2.1. The model

Scale-free percolation in continuous space is a random variable on the space of all simple spatial undirected random graphs with vertices in $\mathbb{R}^d$. To construct an instance of the graph $G=(V,E)$ we proceed in three steps.

1. • We sample the nodes V of the graph according to a homogeneous Poisson point process in $\mathbb{R}^d$ of intensity ${\nu}>0$. We let P denote its law and let E denote the expectation with respect to P. A configuration of the point process will be denoted by $\eta\in(\mathbb{R}^d)^{\mathbb{N}}$.

2. • We assign to each vertex $x\in\eta$ a random weight $W_x>0$. The weights are independent and identically distributed (i.i.d.) with law P. The distribution function F associated with P is regularly varying at infinity with exponent $\tau-1$, that is,

   \begin{equation*} 1-F(w)=P(W>w)=w^{-(\tau-1)}L(w),\end{equation*}
   where $ L(\!\cdot\!)$ is a function that is slowly varying at $+\infty$.

3. • We finally draw the edges E of the random graph via a percolation process. For every pair of points $x,y\in\eta$ with weights $W_x,W_y$, the undirected edge (x, y) is present with probability

   (1) \begin{equation}1-{\textrm{e}}^{{-{{W_x W_y}/{\|x-y\|^{\alpha}}}}} ,\end{equation}
   where $\|\cdot\|$ denotes the Euclidean norm in $\mathbb{R}^d$ and $\alpha>0$ is a parameter. When two vertices x, y are connected by an edge we write $x\leftrightarrow y$ (if they are not connected we write $x\not\leftrightarrow y$). For the event that a vertex x is connected to some other point in a region $A\subseteq\mathbb{R}^d$, we write $x\leftrightarrow A$.

For a given realization $\eta$ of the Poisson point process we will often perform the last two steps at once under the quenched law $ P^\eta$, with $E^\eta$ denoting the associated expectation. This is the law of the graph once we have fixed $\eta$. We write $\mathbb{P}$ for the annealed law of the graph, i.e. $\mathbb{P}=\textrm{P}\times P^\eta$, and $\mathbb{E}$ for the corresponding expectation. For $x\in\mathbb{R}^d$, we let $\textrm{P}_x$ indicate the law of the Poisson point process conditioned on having a point in x; analogously, for $x,y\in\mathbb{R}^d$, the measure $\textrm{P}_{x,y}$ is obtained by conditioning on having one point in x and one in y. We will not enter into the details of (n-fold) Palm measures, but we point out that in both cases the conditioning does not influence the rest of the Poisson point process; see e.g. [Reference Daley and Vere-Jones12, Chapter 13]. Consequently we will let $\mathbb{P}_x =\textrm{P}_x\times P^\eta=\textrm{P}\times P^{\eta\cup \{x\}}$ be the law of the graph conditioned on having a point at $x\in\mathbb{R}^d$ and $\mathbb{E}_x$ the associated expectation. Analogously we will write $\mathbb{P}_{x,y}=\textrm{P}_{x,y}\times P^\eta=\textrm{P}\times P^{\eta\cup \{x\}\cup\{y\}}$ and $\mathbb{E}_{x,y}$ for the expectation with respect to $\mathbb{P}_{x,y}$.

We let 0 denote the origin of $\mathbb{R}^d$. We let $B_n$ be the d-dimensional box of side-length $n>0$ centered at 0. We let $V_n=V_n(\eta)$ be the set of points of $\eta$ that are in $B_n$ and let $N_n=N_n(\eta)$ be their number. Finally, $\mathcal{B}_r(x)$ denotes the Euclidean ball of radius $r>0$ centered at $x\in\mathbb{R}^d$.

2.2. Results on the degree

For each $x\in\eta$ we let $D_x\,:\!=\#\{y\in\eta\colon y\leftrightarrow x\}$ be the degree of a vertex x in the random graph, i.e. the random variable counting the number of neighbors of x. We start by showing that for certain values of the parameters the graph is almost surely degenerate, in the sense that all of its vertices have infinite degree.

We point out that the proof of Theorem 2.1 follows immediately from the analogous result of [Reference Deprez and Wüthrich14] when the weights have a Pareto distribution. We will provide an alternative proof that, besides covering more general distribution functions F, serves as a warm-up for the kind of techniques we will use more intensively later on, based on Campbell’s theorem.

We move to the analysis of the regime where the degree of the nodes is almost surely finite. In this case, we show that for almost all $\eta$ the graph is scale-free and the degree of all points follows a power law with the same exponent.

Theorem 2.2. Suppose that $\alpha>d$ and $\gamma\,:\!=\alpha(\tau-1)/d>1$. Then, for P-almost every $\eta$, we have that $s^\gamma P^\eta(D_x>s)$ is slowly varying for each $x\in\eta$. That is, there exists a slowly varying function $\ell(\!\cdot\!)=\ell(\cdot,\eta,x)$ such that

(2) \begin{equation}P^\eta(D_x>s)=s^{-\gamma}\ell(s) .\end{equation}

2.3. Results on the clustering coefficient

The average clustering coefficient is usually defined for finite graphs as the average of the local clustering coefficients. More precisely, for a finite undirected graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ and a node $x\in \mathcal{V}$, we define the local clustering coefficient at x as

\begin{equation*}{\textrm{CC}}(x)\,:\!=\dfrac{2\Delta_x}{D_x(D_x-1)} ,\end{equation*}

where $\Delta_x$ is the number of closed triangles in the graph that have x as one of their vertices (a closed triangle is a triplet of edges in $\mathcal{E}$ of the type $\{(y,z), (z,w), (w,y)\}$), with the convention that ${\textrm{CC}}(x)$ is 0 if $D_x$ is equal to 0 or 1. The quantity $D_x(D_x-1)/2$ can be interpreted as the number of open triangles in x (an open triangle in a vertex z is a pair of edges of the kind $\{(y,z),(z,w)\}$, without any requirement for the presence of the third edge that would close the triangle), so ${\textrm{CC}}(x)$ takes values in [0,1]. The averaged clustering coefficient of the graph $\mathcal{G}$ is

\begin{equation*}{\textrm{CC}_{\textrm{av}}}(\mathcal{G})\,:\!=\dfrac{1}{|\mathcal{V}|}\sum_{x\in \mathcal{V}} {\textrm{CC}}(x) .\end{equation*}

