# Condensation in scale-free geometric graphs
with excess edges

Remco van der Hofstad  Department of Mathematics and Computer Science,
Eindhoven University of Technology, 5612 AZ Eindhoven, The Netherlands
<EMAIL_ADDRESS>,  Pim van der Hoorn  Department of Mathematics and
Computer Science, Eindhoven University of Technology, 5612 AZ Eindhoven, The
Netherlands<EMAIL_ADDRESS>,  Céline Kerriou  Universität zu Köln,
Department of Mathematics and Computer Science, Weyertal 86-90, 50931 Cologne,
Germany<EMAIL_ADDRESS>,  Neeladri Maitra  Department of
Mathematics and Computer Science, Eindhoven University of Technology, 5612 AZ
Eindhoven, The Netherlands<EMAIL_ADDRESS>and  Peter Mörters  Universität zu
Köln, Department of Mathematics and Computer Science, Weyertal 86-90, 50931
Cologne, Germany<EMAIL_ADDRESS>
###### Abstract.

We identify the upper large deviation probability for the number of edges in
scale-free geometric random graph models as the space volume goes to infinity.
Our result covers the models of scale-free percolation, the Boolean model with
heavy-tailed radius distribution, and the age-dependent random connection
model. In all these cases the mechanism behind the large deviation is based on
a condensation effect. Loosely speaking, the mechanism randomly selects a
finite number of vertices and increases their power, so that they connect to a
macroscopic number of vertices in the graph, while the other vertices retain a
degree close to their expectation and thus make no more than the expected
contribution to the large deviation event. We verify this intuition by means
of limit theorems for the empirical distributions of degrees and edge-lengths
under the conditioning. We observe that at large finite volumes, the edge-
length distribution splits into a bulk and travelling wave part of
asymptotically positive proportions.

###### Contents

  1. 1 Introduction and main results
  2. 2 Overview and proof strategies
  3. 3 Concentration of the bulk: Proof of Proposition 2.1
  4. 4 Concentration of the long edges: Proof of Proposition 2.2
  5. 5 Large deviations of the high-weight edges: Proof of Proposition 2.3
  6. 6 Bulk in edge-length distribution: Proof of Proposition 2.5
  7. 7 Travelling wave in edge-length distribution: Proof of Proposition 2.6
  8. 8 Limit for the weight order statistics: Proof of Proposition 2.7
  9. 9 The empirical degree distribution: Proof of Proposition 2.9

##  1\. Introduction and main results

###  1.1. Motivation

The physical phenomenon of condensation corresponds to the formation of drops
of liquid when a gas is cooled down or put under high pressure. The
mathematics behind this phenomenon however turns out not to be constrained to
mathematical models of gases, it is ubiquitous throughout probability and the
theory of stochastic processes, ranging from random walks [2], random trees
[16], interacting particle systems [24] and Gaussian processes [25] to
queueing networks [21] and extreme value statistics [9], to name just a few
examples. In this paper we show how a condensation effect arises in the large
deviation theory of scale-free geometric random graphs.

To compare the condensation effects we look at the simplest probabilistic
model capturing a condensation effect, which is the _balls-in-bins_ model, see
[16, Sections 11 and 19]. In this model $m$ particles (or balls) are placed
into $n$ labelled containers (or bins) and the probability of a configuration
$(m_{1},\ldots,m_{n})$ with $m_{i}$ particles in the $i$th container (and
hence $\sum_{i=1}^{n}m_{i}=m$) is

$\frac{1}{Z_{m,n}}\prod_{i=1}^{n}w(m_{i}),$

where $w(i)=ci^{-\beta}$ are the weights of a zeta distribution with
$1/c=\sum_{i=1}^{\infty}i^{-\beta}$ and $Z_{m,n}$ is a normalisation constant.
As $n\to\infty$ and $m/n\to\rho$ for some particle density $0<\rho<\infty$, it
is shown in [16, Theorem 11.4] that the proportion $N_{k}(n)$ of containers
containing $k$ particles converges to a number $\pi_{k}$ with
$\sum_{k=1}^{\infty}\pi_{k}=1$ and

  * $\rhd$

$\sum_{k=1}^{\infty}k\pi_{k}=\rho$ if $\rho\leq\nu:=\sum_{i=1}^{\infty}iw(i)$;

  * $\rhd$

$\sum_{k=1}^{\infty}k\pi_{k}=\nu$ if $\rho>\nu$.

As $\sum_{k=1}^{\infty}kN_{k}(n)=m/n\to\rho$, we observe in the second case
that particle mass is escaping. It turns out, see [16, Theorem 19.34], that
the excess mass condenses in a _single_ container, in other words, the
container containing the largest number of particles contains asymptotically
$(\rho-\nu)n$ balls. The vastly increased particle density in this container
is interpreted as a manifestation of the condensation phenomenon in a gas
under high pressure.

In the balls-in-bins model, if $(\xi_{1},\ldots,\xi_{n})$ are independent and
zeta distributed, then

$Z_{m,n}={\mathbb{P}}\left(\sum_{i=1}^{n}\xi_{i}=m\right)$

and the probability of a configuration $(m_{1},\ldots,m_{n})$ is the
conditional distribution of $(\xi_{1},\ldots,\xi_{n})$ given the event
$\sum_{i=1}^{n}\xi_{i}=m$. If $\rho>\nu$ this event has asymptotically
decreasing probability and hence the condensation stems from conditioning on a
rare event. In our main result a similar phenomenon arises in the context of
large deviations for geometric random graphs. Here, loosely speaking, the bins
correspond to vertices and the number of balls in a bin corresponds to the
vertex degree. But other than in the balls-in-bins model, in the geometric
random graphs we can have more than one vertex of macroscopic degree,
depending on the number of excess edges generated in the large deviation
event.

A further analogy arises in the context of Bose-Einstein condensation. Bose
and Einstein predicted in 1925 that above a certain density a macroscopic
fraction of particles in a Bose gas aggregate into a single quantum state.
Based on Feynman’s representation of the Bose gas as a soup of interacting
loops in a container, see for example [28], the phenomenon takes the form that
as the particle number goes to infinity, loops of macroscopic length occur in
the loop soup. This is studied rigorously under simplifying modelling
assumptions, for example in [3, 1, 8], but a fully satisfactory mathematical
result on a physical model is still missing. The results achieved so far show
that in toy models, above a critical particle density, the empirical
distribution of the loop lengths per particle in the limit as the volume of
the container tends to infinity converges to a distribution of total mass
strictly less than one. The missing mass escapes to infinity because a
proportion of particles is associated with loops of a macroscopic length
scale, see for example [23]. In our model of a random graph embedded in a
$d$-dimensional torus of volume $n$, we look at the length of edges instead of
loops and study the empirical distribution of the Euclidean length of all
edges. Conditioning the random graph on having an edge density above the
critical value we prove that a positive fraction of the mass of the empirical
distribution converges to a limit distribution of mass strictly smaller than
one, while the rest of the mass forms a travelling wave of macroscopic speed
and width. In particular, asymptotically, a positive fraction of edge-lengths
is of macroscopic order $n^{1/d}$, loosely analogous to the behaviour of loop
lengths in the models above.

Beyond the relation to condensation phenomena, our results are of interest
from a purely random graph perspective, providing insight in the nature of the
most important models of scale-free random geometric graphs by giving precise
asymptotics for the probability that such a graph has a large number of edges.
We also give limit theorems for the empirical edge-length distribution and the
empirical degree distribution of the graph conditioned on having a large
number of edges, which expose the condensation behaviour. Results of this
nature have been found for non-geometric graph models like the Chung-Lu graphs
[26], very simple geometric models [19], but also for Boolean models which are
not scale-free, see the seminal paper of Chatterjee and Harel [4]. The
behaviour of scale-free Boolean models, and more general scale-free random
geometric graphs, is however fundamentally different from the latter and is
explored in the present paper. Our results are set up in a general framework
covering a wide range of scale-free geometric graphs.

Outline. We formulate our framework of scale-free geometric graphs and give
several examples in Section 1.2, before presenting the main results in Section
1.3. In Section 2 we describe the proof strategies behind our theorems and
reduce the proof to seven key propositions, which are proved in Sections 3 to
9.

Notation. We write $X_{n}\stackrel{{\\!{}_{\mathbb{P}}}}{{\rightarrow}}X$,
respectively $X_{n}\stackrel{{\\!{}_{\mathrm{d}}}}{{\rightarrow}}X$, to denote
that $X_{n}$ converges in probability, respectively, in distribution, to $X$
as $n\to\infty$. We write $\mathcal{B}(x,r)$ to denote the open ball of radius
$r$ centred at $x$ in a given metric space. For real sequences $(x_{n})_{n\geq
1}$ and $(y_{n})_{n\geq 1}$ with $(y_{n})_{n\geq 1}$ being positive, we write
$x_{n}=o(y_{n})$ if $\limsup_{n\to\infty}x_{n}/y_{n}=0$ and $x_{n}=O(y_{n})$
if $\limsup_{n\to\infty}x_{n}/y_{n}<\infty$.

###  1.2. Geometric random graph models

In our general setup, the vertex set of the graph $G_{n}$ is contained in the
$d$-dimensional torus $\mathbb{T}_{n}^{d}$ of volume $n$, which we equip with
the torus metric $d_{n}$. The vertex set $V_{n}$ may be of two different
types:

  * $\rhd$

_Lattice case:_ $n^{\frac{1}{d}}$ is an integer and $V_{n}$ the integer
lattice on the torus,

  * $\rhd$

_Poisson case:_ $n$ is real and $V_{n}$ a Poisson point process of intensity
one on $\mathbb{T}_{n}^{d}$.

Fix $\beta>2$. To define the edge set $E_{n}$, given the vertex set $V_{n}$,
we equip each vertex $x\in V_{n}$ with a random weight $W_{x}$ such that the
random variables $(W_{x})_{x\in V_{n}}$ are independent and identically
distributed to some $W$ with density

$\displaystyle f_{W}(t):=(\beta-1)t^{-\beta}{\mathbbm{1}}_{\\{t\geq 1\\}}.$
(1.1)

We fix a profile function $\varphi\colon[0,\infty)\rightarrow[0,1]$, which is
nonincreasing and integrable, and a connection kernel
$\kappa\colon[1,\infty)\times[1,\infty)\rightarrow(0,\infty)$, which is
symmetric, continuous and nondecreasing in each argument. For every unordered
pair $\\{x,y\\}\subseteq\mathbb{T}_{n}^{d}$, we define a function
$p_{x,y}\colon[1,\infty)\times[1,\infty)\to[0,1]$ by $p_{x,y}:=0$ if $x=y$,
and

$\displaystyle
p_{x,y}(v,w):=\varphi\left(\frac{d_{n}(x,y)^{d}}{\kappa(v,w)}\right)\text{
otherwise.}$ (1.2)

Given the vertex set and the weights, we include the edge $\\{x,y\\}$ in
$E_{n}$ with probability $p_{x,y}(W_{x},W_{y})$, independently for every
unordered pair $\\{x,y\\}$ of distinct vertices in $V_{n}$ with weights
$W_{x},W_{y}$. More precisely, let $(A_{\\{x,y\\}})_{x,y}$ be conditionally
independent Bernoulli-distributed random variables with success parameter
$p_{x,y}(W_{x},W_{y})$. For notational simplicity write $A_{x,y}$ for
$A_{\\{x,y\\}}$. The edge set $E_{n}$ is given by

$\displaystyle E_{n}:=\\{\\{x,y\\}\subseteq V_{n}\colon A_{x,y}=1\\}.$ (1.3)

The degree of a vertex $x\in V_{n}$ is the number of neighbours of $x$ in
$G_{n}$ defined as

$\displaystyle D_{x}:=D_{x}(n):=\sum_{y\in V_{n}}A_{x,y}.$ (1.4)

This model was introduced under the name _weight-dependent random connection
model_ in [12, 11] and under the name _kernel-based spatial random graph_ in
[18]. We now explain how our principal examples fit in this framework. In all
our examples the empirical degree distributions of the graphs converge to a
deterministic limit, which is a heavy-tailed distribution with the same tail
index $\beta$ as the weight distribution.

#### (i) Scale-free percolation

Take the profile function $\varphi(x)=1-{\mathrm{e}}^{-x^{-\alpha}}$, for some
$\alpha>1$, and connection kernel $\kappa(v,w)=vw$. For the lattice case this
model was first introduced and studied in [6], the Poisson case has been
introduced in [7].

#### (ii) Poisson Boolean model with heavy-tailed radii

Consider the Poisson case with profile function
$\varphi(x)={\mathbbm{1}}_{[0,1]}(x)$ and connection kernel
$\kappa(v,w)=(v^{1/d}+w^{1/d})^{d}$. Thinking of $r_{x}:=W_{x}^{1/d}$ as the
radius of a ball $\mathcal{B}(x,r_{x})$ centred in $x$, in this model two
vertices $x,y$ are connected by an edge if and only if
$\mathcal{B}(x,r_{x})\cap\mathcal{B}(y,r_{y})\not=\emptyset$. Therefore this
graph model shares the name with the union of the balls $\mathcal{B}(x,r_{x})$
over all points $x\in V_{n}$, which is studied in stochastic geometry [27].

#### (iii) Age-based preferential attachment graphs

We define a dynamical graph model $(G^{t}\colon t\geq 0)$ with vertices
embedded in the torus $\mathbb{T}_{1}^{d}$ of volume one as in [10]. Let
$\varphi(x)=1\wedge x^{-\alpha}$, for some $\alpha>1$, and fix $0<\gamma<1$.
We take a standard Poisson process of arrival times for the vertices. Given
the graph $G^{t-}$, at the arrival time $t$ of a new vertex, we give this
vertex a random location $x\in\mathbb{T}_{1}^{d}$ uniformly on the torus of
volume one. Independently for every vertex of $G^{t-}$, we establish an edge
to the new vertex with probability $\varphi(td_{1}(x,y)^{d}(s/t)^{\gamma}),$
where $y$ is its location and $s<t$ its birth-time. As $(t/s)^{\gamma}$ is the
order of the expected degree of the vertex at time $t$, the new vertex prefers
to attach itself to existing vertices with large expected degree or,
equivalently, old age.

For any fixed time $n$, we rescale the graph $G^{n}$ to the torus
$\mathbb{T}_{n}^{d}$ by multiplying all locations with the factor $n^{1/d}$.
We denote the rescaled graph by $G_{n}$ and explain how it fits our framework.
If $s\in(0,n)$ is the birth time of a vertex at the new location $x$, then we
let its weight be $W_{x}:=(s/n)^{-\gamma}$. Then the new vertex locations are
a standard Poisson point process on $\mathbb{T}_{n}^{d}$, the weights are
independent Pareto-distributed with parameter $\beta=1+\frac{1}{\gamma}$, and
the probability of two vertices at locations $x,y$ in $G_{n}$ being connected
is

$\varphi\Big{(}n\big{(}W_{x}^{-1/\gamma}\vee
W_{y}^{-1/\gamma}\big{)}\,d_{1}\big{(}xn^{-1/d},yn^{-1/d}\big{)}^{d}\big{(}\tfrac{W_{x}^{-1/\gamma}\vee
W_{y}^{-1/\gamma}}{W_{y}^{-1/\gamma}\wedge
W_{x}^{-1/\gamma}}\big{)}^{\gamma}\Big{)}=\varphi\Big{(}\frac{d_{n}(x,y)^{d}}{\kappa(W_{x},W_{y})}\Big{)},$

for $\kappa(v,w)=(v\wedge w)^{\frac{1}{\gamma}-1}(v\vee w)$. Hence results for
the number of edges in $G^{n}$ as _time_ goes to infinity can be transferred
directly from results for the number of edges for the graph $G_{n}$ as the
_volume_ tends to infinity. The latter fits our framework with the given
$\kappa$.

###  1.3. Main results

We now formulate the assumptions on the kernel $\kappa$ and profile function
$\varphi$ in our framework for our main results to hold.

Assumption A. There exist constants $0<{\sf c}<{\sf C}<\infty$ such that

${\sf c}v\leq\kappa(v,w)\leq{\sf C}vw^{(\beta-2)\vee 1},\text{ for all }v\geq
w\geq 1,$ (1.5)

and the limit

$\mathcal{K}(v,w):=\lim_{x\uparrow\infty}\frac{1}{x}\kappa(xv,w)$ (1.6)

exists. The profile function $\varphi$ is continuous at $0$ and, for some
$\alpha>1$,

${\sf c}{\mathbbm{1}}_{\\{x\leq{\sf c}\\}}\leq\varphi(x)\leq 1\wedge{\sf
C}x^{-\alpha},\text{ for all }x\geq 0.$ (1.7)

The graph of $\varphi$ has no flat pieces except possibly at its essential
supremum or infimum. $\blacktriangleleft$

Observe that, in contrast to $\kappa$, the derived kernel $\mathcal{K}$ is not
symmetric, but linear in the first argument. Recall that $|E_{n}|$ denotes the
total number of edges in $G_{n}$. Then we have, as $n\to\infty$, by an
application of a law of large numbers for geometric functionals,

$\displaystyle\frac{|E_{n}|}{n}=\frac{1}{2n}\sum_{x\in
V_{n}}D_{x}(n)\stackrel{{\scriptstyle\mathbb{P}}}{{\to}}\mu,$ (1.8)

for some $0<\mu<\infty$. More details and an explicit formula for $\mu$ are
given in Remark 1.7 below.

