\\{\varphi_{u,v}:u,v\in\Xi\\}$. We say
that a function $f(\Xi,\Psi_{\Xi})$ defined on the marked vertex set $\Xi$ and
edge-variable set $\Psi_{\Xi}$ is _increasing_ if it is non-decreasing in
$\Xi$ with respect to set inclusion (formally, if
$\Xi^{\prime}\supseteq\Xi,\Psi_{\Xi^{\prime}}\supseteq\Psi_{\Xi}$, then
$f(\Xi^{\prime},\Psi_{\Xi^{\prime}})\geq f(\Xi,\Psi_{\Xi})$ holds), as well as
coordinate-wise non-increasing with respect to the edge variables (formally,
if $\Psi^{\prime}_{\Xi}$ satisfies $\varphi^{\prime}_{u,v}\leq\varphi_{u,v}$
for all $u,v\in{\binom{\Xi}{2}}$, then $f(\Xi,\Psi^{\prime}_{\Xi})\geq
f(\Xi,\Psi_{\Xi})$ holds). Intuitively this means that more vertices and edges
increase the value of $f$. Similarly to [18], we obtain that for two
increasing functions $f_{1}$, $f_{2}$,

$\displaystyle\mathbb{E}[f_{1}(\Xi,\Psi_{\Xi})\cdot
f_{2}(\Xi,\Psi_{\Xi})]=\mathbb{E}\big{[}\mathbb{E}[f_{1}(\Xi,\Psi_{\Xi})\cdot
f_{2}(\Xi,\Psi_{\Xi})\mid\Xi]\big{]}$
$\displaystyle\geq\mathbb{E}\big{[}\mathbb{E}[f_{1}(\Xi,\Psi_{\Xi})\mid\Xi]\cdot\mathbb{E}[f_{2}(\Xi,\Psi_{\Xi})\mid\Xi]\big{]}$
$\displaystyle\geq\mathbb{E}[f_{1}(\Xi,\Psi_{\Xi})\big{]}\cdot\mathbb{E}[f_{2}(\Xi,\Psi_{\Xi})],$

by applying FKG to the random graph conditioned to have $\Xi$ as its vertex
set for the first inequality, and then FKG for point processes for the second
inequality [52, Theorem 20.4]. We say that an event ${\mathcal{A}}$ is
decreasing iff the function $-\mathbbm{1}_{\mathcal{A}}$ is increasing. It
follows that for decreasing events ${\mathcal{A}},{\mathcal{A}}^{\prime}$

$\mathbb{P}\big{(}{\mathcal{A}}\cap{\mathcal{A}}^{\prime}\big{)}=\mathbb{E}[(-\mathbbm{1}_{{\mathcal{A}}}(\Xi,\Psi_{\Xi}))\cdot(-\mathbbm{1}_{{\mathcal{A}}^{\prime}}(\Xi,\Psi_{\Xi}))]\geq\mathbb{E}[\mathbbm{1}_{{\mathcal{A}}}(\Xi,\Psi_{\Xi})]\cdot\mathbb{E}[\mathbbm{1}_{{\mathcal{A}}^{\prime}}(\Xi,\Psi_{\Xi})]=\mathbb{P}\big{(}{\mathcal{A}}\big{)}\cdot\mathbb{P}\big{(}{\mathcal{A}}^{\prime}\big{)}.$
(7.42)

Observe that the events on the right-hand side in (7.41) are all decreasing
(adding vertices/edges make the events less likely to occur) so that (7.42)
applies. Hence,

$\mathbb{P}\big{(}|{\mathcal{C}}_{n}^{\scriptscriptstyle(1)}|<\rho
n\big{)}\geq\prod_{i\leq\lceil
C/\rho\rceil}\mathbb{P}\big{(}|{\mathcal{E}}(\Xi^{\scriptscriptstyle(i)},\Xi\setminus\Xi^{\scriptscriptstyle(i)})|=0\big{)}\cdot\mathbb{P}\big{(}|\Xi^{\scriptscriptstyle(i)}|<\rho
n\big{)}.$ (7.43)

Since each ball has volume $n\rho/2$, the event
$\\{|\Xi^{\scriptscriptstyle(i)}|<\rho n\\}$ holds with probability at least
$1/2$ by concentration inequalities for Poisson random variables (Lemma C.1
for $x=2$). To bound
$\mathbb{P}\big{(}|{\mathcal{E}}(\Xi^{\scriptscriptstyle(i)},\Xi\setminus\Xi^{\scriptscriptstyle(i)})|=0\big{)}$,
we consider the optimally-suppressed mark profile translated to the center of
the $i$-th ball, with $kM_{\mathrm{in}}$ replaced by $n\rho/2$. We restrict
$\Xi$ to be below the mark profile, and to have no edges between
$\Xi^{\scriptscriptstyle(i)}$ and $\Xi\setminus\Xi^{\scriptscriptstyle(i)}$.