Clearly also ${\textrm{CC}_{\textrm{av}}}(\mathcal{G})$ takes values in [0,1]. It is not completely obvious what the definition of averaged clustering coefficient should be for an infinite graph. We say that an infinite spatial graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ embedded in $\mathbb{R}^d$ with all vertices having finite degree has averaged clustering coefficient ${\textrm{CC}_{\textrm{av}}}(\mathcal{G})$ if the following limit exists:

\begin{equation*}\lim_{n\to\infty} \dfrac{1}{| \mathcal{V}\cap B_n |}\sum_{x\in \mathcal{V}\cap B_n} {\textrm{CC}}(x)=\!:\,{\textrm{CC}_{\textrm{av}}}(\mathcal{G})\end{equation*}

(we work with integers for simplicity). It is possible to construct infinite graphs for which this limit does not exist. We also note that this definition is in principle different from considering the limit of the clustering coefficients of the subgraphs of $\mathcal{G}$ obtained by considering only vertices in $B_n$, since ${\textrm{CC}}(x)$ also takes into account the edges connecting points that lie out of $B_n$.

Going back to our model, we will let

\[{\textrm{CC}}_n\,:\!=\dfrac{1}{N_n}\sum_{x\in V_n} {\textrm{CC}}(x)\]

be the random variable describing the averaged clustering coefficient inside the box of side n. Note that under the assumptions $\alpha >d$ and $\gamma >1$, for P-almost every $\eta$, all the nodes $x\in\eta$ have finite degree, $P^\eta$-almost surely (see Theorem 2.2). Thus, the $\textrm{CC}(x)$ are well-defined and finite, except perhaps on a set of measure 0. Our main result of this section states that the averaged clustering coefficient of the scale-free percolation in continuous space exists, is self-averaging, and is strictly positive.

Theorem 2.3. Suppose that $\alpha>d$ and $\gamma\,:\!=\alpha(\tau-1)/d>1$. For P-almost all configurations $\eta$, the average clustering coefficient exists $ P^\eta$-almost surely and does not depend on $\eta$. It is given by

\begin{equation*}{\textrm{CC}_{\textrm{av}}}(G)\,:\!=\lim_{n\to\infty}{\textrm{CC}}_n\,{=}\,\mathbb{E}_0[{\textrm{CC}}(0)]>0 .\end{equation*}

3.1. Infinite degree

In this section we prove Theorem 2.1. We start with the following proposition about $D_0$.

Proof of Proposition 3.1. The statement will be proved by showing that, for each value $w>0$ of the weight in 0 and for P-almost every $\eta$, the following sum is infinite:

\[\sum_{x\in\eta} P^\eta(0\leftrightarrow x \mid W_0=w)=\sum_{x\in\eta}E [1-{\textrm{e}}^{-wW\|x\|^{-\alpha}}] ,\]

where the expectation on the right-hand side is taken with respect to W. Indeed, if we condition on the value of $W_0$, the presence of each edge (0,x) becomes independent from the others. We can therefore use the second Borel–Cantelli lemma to imply that there are infinitely many points in the configuration $\eta$ connected to 0. By Campbell’s theorem (see Theorem A.1), the above sum is infinite for almost every $\eta$ if the following integral is

\[\int_{\mathbb{R}^d}E[1-{\textrm{e}}^{-wW\|y\|^{-\alpha}}]\,\textrm{d}y .\]

However, in view of the inequality $1-{\textrm{e}}^{-u}\geq(u\wedge 1)/2$ for $u \geq0$, it is enough to show the divergence of the integral

(3) \begin{equation}\int_{\mathbb{R}^d}E [{wW}{\|y\|^{-\alpha}}\wedge 1]\,\textrm{d}y\geq\int_{\mathbb{R}^d}E [ \mathds{1}_{\{\|y\|^\alpha>wW\}}{wW}{\|y\|^{-\alpha}} ]\,\textrm{d}y .\end{equation}

By first applying Tonelli’s theorem and then passing to polar coordinates, we can rewrite the right-hand side of (3) as

\[E\biggl[wW\int_{y\colon \|y\|>(wW)^{{1/\alpha}}} \|y\|^{-\alpha}\,\textrm{d}y \biggr]\,= E\biggl[wW\sigma(S_{d-1})\int_{(wW)^{{1/\alpha}}}^{\infty}r^{-\alpha+d-1}\,\textrm{d}r \biggr] ,\]

where $\sigma(S_{d-1})$ denotes the surface area of the $(d-1)$-sphere of radius 1. The last integral is divergent for $\alpha\leq d$, so case (a) is proved.

When instead $\alpha>d$, we note that the integral inside the expectation is finite and we can continue the manipulation, obtaining

\begin{equation*}E\biggl[wW\sigma(S_{d-1})\dfrac{(wW)^{(d-\alpha)/\alpha}}{\alpha-d} \biggr]=cw^{d/\alpha}E [W^{d/\alpha} ]\end{equation*}

for some constant $c>0$. The last expression is again infinite under the assumptions of case (b).

Proof of Theorem 2.1. In view of the Proposition 3.1 we know that the set of $\eta$ for which $\smash{\sum_{x\in\eta} P^\eta(0\leftrightarrow x \mid W_0=w)=\infty}$ has P-measure 1. Take an $\eta$ in this set and any point $y\in\eta$. We have

\[\sum_{x\in\eta\setminus\lbrace y\rbrace} P^\eta(y\leftrightarrow x \mid W_y=w)=\sum_{x\in\eta\setminus\lbrace y\rbrace} P^\eta(0\leftrightarrow x \mid W_0=w)\dfrac{E [1-{\textrm{e}}^{-wW\|x-y\|^{-\alpha}}]}{E[1-{\textrm{e}}^{-wW\|x\|^{-\alpha}}]}=\infty ,\]

since the fraction goes to 1 as $\|x\|$ goes to infinity. By Borel–Cantelli this shows that y also has infinite degree $ P^\eta$-almost surely. Since there is a countable number of points in $\eta$, we are done.

3.2. Polynomial degree

As in the previous section we will first prove a statement about the random variable $D_0$. The proof of Theorem 2.2 will be inferred as a consequence.