For $z\in[-\tfrac{1}{2},\tfrac{1}{2}]^{d}$ and $w>0$, let

$\displaystyle{{\Lambda}}(w,z):={\mathbb{E}}\left[\varphi\left(\frac{\|z\|^{d}}{\mathcal{K}(w,W)}\right)\right],$
(1.9)

and

$\displaystyle{{\Lambda}}(w):=\int_{\big{[}-\tfrac{1}{2},\tfrac{1}{2}\big{]}^{d}}\Lambda(w,z)\,dz.$
(1.10)

Further, for $\rho>0$ set $k:=\lceil\rho\rceil$ and

$\displaystyle
F(\rho):=\frac{(\beta-1)^{k}}{{k!}}\int_{0}^{\infty}\cdots\int_{0}^{\infty}{\mathbbm{1}}_{\\{{\Lambda}(y_{1})+\cdots+{\Lambda}(y_{k})>\rho\\}}\prod_{i=1}^{k}y_{i}^{-\beta}dy_{i}.$
(1.11)

Our main result for the large deviations of the number of edges in the random
graph models $(G_{n})_{n\in{\mathbb{N}}}$ described in Section 1.2 is
summarized in the following theorem.

######  Theorem 1.1 (Upper large deviations).

Let $|E_{n}|$ be the number of edges in the weight-dependent random connection
model $G_{n}$ with kernel $\kappa$ and profile $\varphi$ satisfying Assumption
A. Let $\rho>0$ be non-integer and $k$ the unique integer such that
$k-1<\rho<k$. Then, as $n\rightarrow\infty$,

${\mathbb{P}}\left(|E_{n}|\geq
n(\rho+\mu)\right)=\big{(}F(\rho)+o(1)\big{)}n^{-k(\beta-2)}.$

######  Remark 1.2 (Properties of $\rho\mapsto F(\rho)$).

We always have $F(\rho)<\infty$. If $\sup_{x>0}\varphi(x)=1$, then $F(\rho)>0$
and $F$ is a continuous function on all open intervals not containing an
integer; see Lemma 2.4 for a proof. Therefore, under this assumption, Theorem
1.1 identifies the precise asymptotic behaviour of the large deviation
probability. Otherwise, if $\sup_{x>0}\varphi(x)=p<1$, the graph can be
identified with the graph with profile function $\varphi/p$ after Bernoulli
percolation with retention probability $p$. Given the graph prior to
percolation and denoting its edge set by $E^{p}_{n}$, the number $|E_{n}|$ of
edges in the percolated graph is binomial with parameters $|E^{p}_{n}|$ and
$p$. By Cramér’s theorem this random variable has the same upper large
deviations as its conditional mean $|E^{p}_{n}|p$ and so the upper large
deviation result of Theorem 1.1 holds for $p(k-1)<\rho<pk$ verbatim and also
identifies the precise asymptotic behaviour. $\blacktriangleleft$

######  Remark 1.3 (Our assumptions).

Assumption A holds in our examples (i)\- (iii). Also, in all our examples we
have $\sup_{x>0}\varphi(x)=1$, which ensures that Theorem 1.1 identifies the
asymptotic behaviour of the large deviation probability.  $\blacktriangleleft$

######  Remark 1.4 (Non-integer restriction on $\rho$).

Observe that the exponent of $n$ in the power-law decay of the large deviation
probability in Theorem 1.1 jumps at integer values of $\rho$. At these values
the asymptotic behaviour of ${\mathbb{P}}\left(|E_{n}|\geq n(\rho+\mu)\right)$
depends on specific model details and a universal result covering a wide range
of profiles and kernels in a single limit theorem with explicit limit as above
does not hold. A deeper discussion of this fact is given in Section 2.4 below.
$\blacktriangleleft$

From now on we assume, without loss of generality, that
$\sup_{x>0}\varphi(x)=1$ and consider the behaviour of the graph conditioned
on the large deviation event above.

######  Remark 1.5 (Conditional limit distribution of the number of edges).

It follows immediately from Theorem 1.1 that, conditionally on $|E_{n}|\geq
n(\mu+\rho)$,

$\frac{|E_{n}|}{n}\stackrel{{\\!{}_{\mathrm{d}}}}{{\rightarrow}}\mu+S,$ (1.12)

where the random variable $S$ is supported on $[\rho,k]$, and has tail
distribution function

$\mathbb{P}(S>s)=\frac{F(s)}{F(\rho)}\quad\mbox{ for all $\rho\leq s<k$.}$
(1.13)

This follows as $F$ is continuous on $(k-1,k)$. Observe that the distribution
of $S$ has an atom at $k$ if and only if $\lim_{\rho\nearrow k}F(\rho)>0$,
which happens, for example, in the case of the Poisson Boolean model, Example
(ii), in which $\Lambda$ is constant on the interval
$[\frac{\sqrt{d}}{2},\infty)$. $\blacktriangleleft$

Next we describe the empirical distribution of the edge-lengths, which is the
(random) measure $\mu_{n}$ on $(0,\infty)$ given by

$\mu_{n}:=\frac{1}{n}\sum_{\\{x,y\\}\in E_{n}}\delta_{d_{n}(x,y)},$ (1.14)

where $\delta_{a}$ is the unit mass at $a\in\mathbb{R}$. Now we introduce the
objects appearing in the limit theorem below. For
$(y_{1},\ldots,y_{k})\in(0,\infty)^{k}$, define a measure
$\mu_{y_{1},\ldots,y_{k}}$ on $\mathbb{R}$ by

$\displaystyle\int
fd\mu_{y_{1},\ldots,y_{k}}=\int_{[-\frac{1}{2},\frac{1}{2}]^{d}}f(\|z\|)\sum_{i=1}^{k}{{\Lambda}}(y_{i},z)\,dz.$
(1.15)

Consider the random measure $\mu_{Y_{1},\ldots,Y_{k}}$ where the law of
$(Y_{1},\ldots,Y_{k})$ is given by

$\displaystyle{\mathbb{P}}\left(Y_{1}\geq x_{1},\ldots,Y_{k}\geq x_{k}\right)$
(1.16)
$\displaystyle\qquad=\frac{(\beta-1)^{k}}{F(\rho)}\int_{0}^{\infty}\\!\\!\cdots\\!\\!\int_{0}^{\infty}{\mathbbm{1}}_{\\{\sum_{i=1}^{k}{{\Lambda}}(y_{i})>\rho\\}}\prod_{i=1}^{k}{\mathbbm{1}}_{\\{y_{i}\geq
x_{i}\vee y_{i+1}\\}}y_{i}^{-\beta}dy_{i},$

setting $y_{k+1}=0.$ In terms of this notation, $S$ in (1.12)–(1.13) has
distribution

$S=\sum_{i=1}^{k}\Lambda(Y_{i}).$ (1.17)

For $z\in\mathbb{R}^{d}$ and $w\geq 1$, let

$\displaystyle\lambda(w,z)$
$\displaystyle:={\mathbb{E}}\left[\varphi\left(\frac{\|z\|^{d}}{\kappa(w,W)}\right)\right].$
(1.18)

Let $f\colon(0,\infty)\to[0,\infty)$ be continuous and define, for $w\geq 1$,

$\displaystyle\lambda_{f}(w)$
$\displaystyle=\begin{cases}\sum_{z\in\mathbb{Z}^{d}}f(\|z\|)\,{{\lambda}}(w,z)&\mbox{
in the lattice case},\\\\[8.53581pt]
\int_{\mathbb{R}^{d}}f(\|z\|)\,{{\lambda}}(w,z)\,dz&\mbox{ in the Poisson
case}.\end{cases}$ (1.19)

######  Theorem 1.6 (Convergence of empirical edge-length distribution).

Let $\mu_{n}$ be the empirical edge-length distribution in (1.14) of the
weight-dependent random connection model with kernel $\kappa$ and profile
$\varphi$ satisfying Assumption A. Let $\rho>0$ be a non-integer and $k$ be
the unique integer with $k-1<\rho<k$. For every
$f\colon(0,\infty)\to[0,\infty)$ that is continuous with compact support,
conditionally on $|E_{n}|\geq n(\mu+\rho)$,

  1. (a)

$\displaystyle\int
f\big{(}x\big{)}\,d\mu_{n}(x)\stackrel{{\\!{}_{\mathbb{P}}}}{{\rightarrow}}\frac{1}{2}\mathbb{E}\big{[}{\lambda}_{f}(W)\big{]};$

  2. (b)

$\displaystyle\int
f\Big{(}\frac{x}{n^{1/d}}\Big{)}\,d\mu_{n}(x)\stackrel{{\\!{}_{\mathrm{d}}}}{{\rightarrow}}\int
f\,d\mu_{Y_{1},\ldots,Y_{k}}.$

######  Remark 1.7 (Law of large numbers for edge functionals).

The convergence in probability in (a) also holds when not conditioning on
$|E_{n}|\geq n(\mu+\rho)$, in this case even for all bounded and continuous
functions $f$. A proof based on a local limit theorem, given for a slightly
different vertex set, can be found in [14] and is easily adapted to our
situation. Therefore the right-hand side in (a), with the choice $f=1$,
provides an explicit formula for the asymptotic edge density $\mu$ that
appears in (1.8).  $\blacktriangleleft$

Theorem 1.6 shows that the empirical edge-length distribution splits into two
parts, a _bulk_ part of total mass $\mu$ which converges to the asymptotic
edge-length distribution in the unconditional random graph, and a _travelling
wave_ part of mass at least $\rho$ and random shape, moving to infinity at
speed $n^{1/d}$, see Figure 1 for a schematic picture.

Figure 1. The schematic figure shows the density of the empirical edge-length
distribution, which splits into a bulk of total mass $\mu$ and a travelling
wave of total mass at least $\rho$, which has speed and width of order
$n^{1/d}$. The asymptotic shape of bulk and wave are given explicitly in
Theorem 1.6.

To state the next theorem, denote the order statistics of the weights of
vertices in $V_{n}$ by

$W_{\scriptscriptstyle(1)}>W_{\scriptscriptstyle(2)}>\dots>W_{\scriptscriptstyle(|V_{n}|)}.$

Let $X_{\scriptscriptstyle(i)}$ be the location of the vertex whose weight is
$W_{\scriptscriptstyle(i)}$, that is $W_{X_{(i)}}=W_{\scriptscriptstyle(i)}$.
For a vertex $x\in V_{n}$, define

$\displaystyle D^{\mathbf{b}}_{x}:=\sum_{j>k}A_{x,X_{(j)}},\qquad\text{ and
}\qquad D^{\mathbf{h}}_{x}:=\sum_{j=1}^{k}A_{x,X_{(j)}}.$ (1.20)

We consider the joint empirical distribution of these two types of degrees.
For $a\in{\mathbb{N}}\cup\\{0\\}$ and $b\in\\{0,1,\dots,k\\}$, define

$\displaystyle\pi_{a,b}^{\scriptscriptstyle(n)}:=\frac{1}{|V_{n}|}\sum_{x\in
V_{n}}{\mathbbm{1}}_{\\{D_{x}^{\mathbf{b}}=a,D_{x}^{\mathbf{h}}=b\\}}.$ (1.21)

In words, $\pi_{a,b}^{\scriptscriptstyle(n)}$ is the proportion of vertices
that have exactly $a$ neighbours in the set
$V_{n}\setminus\\{X_{(1)},\dots,X_{(k)}\\}$, and $b$ neighbours in the set
$\\{X_{(1)},\dots,X_{(k)}\\}$. Let $(Y_{1},\dots,Y_{k})$ be a sample from the
law (1.16). We introduce the local limit $G_{\scriptscriptstyle\infty}$ of the
graph $G_{n}$, which is based on the same connection probabilities but with an
infinite vertex set. For the lattice case, $G_{\infty}$ is obtained by
considering as vertex set all the points of the lattice $\mathbb{Z}^{d}$, for
the Poisson case $G_{\scriptscriptstyle\infty}$ is constructed by taking as
vertex set a Poisson point process of intensity one on $\mathbb{R}^{d}$, with
an atom inserted at $0\in\mathbb{R}^{d}$. In both cases, the infinite random
graph, rooted at $0\in\mathbb{R}^{d}$ arises as the local (Benjamini-Schramm)
limit in probability of $G_{n}$, see [14, Theorem 1.11]. Denote the degree of
the root in $G_{\scriptscriptstyle\infty}$ by $D_{\scriptscriptstyle\infty}$
and the degree of the root in $G_{\scriptscriptstyle\infty}$ conditioned on
having weight $w$ by $D_{\scriptscriptstyle\infty}(w)$.

Our third main result describes the degree distribution of the graph $G_{n}$
conditioned on the event $|E_{n}|\geq n(\mu+\rho)$. We start with a version
describing degree sequences of hubs and other vertices separately.

######  Theorem 1.8 (Degree sequences of hubs and normal vertices).

Consider the weight-dependent random connection model with kernel $\kappa$ and
profile $\varphi$ satisfying Assumption A. Let $\rho>0$ be a non integer, and
$k$ be the unique integer satisfying $k-1<\rho<k$. Conditionally on the rare
event $|E_{n}|\geq n(\mu+\rho)$, as $n\rightarrow\infty$,

$\displaystyle\left(\tfrac{1}{n}(D_{X_{\scriptscriptstyle(i)}})_{i\in[k]},(\pi_{a,b}^{\scriptscriptstyle(n)})_{0\leq
a,0\leq b\leq k}\right)\stackrel{{\scriptstyle
d}}{{\to}}\left((\Lambda(Y_{i}))_{i\in[k]},(\pi_{a,b})_{0\leq a,0\leq b\leq
k}\right),$ (1.22)

in distribution on sequence space with the product topology, while

$\displaystyle\tfrac{1}{n}{D_{X_{\scriptscriptstyle(i)}}}\stackrel{{\\!{}_{\mathbb{P}}}}{{\rightarrow}}0\qquad\mbox{
for each $i>k$,}$ (1.23)

where

$\displaystyle\pi_{a,b}=\pi_{a,b}((U_{i},Y_{i})_{i\in[k]}):=\mathbb{E}\Big{[}D_{\scriptscriptstyle\infty}(W)=a,\sum_{i=1}^{k}B^{\scriptscriptstyle{(i)}}_{U,W}=b\,\Big{|}\,(U_{i},Y_{i})_{i\in[k]}\Big{]},$
(1.24)

and where the vector $(Y_{1},\dots,Y_{k})$ is chosen according to the law
given by (1.16) and $U_{1},\ldots,U_{k}$ are i.i.d. uniformly distributed
random variables on $\mathbb{T}_{1}^{d}$ independent of
$(Y_{1},\ldots,Y_{k})$. Moreover, the expectation is over $U$ which is
uniformly distributed on $\mathbb{T}_{1}^{d}$, and an independent Pareto-
distributed weight $W$ and, for $u\in\mathbb{T}_{1}^{d}$ and $w\geq 1$, over
conditionally independent Bernoulli random variables
$B^{\scriptscriptstyle(1)}_{u,w},\ldots,B^{\scriptscriptstyle(k)}_{u,w}$ with
parameter

$\varphi\left(\frac{d_{1}(u,U_{i})^{d}}{\mathcal{K}(Y_{i},w)}\right).$

Combining the results in Theorem 1.8, we get convergence of the degree
distribution.

######  Corollary 1.9 (Degree distribution).

Conditionally on the rare event $|E_{n}|\geq n(\mu+\rho)$, the empirical
degree distribution

$\frac{1}{|V_{n}|}\sum_{x\in V_{n}}\delta_{D_{x}}$

converges in distribution on the Polish space of probability measures with the
weak topology to the random probability measure

$\sum_{a=0}^{\infty}\sum_{b=0}^{k}\pi_{a,b}\,\delta_{a+b}\,.$

######  Remark 1.10 (The condensate).

Theorem 1.8 identifies where the surplus edges, whose limit in distribution
equals $S$ by Remark 1.5, are located. Indeed, using that the sum of the
degrees equals $2|E_{n}|$, we see that the vertices of high-weight of order
$n$ form the condensate attracting roughly $nS$ incident edges. The other ends
of the edges are at vertices whose weights have the unconditional weight
distribution $W$. Thus, the condensate consists of $k$ vertices, but other
vertices can be incident to this condensate. This makes the condensation
phenomenon different, and arguably richer, than the one described in Section
1.1.  $\blacktriangleleft$

##  2\. Overview and proof strategies

###  2.1. Upper large deviations: Strategy and proof of Theorem 1.1

We call an edge high-weight if at least one of its endpoints has high weight,
which we make precise below. Further, light-weight edges are edges where
neither endpoints have high weight. We also divide edges into long and short
edges, where long edges are those that connect vertices far away from each
other. More precisely, let $(\varepsilon_{n})_{n\geq 1}$ be some sequence
tending to zero, let $a\in(0,1)$ and split the edge set $E_{n}$ into the three
disjoint sets

$\displaystyle E_{\rm main}$ $\displaystyle:=\\{\\{x,y\\}\in E_{n}\colon
d_{n}(x,y)\leq\varepsilon_{n}n^{1/d},\,W_{x}\vee W_{y}\leq n^{a}\\},$ (2.1)
$\displaystyle E_{\rm long}$ $\displaystyle:=\\{\\{x,y\\}\in E_{n}\colon
d_{n}(x,y)>\varepsilon_{n}n^{1/d},\,W_{x}\vee W_{y}\leq n^{a}\\},$ (2.2)
$\displaystyle E_{\rm high}$ $\displaystyle:=\\{\\{x,y\\}\in E_{n}\colon
W_{x}\vee W_{y}>n^{a}\\},$ (2.3)

where $x\vee y$ denotes the maximum of $x$ and $y$. These sets correspond to
the sets containing short and light-weight edges, long and light-weight edges
and lastly high-weight edges. Theorem 1.1 follows from three key steps
estimating the number of edges in these sets.

#### Step 1. Short and light-weight edges form the bulk.

Consider short, light-weight edges. These constitute the ‘bulk’ edge count,
i.e., a typical edge of the graph falls into this category. Proposition 2.1
shows that the number of these edges are well concentrated about the mean
$n\mu$.

######  Proposition 2.1.

Let $\alpha>1$ be the parameter in (1.7) and let $E_{\rm main}$ be as in (2.1)
with $\varepsilon_{n}$ and $a$ chosen such that

  * $\rhd$

$\varepsilon_{n}^{d}=n^{-\gamma}$ for some
$0<\gamma<(1-\tfrac{1}{\alpha})\wedge\tfrac{1}{d}$, and

  * $\rhd$

$0<a(2\vee(\beta-1))<(1-\gamma-\tfrac{1}{\alpha})\wedge\tfrac{1}{2}$.