Proposition 3.2. Suppose that $\alpha>d$ and $\gamma\,:\!=\alpha(\tau-1)/d>1$. Then, for P-almost every $\eta$, there exists a slowly varying function $\ell(\!\cdot\!)=\ell(\cdot,\eta)$ such that

(4) \begin{equation}P^\eta(D_0>s)=s^{-\gamma}\ell(s) .\end{equation}

The strategy for demonstrating Proposition 3.2 follows the lines of the proof of Theorem 2.2 in [Reference Deijfen, van der Hofstad and Hooghiemstra13], which in turn follows [Reference Yukich41]. Analogously to Proposition 2.3 of [Reference Deijfen, van der Hofstad and Hooghiemstra13], we first analyze the properties of the expectation of the degree of a point conditional to its weight $w>0$. One of the main problems when dealing with vertices that are randomly distributed in space is the following: while in scale-free percolation on the lattice one can show by direct computations that the expectation of the degree is equal to a constant times $w^{d/\alpha}$, in the continuous-space case one has to finely control the fluctuations of the degree due to the irregularity of the point process. We show that these fluctuations are of order smaller than $w^{d/2\alpha}\log w$.

Proposition 3.3. Suppose that $\alpha>d$ and $\gamma\,:\!=\alpha(\tau-1)/d>1$. Then, for P-almost every realization $\eta$, we have

(5) \begin{equation}E^\eta[D_0 \mid W_0=w]= {\nu} c_0\,w^{d/\alpha}+\textrm{O}(w^{d/(2\alpha)}\log w) ,\end{equation}

with $c_0\,:\!=v_d\Gamma(1-d/\alpha)E[W^{d/\alpha}]$. Here $v_d$ indicates the volume of the d-dimensional unitary ball, $\Gamma(\!\cdot\!)$ is the gamma function, and the constant in $\textrm{O}(\!\cdot\!)$ is uniformly bounded in $\eta$ from below and above.

Proof of Proposition 3.3. We let $Z_w=Z_w(\eta)\,:\!=E^\eta[D_0 \mid W_0=w]$ and note that $Z_w$ is a random variable that depends only on the Poisson process. Since $D_0=\sum_{x\in\eta}\mathds 1_{\{{0\leftrightarrow x}\}}$, we can use Campbell’s theorem (see (27) in Theorem A.1) applied to the function $f(x)\,:\!=E[1-{\textrm{e}}^{-wW/\|x\|^\alpha}]$, to explicitly compute

(6) \begin{equation}\textrm{E}[Z_w]={\nu}\int_{\mathbb{R}^d}E [1-{\textrm{e}}^{-wW\|x\|^{-\alpha}}]\,\textrm{d}x=w^{d/\alpha}E[W^{d/\alpha}]{\nu}\int_{\mathbb{R}^d}(1-{\textrm{e}}^{-\|y\|^{-\alpha}})\,\textrm{d}y={\nu} c_0 w^{d/\alpha} .\end{equation}

For the second equality we used Fubini and then made the change of variables $y=x(wW)^{-1/\alpha}$. The constant $c_0$ is the one appearing in the statement of the theorem; it is obtained by passing to polar coordinates and then using integration by parts. We can instead use (28) in order to calculate the variance $ \textrm{V}$:

(7) \begin{equation}\textrm{V}(Z_w)={\nu}\int_{\mathbb{R}^d}E[1-{\textrm{e}}^{-wW\|x\|^{-\alpha}}]^2\,\textrm{d}x={\nu} c_1 w^{d/\alpha} .\end{equation}

To see the last equality it is sufficient to make the change of variables $y=xw^{-1/\alpha}$. We also note that $c_1\leq c_0$ because the expectation inside the integral is smaller than 1, so the variance has to be smaller than the expectation.

We will show that for P-almost every $\eta$ we have

\begin{equation*} -1\leq\liminf_{w\to\infty}\dfrac{Z_w-\textrm{E}[Z_w]}{\sqrt{\textrm{V}(Z_w)}\log w}\leq\limsup_{w\to\infty}\dfrac{Z_w-\textrm{E}[Z_w]}{\sqrt{\textrm{V}(Z_w)}\log w}\leq 1 ,\end{equation*}

which implies (5). We will only show the upper bound, the lower bound being very similar. Let $\theta>0$. We use the exponential Chebyshev inequality to bound

(8) \begin{align}\textrm{P}(Z_w - \textrm{E}[Z_w]\geq \sqrt{\textrm{V}(Z_w)}\log w)&=\textrm{P}({\textrm{e}}^{\theta Z_w }\geq {\textrm{e}}^{\theta ( \sqrt{\textrm{V}(Z_w)}\log w + \textrm{E}[Z_w])})\nonumber\\*&\leq \exp\{-\theta (\sqrt{\textrm{V}(Z_w)}\log w+\textrm{E}[Z_w])+\log \textrm{E}[{\textrm{e}}^{\theta Z_w}]\} .\end{align}

Campbell’s theorem in its exponential form (see (26)) applied to the function $f(x)\,:\!=\theta E[1-{\textrm{e}}^{-wW\|x\|^{-\alpha}}]$ gives

\begin{equation*} \log \textrm{E}[{\textrm{e}}^{\theta Z_w}]= {\nu}\int_{\mathbb{R}^d} (\!\exp\{\theta E[1-{\textrm{e}}^{-wW\|x\|^{-\alpha}}]\}-1) \,\textrm{d}x .\end{equation*}

If we restrict to values of $\theta$ smaller than 1, a third-order Taylor expansion shows that

\begin{align*}&\exp\{\theta E[1-{\textrm{e}}^{-wW\|x\|^{-\alpha}}]\}-1\\*&\quad \leq \theta E[1-{\textrm{e}}^{-wW\|x\|^{-\alpha}}] +\dfrac{ \theta^2}{2} E[1-{\textrm{e}}^{-wW\|x\|^{-\alpha}}]^2 + \textrm{O}(\theta^3 E[1-{\textrm{e}}^{-wW\|x\|^{-\alpha}}]) ,\end{align*}

where we used the trivial bound $E[1-{\textrm{e}}^{-wW\|x\|^{-\alpha}}]\leq 1$ for the last term. Integrating over x in the above inequality and in view of the expressions for the expectation (6) and the variance (7) of $Z_w$, we obtain

\begin{equation*}\log \textrm{E}[{\textrm{e}}^{\theta Z_w}]\leq \theta \textrm{E}[Z_w]+\dfrac{ \theta^2}{2}\textrm{V}(Z_w)+\textrm{O}(\theta^3\textrm{E}[Z_w]) .\end{equation*}