Then, for all $\delta>0$, we have

${\mathbb{P}}\left(||E_{\rm main}|-n\mu|\geq
n\delta\right)=o({n^{-k(\beta-2)}}).$

We prove Proposition 2.1 in Section 3. In the proof we first fix the vertex
weights (and in the Poisson case both the vertex locations and weights) and
apply a conditional version of McDiarmid’s concentration inequality, using
that the edges are included independently. Then we use McDiarmid’s inequality
again to show concentration of the resulting functionals of the independent
weights (and in the Poisson case also the vertex locations).

#### Step 2. Long and light-weight edges are negligible.

Proposition 2.2 shows that this contribution is asymptotically negligible.

######  Proposition 2.2.

For all $\delta>0$, there exist $\varepsilon_{n}$ and $a$ satisfying the
conditions of Proposition 2.1 such that $E_{\rm long}$ in (2.2) satisfies

${\mathbb{P}}\left(|E_{\rm long}|\geq n\delta\right)=o({n^{-k(\beta-2)}}).$

We prove Proposition 2.2 in Section 4 for which we use that the connection
probability of a pair of vertices is increasing with respect to the weights of
the vertices, and decreasing with respect to the spatial distance between the
vertices. This implies that the total number of long and light-weight edges
can be stochastically dominated by a suitable binomial random variable and we
can use a standard binomial concentration argument.

#### Step 3. High-weight edges form the condensate.

Using the first two steps, it remains to understand the probability that the
total number of high-weight edges is larger than $n\rho$. This is accomplished
in Proposition 2.3.

######  Proposition 2.3.

Let $E_{\rm high}$ be as in (2.3). Then, for any $0<a<1$, we have

${\mathbb{P}}\left(|E_{\rm high}|\geq
n\rho\right)=(F(\rho)+o(1))n^{-k(\beta-2)},$

where $F(\rho)$ is given by (1.11).

We prove Proposition 2.3 in Section 5 for which we use that the number of
high-weight edges is well approximated by the sum of the degrees of the high-
weight vertices. The degree of a vertex given its weight has Poisson-like
statistics with the mean being an increasing function (depending on $n$) of
the weight, see for example [14, Proposition 1] for a proof of this fact when
the locations of the vertices are uniform, as in the Poisson models. As a
consequence, again using concentration arguments, when the weight of a vertex
diverges, its degree behaves like this function evaluated at its weight. This
observation allows us to approximate the total number of high-weight edges by
an i.i.d. sum of a deterministic (but $n$-dependent) function of the weights.
To understand the probability that this sum is at least $n\rho$, we do a large
deviation analysis similar to the one given in [19], and show that this event
occurs precisely when there are $k=\lceil\rho\rceil$ vertices with weight of
order $n$. This shows that ${\mathbb{P}}\left(|E_{\rm high}|\geq n\rho\right)$
behaves as stated in Proposition 2.3.

We now prove continuity of $F$, which is the last ingredient in the proof of
Theorem 1.1, which then follows straightforwardly from the above propositions.

######  Lemma 2.4.

For all $b\geq 0$ and $k\in\mathbb{N}$,

$\rho\mapsto
F_{b}(\rho):=\frac{(\beta-1)^{k}}{{k!}}\int_{b}^{\infty}\cdots\int_{b}^{\infty}{\mathbbm{1}}_{\\{{{\Lambda}(y_{1})+\cdots+{\Lambda}(y_{k})>\rho}\\}}\prod_{i=1}^{k}y_{i}^{-\beta}dy_{i}$

defines a continuous function $F_{b}\colon(k-1,k)\to(0,\infty)$.

###### Proof.

Finiteness for $b>0$ follows from $\beta>2$. For $b=0$, we note that
$\lim_{y\searrow 0}\Lambda(y)=0$. Choosing $b>0$ such that
$\Lambda(y)<\rho-(k-1)$ for all $y\in(0,b)$, and using that $\Lambda(y)\leq
1$, we see that $F_{0}(\rho)=F_{b}(\rho)$. To see positivity, we lower bound
$F_{b}(\rho)$ by

$\left(\int_{b}^{\infty}{\mathbbm{1}}_{\\{{\Lambda}(y)>\rho/k\\}}y^{-\beta}dy\right)^{k}.$

Note that $\lim_{y\uparrow\infty}{\Lambda}(y)=1$ and hence the set
$\\{{\Lambda}(y)>\rho/k\\}$ contains an interval unbounded to the right and
hence has positive measure. Hence $F_{b}(\rho)>0$.

To see continuity, fix $\rho\in(k-1,k)$. Given $\varepsilon>0$, we need to
find $\delta>0$ with

$\int_{0}^{\infty}\cdots\int_{0}^{\infty}{\mathbbm{1}}_{\\{\rho+\delta>\Lambda(y_{1})+\cdots+{\Lambda}(y_{k})>\rho-\delta\\}}\prod_{i=1}^{k}y_{i}^{-\beta}dy_{i}<\varepsilon.$

For this, we note that if $\delta^{\prime}>0$ is small enough, then
$\rho+\delta^{\prime}>\Lambda(y_{1})+\cdots+{\Lambda}(y_{k})>\rho-\delta^{\prime}$
implies that there exists $i$ with
$\delta^{\prime}<\Lambda(y_{i})<1-\delta^{\prime}$. Using that $\varphi$ has
no flat pieces except possibly at zero and one, we find
$\delta^{\prime}>\delta^{\prime\prime},\delta>0$ such that
$|\Lambda(y)-\Lambda(y^{\prime})|>2\delta$ if
$\delta^{\prime}<\Lambda(y)<1-\delta^{\prime}$ and
$|y-y^{\prime}|>\delta^{\prime\prime}$. Hence one of the integrations with
respect to $dy_{1},\ldots,dy_{k}$ is over an interval of length at most
$\delta^{\prime\prime}$ and choosing $\delta^{\prime}>0$ small we can upper
bound the integral by $\varepsilon$. ∎

###### Proof of Theorem 1.1.

Take $\delta>0$ sufficiently small such that
$(\rho-\delta,\rho+\delta)\subseteq(k-1,k)$. Then fix $0<a<1$ and
$(\varepsilon_{n})_{n\geq 1}$ such that Propositions 2.1, 2.2 and 2.3 can be
applied. Using the definitions of $E_{\rm main}$, $E_{\rm long}$ and $E_{\rm
high}$ in (2.1)-(2.3), we can write

$\displaystyle|E_{n}|$ $\displaystyle=|E_{\rm main}|+|E_{\rm long}|+|E_{\rm
high}|.$ (2.4)

The upper bound of Theorem 1.1 follows by noting that

$\displaystyle{\mathbb{P}}\left(|E_{n}|\geq n(\mu+\rho)\right)$
$\displaystyle\leq{\mathbb{P}}\left(|E_{\rm main}|\geq
n(\mu+\delta/2)\right)+{\mathbb{P}}\left(|E_{\rm long}|\geq
n\delta/2\right)+{\mathbb{P}}\left(|E_{\rm high}|\geq n(\rho-\delta)\right)$
(2.5) $\displaystyle=(F(\rho-\delta)+o(1))n^{-k(\beta-2)},$

where the last equality holds by applying Propositions 2.1, 2.2 and 2.3. For
the lower bound, remark that

$\displaystyle{\mathbb{P}}\left(|E_{n}|\geq n(\mu+\rho)\right)$
$\displaystyle\geq{\mathbb{P}}\left(|E_{\rm high}|+|E_{\rm main}|\geq
n(\mu+\rho)\right)$ (2.6) $\displaystyle\geq{\mathbb{P}}\left(|E_{\rm
high}|\geq n(\rho+\delta),\,|E_{\rm main}|\geq n(\mu-\delta)\right)$
$\displaystyle\geq{\mathbb{P}}\left(|E_{\rm high}|\geq
n(\rho+\delta)\right)-{\mathbb{P}}\left(|E_{\rm main}|<n(\mu-\delta)\right).$

By Propositions 2.1 and 2.3, the last line equals
$(F(\rho+\delta)+o(1))n^{-k(\beta-2)}$. Combining the upper and lower bounds
and, recalling from Lemma 2.4 that $F$ is continuous on $(k-1,k)$, we complete
the proof by letting $\delta$ tend to zero. ∎

###  2.2. Convergence of the empirical edge-lengths: Strategy and proof of
Theorem 1.6

Throughout this section $k$ is the unique integer with $k-1<\rho<k$. The key
step in the proof of Theorem 1.6 (a) is the following law of large numbers.

######  Proposition 2.5.

Let $f\colon(0,\infty)\to[0,\infty)$ be continuous with compact support and
recall $\lambda_{f}$ from (1.19). Then, for all $\delta>0$,

$\mathbb{P}\Big{(}\Big{|}\frac{1}{n}\sum_{{}_{\\{x,y\\}\in
E_{n}}}f(d_{n}(x,y))-\tfrac{1}{2}{\mathbb{E}}\left[\lambda_{f}(W)\right]\Big{|}>\delta,|E_{n}|\geq
n(\mu+\rho)\Big{)}=o({n^{-k(\beta-2)}}).$

In the proof of Proposition 2.5, given in Section 6, we split the sum over all
edges into the sum over the edges in $E_{\rm main}$, for which we can argue
similarly as in Proposition 2.1, and the sum over the edges in $E_{\rm long}$
and $E_{\rm high}$, which are negligible as most of these edges are too long
to be in the support of $f$.

###### Proof of Theorem 1.6(a).

Proposition 2.5 together with Theorem 1.1 imply that for all $\delta>0$, we
have

$\displaystyle{\mathbb{P}}\left(\left|\int
f(x)d\mu_{n}(x)-\tfrac{1}{2}{\mathbb{E}}\left[\lambda_{f}(W)\right]\right|>\delta,|E_{n}|\geq
n(\mu+\rho)\right)=o(1){\mathbb{P}}\left(|E_{n}|\geq n(\rho+\mu)\right).$

Theorem 1.6(a) then follows. ∎

For the proof of Theorem 1.6(b), we first approximate the empirical edge-
length distribution at macroscopic scale by a deterministic function of the
$k$ largest weights in the graph. Let $f\colon(0,\infty)\to[0,\infty)$ be
continuous with compact support and define

$\displaystyle\Lambda_{f}(w):=\int_{[-\frac{1}{2},\frac{1}{2}]^{d}}f(\|z\|){{\Lambda}}(w,z)\,dz,$
(2.7)

where we recall $\Lambda(w,z)$ from (1.9). Our next result identifies the
limiting distribution of long edges in terms of the largest weights.

######  Proposition 2.6.

Conditionally on $|E_{n}|\geq n(\rho+\mu)$,

$\displaystyle\frac{1}{n}\sum_{{}_{\\{x,y\\}\in
E_{n}}}f\left(\frac{d_{n}(x,y)}{n^{1/d}}\right)-\sum_{i=1}^{k}\Lambda_{f}\left(\frac{W_{\scriptscriptstyle(i)}}{n}\right)\stackrel{{\\!{}_{\mathbb{P}}}}{{\rightarrow}}0,$

where $W_{\scriptscriptstyle(1)},\ldots,W_{\scriptscriptstyle(k)}$ are the $k$
largest weights in the graph ordered by decreasing size.

We prove Proposition 2.6 in Section 7 by using a concentration argument to
approximate, for all vertices $x$ with weight $W_{x}$ exceeding $n^{a}$, the
term $\sum_{y:\,\\{x,y\\}\in E_{n}}f(\tfrac{d_{n}(x,y)}{n^{1/d}})$ by its
conditional expectation given $W_{x}$. For $W_{x}\approx wn$ (here $\approx$
meaning the ratio of the two sides going to $1$), the conditional expectation
converges to $\Lambda_{f}(w)$. Together with the fact that the contribution of
light-weight edges to the number of long edges is negligible, this allows us
to approximate $\int f(\frac{x}{n^{1/d}})\,d\mu_{n}$ by a deterministic
function of the $k$ largest weights.

We then show convergence in distribution of the $k$ largest weights in the
graph.

######  Proposition 2.7.

Conditionally on $|E_{n}|\geq n(\mu+\rho)$,

$\displaystyle\frac{1}{n}(W_{\scriptscriptstyle(1)},\dots,W_{\scriptscriptstyle(k)})\stackrel{{\\!{}_{\mathrm{d}}}}{{\rightarrow}}(Y_{{1}},\dots,Y_{{k}}),$

where the vector $(Y_{1},\dots,Y_{k})$ is chosen according to the law given by
(1.16).

We prove Proposition 2.7 in Section 8 by refining the analysis of the high-
weight edges in Section 5. We show that the event $|E_{n}|\geq n(\mu+\rho)$ is
asymptotically equivalent to the event
$\sum_{i=1}^{k}{{\Lambda}}(W_{\scriptscriptstyle(i)}/n)>\rho$. The result
follows because, given the vertex set $V_{n}$, the random variables
$(W_{x})_{x\in V_{n}}$ are independent Pareto distributed.

###### Proof of Theorem 1.6(b).

Note that $\Lambda_{f}$ is bounded from above and continuous by means of the
linearity of $\mathcal{K}$ in the first component. Thus the function
$(x_{1},\dots,x_{k})\mapsto\sum_{i=1}^{k}\Lambda_{f}(x_{i})$ is continuous and
bounded. Combining Propositions 2.6 and 2.7, we get that

$\frac{1}{n}\sum_{{}_{\\{x,y\\}\in
E_{n}}}f\left(\frac{d_{n}(x,y)}{n^{1/d}}\right)\stackrel{{\\!{}_{\mathrm{d}}}}{{\rightarrow}}\sum_{i=1}^{k}\Lambda_{f}\left(Y_{i}\right),$

as claimed in the theorem. ∎

######  Remark 2.8 (A hub clique).

As a consequence of Proposition 2.7, conditionally on the event
$|E_{n}|\geq(\mu+\rho)n$, there are $k$ vertices with weight at least
$\varepsilon n$, with probability at least $1-f(\varepsilon)$, where
$f(\varepsilon)\to 0$ as $\varepsilon\searrow 0$. We call these high-weight
vertices  _hubs_. As the length of any edge is at most
$\frac{\sqrt{d}}{2}n^{1/d}$, and the connection probability is
$\varphi({d_{n}(X_{i},X_{j})^{d}}/{\kappa(W_{i},W_{j})})$, we use a union
bound to see that the probability of the event that there is a pair of hubs
which does not share an edge is at most $\binom{k}{2}(1-\varphi(2^{-d}/({\sf
c}\varepsilon n^{2}n)))$. Hence if $\lim_{r\to 0}\varphi(r)=1$ the $k$ hubs
form a clique with high probability. $\blacktriangleleft$

###  2.3. Convergence of degree distribution: Strategy and proof of Theorem
1.8

In this section, we reduce the proof of Theorem 1.8 to the following
proposition. Recall $D_{\infty}(w)$ denotes the degree of the root in the
local limit $G_{\infty}$ of the graph $G_{n}$ conditioned on having weight
$w$.

######  Proposition 2.9.

Let $y_{1}>y_{2}>\cdots>y_{k}>0$ and
$u_{1},\ldots,u_{k}\in\mathbb{T}_{1}^{d}$. Let

$\displaystyle\pi_{a,b}\big{(}(y_{i},u_{i})_{i\in[k]})=\mathbb{P}\Big{(}D_{\scriptscriptstyle\infty}(W)=a,\sum_{i=1}^{k}B^{\scriptscriptstyle(i)}_{U,W}=b\Big{)},$
(2.8)

where $(B^{\scriptscriptstyle(i)}_{U,W})_{i\in[k]}$ are, conditionally on
$U,W$, independent Bernoulli random variables with parameter

$\varphi\bigg{(}\frac{d_{1}(U,u_{i})^{d}}{\mathcal{K}(y_{i},W)}\bigg{)}.$

Conditionally on $W_{\scriptscriptstyle(i)}=ny_{i}$ and
$X_{\scriptscriptstyle(i)}=n^{1/d}u_{i}$ (in the lattice case projected to the
nearest lattice point of $\mathbb{T}_{n}^{d}$) for all $i\in[k]$,

  1. (i)

$\frac{1}{n}(D_{X_{\scriptscriptstyle(1)}},\dots,D_{X_{\scriptscriptstyle(k)}})\stackrel{{\\!{}_{\mathbb{P}}}}{{\rightarrow}}(\Lambda(y_{1}),\dots,\Lambda(y_{k})),$

  2. (ii)

$\pi_{a,b}^{\scriptscriptstyle(n)}\stackrel{{\\!{}_{\mathbb{P}}}}{{\rightarrow}}\pi_{a,b}\big{(}(y_{i},u_{i})_{i\in[k]}\big{)},$

  3. (iii)

$\frac{1}{n}D_{X_{\scriptscriptstyle(j)}}\stackrel{{\\!{}_{\mathbb{P}}}}{{\rightarrow}}0,\mbox{
for each $j>k$.}$

We prove Proposition 2.9 in Section 9, and now prove Theorem 1.8 subject to
it.

###### Proof of Theorem 1.8.

We present the technicalities in the language of the Poisson case, the lattice
case is analogous. To prepare this proof, take
$g\colon(0,\infty)^{k}\to[0,\infty)$ an arbitrary, measurable function. Write
$\Pi_{n}=\sum_{x\in V_{n}}\delta_{W_{x}}$ for the point process of weights and
$\mathcal{W}_{n}=\\{W_{x}\colon x\in V_{n}\\}$ for the corresponding point
set. Then

$g(W_{\scriptscriptstyle(1)},\ldots,W_{\scriptscriptstyle(k)})=\sum_{{w_{1},\ldots,w_{k}\in\mathcal{W}_{n}}\atop{w_{1}>\cdots>w_{k}}}{\mathbbm{1}}_{\\{\Pi_{n}(w_{1},\infty)=0,\Pi_{n}(w_{2},w_{1})=0,\ldots,\Pi_{n}(w_{k},w_{k-1})=0\\}}g(w_{1},\ldots,w_{k}).$

Applying the multivariate Mecke formula [20, Theorem 4.4] and denoting the
intensity measure of $\Pi_{n}$ by $\pi_{n}$, we get

$\displaystyle\mathbb{E}[$ $\displaystyle
g(W_{\scriptscriptstyle(1)},\ldots,W_{\scriptscriptstyle(k)})]$
$\displaystyle=\int\\!\\!\pi_{n}(dw_{1})\cdots\\!\int\\!\\!\pi_{n}(dw_{k}){\mathbbm{1}}_{\\{w_{1}>\cdots>w_{k}\\}}\mathbb{P}(\Pi_{n}(w_{k},\infty)=0)g(w_{1},\ldots,w_{k}).$

Because $\pi_{n}(dt)=n(\beta-1)t^{-\beta}{\mathbbm{1}}_{\\{t\geq 1\\}}dt$ and
hence $\pi_{n}(w_{k},\infty)=nw_{k}^{1-\beta}$, we get

$\displaystyle\mathbb{E}$
$\displaystyle[g(W_{\scriptscriptstyle(1)},\ldots,W_{\scriptscriptstyle(k)})]$
$\displaystyle=n^{k}(\beta-1)^{k}\int_{1}^{\infty}dt_{1}\cdots\int_{1}^{\infty}dt_{k}(t_{1}\cdots
t_{k})^{-\beta}{\mathrm{e}}^{-nt_{k}^{1-\beta}}{\mathbbm{1}}_{\\{t_{1}>\cdots>t_{k}\\}}g(t_{1},\ldots,t_{k}).$
(2.9)

Now fix an integer $\ell\geq 1$ and a bounded, continuous function
$h\colon\mathbb{R}^{k+(k+1)(\ell+1)}\to\mathbb{R}$. To simplify notation we
write $(\pi_{a,b}^{\scriptscriptstyle(n)})$ for
$(\pi_{a,b}^{\scriptscriptstyle(n)})_{0\leq a\leq\ell,0\leq b\leq k}$ and
$(y_{i},u_{i})$ for $(y_{i},u_{i})_{i\in[k]}$, dropping the index range when
there is no ambiguity. To prove (1.22), it suffices to show, as $n\to\infty$,

$\displaystyle\mathbb{E}\Big{[}h\big{(}\tfrac{1}{n}(D_{X_{\scriptscriptstyle(i)}})_{i\in[k]},$
$\displaystyle(\pi_{a,b}^{\scriptscriptstyle(n)})\big{)}~{}\Big{|}~{}|E_{n}|\geq(\mu+\rho)n\Big{]}\rightarrow\mathbb{E}\Big{[}h\big{(}(\Lambda({Y_{i}})),\big{(}\pi_{a,b}({(Y_{i},U_{i})})\big{)}\big{)}\Big{]}.$
(2.10)

Fix $\varepsilon>0$ small enough so that $\Lambda(\varepsilon)<\rho-(k-1)$.
Abbreviate $B=\\{|E_{n}|\geq(\mu+\rho)n\\}$ and

$A=\Big{\\{}(t_{1},\ldots,t_{k})\in(\varepsilon,\infty)^{k}\colon\sum_{i\in[k]}\Lambda(t_{i})\geq\rho\Big{\\}}.$

The left-hand side of (2.10) can be written as

$\displaystyle\frac{1}{{\mathbb{P}}\left(B\right)}{\mathbb{E}}\left[{\mathbbm{1}}_{A}(({W_{\scriptscriptstyle(i)}}/{n}))h\big{(}\tfrac{1}{n}(D_{X_{\scriptscriptstyle(i)}}),(\pi_{a,b}^{\scriptscriptstyle(n)})\big{)}\right]$
(2.11)
$\displaystyle-\frac{1}{{\mathbb{P}}\left(B\right)}{\mathbb{E}}\left[{\mathbbm{1}}_{B^{c}}{\mathbbm{1}}_{A}(({W_{\scriptscriptstyle(i)}}/{n}))h\big{(}\tfrac{1}{n}(D_{X_{\scriptscriptstyle(i)}}),(\pi_{a,b}^{\scriptscriptstyle(n)})\big{)}\right]$
(2.12)
$\displaystyle+\frac{1}{{\mathbb{P}}\left(B\right)}{\mathbb{E}}\left[{\mathbbm{1}}_{B}{\mathbbm{1}}_{A^{c}}(({W_{\scriptscriptstyle(i)}}/{n}))h\big{(}\tfrac{1}{n}(D_{X_{\scriptscriptstyle(i)}}),(\pi_{a,b}^{\scriptscriptstyle(n)})\big{)}\right].$
(2.13)

We will show that (2.11) converges to the right-hand side of (2.10) while
(2.12) and (2.13) tend to zero as $n\to\infty$.

Since $h$ is bounded, (2.13) can be bounded from above by a constant multiple
of

$\displaystyle\mathbb{P}\Big{(}$
$\displaystyle(W_{\scriptscriptstyle(i)}/n)_{i\in[k]}\in A^{c}\mid{B}\Big{)}$
$\displaystyle\leq$
$\displaystyle\,\mathbb{P}\Big{(}\sum_{i\in[k]}\Lambda(W_{\scriptscriptstyle(i)}/n)<\rho\mid
B\Big{)}+\mathbb{P}\left(\left.W_{\scriptscriptstyle(i)}/n\leq\varepsilon\mbox{
for some }i\in[k]\;\right|\;B\right).$ (2.14)

By Proposition 2.7 and the Portmanteau theorem,

$\mathbb{P}\Big{(}\sum_{i\in[k]}\Lambda\big{(}W_{\scriptscriptstyle(i)}/n\big{)}<\rho\mid
B\Big{)}\to\mathbb{P}\Big{(}\sum_{i\in[k]}\Lambda\big{(}Y_{i}\big{)}<\rho\Big{)}=0.$

For the second summand of (2.3), observe that by Proposition 2.7,

$\displaystyle\mathbb{P}\left(\left.W_{\scriptscriptstyle(i)}/n\leq\varepsilon\mbox{
for some
}i\in[k]\;\right|\;B\right)\to{\mathbb{P}}\left(Y_{i}\leq\varepsilon\mbox{ for
some }i\in[k]\right).$

Recall that $\varepsilon$ is chosen such that
$\Lambda(\varepsilon)<\rho-(k-1)$ and so
$\sum_{i\in[k]}\Lambda(Y_{i})\geq\rho$ cannot hold if there is $i\in[k]$ with
$Y_{i}<\epsilon$. Hence the law of $(Y_{1},\ldots,Y_{k})$ given in (1.16) is
such that $Y_{i}>\varepsilon$ for all $i\in[k]$, almost surely. It follows
that (2.3), and hence (2.13) tend to zero.

Next, (2.12) can be bounded by a constant multiple of

$|{{\mathbb{P}}\left((W_{\scriptscriptstyle(i)}/n)_{i\in[k]}\in
A\right)}/{{\mathbb{P}}\left(B\right)}-\mathbb{P}((W_{\scriptscriptstyle(i)}/n)_{i\in[k]}\in
A\mid B)|.$

By the above, the subtrahend goes to one. The minuend does the same as, by
(2.9),

$\displaystyle{\mathbb{P}}\left((W_{\scriptscriptstyle(i)}/n)_{i\in[k]}\in
A\right)$
$\displaystyle=n^{k}(\beta-1)^{k}\int_{1}^{\infty}dt_{1}\cdots\int_{1}^{\infty}dt_{k}(t_{1}\cdots
t_{k})^{-\beta}{\mathrm{e}}^{-nt_{k}^{1-\beta}}{\mathbbm{1}}_{\\{t_{1}>\cdots>t_{k},(t_{1}/n,\ldots,t_{k}/n)\in
A\\}}$ $\displaystyle\sim
n^{k(2-\beta)}(\beta-1)^{k}\int_{\varepsilon}^{\infty}dy_{1}\cdots\int_{\varepsilon}^{\infty}dy_{k}(y_{1}\cdots
y_{k})^{-\beta}{\mathbbm{1}}_{\\{y_{1}>\cdots>y_{k},\sum_{i\in[k]}\Lambda(y_{i})\geq\rho\\}},$
(2.15)

using the substitution $t_{i}=y_{i}n$ and using that $2-\beta<0$. Again by
choice of $\varepsilon$ if there is $i\in[k]$ with $y_{i}<\epsilon$ then
$\sum_{i\in[k]}\Lambda(y_{i})<\rho$. Together with Theorem 1.1 this shows that
(2.15) is asymptotically equivalent to $\mathbb{P}(B)$.

It remains to show that (2.11) converges to the right-hand side of (2.10).
Note that

$\displaystyle\frac{1}{{\mathbb{P}}\left(B\right)}{\mathbb{E}}\left[{\mathbbm{1}}_{A}((W_{\scriptscriptstyle(i)}/n)_{i\in[k]})h\big{(}\tfrac{1}{n}(D_{X_{\scriptscriptstyle(i)}})_{i\in[k]},(\pi_{a,b}^{\scriptscriptstyle(n)})\big{)}\right]$
$\displaystyle=\frac{1}{{\mathbb{P}}\left(B\right)}{\mathbb{E}}\left[{\mathbbm{1}}_{A}((W_{\scriptscriptstyle(i)}/n)_{i\in[k]})\mathbb{E}\left[\left.h\big{(}\tfrac{1}{n}(D_{X_{\scriptscriptstyle(i)}})_{i\in[k]},(\pi_{a,b}^{\scriptscriptstyle(n)})\big{)}\;\right|\;(W_{\scriptscriptstyle(i)},X_{\scriptscriptstyle(i)})\right]\right].$

Using (2.9) and the substitution $t_{i}=y_{i}n$ again, this equals

$\displaystyle(\beta-1)^{k}$
$\displaystyle\int_{\varepsilon}^{\infty}dy_{1}\cdots\int_{\varepsilon}^{\infty}dy_{k}\int_{\mathbb{T}_{1}^{d}}du_{1}\cdots\int_{\mathbb{T}_{1}^{d}}du_{k}(y_{1}\cdots
y_{k})^{-\beta}{\mathbbm{1}}_{\\{y_{1}>\cdots>y_{k},\sum_{i\in[k]}\Lambda(y_{i})\geq\rho\\}}$
$\displaystyle\cdot\frac{1}{{\mathbb{P}}\left(B\right)}n^{k(2-\beta)}{\mathrm{e}}^{-n^{2-\beta}y_{k}^{1-\beta}}\mathbb{E}\left[\left.h\big{(}\tfrac{1}{n}(D_{X_{\scriptscriptstyle(i)}})_{i\in[k]},(\pi_{a,b}^{\scriptscriptstyle(n)})\big{)}\;\right|\;(W_{\scriptscriptstyle(i)},X_{\scriptscriptstyle(i)})=(ny_{i},n^{1/d}u_{i})\right].$

By Theorem 1.1 the integrand (in the second line) is bounded, and by
Proposition 2.9(i) and (ii), converges to

$\frac{1}{F(\rho)}h\big{(}(\Lambda({y_{i}})),\big{(}\pi_{a,b}({(y_{i},u_{i})})\big{)}.$

As the integrating measure (in the first line) is finite, dominated
convergence implies convergence to

$\mathbb{E}\Big{[}h\big{(}(\Lambda({Y_{i}})),\big{(}\pi_{a,b}({(Y_{i},U_{i})})\big{)}\big{)}\Big{]},$

as required to show (2.10) and hence (1.22).

Finally, to prove (1.23), it suffices to show that, for $j>k$ fixed,
$\mathbb{E}[\frac{1}{n}D_{X_{\scriptscriptstyle(j)}}|B]\to 0$ . Similar to
above, this can be reduced to the statement of Proposition 2.9(iii). ∎

###  2.4. Discussion: Why we assume $\rho$ to be non-integer.

We now briefly discuss the nature of the problem when $\rho>0$ is an integer
in Theorem 1.1. The proof can be reduced to studying the deviations of a
truncated i.i.d. sum of the form
$\sum_{i=1}^{n}\lambda_{n}(W_{i}){\mathbbm{1}}_{\\{W_{i}>n^{a}\\}}$ for a
suitable $a>0$, where we can think of
$\lambda_{n}(w)=\mathbb{E}\left[\left.D_{x}\;\right|\;W_{x}=w\right]$. It
turns out that, for any $b>0$, we have $\lambda_{n}(bn)/n\to{\Lambda}(b)$ for
some function ${\Lambda}(b)$, which converges to $1$ as $b\to\infty$. This
leads us to study the large deviations of
$S_{n}:=\sum_{i=1}^{n}n{\Lambda}(W_{i}/n)$, with $(W_{i})_{i\geq 1}$ an i.i.d.
sequence with law (1.1) and ${\Lambda}(w)\leq 1$ with ${\Lambda}(w)\to 1$ as
$w\to\infty$. We claim that these large deviations are not universal at the
integers.

We discuss the probability of the event $S_{n}\geq(\mu+k)n$, where
$S_{n}/n\stackrel{{\\!{}_{\mathbb{P}}}}{{\rightarrow}}\mu=\mathbb{E}[W_{1}]$,
for three illustrative examples of ${\Lambda}$, which differ in the approach
to $1$ in the large $x$ limit:

$\displaystyle\text{(a)}\;{\Lambda}(x)=1\wedge
x;\;\;\;\text{(b)}\;{\Lambda}(x)=1-{\mathrm{e}}^{-x};\;\;\;\text{(c)}\;{\Lambda}(x)=\frac{x}{1+x}.$

In case (a), we note that if $W_{i}>n$, the corresponding summand contributes
$n$ to the sum $S_{n}$. Thus, we get $S_{n}\geq(\mu+k)n$ if there are $k$
values of $i$ for which $W_{i}>n$. The remaining contribution comes from the
‘small’ weights remaining tightly concentrated around $\mu n$. This strategy
turns out to be optimal and has a probability of order $n^{-k(\beta-2)}$,
similar in flavour to Theorem 1.1.

In case (b), we can consider a strategy where there are exactly $k$ weights
being at least $cn\log{n}$ for some large constant $c>0$. The contribution
coming from these large weights is at least $nk(1-n^{-c})$. The contribution
coming from the remaining small weights is again tightly concentrated around
$\mu n$, so they contribute with good probability at least $\mu n+n^{1-c}$ to
the sum. This strategy has probability of order $(cn\log{n})^{-k(\beta-2)}$,
which vanishes faster than $n^{-k(\beta-2)}$, but is still much cheaper than
having $k+1$ vertices with weight of order $n$, and we believe it to be
optimal.

In case (c), as before, we can always choose $k+1$ vertices of weight of order
$n$. However, another strategy similar to case (b) can be obtained by choosing
$k$ weights to be at least $n^{b}$ for some appropriate $b>1$, and arguing as
before that the contribution from the small weights is highly concentrated.
Depending on $k$ and $\beta$, either of these strategies can be optimal, and
we expect to see a phase transition depending on $k$ and $\beta$ in this case.

##  3\. Concentration of the bulk: Proof of Proposition 2.1

In this section, we show that the short edges are essentially those in $E_{\rm
main}$ and the measure $\mu_{n}$ is asymptotically equivalent to the empirical
measure of the edge lengths in the unconditioned model. We define
$\mathbb{Q}_{n}$ to be the cube of volume $n$ in $\mathbb{R}^{d}$ centred at
zero.

###  3.1. Setup and main techniques

Recall the definition of $E_{\rm main}$ given by (2.1),

$E_{\rm main}(G_{n})=\\{\\{x,y\\}\in E_{n}\colon
d_{n}(x,y)\leq\varepsilon_{n}n^{1/d},\,W_{x}\vee W_{y}\leq n^{a}\\},$

where $\varepsilon_{n}^{d}=n^{-\gamma}$ for some
$0<\gamma<(1-\tfrac{1}{\alpha})\wedge\tfrac{1}{d}$ and $a>0$ such that
$a(2\vee(\beta-1))<(1-\gamma-\tfrac{1}{\alpha})\wedge\tfrac{1}{2}$. Our main
goal is to prove that $|E_{\rm main}|$ is well-concentrated around $n\mu$. The
number of vertices in our graph, denoted by $N_{n}$, is equal to $n$ in the
lattice case, and a $\mathrm{Poi}(n)$-distributed random variable in the
Poisson case. In this case it is helpful to condition on the number of
vertices. Let $\mathbb{P}_{N_{n}}$ denote the probability of our graph models
conditioned on the number of vertices $N_{n}$ and write $\mathbb{E}_{N_{n}}$
for the associated expectation.

For $M>0$, define the interval

$\displaystyle I_{M}(n):=[n-M\sqrt{n\log n},n+M\sqrt{n\log n}].$ (3.1)

Standard Chernoff concentration bounds [17, Remark 2.6] for Poisson random
variables give us, for Poisson models,

$\displaystyle{\mathbb{P}}\left(N_{n}\notin I_{M}(n)\right)$
$\displaystyle\leq 2\exp\left(-\frac{M^{2}n\log n}{2(n+M\sqrt{n\log
n})}\right)=n^{-\frac{M^{2}}{2+M\sqrt{\log n/n}}},$ (3.2)

for all $M>0$. In particular, choosing $M$ sufficiently large, for example
such that $M^{2}>3k(\beta-2)$, we get that

$\displaystyle{\mathbb{P}}\left(N_{n}\notin
I_{M}(n)\right)=o(n^{-k(\beta-2)}),$ (3.3)

for Poisson models. Once we condition on the number of vertices $N_{n}$, the
graph $G_{n}$ is completely determined by a vector
${\mathbf{W}}_{N_{n}}=(W_{i})_{1\leq i\leq N_{n}}$ of i.i.d. weights with law
(1.1), the vector ${\mathbf{X}}_{N_{n}}=(X_{i})_{1\leq i\leq N_{n}}$ of
conditionally i.i.d. uniform positions in $\mathbb{T}_{n}^{d}$ in the Poisson
case, resp. locations of all the lattice points in $\mathbb{T}_{n}^{d}$ in the
lattice case, and the edge indicators

${\mathbf{A}}_{N_{n}}=(A_{i,j})_{1\leq i<j\leq
N_{n}}\in\\{0,1\\}^{\binom{N_{n}}{2}}.$

We denote the degree of the $i$th vertex by $D_{i}=D_{X_{i}}$. This setup
allows us to provide a single proof that covers both cases simultaneously. For
notational simplicity we will abbreviate $N_{n}$ to $N$ for the rest of the
paper.