Inserting this estimate back into (8) yields, for all $\theta\in[0,1]$,

\begin{equation*}\textrm{P}(Z_w - \textrm{E}[Z_w]\geq \sqrt{\textrm{V}(Z_w)} \log w)\leq \exp\biggl\{-\theta \sqrt{\textrm{V}(Z_w)}\log w+\dfrac{ \theta^2}{2}\textrm{V}(Z_w)+\textrm{O}(\theta^3 \textrm{E}[Z_w])\biggr\} .\end{equation*}

In particular, by choosing $\theta=\log w/\sqrt{\textrm{V}(Z_w)}$, we obtain for w large enough

(9) \begin{equation}\textrm{P}(Z_w - \textrm{E}[Z_w]\geq \sqrt{\textrm{V}(Z_w)}\log w)\leq {\textrm{e}}^{-\log^2 w/4} .\end{equation}

If we consider the sequence $w_n=n$, we see that the quantity on the right-hand side of (9) is summable in n and Borel–Cantelli tells us that

\begin{equation*}\limsup_{n\to\infty}\dfrac{Z_{n}-\textrm{E}[Z_{n}]}{\sqrt{\textrm{V}(Z_{n})}\log n}\leq 1 .\end{equation*}

The random variable $Z_w$ is increasing in w, so that $Z_{\lfloor w\rfloor} \leq Z_w\leq Z_{\lfloor w\rfloor+1}$. Moreover, in view of (6) and (7), we see that the sequences $\textrm{E}(Z_n), \sqrt{\textrm{V}(Z_n)}$ and $\log n$ all satisfy

\[\lim_{n\to\infty}\dfrac{a_{n+1}}{a_n}=1 .\]

We can thus conclude that

\begin{align*} \limsup_{n\to \infty}\dfrac{Z_w-\textrm{E}(Z_w)}{\sqrt{\textrm{V}(Z_w)}\log w}&\leq \limsup_{n\to\infty}\dfrac{Z_{\lfloor w\rfloor +1}-\textrm{E}(Z_{\lfloor w\rfloor}\!)}{\sqrt{\textrm{V}(Z_{\lfloor w \rfloor})}\log \lfloor w \rfloor}\\*&=\limsup_{n\to\infty}\dfrac{Z_{\lfloor w\rfloor +1}-\textrm{E}(Z_{\lfloor w\rfloor+1}\!)}{\sqrt{\textrm{V}(Z_{\lfloor w \rfloor+1})}\log (\lfloor w \rfloor+1)}\\*&\leq 1 .\end{align*}

Proof of Proposition 3.2. We let $Y_w$ be the random variable describing the value of $D_0$ conditioned on the event $\{W_0=w\}$:

\[Y_w\sim (D_0 \mid W_0=w) .\]

We split the integral

(10) \begin{equation}P^\eta(D_0>s)=\int_0^{m(s)} P^\eta(Y_w>s)\,\textrm{d}F(w)+\int_{m(s)}^\infty P^\eta(Y_w>s)\,\textrm{d}F(w)=\!:\,A_1(s)+A_2(s) ,\end{equation}

where

(11) \begin{equation}m(s)\,:\!=\biggl(\dfrac{s-\sqrt s\log^{ 2} s}{{\nu} c_0}\biggr)^{\alpha/d} ,\end{equation}

$c_0$ being the same constant appearing in (5). We point out that this definition is slightly different from the equivalent appearing in [Reference Deijfen, van der Hofstad and Hooghiemstra13, eq. (2.11)] in order to control the fluctuations of the expected degree of 0. Since $P^\eta(D_0>s)$ is a monotone function, (4) follows if we can prove that

(12) \begin{equation}\lim_{t\to\infty}\dfrac{P^\eta(D_0>st)}{P^\eta(D_0>t)}=s^{-\gamma}\end{equation}

on a dense set of points (see [Reference Feller17, Section VIII.8]). We claim that $A_1(s)=A_1(\eta,s)$ does not contribute to the regular variation of $P^\eta(D_0>s)$ for P-almost all $\eta$, that is,

(13) \begin{equation}\lim_{s\to\infty} s^aA_1(s)=0\quad \text{for all $a>0$, \textrm{P}-almost surely.}\end{equation}

We will show how to get (13) at the end of the proof. Thanks to (13), we see that in order to prove (12) it is enough to verify that, for $s\in(0,\infty)$,

(14) \begin{equation}\lim_{t\to\infty}\dfrac{A_2(st)}{A_2(t)}=s^{-\gamma} .\end{equation}

On the one hand, for all $t>0$ we clearly have

\begin{equation*}A_2(t)\leq 1-F(m(t)) .\end{equation*}

On the other hand, for all $\varepsilon>0$ we have

\begin{align*}A_2(t)&\geq 1-F((1+\varepsilon)m(t))-\int_{(1+\varepsilon)m(t)}^\infty P^\eta(Y_w\leq t)\,\textrm{d}F(w)\nonumber\\*&\geq(1-F((1+\varepsilon)m(t))(1+\textrm{o}_t(1)) .\end{align*}

For the last line, we used the following fact. By (5), $E^\eta[Y_{w}]>t(1+\varepsilon d/2\alpha)$ for all $w\geq (1+\varepsilon)m(t)$ when t is sufficiently large (depending on $\eta$); this allows us to use the Chebyshev inequality and bound, uniformly in $w>(1+\varepsilon)m(t)$,

\begin{equation*}P^\eta(Y_{w}\leq t)\leq\dfrac{V^\eta(Y_{w})}{(E^\eta[Y_{w}]-t)^2}\leq\dfrac{E^\eta[Y_{w}]}{(E^\eta[Y_{w}]-t)^2}\leq \dfrac{C}{\varepsilon^2 t}=\textrm{o}_t(1) ,\end{equation*}

where $V^\eta$ represents the variance with respect to $ P^\eta$ and the second inequality follows from the fact that $Y_w$ is a sum of independent indicator functions.

Putting it all together and using the fact that $\lim_{t\to\infty}m(st)/m(t)=s^{\alpha/d}$, we obtain

\begin{equation*}\lim_{t\to\infty}\dfrac{A_2(st)}{A_2(t)}=\lim_{t\to\infty}\dfrac{1-F(m(st))}{1-F(m(t))}=s^{-\alpha(\tau-1)/d}\end{equation*}

as we wanted to prove.

We now only need to show (13). We start by upper-bounding $A_1(s)$ by

(15) \begin{align} &P^\eta(Y_{m(s)}>s) \notag \\* &\quad =P^\eta\biggl(\sum_{y\in\eta}\mathds 1_{\{{y\leftrightarrow 0}\}}-E^\eta[\mathds 1_{\{{y\leftrightarrow 0}\}} \mid W_0=m(s)]>s-E^\eta[Y_{m(s)}] \mid W_0=m(s)\biggr).\end{align}

Taking s sufficiently large and thanks to Proposition 3.3, we can make $s-E^\eta[Y_{m(s)}]$ positive, allowing us to apply Bernstein’s inequality. We obtain

\begin{equation*}A_1(s)\leq \exp\biggl\{-\dfrac 12\dfrac{(s-E^\eta[Y_{m(s)}])^2}{E^\eta[Y_{m(s)}]+(s-E^\eta[Y_{m(s)}])/3}\biggr\} ,\end{equation*}

where for the first term in the denominator of the exponent we used the inequality

\begin{equation*}E^\eta[(\mathds 1_{\{{y\leftrightarrow 0}\}}-E^\eta[\mathds 1_{\{{y\leftrightarrow 0}\}} \mid W_0=m(s)])^2\mid W_0=m(s)]\leq E^\eta[\mathds 1_{\{{y\leftrightarrow 0}\}}\mid W_0=m(s)] .\end{equation*}

Since

\[E^\eta[Y_{m(s)}]=s-\sqrt s \log^2s+\textrm{O}(\sqrt s \log s),\]

we get $A_1(s)\leq {\textrm{e}}^{-\log^4s/4}$ and (13) is proved.

Proof of Theorem 2.2. Take any $x\in\eta$. We first prove a lower bound on $ P^\eta(D_x>s)$. First of all we claim that

\begin{equation*} P^\eta(D_x>s \mid W_x=w)\geq P^\eta(D_0>s+1 \mid W_0= c(x,\eta)\cdot w) ,\end{equation*}

where $c(x,\eta)\,:\!=(1+\|x\|/\|y\|)^{-\alpha}$ and $y\in\eta$ is the closest point to the origin in $\eta$. In order to prove the claim we write

\[D_x=\sum_{z\in\eta,\,z\neq x}\mathds 1_{\{{x\leftrightarrow z}\}}\quad\text{and}\quad D_0\leq \sum_{z\in\eta,\,z\neq x}\mathds 1_{\{{0\leftrightarrow z}\}}+1.\]

Since the indicator functions in the first sum are mutually independent under $ P^\eta(\cdot \mid W_x=w)$ and those in the second sum are mutually independent under $ P^\eta(\cdot \mid W_0=c(x,\eta)\cdot w)$, it will be sufficient to show that, for all $z\in\eta\setminus \{x\}$,

\begin{equation*} P^\eta(\mathds 1_{\{{x\leftrightarrow z}\}}=1 \mid W_x=w)\geq P^\eta(\mathds 1_{\{{0\leftrightarrow z}\}}=1 \mid W_0=c(x,\eta)\cdot w) ,\end{equation*}

since this ensures that $D_x\geq D_0-1$ stochastically. We calculate

\begin{align*} P^\eta(\mathds 1_{\{{x\leftrightarrow z}\}}=1 \mid W_x=w)&=E[1-{\textrm{e}}^{-wW_z\|z-x\|^{-\alpha} }]\\*&\geq E[1-{\textrm{e}}^{-c(x,\eta)\cdot w W_z\|z\|^{-\alpha} }]\\*&= P^\eta(\mathds 1_{\{{0\leftrightarrow z}\}}=1 \mid W_0=c(x,\eta)\cdot w) ,\end{align*}

where for the inequality we used the fact that $\|z-x\|/\|z\|\leq (\|y\|+\|x\|)/\|y\|$.

Thanks to the claim we can now bound

\begin{align*} P^\eta(D_x>s)&=\int_0^\infty P^\eta(D_x>s \mid W_x=w)\,\textrm{d}F(w)\\*&\geq \int_0^\infty P^\eta(D_0>s+1 \mid W_0=c(x,\eta)\cdot w )\,\textrm{d}F(w) .\end{align*}

If the weights simply follow a Pareto distribution with exponent $\tau$, then with a change of variables $u=c(x,\eta)\cdot w$ we would obtain $ P^\eta(D_x>s)\geq c(x,\eta)^{\tau-1} P^\eta(D_0>s+1)$, and we would be done thanks to Proposition 3.2. In the general case we have to proceed as in the proof of Proposition 3.2 by splitting the integral on the right-hand side of the last display into the integral between 0 and $m(s)/c(x,\eta)$ plus the integral between $m(s)/c(x,\eta)$ and $+\infty$, where m(s) is defined in (11). We call the first part $\tilde A_1(s)$ and the second part $\tilde A_2(s)$ similarly to (10). Following step by step what we did in the proof of Proposition 3.2 (see (15) and below), we note that

\[\tilde A_1(s)\leq P^\eta(D_0>s+1 \mid W_0=m(s))\leq {\textrm{e}}^{-(\!\log s)^4/4},\]

which does not contribute to the regular variation of the sum. Finally (see (14) and the argument below) we have that $\lim_{t\to\infty}\tilde A_2(st)/\tilde A_2(t)=s^{-\gamma}$, since $\tilde A_2(t)$ is upper-bounded by $1-F(m(t)/c(x,\eta))$ and lower-bounded by $1-F((1+\varepsilon)m(t)/c(x,\eta))$ minus a negligible term. We therefore conclude that there exists a slowly varying function $\ell_1(\!\cdot\!)=\ell_1(\cdot,x,\eta)$ such that

\begin{equation*} P^\eta(D_x>s)\geq \ell_1(s)s^{-\gamma} .\end{equation*}

An upper bound $ P^\eta(D_x>s)\leq \ell_2(s)s^{-\gamma}$ for some other slowly varying function $\ell_2=\ell_2(x,\eta)$ can be obtained in a completely similar way. These two bounds yield the desired result.