In this section we actually prove a more general result than just for $E_{\rm
main}$, which involves a continuous bounded function
$f\colon[0,\infty)\to[0,\infty)$ evaluated on the edge-lengths. The first step
is to define a function $h_{n}$ that depends on the edge indicators as well as
the positions and weights of vertices and the value of $f$ on the distances
between the positions. We let

${\mathbf{a}}_{N}={(a_{i,j})_{1\leq i<j\leq
N}}\in\\{0,1\\}^{\binom{N}{2}},\phantom{X}{\bf
w}_{N}\in[1,\infty)^{N},\phantom{X}\text{ and
}{\mathbf{x}}_{N}\in\mathbb{T}_{n}^{d}.$

We define the function
$h_{n}:\\{0,1\\}^{\binom{N}{2}}\times[1,\infty)^{N}\times(\mathbb{T}_{n}^{d})^{N}\to\mathbb{R}_{+}$,
which depends on $f$ as

$h_{n}({\mathbf{a}}_{N},{\mathbf{w}}_{N},{\mathbf{x}}_{N})=\sum_{1\leq i<j\leq
N}a_{i,j}f(d_{n}(x_{i},x_{j})){\mathbbm{1}}_{\\{d_{n}(x_{i},x_{j})\leq\varepsilon_{n}n^{1/d}\\}}{\mathbbm{1}}_{\\{w_{i}\vee
w_{j}\leq n^{a}\\}}.$ (3.4)

The main result of this section is a concentration result for
$h_{n}({\mathbf{A}}_{N},{\mathbf{W}}_{N},{\mathbf{X}}_{N})$ around the value
$n{\mathbb{E}}\left[\lambda_{f}(W)\right]/2$.

######  Proposition 3.1.

Let $W$ be a random variable as in (1.1) and $f\colon[0,\infty)\to[0,\infty)$
a continuous bounded function. Then, for any $k\in\mathbb{N}$ and $\delta>0$,
as $n\to\infty$,

${\mathbb{P}}\left(\left|h_{n}({\mathbf{A}}_{N},{\mathbf{W}}_{N},{\mathbf{X}}_{N})-\frac{n}{2}{\mathbb{E}}\left[\lambda_{f}(W)\right]\right|>\delta
n\right)=o(n^{-k(\beta-2)}).$

Note that if $f=1$, then
$h_{n}({\mathbf{A}}_{N},{\mathbf{W}}_{N},{\mathbf{X}}_{N})=|E_{\rm main}|$
while $\frac{n}{2}{\mathbb{E}}\left[\lambda_{f}(W)\right]=n\mu$. Hence
Proposition 2.1 is a direct corollary of Proposition 3.1.

###  3.2. Proof of Proposition 3.1

The key tool behind proving the concentration result is two applications of
McDiarmid’s inequality on $h_{n}$. The first application conditions on the
weights and positions, so that the edge indicators are independent Bernoulli
random variables, and the second application is for the conditional
expectation of $h_{n}$ given the weights and positions.

It turns out to be convenient to include the indicator of a _good event_ in
the function $h_{n}$. For this, let $\varepsilon_{n}=n^{-\gamma}$ for some
$0<\gamma<1/d$, and define $\mathcal{P}_{n}(\varepsilon_{n})$ as the partition
of $\mathbb{T}_{n}^{d}$ of equally-sized cubes, each of side length
$\varepsilon_{n}n^{1/d}$. Note that $\mathcal{P}_{n}(\varepsilon_{n})$
contains at most $\varepsilon_{n}^{-d}$ many such cubes. For $M>0$, consider
the event

$\displaystyle\mathcal{E}_{\varepsilon_{n},M}:=\\{\forall\,\mathcal{C}\in\mathcal{P}_{n}(\varepsilon_{n}):\
|\mathcal{C}\cap\mathbf{X}_{N}|\in I_{M}(N\varepsilon_{n}^{d})\\},$ (3.5)

where we recall the interval $I_{M}(N\varepsilon_{n}^{d})$ from (3.1). In
words, $\mathcal{E}_{\varepsilon_{n},M}$ corresponds to the event that the
number of points of $\mathbf{X}_{N}$ contained in each cube in the partition
$\mathcal{P}_{n}(\varepsilon_{n})$ lies in the given interval
$I_{M}(N\varepsilon_{n}^{d})$. Note that for a cube
$\mathcal{C}\in\mathcal{P}_{n}(\varepsilon_{n})$, and conditionally on $N$,
the number of points in $\mathcal{C}\cap\mathbf{X}_{N}$ is ${\rm
Bin}(N,\varepsilon_{n}^{d})$-distributed. Using standard Chernoff bounds, and
a union bound for the number of cubes, it is easy to see that

$\displaystyle 1-{\mathbb{P}}_{N}\left(\mathcal{E}_{\varepsilon_{n},M}\right)$
$\displaystyle\leq\varepsilon_{n}^{d}\max_{\mathcal{C}\in\mathcal{P}_{n}(\varepsilon_{n})}{\mathbb{P}}_{N}\left(|\mathcal{C}\cap\mathbf{X}_{N}|\notin
I_{M}(N\varepsilon_{n}^{d})\right)$ $\displaystyle\leq
2\varepsilon_{n}^{d}\exp\left(-M^{2}\log(\varepsilon_{n}^{d}N)/3\right)\leq
cN^{-M^{2}/3}n^{\gamma d(1+M^{2}/3)},$ (3.6)

for some constant $c>0$. Hence, by taking an expectation over $N$,

$\displaystyle 1-{\mathbb{P}}\left(\mathcal{E}_{\varepsilon_{n},M}\right)$
$\displaystyle\leq{\mathbb{E}}\left[{\mathbbm{1}}_{\\{N\in
I_{M}(n)\\}}(1-{\mathbb{P}}_{N}\left(\mathcal{E}_{\varepsilon_{n},M}\right))\right]+{\mathbb{P}}\left(N\notin
I_{M}(n)\right)$ $\displaystyle\leq c\,2^{M^{2}/3}n^{-M^{2}/3}n^{\gamma
d(1+M^{2}/3)}+{\mathbb{P}}\left(N\notin I_{M}(n)\right).$

Using (3.3) and the fact that $0<\gamma<1/d$, we can choose $M$ sufficiently
large such that

$\displaystyle 1-{\mathbb{P}}\left(\mathcal{E}_{\varepsilon_{n},M}\right)$
$\displaystyle=o(n^{-k(\beta-2)}).$ (3.7)

Therefore it suffices to work conditionally on the event
$\mathcal{E}_{\varepsilon_{n},M}$. Define the function

$\displaystyle
g_{n}\colon\\{0,1\\}^{\binom{N}{2}}\times[1,\infty)^{N}\times(\mathbb{T}_{n}^{d})^{N}$
$\displaystyle\to[0,\infty),$ (3.8) $\displaystyle
g_{n}(\mathbf{a}_{N},\mathbf{w}_{N},\mathbf{x}_{N})$
$\displaystyle={\mathbbm{1}}_{\mathcal{E}_{\varepsilon_{n},M}}h_{n}(\mathbf{a}_{N},\mathbf{w}_{N},\mathbf{x}_{N}).$

The main technical result needed to prove Proposition 3.1 is the following
concentration bound on $g_{n}$.

######  Lemma 3.2.

Let $\delta>0$. For both the lattice and Poisson case, there exist constants
${\sf K}_{1},{\sf K}_{2}>0$ such that for sufficiently large $n$,
conditionally on the event $N\in I_{M}(n)$,

$\displaystyle{\mathbb{P}}_{N}\left(\left|g_{n}({\mathbf{A}}_{N},{\mathbf{W}}_{N},{\mathbf{X}}_{N})-\mathbb{E}_{N}\left[g_{n}({\mathbf{A}}_{N},{\mathbf{W}}_{N},{\mathbf{X}}_{N})\right]\right|\geq\delta
n\right)$ (3.9) $\displaystyle\leq
2(1-{\mathbb{P}}_{N}\left(\mathcal{E}_{\varepsilon_{n},M}\right))+2\exp\left(-{\sf
K}_{1}\delta^{2}\varepsilon_{n}^{-d}\right)+2\exp\left(-{\sf
K}_{2}\delta^{2}n^{1-2a(2\vee(\beta-1))}\right).$

Another technical lemma we need provides bounds on the conditional expectation
of the degree of a vertex, given the number of vertices $N$ and the weight of
the vertex.

######  Lemma 3.3.

For both the lattice and Poisson case, there exist positive constants ${\sf
k},{\sf K}>0$ such that, for any $1\leq i\leq N$ and $w\geq 1$,

${\sf k}\left(\tfrac{N-1}{n}\wedge
1\right)w\leq\mathbb{E}_{N}\left[\left.D_{i}\;\right|\;W_{i}=w\right]\leq{\sf
K}\left(\tfrac{N-1}{n}\vee 1\right)w.$

We postpone the proofs of both lemmas till the end of this section and proceed
with the proof of Proposition 3.1 based on these technical results.

###### Proof of Proposition 3.1.

Throughout this proof we let $\sf F$ denote the upper bound for the function
$f$. We start by splitting the difference between $h_{n}$ and the expectation
into four parts as follows:

$\displaystyle\left|h_{n}({\mathbf{A}}_{N},{\mathbf{W}}_{N},{\mathbf{X}}_{N})-\frac{n}{2}{\mathbb{E}}\left[\lambda_{f}(W)\right]\right|$
$\displaystyle\leq\left|h_{n}({\mathbf{A}}_{N},{\mathbf{W}}_{N},{\mathbf{X}}_{N})-g_{n}({\mathbf{A}}_{N},{\mathbf{W}}_{N},{\mathbf{X}}_{N})\right|$
(3.10) $\displaystyle\hskip
10.0pt+\left|g_{n}({\mathbf{A}}_{N},{\mathbf{W}}_{N},{\mathbf{X}}_{N})-\mathbb{E}_{N}\left[g_{n}({\mathbf{A}}_{N},{\mathbf{W}}_{N},{\mathbf{X}}_{N})\right]\right|$
(3.11) $\displaystyle\hskip
10.0pt+\Big{|}\mathbb{E}_{N}\left[g_{n}({\mathbf{A}}_{N},{\mathbf{W}}_{N},{\mathbf{X}}_{N})\right]-\mathbb{E}_{N}[\sum_{{}^{1\leq
i<j\leq N}}A_{i,j}f(d_{n}(X_{i},X_{j}))]\Big{|}$ (3.12) $\displaystyle\hskip
10.0pt+\Big{|}\mathbb{E}_{N}[\sum_{{}^{1\leq i<j\leq
N}}A_{i,j}f(d_{n}(X_{i},X_{j}))]-\frac{n}{2}{\mathbb{E}}\left[\lambda_{f}(W)\right]\Big{|}.$
(3.13)

We aim to show that the probability of exceeding $\delta n/4$ is
$o(1)n^{-k(\beta-2)}$ for each of these four terms. Proposition 3.1 directly
follows from this.

Bound on (3.10). For this term we have

$\displaystyle\left|h_{n}({\mathbf{A}}_{N},{\mathbf{W}}_{N},{\mathbf{X}}_{N})-g_{n}({\mathbf{A}}_{N},{\mathbf{W}}_{N},{\mathbf{X}}_{N})\right|$
$\displaystyle=h_{n}({\mathbf{A}}_{N},{\mathbf{W}}_{N},{\mathbf{X}}_{N})(1-{\mathbbm{1}}_{\mathcal{E}_{\varepsilon_{n},M}}).$

Moreover, because $h_{n}$ is uniformly bounded by ${\sf F}N^{2}$ we get, using
(3.2) and Markov’s inequality,

$\displaystyle{\mathbb{P}}_{N}\left(h_{n}({\mathbf{A}}_{N},{\mathbf{W}}_{N},{\mathbf{X}}_{N})(1-{\mathbbm{1}}_{\mathcal{E}_{\varepsilon_{n},M}})\geq\delta
n/{4}\right)$ $\displaystyle\leq\frac{{4}{\sf F}N^{2}}{\delta
n}\left(1-{\mathbb{P}}_{N}\left(\mathcal{E}_{\varepsilon_{n},M}\right)\right)$
$\displaystyle\leq\frac{4c{\sf F}}{\delta}N^{2-M^{2}/3}n^{\gamma
d(1+M^{2}/3)-1}.$

For Poisson models, picking $M$ sufficiently large gives us

$\displaystyle{\mathbb{P}}\left(\left|h_{n}({\mathbf{A}}_{N},{\mathbf{W}}_{N},{\mathbf{X}}_{N})-g_{n}({\mathbf{A}}_{N},{\mathbf{W}}_{N},{\mathbf{X}}_{N})\right|\geq\delta
n/4\right)$ (3.14)
$\displaystyle\leq{\mathbb{E}}\left[{\mathbb{P}}_{N}\left(h_{n}({\mathbf{A}}_{N},{\mathbf{W}}_{N},{\mathbf{X}}_{N})(1-{\mathbbm{1}}_{\mathcal{E}_{\varepsilon_{n},M}})>\delta
n/4\right){\mathbbm{1}}_{\\{N\in I_{M}(n)\\}}\right]+{\mathbb{P}}\left(N\notin
I_{M}(n)\right)$ $\displaystyle\leq\frac{4cF}{\delta}n^{\gamma
d(1+M^{2}/3)-1}{\mathbb{E}}\left[N^{2-M^{2}/3}{\mathbbm{1}}_{\\{N\in
I_{M}(n)\\}}\right]+{\mathbb{P}}\left(N\notin I_{M}(n)\right).$

As $N\in I_{M}(n)$ implies $N=n(1+o(1))$, the first term of (3.14) can further
be bounded from above by a constant multiple of $n^{-M^{2}(1-\gamma d)/3}$.
Using that $0<\gamma<1/d$ and (3.3), we conclude that (3.14) equals
$o(n^{-k(\beta-2)})$. The same conclusion holds for lattice models, as here
$N=n$.

Bound on (3.11). For this term we use Lemma 3.2 together with (3.3) to get,
for sufficiently large $M$,

$\displaystyle{\mathbb{P}}\left(\left|g_{n}({\mathbf{A}}_{N},{\mathbf{W}}_{N},{\mathbf{X}}_{N})-\mathbb{E}_{N}\left[g_{n}({\mathbf{A}}_{N},{\mathbf{W}}_{N},{\mathbf{X}}_{N})\right]\right|\geq\delta
n/4\right)$ $\displaystyle\leq{\mathbb{E}}\left[2\exp\left(-\frac{{\sf
K}_{1}\delta^{2}}{16}\varepsilon_{n}^{-d}\right){\mathbbm{1}}_{\\{{N\in
I_{M}(n)}\\}}\right]$ $\displaystyle\hskip
10.0pt+{\mathbb{E}}\left[2\exp\left(-\frac{{\sf
K}_{2}\delta^{2}}{16}n^{1-2a(2\vee(\beta-1))}\right){\mathbbm{1}}_{\\{{N\in
I_{M}(n)}\\}}\right]$ $\displaystyle\hskip 10.0pt+{\mathbb{P}}\left(N\notin
I_{M}(n)\right)+2(1-{\mathbb{P}}\left(\mathcal{E}_{\varepsilon_{n},M}\right)).$

Observe that $\varepsilon_{n}^{d}=n^{-\gamma}$ where $0<\gamma<1/d$, so that
the first term is $o(n^{-k(\beta-2)})$. Moreover, since by assumption
$a(2\vee(\beta-1))<1/2$, the same holds for the second term. Finally,
${\mathbb{P}}\left(N\notin I_{M}(n)\right)=o(n^{-k(\beta-2)})$ when we pick
$M$ large enough, while this also holds for the last term due to (3.7). We
thus conclude that all four terms in the upper bound go to zero sufficiently
fast.