4.1. Proofs of the lemmas

Proof of Lemma 4.1. We write $n=mq+r$, with $q\in \mathbb{N}$ and $0\leq r<m$. We let $Q(1),...,Q({q^d})$ be the $q^d$-boxes of side-length m fully contained in $B_n$ and $H=\bigl(\bigcup_{j}Q(j)\bigr)^c$. We also define

\[X(j)\,:\!=(1/\nu m^d)\sum_{x\in Q(j)\cap \eta}\widehat{\textrm{CC}}^m(x)\quad\text{for $j=1,\dots,q^d$.}\]

Then

\begin{equation*}\widehat{\textrm{CC}}^m_n=\biggl(\dfrac{mq}{n}\biggr)^d\dfrac{1}{q^d}\sum_{j=1}^{q^d}X(j)+\dfrac{1}{\nu n^d}\sum_{x\in H\cap V_n}\widehat{\textrm{CC}}^m(x) .\end{equation*}

Since the X(j) are i.i.d. random variables (note that the overlap of the Q-boxes does not represent a problem: as their intersection is a set of Lebesgue measure 0, with probability 1 there will not be any point lying in their intersection) with finite $\mathbb{P}$-expectation, as n goes to infinity, the first summand on the right-hand side converges $\mathbb{P}$-almost surely to $\mathbb{E}[X(1)]=\mathbb{E}[\widehat{\textrm{CC}}^m_m]$ by translation invariance. The second summand converges almost surely to 0 since $\widehat{\textrm{CC}}^m(x)\leq 1$ and the number of points in $H\cap V_n$ follows a Poisson distribution with parameter $n^d-(mq)^d=\textrm{O}(n^{d-1})$.

Proof of Lemma 4.2. By the Slivnyak–Mecke theorem (see Theorem A.2) with $n=1$ we have

\begin{equation*}\mathbb{E}\biggl[\dfrac{1}{{\nu} m^d}\sum_{x\in V_m}{\textrm{CC}}(x)\biggr]=\dfrac {1}{{\nu} m^d} \int_{B_m}\mathbb{E}_x[{\textrm{CC}}(x)]{\nu}\,\textrm{d}x=\mathbb{E}_0[{\textrm{CC}}(0)] ,\end{equation*}

where we used translation invariance for the last equality. On the other hand,

\begin{equation*}\biggl|\mathbb{E}[\widehat{\textrm{CC}}^m_m]- \mathbb{E}\biggl[\dfrac{1}{{\nu} m^d}\sum_{x\in V_m}{\textrm{CC}}(x)\biggr] \biggr|= \dfrac{\mathbb{E}[W_m+U_m]}{{\nu} m^d} ,\end{equation*}

where

\[W_m\,:\!=\#\{x\in\partial B_m(x)\cap \eta\}\quad\text{and}\quad U_m\,:\!=\#\{x\in\overline B_m(x)\cap\eta\colon x\leftrightarrow (B_m(x))^c\}.\]

Since $W_m$ follows a Poisson law of parameter $(1-(1-2\delta)^d){\nu} m^d$, $\mathbb{E}[W_m]/({\nu} m^d)$ is smaller than $2d\delta$. The fact that $\mathbb{E}[U_m]/({\nu} m^d)$ is bounded from above by $c (\delta m)^{d-\alpha}$ will be proved in Proposition 4.2. Putting it all together we obtain

\begin{align*}|\mathbb{E}[\widehat{\textrm{CC}}^m_m-\mathbb{E}_0[{\textrm{CC}}(0)]]|\leq c_1\delta +c_2 (\delta m)^{d-\alpha}.\\[-40pt]\end{align*}

4.2 Proof of Proposition 4.1

We begin with an elementary lemma.

Lemma 4.3. For P-almost all $\eta$ there exists $ \bar n\in\mathbb{N}$ such that, for all $n\geq \bar n$,

\begin{equation*}N_n(\eta)\in[{\nu} n^d-\sqrt{{\nu} n^d}\log n,\,{\nu} n^d+\sqrt{{\nu} n^d}\log n] .\end{equation*}

Proof. For a Poisson random variable X of parameter $\mu>0$, the bound $P(|X-\mu|>\varepsilon)< 2\exp\{-\varepsilon^2/(4\mu)\}$ holds for all $0<\varepsilon<\mu$; see e.g. [Reference Mitzenmacher and Upfal29, Theorem 5.4]. Since under P the number of points falling in $B_n$ follows a Poisson distribution of parameter ${\nu} n^d$, the conclusion follows by a simple Borel–Cantelli argument (in fact it is possible to improve the statement further).

Proof of Proposition 4.1. For simplicity we will consider a sequence of n of the form $n=qm$, where $q\in\mathbb{N}$. It is in fact highly plausible that for all other n of the form $n=qm+r$ with $r\in[0,m)$, what happens in the area $B_n\setminus B_{qm}$ is negligible in the limit $n\to\infty$. We bound

(16) \begin{align}|{\textrm{CC}}_n-\widehat{\textrm{CC}}_n^m|&=\biggl|\dfrac{1}{N_n}-\dfrac{1}{{\nu} n^d}\biggr|\sum_{x\in V_n}{\textrm{CC}}(x)+\dfrac{1}{{\nu} n^d}\biggl(\sum_{x\in V_n}{\textrm{CC}}(x)-\widehat{\textrm{CC}}^m(x)\biggr)\nonumber\\*&\leq \biggl|1-\dfrac{N_n}{{\nu} n^d}\biggr|+\dfrac{1}{{\nu} n^d}(W_n+U_n) ,\end{align}

where

\[W_n=W_n(m)\,:\!=\#\{x\in V_n\colon x\in\partial Q_m(x)\}\]

and

\[U_n=U_n(m)\,:\!=\#\{x\in V_n\colon x\in\overline Q_m(x),\,x\leftrightarrow (Q_m(x))^c\} .\]

The first summand in (16) tends $\mathbb{P}$-almost surely to 0 as $n\to \infty$ by Lemma 4.3. We then note that under $\mathbb{P}$ the random variable $W_n$ has a Poisson distribution of parameter $(1 - {(1 - 2\delta )^d})v{n^d}$, which can be dominated by a Poisson distribution of parameter $ 2d\delta{\nu} n^d$. Reasoning as in the the proof of Lemma 4.3, it is possible to show that, for all n large enough, $W_n$ is smaller than $4d\delta {\nu} n^d$, for example, so $\limsup_{n\to\infty}n^{-d}W_n\leq c_1{\delta}$.