Bound on (3.12). We will prove that, for all $\delta>0$,

$\displaystyle\mathbb{P}\Big{(}\Big{|}\mathbb{E}_{N}\big{[}\\!\\!\\!\\!\sum_{{}_{1\leq
i<j\leq
N}}\\!\\!A_{i,j}f(d_{n}(X_{i},X_{j}))\Big{]}-{\mathbb{E}}_{N}\left[g_{n}({\mathbf{A}}_{N},{\mathbf{W}}_{N},{\mathbf{X}}_{N})\right]\Big{|}>\delta
n\Big{)}=o(n^{-k(\beta-2)}).$ (3.15)

Observe that
$g_{n}({\mathbf{A}}_{N},{\mathbf{W}}_{N},{\mathbf{X}}_{N})\leq\sum_{1\leq
i<j\leq N}A_{i,j}f(d_{n}(X_{i},X_{j})).$ Therefore,

$\displaystyle\Big{|}\mathbb{E}_{N}\Big{[}\sum_{{}_{1\leq i<j\leq
N}}A_{i,j}f(d_{n}(X_{i},X_{j}))\Big{]}-{\mathbb{E}}_{N}\left[g_{n}({\mathbf{A}}_{N},{\mathbf{W}}_{N},{\mathbf{X}}_{N})\right]\Big{|}$
(3.16) $\displaystyle=\mathbb{E}_{N}\Big{[}\sum_{{}_{1\leq i<j\leq
N}}A_{i,j}f(d_{n}(X_{i},X_{j}))-g_{n}({\mathbf{A}}_{N},{\mathbf{W}}_{N},{\mathbf{X}}_{N})\Big{]}$
$\displaystyle\leq\mathbb{E}_{N}\Big{[}(1-{\mathbbm{1}}_{\mathcal{E}_{\varepsilon_{n},M}}){\sf
F}\sum_{{}_{1\leq i<j\leq N}}A_{i,j}\Big{]}+{\sf F}\sum_{{}_{1\leq i<j\leq
N}}{\mathbb{E}}_{N}\left[A_{i,j}{\mathbbm{1}}_{\\{d_{n}(X_{i},X_{j})>\varepsilon_{n}n^{1/d}\\}}{\mathbbm{1}}_{\\{W_{i}\vee
W_{j}\leq n^{a}\\}}\right]$ $\displaystyle\hskip 10.0pt+{\sf F}\sum_{{}_{1\leq
i<j\leq N}}{\mathbb{E}}_{N}\left[A_{i,j}{\mathbbm{1}}_{\\{W_{i}\vee
W_{j}>n^{a}\\}}\right].$

The first term on the right-hand side can be bounded by ${\sf
F}N^{2}[1-{\mathbb{P}}\left(\mathcal{E}_{\varepsilon_{n},M}\right)]$, using
that $\sum_{1\leq i<j\leq N}A_{i,j}\leq N^{2}$. For the second term on the
right-hand side, we note that by (1.5) and (1.7), when $W_{1}\vee W_{2}\leq
n^{a}$ and $d_{n}(X_{1},X_{2})>\varepsilon_{n}n^{1/d}$,

$\displaystyle p_{X_{1},X_{2}}(W_{1},W_{2})\leq{\sf
C}\left(\frac{\kappa(W_{1},W_{2})}{\varepsilon_{n}^{d}n}\right)^{\alpha}\leq{\sf
C}^{1+\alpha}\left(\frac{n^{\xi-1}}{\varepsilon_{n}^{d}}\right)^{\alpha}\leq{\sf
C}^{1+\alpha}n^{-\alpha(1-\xi-\gamma)},$

where $\xi:=a(2\vee(\beta-1))$. It follows that

$\mathbb{E}_{N}\Big{[}{\sf F}\sum_{{}_{1\leq i<j\leq
N}}A_{i,j}{\mathbbm{1}}_{\\{d_{n}(X_{i},X_{j})>\varepsilon_{n}n^{1/d}\\}}{\mathbbm{1}}_{\\{W_{i}\vee
W_{j}\leq n^{a}\\}}\Big{]}\leq{\sf F}{\sf
C}^{1+\alpha}N^{2}n^{-\alpha(1-\xi-\gamma)}.$ (3.17)

Finally, we have

$\displaystyle\mathbb{E}_{N}\Big{[}\sum_{{}_{1\leq i<j\leq
N}}A_{i,j}{\mathbbm{1}}_{\\{W_{i}\vee
W_{j}>n^{a}\\}}\Big{]}\leq\mathbb{E}_{N}\Big{[}\sum_{i=1}^{N}D_{i}{\mathbbm{1}}_{\\{W_{i}>n^{a}\\}}\Big{]}=N{\mathbb{E}}_{N}\left[D_{1}{\mathbbm{1}}_{\\{W_{1}>n^{a}\\}}\right].$

By Lemma 3.3, it follows that
$\mathbb{E}_{N}[D_{1}{\mathbbm{1}}_{\\{W_{1}>n^{a}\\}}|W_{1}]\leq{\sf
K}_{1}\frac{N}{n}W_{1}{\mathbbm{1}}_{\\{W_{1}>n^{a}\\}}$ and hence

${\mathbb{E}}_{N}\left[D_{1}{\mathbbm{1}}_{\\{W_{1}>n^{a}\\}}\right]\leq{\sf
K}_{1}\frac{N}{n}\int_{n^{a}}^{\infty}wf_{W}(w)dw=\frac{{\sf
K}_{1}(\beta-1)}{\beta-2}\frac{N}{n}n^{-a(\beta-2)}.$

Therefore,

$\mathbb{E}_{N}\Big{[}2{\sf F}\sum_{{}_{1\leq i<j\leq
N}}A_{i,j}{\mathbbm{1}}_{\\{W_{i}\vee W_{j}>n^{a}\\}}\Big{]}\leq\frac{2{\sf
F}{\sf K}_{1}(\beta-1)}{\beta-2}N^{2}n^{-1-a(\beta-2)}.$ (3.18)

Putting (3.17) and (3.18) back into the upper bound for (3.16) we get

$\displaystyle\Big{|}\mathbb{E}_{N}\Big{[}\sum_{{}_{1\leq i<j\leq
N}}A_{i,j}f(d_{n}(X_{i},X_{j}))\Big{]}-\mathbb{E}_{N}\left[g_{n}({\mathbf{A}}_{N},{\mathbf{W}}_{N},{\mathbf{X}}_{N})\right]\Big{|}$
$\displaystyle\leq\frac{{\sf F}}{2}nN^{2-M/4}+\frac{{\sf F}{\sf
C}^{1+\alpha}}{2}N^{2}n^{-\alpha(1-\xi-\gamma)}+\frac{2{\sf F}{\sf
K}_{1}(\beta-1)}{\beta-2}N^{2}n^{-1-a(\beta-2)}.$

Now recall that on the event $N\in I_{M}(n)$, it holds that $N=(1+o(1))n$.
Therefore, on this event, the right-hand side can be bounded from above by a
constant multiple of
$n\left(n^{2-M/4}+n^{1-\alpha(1-\xi-\gamma)}+n^{-a(\beta-2)}\right)$. By
selecting $M$ large enough, this is $o(n)$, using our assumptions that
$\beta-2>0$ and $\xi<(1-\gamma-1/\alpha)$ and therefore
$\alpha(1-\xi-\gamma)>1$. Hence, by picking a sufficiently large $M$, we use
(3.3) to conclude that

$\displaystyle\mathbb{P}\Big{(}\Big{|}\mathbb{E}_{N}\Big{[}g_{n}({\mathbf{A}}_{N},{\mathbf{W}}_{N},{\mathbf{X}}_{N})\Big{]}-\mathbb{E}_{N}\Big{[}\sum_{{}_{1\leq
i<j\leq N}}A_{i,j}f(d_{n}(X_{i},X_{j}))\Big{]}\Big{|}\geq\delta n/4\Big{)}$
$\displaystyle\leq{{\mathbb{P}}\left(N\notin
I_{M}(n)\right)}=o(n^{-k(\beta-2)}).$

Bound on (3.13). For the final term, recall $\lambda_{f}(w)$ from (1.19) and
$\lambda(w,z)$ from (1.18), for $w\geq 1$ and $z\in\mathbb{R}^{d}$. Let us
write

$\lambda_{f}^{n}(w):=\begin{cases}\sum_{z\in\mathbb{Q}_{n}\cap\mathbb{Z}^{d}}f(\|z\|)\lambda(z,w)&\text{in
the lattice case,}\\\ \int_{\mathbb{Q}_{n}}f(\|z\|)\lambda(z,w)\,dz&\text{in
the Poisson case.}\end{cases}$

Recalling that $\mathbb{Q}_{n}$ is the centred cube of volume $n$, we observe

$\mathbb{E}_{N}\Bigg{[}\sum_{{}_{1\leq i<j\leq
N}}A_{i,j}f(d_{n}(X_{i},X_{j}))\Bigg{]}=\begin{cases}\frac{n-1}{2}{\mathbb{E}}\left[\lambda_{f}^{n}(W)\right]&\text{for
lattice models,}\\\
\frac{N(N-1)}{2n}{\mathbb{E}}\left[\lambda_{f}^{n}(W)\right]&\text{for Poisson
models,}\end{cases}$

where in the lattice case we use that $N=n$. Note that $\lambda_{f}^{n}(w)$
converges pointwise to $\lambda_{f}(w)$ as $n$ tends to infinity. Together
with the fact that $\mathbb{E}[\lambda_{f}(W)]<\infty$, the dominated
convergence theorem implies that
$\lim_{n\rightarrow\infty}\mathbb{E}[\lambda_{f}^{n}(W)]={\mathbb{E}}\left[\lambda_{f}(W)\right]$.
It follows that for the Poisson case, on the event $N\in I_{M}(n)$,

$\displaystyle\frac{1}{n}\mathbb{E}_{N}\Big{[}\sum_{{}_{1\leq i<j\leq
N}}A_{i,j}f(d_{n}(X_{i},X_{j}))\Big{]}=(1+o(1))\frac{1}{2}{\mathbb{E}}\left[\lambda_{f}(W)\right],$

since $N=(1+o(1))n$. The same equality holds for the lattice case. For Poisson
models, this implies that for all $\delta>0$ and sufficiently large $n$,

$\displaystyle\mathbb{P}\Big{(}\Big{|}\frac{1}{n}\mathbb{E}_{N}\Big{[}\sum_{{}_{1\leq
i<j\leq
N}}A_{i,j}f(d_{n}(X_{i},X_{j}))\Big{]}-\frac{1}{2}{\mathbb{E}}\left[\lambda_{f}(W)\right]\Big{|}>\delta/4,\,N\in
I_{M}(n)\Big{)}=0.$

Using (3.3) we can conclude that

$\displaystyle\mathbb{P}\Big{(}\Big{|}\frac{1}{n}\mathbb{E}_{N}\Big{[}\sum_{{}_{1\leq
i<j\leq
N}}A_{i,j}f(d_{n}(X_{i},X_{j}))\Big{]}-\frac{1}{2}{\mathbb{E}}\left[\lambda_{f}(W)\right]\Big{|}>\delta/4\Big{)}$
$\displaystyle\leq{\mathbb{P}}\left(N\notin
I_{M}(n)\right)=o(n^{-k(\beta-2)}).$

by choosing $M$ sufficiently large. The bound above also holds for lattice
models. This deals with the last term (3.13) and therefore finishes the proof
of Proposition 3.1 subject to Lemmas 3.2 and 3.3. ∎

###  3.3. Proofs of technical lemmas

The rest of this section is dedicated to the proof of Lemmas 3.2 and 3.3.
Since we condition on the number of vertices $N$, the graph $G_{n}$ will be
based on a vector ${\mathbf{W}}_{N}=(W_{i})_{1\leq i\leq N}$ of i.i.d. Pareto
weights and ${\mathbf{X}}_{N}=(X_{i})_{1\leq i\leq N}$ of uniform positions in
$\mathbb{T}_{n}^{d}$ for the Poisson case and $(X_{i})_{1\leq i\leq N}$ being
just the locations of the lattice points of $\mathbb{T}_{n}^{d}$ for the
lattice case. The key tool in proving Lemma 3.2 lies in applying McDiarmid’s
inequality [22] twice.

We begin by recalling the version of McDiarmid’s inequality we shall use,
given in [5]. Consider a function
$g:\mathcal{Y}_{1}\times\dots\times\mathcal{Y}_{m}\to\mathbb{R}$, where each
$\mathcal{Y}_{i}$ is a probability space. Moreover, let
$\mathcal{G}\subseteq\mathcal{Y}_{1}\times\dots\times\mathcal{Y}_{m}$ be in
the domain of $g$. We call this the _good event_ and say that $g$ satisfies
the _bounded difference inequality on $\mathcal{G}$_ with non-negative
constants $(c_{1},\dots,c_{m})$ if for every $i\in\\{1,\ldots,m\\}$ and for
all
$(y_{1},\ldots,y_{m}),(y_{1}^{\prime},\ldots,y_{m}^{\prime})\in\mathcal{G}$
with $y_{j}=y_{j}^{\prime}$ for $j\neq i$,

$\displaystyle|g(y_{1},\dots,y_{m})-g(y_{1}^{\prime},\dots,y_{m}^{\prime})|\leq
c_{i}.$ (3.19)

For such a function $g$, for any independent random variables
$Y_{1},\dots,Y_{m}$ on the spaces $\mathcal{Y}_{1},\dots,\mathcal{Y}_{m}$ and
for all $\delta>{\mathbb{P}}\left(\mathcal{G}^{c}\right)\sum_{i=1}^{m}c_{i}$,
it holds that

$\displaystyle{\mathbb{P}}\left(|g(Y_{1},\dots,Y_{m})-{\mathbb{E}}\left[g(Y_{1},\dots,Y_{m})\mid(Y_{1},\dots,Y_{m})\in\mathcal{G}\right]|>\delta\right)$
(3.20) $\displaystyle\leq
2{\mathbb{P}}\left(\mathcal{G}^{c}\right)+2\exp\Big{(}-\tfrac{2\left(\delta-{\mathbb{P}}\left(\mathcal{G}^{c}\right)\sum_{i=1}^{m}c_{i}\right)^{2}}{\sum_{i=1}^{m}c_{i}^{2}}\Big{)}.$

For the proof we apply McDiarmid’s inequality to
$|g_{n}-\mathbb{E}_{N}\left[\left.g_{n}\;\right|\;\mathbf{W}_{N},\mathbf{X}_{N}\right]|$
and also to
$|\mathbb{E}_{N}\left[\left.g_{n}\;\right|\;\mathbf{W}_{N},\mathbf{X}_{N}\right]-{\mathbb{E}}_{N}\left[g_{n}\right]|$,
where we write $g_{n}$ for
$g_{n}({\mathbf{A}}_{N},{\mathbf{W}}_{N},{\mathbf{X}}_{N})$ for notational
simplicity. The first term considers the probability conditioned on the
weights and edges. Under this conditioning the edge indicators are independent
so that McDiarmid’s inequality can be applied without specifying a good event.
The good event is however needed to apply McDiarmid’s inequality to the second
term, where the good event will be $\mathcal{E}_{\varepsilon_{n},M}$.

Both these steps will yield an upper bound in terms of $N$ that is
$o(n^{-k(\beta-2)})$ when $N=(1+o(1))n$, which holds for the lattice as well
as the Poisson case with a probability that converges to $1$, by (3.3) with an
error of order $o(n^{-k(\beta-2)})$.

###### Proof of Lemma 3.2.

Recall $g_{n}(\cdot)$ from (3.8). Following the discussion above, we bound the
probability on the left hand side of (3.9), for $\delta>0$, by

$\displaystyle{\mathbb{P}}_{N}\left(\left|g_{n}(\mathbf{A}_{N},\mathbf{W}_{N},\mathbf{X}_{N})-\mathbb{E}_{N}[g_{n}(\mathbf{A}_{N},\mathbf{W}_{N},\mathbf{X}_{N})]\right|\geq
n\delta\right)$
$\displaystyle\leq{\mathbb{P}}_{N}\left(\left|g_{n}(\mathbf{A}_{N},\mathbf{W}_{N},\mathbf{X}_{n})-\mathbb{E}_{N}[g_{n}(\mathbf{A}_{N},\mathbf{W}_{N},\mathbf{X}_{N})\,|\,{\mathbf{W}}_{N},{\mathbf{X}}_{N}]\right|\geq
n\delta/2\right)$ (3.21) $\displaystyle\hskip
10.0pt+{\mathbb{P}}_{N}\left(\left|\mathbb{E}_{N}[g_{n}(\mathbf{A}_{N},\mathbf{W}_{N},\mathbf{X}_{N})\,|\,{\mathbf{W}}_{N},{\mathbf{X}}_{N}]-\mathbb{E}_{N}[g_{n}(\mathbf{A}_{N},\mathbf{W}_{N},\mathbf{X}_{N})]\right|\geq
n\delta/2\right).$ (3.22)

#### Concentration bulk edges given the weights and positions.

We begin with a bound on the term (3.21). Choose $m=\binom{N}{2}$ and let
${\mathcal{Y}}_{(i,j)}$ be the canonical probability space $\\{0,1\\}$ for
$1\leq i<j\leq N$. Conditionally on the weights $\mathbf{W}_{N}$ and the
positions $\mathbf{X}_{N}$, we consider $g_{n}$ as a function on
$\\{0,1\\}^{m}$.