It remains to show that $\mathbb{P}$-almost surely.

(17) \begin{equation}\limsup_{n\to\infty}\dfrac{1}{{\nu} n^d}U_n\leq c_2 (\delta m)^{d-\alpha} .\end{equation}

Assume for now that there exists a constant $c>0$ such that the following estimates hold:

\[\mathbb{E}[U_n]\leq c\,n^d(\delta m)^{d-\alpha},\quad\mathbb{V}(U_n)\leq c\, n^{3d-\alpha} ,\]

where $\mathbb{V}$ denotes the variance with respect to $\mathbb{P}$. The proof of these two bounds will be postponed to Proposition 4.2 below. By the Chebyshev inequality,

\begin{equation*}\mathbb{P}(U_n>2c\,n^d(\delta m)^{d-\alpha})\leq \dfrac{\mathbb{V}(U_n)}{c^2n^{2d}(\delta m)^{2(d-\alpha)}}\leq \tilde c\,(\delta m)^{2(\alpha-d)} n^{d-\alpha} ,\end{equation*}

for some $\tilde c>0$. We would like to finish the proof via Borel–Cantelli, but the right-hand side of the last display is not summable in n. Nevertheless $U_n$ is an increasing sequence, so we can proceed as follows. Recall that $n=qm$. By taking the sequence $n_k\,:\!=2^km$, we have

\begin{equation*}\mathbb{P}(U_{n_k}>2c\,{n_k}^d(\delta m)^{d-\alpha})\leq \tilde c\,(\delta m)^{2(\alpha-d)} (m2^k)^{d-\alpha} ,\end{equation*}

which is summable in k. Hence there exists almost surely a $\bar k$ such that $U_{n_k}\leq 2c\,{n_k}^d(\delta m)^{d-\alpha}$ for all $k\geq \bar k$. Now take any $n>2^{\bar k}m$ and let K be the integer such that $2^Km\leq n<2^{K+1}m$. By monotonicity we get

\begin{equation*}U_n\leq U_{2^{K+1}m}\leq 2 c (2^{K+1}m)^d(\delta m)^{d-\alpha}\leq 2^{d+1}c\,(\delta m)^{d-\alpha}n^d ,\end{equation*}

and (17) follows.

Proposition 4.2. There exists a constant $c>0$ such that

(18) \begin{align}\mathbb{E}[U_n]&\leq c\,n^d(\delta m)^{d-\alpha},\end{align}

(19) \begin{align}\mathbb{V}(U_n)&\leq c\, n^{3d-\alpha} ,\\[10pt]\nonumber\end{align}

where $\mathbb{V}$ denotes the variance with respect to $\mathbb{P}$.

Proof of Proposition 4.2. Let $Q(1),\ldots,Q(q^d)$ be the boxes of side-length m contained in $B_n$. For the expectation we have

(20) \begin{align}\mathbb{E}[U_n]&=\mathbb{E}\Biggl[\sum_{j=1}^{q^d}\sum_{x\in \overline Q(j)\cap\eta}\mathds 1_{\{{x\leftrightarrow(Q_m(x))^c}\}}\Biggr]\nonumber\\*&= q^d\mathbb{E}\Biggl[ \mathbb{E}\Biggl[ \sum_{x\in V_{m(1-2\delta)}} \mathds 1_{\{{x\leftrightarrow B_m^c}\}} \biggm| V_m \Biggr] \Biggr]\notag \\*&\leq q^d \mathbb{E}[ N_{m(1-2\delta)}\mathbb{P}_0(0\leftrightarrow B_{\delta m}^c) ] .\end{align}

In the second equality we used the translation invariance of the model and then the tower property of expectation by conditioning on the position of the points of the Poisson process inside $B_m$. For the inequality we upper-bounded the probability that a point $x\in V_{m(1-2\delta)}$ is connected to the exterior of $B_m$ by the probability that, under the Palm measure, the origin is connected to some point in $B_{\delta m}^c$ (since for each $x\in V_{m(1-2\delta)}$ all the points in $B_m^c$ are outside a box of side $\delta m$ centered at x). We estimate this probability as follows:

(21) \begin{align}\mathbb{P}_0(0\leftrightarrow B_{\delta m}^c)&=1-\mathbb{E}_0\Biggl[ \prod_{y\in V\setminus V_{\delta m}}E[{\textrm{e}}^{-{{W_0W_y}/{\|y\|^\alpha}}}\mid W_0] \Biggr]\nonumber\\*&\leq 1-\exp\Biggl\{-\mathbb{E}_{0}\Biggl[\sum_{y\in V\setminus V_{\delta m}}\dfrac{W_0E[W_y]}{\|y\|^\alpha}\Biggr]\Biggr\} \notag \\&\leq \int_{y\in B_{\delta m}^c}\dfrac{E[W]^2}{\|y\|^{\alpha}}\,\textrm{d} y \notag \\*&= c'(\delta m)^{d-\alpha}\end{align}

for some $c'>0$ that does not depend on either m or $\delta$. In the first line we used the fact that, conditioned on the weight of point 0, the presence of connections between 0 and the other points of the Poisson point process becomes independent. For the second line we applied Jensen’s inequality twice. For the third line we first used the inequality $1-{\textrm{e}}^{-x}\leq x$ for $x>0$, then we applied Campbell’s theorem (see Theorem A). Finally we used polar coordinates in order to evaluate the integral. Plugging this back into (20), we obtain (18).