For the application of McDiarmid’s inequality we take the good event to be the
entire space, i.e., $\mathcal{G}=\\{0,1\\}^{m}$ so that in particular
${\mathbb{P}}\left(\mathcal{G}^{c}\right)=0$. Denote by
${\mathbf{A}}_{N}^{{}_{(i,j)}}$ the vector obtained from ${\mathbf{A}}_{N}$
where we replace $A_{i,j}$ by $A^{\prime}_{i,j}=1-A_{i,j}$ and leave all other
coordinates unchanged. Then, conditionally on $\mathbf{W}_{N}$ and
$\mathbf{X}_{N}$, the function $g_{n}$ satisfies, for all $1\leq i<j\leq N$,
the bounded difference inequality

$\displaystyle|g_{n}(\mathbf{A}_{N},\mathbf{W}_{N},\mathbf{X}_{N})-$
$\displaystyle g_{n}(\mathbf{A}_{N}^{(i,j)},\mathbf{W}_{N},\mathbf{X}_{N})|$
$\displaystyle\leq{\sf
F}{\mathbbm{1}}_{\mathcal{E}_{\varepsilon_{n},M}}|A_{i,j}-A_{i,j}^{\prime}|{\mathbbm{1}}_{\\{d_{n}(X_{i},X_{j})\leq\varepsilon_{n}n^{1/d}\\}}{\mathbbm{1}}_{\\{W_{i}\vee
W_{j}\leq n^{a}\\}}$ $\displaystyle\leq{\sf
F}{\mathbbm{1}}_{\mathcal{E}_{\varepsilon_{n},M}}{\mathbbm{1}}_{\\{d_{n}(X_{i},X_{j})\leq\varepsilon_{n}n^{1/d}\\}}=:c_{ij}.$

The sum of the square of differences is given by

$\sum_{{}_{1\leq i<j\leq N}}c_{ij}^{2}={\sf F}^{2}\sum_{{}_{1\leq i<j\leq
N}}{\mathbbm{1}}_{\mathcal{E}_{\varepsilon_{n},M}}{\mathbbm{1}}_{\\{d_{n}(X_{i},X_{j})\leq\varepsilon_{n}n^{1/d}\\}}.$

On the event $\mathcal{E}_{\varepsilon_{n},M}$, the number of points in every
cube of side length $\varepsilon_{n}n^{1/d}$ is at most
$2N\varepsilon_{n}^{d}$, and any ball of radius $R$ is covered by at most
$(2Rn^{-1/d}\varepsilon_{n}^{-1})^{d}$ such cubes. This implies that, for any
$y\in\mathbb{T}_{n}^{{d}}$,

${\mathbbm{1}}_{\mathcal{E}_{\varepsilon_{n},M}}\sum_{j=1}^{N}{\mathbbm{1}}_{\\{d_{n}(X_{j},y)\leq
R\\}}\leq(2Rn^{-1/d}\varepsilon_{n}^{-1})^{d}\cdot
2N\varepsilon_{n}^{d}=2^{d+1}R^{d}n^{-1}N.$ (3.23)

Taking $R=\varepsilon_{n}n^{1/d}$ it follows that $\sum_{1\leq i<j\leq
N}c_{ij}^{2}\leq{{\sf F}^{2}}2^{d+1}N^{2}\varepsilon_{n}^{d}$ and hence
McDiarmid’s inequality (3.20) implies that

$\displaystyle\mathbb{P}_{N}(\left|g_{n}({\mathbf{A}}_{N},\mathbf{W}_{N},\mathbf{X}_{N})-\mathbb{E}_{N}[g_{n}({\mathbf{A}}_{N},\mathbf{W}_{N},\mathbf{X}_{N})\,|\,{\mathbf{W}}_{N},{\mathbf{X}}_{N}]\right|\geq
n\delta/2\,|\,{\mathbf{W}}_{N},{\mathbf{X}}_{N})$ (3.24)
$\displaystyle\leq\exp\left(-\tfrac{\delta^{2}n^{2}}{2^{d+2}{\sf
F}^{2}\varepsilon_{n}^{d}N^{2}}\right).$

#### Concentration of bulk for weights and positions.

We next establish an upper bound for (3.22) by again an application of
McDiarmid’s inequality. For the setup, let $m=N$,
$\mathcal{Y}_{i}=[1,\infty)\times\mathbb{R}_{+}$ for all
$i\in\\{1,\ldots,N\\}$, and take $\mathcal{E}_{\varepsilon_{n},M}$ as the good
event. We define the function
$\hat{g}_{n}\colon[1,\infty)^{N}\times\mathbb{R}_{+}^{N}\to\mathbb{R}_{+}$,
for $\mathbf{w}_{N}\in[1,\infty)^{N}$ and
$\mathbf{x}_{N}\in\mathbb{R}_{+}^{N}$, as

$\displaystyle\hat{g}_{n}({\mathbf{w}}_{N},{\mathbf{x}}_{N})$
$\displaystyle:=\mathbb{E}_{N}\left[\left.g_{n}(\mathbf{A}_{N},\mathbf{w}_{N},\mathbf{x}_{N})\;\right|\;\mathbf{W}_{N}=\mathbf{w}_{N},\mathbf{X}_{N}=\mathbf{x}_{N}\right]$
(3.25)
$\displaystyle={\mathbbm{1}}_{\mathcal{E}_{\varepsilon_{n},M}}\sum_{{}_{1\leq
i<j\leq
N}}f(d_{n}(x_{i},x_{j}))p_{x_{i},x_{j}}(w_{i},w_{j}){\mathbbm{1}}_{\\{d_{n}(x_{i},x_{j})\leq\varepsilon_{n}n^{1/d}\\}}{\mathbbm{1}}_{\\{w_{i}\vee
w_{j}\leq n^{a}\\}}.$

and observe that
$\mathbb{E}_{N}\left[\left.g_{n}(\mathbf{A}_{N},\mathbf{W}_{N},\mathbf{X}_{n})\;\right|\;{\mathbf{W}}_{N},{\mathbf{X}}_{N}\right]=\hat{g}_{n}({\mathbf{W}}_{N},{\mathbf{X}}_{N}).$
In addition, we note that

$\displaystyle{\mathbb{E}}_{N}\left[g_{n}(\mathbf{A}_{N},\mathbf{W}_{N},\mathbf{X}_{N})\right]={\mathbb{E}}_{N}\left[\hat{g}_{n}(\mathbf{W}_{N},\mathbf{X}_{N})\right]$
$\displaystyle={\mathbb{E}}_{N}\left[\hat{g}_{n}(\mathbf{W}_{N},\mathbf{X}_{N}){\mathbbm{1}}_{\mathcal{E}_{\varepsilon_{n},M}}\right]$
$\displaystyle={\mathbb{E}}_{N}\left[\hat{g}_{n}(\mathbf{W}_{N},\mathbf{X}_{N})\mid\mathcal{E}_{\varepsilon_{n},M}\right]{\mathbb{P}}_{N}\left(\mathcal{E}_{\varepsilon_{n},M}\right).$

We claim that $\hat{g}_{n}$ satisfies the bounded difference inequality on the
event $\mathcal{E}_{\varepsilon_{n},M}$. To see this, for any two vectors
${\mathbf{w}}_{N}$ and ${\mathbf{x}}_{N}$ of weights and positions, denote by
${\mathbf{w}}_{N}^{{}_{(i)}}$ and ${\mathbf{x}}_{N}^{{}_{(i)}}$ the vectors
where the $i^{\rm th}$ entry is replaced by any value
$w_{i}^{\prime}\in[1,\infty)$ and $x_{i}^{\prime}\in\mathbb{R}_{+}$, and all
other coordinates are unchanged. Then,

$\displaystyle\left|\hat{g}_{n}({\mathbf{w}}_{n},{\mathbf{x}}_{n})-\hat{g}_{n}({\mathbf{w}}_{n}^{(i)},{\mathbf{x}}_{n}^{(i)})\right|$
$\displaystyle\leq{\mathbbm{1}}_{\mathcal{E}_{\varepsilon_{n},M}}{\sf
F}\sum_{{{1\leq j\leq N}\atop{j\neq
i}}}p_{x_{i},x_{j}}(w_{i},w_{j}){\mathbbm{1}}_{\\{d_{n}(x_{i},x_{j})\leq\varepsilon_{n}n^{1/d}\\}}{\mathbbm{1}}_{\\{w_{i}\vee
w_{j}\leq n^{a}\\}}$ $\displaystyle\hskip
10.0pt+{\mathbbm{1}}_{\mathcal{E}_{\varepsilon_{n},M}}{\sf F}\sum_{{1\leq
j\leq N}\atop{j\neq
i}}p_{x_{i}^{\prime},x_{j}}(w_{i}^{\prime},w_{j}){\mathbbm{1}}_{\\{d_{n}(x_{i}^{\prime},x_{j})\leq\varepsilon_{n}n^{1/d}\\}}{\mathbbm{1}}_{\\{w_{i}^{\prime}\vee
w_{j}\leq n^{a}\\}}.$

We will show that there is a constant ${\sf K}_{0}>0$ that does not depend on
$w_{i}$ and $x_{i}$ such that

$\displaystyle{\mathbbm{1}}_{\mathcal{E}_{\varepsilon_{n},M}}{\sf
F}\sum_{j\neq i}p_{x_{i},x_{j}}(w_{i},w_{j})$
$\displaystyle{\mathbbm{1}}_{\\{d_{n}(x_{i},x_{j})\leq\varepsilon_{n}n^{1/d}\\}}{\mathbbm{1}}_{\\{w_{i}\vee
w_{j}\leq n^{a}\\}}$ $\displaystyle\leq{\sf K}_{0}n^{2a((\beta-2)\vee
1)}(1+n^{-1}N).$ (3.26)

The other sum satisfies the same upper bound, which implies that
$\hat{g}_{n}(\cdot)$ satisfies a bounded difference inequality on the event
$\mathcal{E}_{\varepsilon_{n},M}$ with bounds $c_{i}=2{\sf
K}_{0}n^{a((\beta-2)\vee 1)}(1+n^{-1}N)$ and thus

$\sum_{i=1}^{N}c_{i}^{2}\leq 4{\sf
K}_{0}^{2}n^{2a(2\vee(\beta-1))}N(1+n^{-1}N)^{2}.$

We will first finish the application of McDiarmid’s inequality (3.20) using
(3.26). For this, we note that on the event $N\in I_{M}(n)$, it holds for $n$
sufficiently large that

$\frac{n\delta}{2}-(1-{\mathbb{P}}\left(\mathcal{E}_{\varepsilon_{n},M}\right))2{\sf
K}_{0}n^{a(2\vee(\beta-1))}(1+n^{-1}N)>\frac{\delta}{4}n,$

which implies
$n\delta/2>(1-{\mathbb{P}}\left(\mathcal{E}_{\varepsilon_{n},M}\right))\sum_{i=1}^{N}c_{i}$.
Moreover, there exists a constant $Q$ such that

$4{\sf K}_{0}^{2}n^{2a(2\vee(\beta-1))}N(1+n^{-1}N)^{2}\leq
Qn^{1+2a(2\vee(\beta-2))}.$

Therefore, using that $\hat{g}\leq{\sf F}N^{2}$, we get

$\displaystyle\mathbb{P}_{N}(\left|\mathbb{E}_{N}[g_{n}(\mathbf{A}_{N},\mathbf{W}_{N},\mathbf{X}_{n})\,|\,{\mathbf{W}}_{N},{\mathbf{X}}_{N}]-\mathbb{E}_{N}[g_{n}(\mathbf{A}_{N},\mathbf{W}_{N},\mathbf{X}_{n})]\right|\geq
n\delta/2)$
$\displaystyle\leq{\mathbb{P}}_{N}\left(\left|\hat{g}_{n}(\mathbf{W}_{N},\mathbf{X}_{N})-{\mathbb{E}}_{N}\left[\hat{g}_{n}(\mathbf{W}_{N},\mathbf{X}_{N})\mid\mathcal{E}_{\varepsilon_{n},M}\right]\right|\geq
n\delta/4\right)$ $\displaystyle\hskip 10.0pt+{\mathbb{P}}_{N}\left({\sf
F}N^{2}(1-{\mathbb{P}}_{N}\left(\mathcal{E}_{\varepsilon_{n},M}\right))\geq
n\delta/4\right).$

Noting that on the event $I_{M}(N)$ we have $N^{-2}\geq n^{-2}/2$ it follows
from (3.2) that for $n$ sufficiently large the last term is zero. Thus, by
applying (3.20) to the first part we get

$\displaystyle\mathbb{P}_{N}(\left|\mathbb{E}_{N}[g_{n}(\mathbf{A}_{N},\mathbf{W}_{N},\mathbf{X}_{n})\,|\,{\mathbf{W}}_{N},{\mathbf{X}}_{N}]-\mathbb{E}_{N}[g_{n}(\mathbf{A}_{N},\mathbf{W}_{N},\mathbf{X}_{n})]\right|\geq
n\delta/2)$ $\displaystyle\hskip 10.0pt\leq
2(1-{\mathbb{P}}_{N}\left(\mathcal{E}_{\varepsilon_{n},M}\right))+2\exp\left(-\frac{\frac{\delta^{2}}{8}n^{2}}{Qn^{1+2a(2\vee(\beta-2))}}\right).$

Combining this with (3.24), the main result (3.9) follows by picking ${\sf
K}_{1}=2^{d+2}{\sf F}$ and ${\sf K}_{2}=(8Q)^{-1}$. It only remains to
establish the bound (3.26). We first use (1.7) to get

$\displaystyle p_{X_{i},X_{j}}(W_{i},W_{j})\leq\left(1\wedge{\sf
C}\left(\tfrac{\kappa(W_{i},W_{j})}{d_{n}(X_{i},X_{j})^{d}}\right)^{\alpha}\right)$
$\displaystyle={\mathbbm{1}}_{\\{d_{n}(X_{i},X_{j})\leq{\sf C}^{{1/(\alpha
d)}}\kappa(W_{i},W_{j})^{1/d}\\}}+{\mathbbm{1}}_{\\{d_{n}(X_{i},X_{j})>{\sf
C}^{{1/(\alpha d)}}\kappa(W_{i},W_{j})^{1/d}\\}}{\sf
C}\left(\tfrac{\kappa(W_{i},W_{j})}{d_{n}(X_{i},X_{j})^{d}}\right)^{\alpha}.$

Thus, the left-hand side of (3.26) can be bounded from above by a constant
multiple of

$\displaystyle{\mathbbm{1}}_{\mathcal{E}_{\varepsilon_{n},M}}\sum_{{1\leq
j\leq N}\atop{j\neq i}}{\mathbbm{1}}_{\\{d_{n}(X_{i},X_{j})\leq{\sf
C}^{1/(\alpha
d)}\kappa(W_{i},W_{j})^{1/d}\wedge\varepsilon_{n}n^{1/d}\\}}{\mathbbm{1}}_{\\{W_{i}\vee
W_{j}\leq n^{a}\\}}$ (3.27)
$\displaystyle+{\mathbbm{1}}_{\mathcal{E}_{\varepsilon_{n},M}}\sum_{{1\leq
j\leq N,}\atop{j\neq i}}{\mathbbm{1}}_{\\{{\sf C}^{1/(\alpha
d)}\kappa(W_{i},W_{j})^{1/d}<d_{n}(X_{i},X_{j})\leq\varepsilon_{n}n^{1/d}\\}}{\sf
C}\left(\tfrac{\kappa(W_{i},W_{j})}{d_{n}(X_{i},X_{j})^{d}}\right)^{\alpha}{\mathbbm{1}}_{\\{W_{i}\vee
W_{j}\leq n^{a}\\}}.$ (3.28)

We begin by studying (3.27). For any $v,w\geq 1$ satisfying $w\vee v\leq
n^{a}$, (1.5) gives us that $\kappa(v,w)\leq{\sf
C}n^{a+a(1\vee(\beta-2))}={\sf C}n^{a(2\vee(\beta-1))}$. Recall that we chose
$\varepsilon_{n}$ and $a$ to satisfy the conditions of Proposition 2.1, that
is $\varepsilon_{n}^{d}=n^{-\gamma}$ and
$a(2\vee(\beta-1))<(1-\gamma-1/\alpha)\wedge\tfrac{1}{2}$. Thus, if $w\vee
v\leq n^{a}$, then for $n$ sufficiently large ${\sf C}^{1/\alpha
d}\kappa(v,w)^{1/d}<\varepsilon_{n}n^{1/d}$. Therefore, for large $n$, (3.27)
can be bounded from above by

$\displaystyle{\mathbbm{1}}_{\mathcal{E}_{\varepsilon_{n},M}}\sum_{{1\leq
j\leq N}\atop{j\neq i}}{\mathbbm{1}}_{\\{d_{n}(X_{i},X_{j})\leq{\sf
C}^{1/(\alpha d)}\kappa(W_{i},W_{j})^{1/d}\\}}{\mathbbm{1}}_{\\{W_{i}\vee
W_{j}\leq n^{a}\\}}$
$\displaystyle\leq{\mathbbm{1}}_{\mathcal{E}_{\varepsilon_{n},M}}\sum_{{1\leq
j\leq N}\atop{j\neq i}}{\mathbbm{1}}_{\\{d_{n}(X_{i},X_{j})\leq{\sf
C}^{\frac{1}{d}(1+\frac{1}{\alpha})}n^{\frac{1}{d}a(2\vee(\beta-1))}\\}}.$

Applying (3.23) with $R={\sf
C}^{\frac{1}{d}(1+\frac{1}{\alpha})}n^{\frac{1}{d}a(2\vee(\beta-1))}$, the
above can be bounded by $2^{d+1}{\sf C}^{1+\frac{1}{\alpha}}$
$n^{a(2\vee(\beta-1))}n^{-1}N$. To conclude (3.26), it therefore remains to
show that the second summand (3.28) is bounded from above by a constant
multiple of $n^{a(1\vee(\beta-2))}$.