We move to the variance of $U_n$. For $x\in \eta$ we set

\begin{equation*}A(x)\,:\!=\mathds 1_{\{{x\in \overline Q_m(x) ,\;x\leftrightarrow(Q_m(x))^c}\}}\,\end{equation*}

so that $U_n=\sum_{x\in V_n}A(x)$. Note that by the Slivnyak–Mecke theorem (see Theorem A.2) we have

\[\mathbb{E}[U_n]={\nu}\int_{B_n}\mathbb{E}_{x}[A(x)]\,\textrm{d}x.\]

On the other hand we can write

(22) \begin{equation}\mathbb{V}(U_n)=\mathbb{E}[U_n^2]-\mathbb{E}[U_n]^2=\mathbb{E}[U_n]+\mathbb{E}\Biggl[\sum_{x\neq y\in V_n}A(x)A(y)\Biggr]-\mathbb{E}[U_n]^2 .\end{equation}

Now applying Theorem A.2 to the function

\[f(x,y,\eta)\,:\!=\mathds 1_{\{{x,y\in B_n}\}}E^{\eta\cup \{x\}\cup\{y\}}[A(x)A(y)],\]

we can evaluate the second summand on the right-hand side:

(23) \begin{align}\mathbb{E}\Biggl[\sum_{x\neq y\in V_n}A(x)A(y)\Biggr]&={\nu}^2\int_{B_n}\int_{B_n}\mathbb{E}_{x,y}[A(x)A(y)]\,\textrm{d}x\,\textrm{d}y\notag\\*&={\nu}^2\int_{B_n}\int_{B_n}C(x,y)\,\textrm{d}x\,\textrm{d}y+{\mathbb{E}[U_n]}^2+{\nu}^2R ,\end{align}

where

\[C(x,y)=\mathbb{E}_{x,y}[A(x)A(y)]-\mathbb{E}_{x,y}[A(x)]\mathbb{E}_{x,y}[A(y)]\]

and R is the rest given by

\[R=\int_{B_n}\int_{B_n}\mathbb{E}_{x,y}[A(x)]\mathbb{E}_{x,y}[A(y)]-\mathbb{E}_{x}[A(x)]\mathbb{E}_{y}[A(y)]\,\textrm{d}x\,\textrm{d}y .\]

Note that for $y\in Q_m(x)$ we have $\mathbb{E}_{x,y}[A(x)]=\mathbb{E}_{x}[A(x)]$. For $y\not\in Q_m(x)$,

\[\mathbb{E}_{x,y}[A(x)]\leq \mathbb{E}_{x}[A(x)]+E[1-\exp\{-W_xW_y\|x-y\|^{-\alpha}\}]\leq \mathbb{E}_{x}[A(x)]+ {\tilde c_1}\|x-y\|^{-\alpha}\]

for some ${\tilde c_1}>0$. For the first inequality we used the fact that for $A(x)=1$, the point x has to be connected to y (which gives the second addendum) or to any other point outside $Q_m(x)$ (first addendum), and in this second case the presence of y in the point process plays no role. This, together with the bound

\[\int_{B_n\setminus Q_m(x)}\|x-y\|^{-\alpha}\,\textrm{d}y\leq {\tilde c_2} (\delta m)^{d-\alpha}\quad\text{for some ${\tilde c_2}>0$,}\]

yields

\begin{equation*}R\leq {\tilde c_3} (\mathbb{E}[U_n](\delta m)^{d-\alpha}+n^d(\delta m)^{-2\alpha+d})\leq {\tilde c_4} n^d(\delta m)^{2(d-\alpha)} .\end{equation*}

We now only need to bound the correlation C(x, y). We introduce the event

\begin{equation*}\mathcal{E}(x,y)\,:\!=\{\text{neither \textit{x} nor \textit{y} have neighbors at distance larger than $\|x-y\|/2$}\}.\end{equation*}

The random variables A(x) and A(y) are independent under the event $\mathcal{E}(x,y)$. Therefore

(24) \begin{align}C(x,y)&\leq \mathbb{E}_{x,y}[A(x)|\mathcal{E}(x,y)] \mathbb{E}_{x,y}[A(y)|\mathcal{E}(x,y)]\mathbb{P}(\mathcal{E}(x,y))\notag\\*&\quad\, +\mathbb{P}_{x,y}(\mathcal{E}(x,y)^c)-\mathbb{E}_{x,y}[A(x)]\mathbb{E}_{x,y}[A(y)]\nonumber\\&\leq \mathbb{E}_{x,y}[A(x)]\mathbb{E}_{x,y}[A(y)]\biggl(\dfrac{1}{\mathbb{P}_{x,y}(\mathcal{E}(x,y))}-1\biggr)+\mathbb{P}_{x,y}(\mathcal{E}(x,y)^c)\nonumber\\*&\leq \mathbb{P}_{x,y}(\mathcal{E}(x,y)^c)\biggl(\dfrac{1}{\mathbb{P}_{x,y}(\mathcal{E}(x,y))}+1\biggr) ,\end{align}

where in the second line we used the inequality $\mathbb{E}_{x,y}[W|Z]\leq \mathbb{E}_{x,y}[W]/\mathbb{P}_{x,y}(Z)$ and in the last line we just bounded $\mathbb{E}_{x,y}[A(x)]$ and $\mathbb{E}_{x,y}[A(y)]$ by 1. We also observe that

(25) \begin{align}\mathbb{P}_{x,y}(\mathcal{E}(x,y)^c)&\leq \mathbb{P}_{x,y}(x\leftrightarrow \mathcal{B}_{\|x-y\|/2}(x))+\mathbb{P}_{x,y}(y\leftrightarrow \mathcal{B}_{\|x-y\|/2}(y))\nonumber\\*&\leq \mathbb{P}_{x,y}(x\leftrightarrow y)+2\mathbb{P}_{0}(0\leftrightarrow \mathcal{B}_{\|x-y\|/2}(0))\notag\\*&\leq {\tilde c_5}\,\|x-y\|^{d-\alpha}\end{align}

for some ${\tilde c_5}>0$, where the last inequality can be obtained as in (21). From (24) and (25) it follows that whenever we consider $x,y\in\mathbb{R}^d$ such that, for example, ${\tilde c_5}\,\|x-y\|^{d-\alpha}\leq 1/2$, we get $C(x,y)\leq {\tilde c_6}\,\|x-y\|^{d-\alpha}$. Hence, setting ${r}\,:\!=(2{\tilde c_5})^{1/(\alpha-d)}$, we can bound

\begin{equation*}\int_{B_n}\int_{B_n}C(x,y)\,\textrm{d}x\,\textrm{d}y\leq \int_{B_n}\biggl[{\tilde c_6} {r}^d+\int_{y\not\in \mathcal{B}_{{r}}(x)}{\tilde c_5}\|x-y\|^{d-\alpha}\,\textrm{d}y\biggr]\,\textrm{d}x\leq c\,n^{3d-\alpha}\,\end{equation*}

for some constant $c>0$. The proof is finished by putting together (22), (18), (23), and this last estimate.