For (3.28) observe that, since $d_{n}$ is translation invariant,

$\displaystyle\sum_{j\neq i}{\mathbbm{1}}_{\\{{\sf C}^{1/(\alpha
d)}\kappa(W_{i},W_{j})^{1/d}<d_{n}(X_{i},X_{j})\leq\varepsilon_{n}n^{1/d}\\}}d_{n}(X_{i},X_{j})^{-d\alpha}$
$\displaystyle=\sum_{j\neq i}{\mathbbm{1}}_{\\{{\sf C}^{1/(\alpha
d)}\kappa(W_{i},W_{j})^{1/d}<\|Z_{j}\|\leq\varepsilon_{n}n^{1/d}\\}}\|Z_{j}\|^{-d\alpha},$

where $Z_{j}=\Phi(X_{j})$ for $\Phi$ the canonical mapping of
$\mathbb{T}_{n}^{d}$ to $\mathbb{Q}_{n}$ with $\Phi(X_{i})=0$ recalling that
$\mathbb{Q}_{n}$ is the $d$-dimensional cube of volume $n$ centred at the
origin. In particular, there exist constants ${\sf Q}_{1},{\sf Q}_{2},{\sf
Q}_{3}>0$ such that for sufficiently large $n$

$\displaystyle\sum_{j\neq i}{\mathbbm{1}}_{\\{{\sf C}^{1/(\alpha
d)}\kappa(W_{i},W_{j})^{1/d}<d_{n}(X_{i},X_{j})\leq\varepsilon_{n}n^{1/d}\\}}d_{n}(X_{i},X_{j})^{-d\alpha}$
$\displaystyle\leq{\sf Q}_{1}\int_{\mathbb{Q}_{n}}{\mathbbm{1}}_{\\{{\sf
C}^{1/(\alpha
d)}\kappa(W_{i},W_{j})^{1/d}<\|z\|\leq\varepsilon_{n}n^{1/d}\\}}\|z\|^{-d\alpha}dz={\sf
Q}_{2}\int_{{\sf C}^{1/(\alpha
d)}\kappa(W_{i},W_{j})^{1/d}}^{\varepsilon_{n}n^{1/d}}r^{-d(\alpha-1)-1}dr$
$\displaystyle\leq{\sf Q}_{3}\kappa(W_{i},W_{j})^{-(\alpha-1)}.$

Plugging this back into the sum (3.28), and using 1.5, we get

$\displaystyle{\mathbbm{1}}_{\mathcal{E}_{\varepsilon_{n},Q}}{\sf
F}\sum_{j\neq i}{\mathbbm{1}}_{\\{{\sf C}^{1/(\alpha
d)}\kappa(W_{i},W_{j})^{1/d}<d_{n}(X_{i},X_{j})\leq\varepsilon_{n}n^{1/d}\\}}{\sf
C}\left(\frac{\kappa(W_{i},W_{j})}{d_{n}(X_{i},X_{j})^{d}}\right)^{\alpha}{\mathbbm{1}}_{\\{W_{i}\vee
W_{j}\leq n^{a}\\}}$ $\displaystyle\leq{\sf F}{\sf Q}_{2}{\sf
C}\kappa(W_{i},W_{j}){\mathbbm{1}}_{\\{W_{i}\vee W_{j}\leq n^{a}\\}}\leq{\sf
F}{\sf Q}_{2}{\sf C}n^{a(2\vee(\beta-1))}.$

Taking ${\sf K}_{0}={\sf F}{\sf Q}_{2}{\sf C}\vee{\sf F}2^{d+1}{\sf
C}^{\alpha+1}$ now yields (3.26). ∎

The proof of the last technical lemma rests on the following easy bound.

######  Lemma 3.4.

There exist constants ${\sf K}\geq{\sf k}>0$ such that

$\displaystyle{\sf
k}w\leq(\beta-1)\int_{1}^{\infty}\int_{\mathbb{Q}_{n}}\varphi\left(\frac{\|z\|^{d}}{\kappa(w,t)}\right)dzt^{-\beta}dt\leq{\sf
K}w.$ (3.29)

###### Proof.

Using (1.7), for $z\in\mathbb{R}^{d}$, we bound

$\displaystyle{\sf c}{\mathbbm{1}}_{\\{\|z\|<{\sf
c}(\kappa(w,t))^{1/d}\\}}\leq\varphi$
$\displaystyle\Big{(}\tfrac{\|z\|^{d}}{\kappa(w,t)}\Big{)}\leq{\mathbbm{1}}_{\\{\|z\|<\kappa(w,t)^{1/d}\\}}+{\sf
C}\left(\tfrac{\kappa(w,t)}{\|z\|^{d}}\right)^{\alpha}{\mathbbm{1}}_{\\{\|z\|\geq(\kappa(w,t))^{1/d}\\}}.$

The integrals
$\int_{\mathbb{Q}_{n}}{\mathbbm{1}}_{\\{\|z\|<(\kappa(w,t))^{1/d}\\}}dz$, and
$\int_{\mathbb{Q}_{n}}({\kappa(w,t)}/{\|z\|^{d}})^{\alpha}{\mathbbm{1}}_{\\{\|z\|\geq(\kappa(w,t))^{1/d}\\}}dz$
for $\alpha>1$, can both be bounded from above and below by constant multiples
of $\kappa(w,t)$, where the constants only depend on the dimension $d$. Next,
by (1.5),

$\int_{1}^{\infty}\kappa(w,t)t^{-\beta}\,dt\leq{\sf
C}w\int_{1}^{w}t^{(\beta-2)\vee 1}t^{-\beta}\,dt+{\sf C}w^{(\beta-2)\vee
1}\int_{w}^{\infty}t^{1-\beta}\,dt\leq{\sf K}^{\prime}w,$

where the last inequality follows from basic computations and holds for some
constant ${\sf K}^{\prime}>1$. For the lower bound, $\kappa(w,t)\geq{\sf
c}(w\vee t)$ by (1.5) and so

$\displaystyle\int_{1}^{\infty}\kappa(w,t)t^{-\beta}\,dt\geq{\sf
c}w\int_{1}^{w}t^{-\beta}dt+{\sf c}\int_{w}^{\infty}t^{1-\beta}dt={\sf
k}^{\prime}w,$

for some constant ${\sf k}^{\prime}>0$. Combining the above implies (3.29). ∎

###### Proof of Lemma 3.3.

We first consider the lattice models, where the number of vertices equals
$N=n$. For $w\geq 1$, by embedding the torus into the integer lattice and
using the translation invariance of $d_{n}$, we get

$\mathbb{E}\left[\left.D_{i}\;\right|\;W_{i}=w\right]=(\beta-1)\int_{1}^{\infty}\sum_{z\in\mathbb{Q}_{n}\cap\mathbb{Z}^{d}\setminus\\{0\\}}\varphi\left(\frac{\|z\|^{d}}{\kappa(w,t)}\right)t^{-\beta}dt.$
(3.30)

When conditioning on $N$ in the Poisson case, the positions of the vertices
are sampled uniformly and independently on the torus. Thus, by embedding the
torus into $\mathbb{Q}_{n}$ and using translation invariance of $d_{n}$,

$\mathbb{E}_{N}\left[\left.D_{i}\;\right|\;W_{i}=w\right]=\frac{N-1}{n}(\beta-1)\int_{1}^{\infty}\int_{\mathbb{Q}_{n}}\varphi\left(\frac{\|z\|^{d}}{\kappa(w,t)}\right)dz\,t^{-\beta}dt.$

Note that for the lattice case, the sum in (3.30) can be interpreted as a
$d$-dimensional Riemann sum. Hence, there exists constants ${\sf q},{\sf Q}>0$
such that

${\sf
q}\int_{\mathbb{Q}_{n}}\varphi\left(\frac{\|z\|^{d}}{\kappa(w,t)}\right)dz\leq\sum_{z\in\mathbb{Q}_{n}\cap\mathbb{Z}^{d}}\varphi\left(\frac{\|z\|^{d}}{\kappa(w,t)}\right)\leq{\sf
Q}\int_{\mathbb{Q}_{n}}\varphi\left(\frac{\|z\|^{d}}{\kappa(w,t)}\right)dz.$

Lemma 3.3 for both lattice and Poisson cases follows from Lemma 3.4. ∎

##  4\. Concentration of the long edges: Proof of Proposition 2.2

Recall that we denote the number of vertices in our graph by $N$. Recall
$I_{M}(n)$ from (3.1) for $M>0$. For both lattice and Poisson models we remark
that, for all $x,y\in V_{n}$ with $x\neq y$, conditionally on $W_{x}\vee
W_{y}\leq n^{a}$ and $d_{n}(x,y)>\varepsilon_{n}n^{1/d}$,

$\displaystyle p_{x,y}(W_{x},W_{y})\leq{\sf
C}\left(\frac{\kappa(W_{x},W_{y})}{\varepsilon_{n}^{d}n}\right)^{\alpha}\leq{\sf
C}^{1+\alpha}\left(\frac{n^{\xi-1}}{\varepsilon_{n}^{d}}\right)^{\alpha},$

where $\xi:=a(2\vee(\beta-1))$, using (1.5) and (1.7). Recall that we suppose
that $\varepsilon_{n}$ and $a$ satisfy the conditions of Proposition 2.1. More
precisely, $\varepsilon_{n}=n^{-\gamma}$ for some
$0<\gamma<(1-\tfrac{1}{\alpha})\wedge\tfrac{1}{d}$ and
$\alpha(1-\xi-\gamma)>1$. Define, for $n\geq 1$,

$\displaystyle
p_{n}:=(n^{\xi-1}/\varepsilon_{n}^{d})^{\alpha}=\delta^{-\alpha}n^{\alpha(\xi-1+\gamma)},$

where $\delta>0$. We begin by proving the proposition for lattice models. Let
$B_{n}$ be a $\mathrm{Bin}(N^{2},p_{n})$-distributed random variable with mean
$\delta^{-\alpha}n^{2+\alpha(\xi-1+\gamma)}$ since $N=n$. Then $|E_{\rm
long}|$ is stochastically dominated by $B_{n}$. Observe that since
$\alpha(1-\xi-\gamma)>1$, there exists some $\delta_{1}>0$ such that

$n^{2}p_{n}/n=(\delta^{-\alpha}n^{2+\alpha(\xi-1+\gamma)})/n\leq\delta^{-\alpha}n^{-\delta_{1}},$

which goes to zero polynomially fast. Consequently, ${\mathbb{P}}\left(|E_{\rm
long}|\geq n\delta\right)\leq{\mathbb{P}}\left(B_{n}\geq n\delta\right)$ and
the upper bound tends to zero stretched-exponentially fast again using a
standard Chernoff bound [17, Theorem 2.1] for binomial random variables, and
hence faster than $n^{-k(\beta-2)}$ for any $k\in\mathbb{N}$. This concludes
the proof of Proposition 2.2 for lattice models.

For Poisson models, conditionally on the number of vertices $N$, let $B_{n}$
be a random variable, which is $\mathrm{Bin}({N^{2}},p_{n})$-distributed and
note again that $|E_{\rm long}|$ is stochastically dominated by $B_{n}$.
Further

$\displaystyle{\mathbb{P}}\left(B_{n}>\delta n\right)$
$\displaystyle\leq{\mathbb{P}}\left(B_{n}>{N^{2}}p_{n}+n\delta/2\right)+{\mathbb{P}}\left({N^{2}}p_{n}\geq
n\delta/2\right).$ (4.1)

The conditional mean of $B_{n}$ is ${N^{2}}p_{n}$, thus standard large
deviation bounds for binomial random variables (see, e.g., [17, Theorem 2.1])
give

$\displaystyle{\mathbb{P}}_{N}\left(B_{n}>{N^{2}}p_{n}+n\delta/2\right)\leq\exp\left({-\frac{\delta^{2}n^{2}}{8({N^{2}}p_{n}+\delta
n/6)}}\right).$

By taking expectations we can bound the first summand of the right-hand side
of (4.1) from above by

$\displaystyle{\mathbb{E}}\left[\exp\left({-\frac{\delta^{2}n^{2}}{8({N^{2}}p_{n}+\delta
n/6)}}\right){\mathbbm{1}}_{\\{N\in I_{M}(n)\\}}\right]$
$\displaystyle+{\mathbb{P}}\left(N\notin I_{M}(n)\right).$

By choosing $M$ sufficiently large, (3.3) gives us that the second summand
above equals $o(n^{-k(\beta-2)})$. The first summand of the above inequality
can be bounded from above by

$\displaystyle\exp{\left(-\frac{\delta^{2}n^{2}}{8{n^{2}p_{n}}+4\delta
n/3)}\right)}=\exp{\left(-\frac{\delta^{2}}{8\delta^{-\alpha}n^{-\alpha(1-\xi-\gamma)}+4\delta/(3n)}\right)}.$

This can be further upper bounded by a constant multiple of
$\exp(-n^{\alpha(1-\xi-\gamma)})$ for large $n$. Since
$\alpha(1-\xi-\gamma)>1$, the above equals $o(n^{-k(\beta-2)})$. We are left
to bound the second summand of (4.1). Note that, for the same $\delta_{1}>0$
as before,

$\displaystyle{\mathbb{P}}\left({N^{2}}p_{n}\geq n\delta/2\right)$
$\displaystyle={\mathbb{P}}\left({N}\geq\sqrt{n\delta/(2p_{n})}\right)\leq{\mathbb{P}}\left({N}\geq
n^{1+\delta_{1}/2}\right),$

using the fact that
$\sqrt{n/p_{n}}=\delta^{\alpha/2}n^{\tfrac{1}{2}+\alpha/2(1-\xi-\gamma)}>n^{1+\delta_{1}/2}$.
Since $N$ is a Poi$(n)$-distributed random variable, standard concentration
bounds [17, Remark 2.6] imply $\mathbb{P}(N\geq n^{1+\delta_{1}/2})$ is
$o(n^{-k(\beta-2)})$, which concludes the proof. ∎

##  5\. Large deviations of the high-weight edges: Proof of Proposition 2.3

###  5.1. Overview of the proof of Proposition 2.3

In this section, we reduce the proof of Proposition 2.3 to a key technical
result, which we state as Proposition 5.3. The proof of Proposition 5.3 is
deferred to Section 5.3. We begin by noting that

$\displaystyle|E_{\rm high}|$ $\displaystyle=\sum_{\\{x,y\\}\subseteq
V_{n}}A_{x,y}{\mathbbm{1}}_{\\{W_{x}\vee W_{y}\geq n^{a}\\}}=\sum_{x\in
V_{n}}D_{x}{\mathbbm{1}}_{\\{W_{x}\geq n^{a}\\}}-\frac{1}{2}\sum_{x\in
V_{n}}\sum_{y\in V_{n}}A_{x,y}{\mathbbm{1}}_{\\{W_{x}\wedge W_{y}\geq
n^{a}\\}}$ $\displaystyle=\sum_{x\in V_{n}}{\mathbbm{1}}_{\\{W_{x}\geq
n^{a}\\}}\tilde{D}_{x},$ (5.1)

where $D_{x}$ is the degree of $x\in V_{n}$ defined in (1.4), and

$\tilde{D}_{x}:=D_{x}-\frac{1}{2}\sum_{y\in
V_{n}}A_{x,y}{\mathbbm{1}}_{\\{W_{y}\geq n^{a}\\}}.$

Recall that for Poisson models, the vertex set $V_{n}$ is a Poisson point
process $\mathcal{P}$ of intensity one on $\mathbb{T}_{n}^{d}$. In this case,
we need to state results for the Palm version obtained by adding the point
$x\in\mathbb{T}_{n}^{d}$ with independent Pareto-distributed weight $W_{x}$ to
$\mathcal{P}$. We denote by $\mathbb{P}^{x}$ the probability and by
$\mathbb{E}^{x}$ the expectation with respect to this new vertex set.

For $w>1$ and $x\in\mathbb{T}_{n}^{d}$ define, for both lattice and Poisson
models,

$\displaystyle\lambda^{x}_{n}(w):=\mathbb{E}^{x}\left[\left.\tilde{D}_{x}\;\right|\;W_{x}=w\right].$
(5.2)

Observe that $\lambda^{x}_{n}(w)=\lambda^{y}_{n}(w)$ for
$x,y\in\mathbb{T}_{n}^{d}$, by the translation invariance of the torus
$\mathbb{T}_{n}^{d}$, we therefore write $\lambda_{n}(w)$ for
$\lambda^{x}_{n}(w)$. Further, for $C>0$, define the events

$\displaystyle\mathcal{F}_{n,C}:={\bigcap_{x\in V_{n}}\big{\\{}W_{x}\geq
n^{a},|\tilde{D}_{x}-\lambda_{n}(W_{x})|<C\sqrt{\lambda_{n}(W_{x})\log{\lambda_{n}(W_{x})}}\big{\\}}\cup\\{W_{x}<n^{a}\\}.}$
(5.3)

We now state a few results that are necessary to prove Proposition 2.3 and
postpone some of their proofs to the end of the section:

######  Lemma 5.1 (Concentration of $\tilde{D}_{x}$).

For any $\xi>0$, there is $C=C(\xi,{a})>0$ such that

${\mathbb{P}}\left(\mathcal{F}_{n,C}^{\,c}\right)=o(n^{-\xi}).$

For $w>1$ and $x\in\mathbb{T}_{n}^{d}$, define

$\displaystyle\pi^{x}_{n}(w):=\mathbb{E}^{x}\left[\left.D_{x}\;\right|\;W_{x}=w\right].$
(5.4)

As before, for $x,y\in\mathbb{T}_{n}^{d}$, we have
$\pi_{n}^{x}(w)=\pi_{n}^{y}(x)$ and we therefore write $\pi_{n}(w)$ for
$\pi_{n}^{x}(w)$. The following lemma bounds $\pi_{n}(w)$ and
$\lambda_{n}(w)$:

######  Lemma 5.2.

Let $w>1$. There exist positive constants ${\sf k},{\sf K}$ such that

${\sf k}w\leq\pi_{n}(w)\leq{\sf K}w,$

for sufficiently large $n$. Since $\frac{1}{2}D_{x}\leq\tilde{D}_{x}\leq
D_{x}$, the above implies $\frac{1}{2}{\sf k}w\leq\lambda_{n}(w)\leq{\sf K}w.$

The proof of Lemma 5.2 follows directly from Lemma 3.3 by noting that for
lattice models $N=n$, while for Poisson models ${\mathbb{E}}\left[N\right]=n$.

The following proposition states a limit law for the sum
$\sum_{x}\lambda_{n}(W_{x}){\mathbbm{1}}_{\\{W_{x}\geq n^{a}\\}}$.

######  Proposition 5.3.

Let $\rho>0$ be non-integer and $k-1<\rho<k$. With $F=F_{0}$ as in (1.11)

$\displaystyle{\mathbb{P}}\left(\sum_{x\in
V_{n}}\lambda_{n}(W_{x}){\mathbbm{1}}_{\\{W_{x}\geq
n^{a}\\}}>n\rho\right)=(F(\rho)+o(1))n^{-k(\beta-2)}.$ (5.5)

The proof of Proposition 5.3 will be given in Section 5.3. We are now ready to
prove Proposition 2.3 using Lemmas 5.1 and 2.4, and Proposition 5.3.

###### Proof of Proposition 2.3.

The idea of the proof is to approximate $|E_{\rm high}|$ by the i.i.d. sum of
Proposition 5.3, and argue that the errors we encounter in this approximation
tend to zero faster than the right-hand side of Proposition 5.3. The proof
holds for both lattice and Poisson models. To this end, we fix $C>0$ to be
sufficiently large such that the probability of the event
$\mathcal{F}_{n,C}^{c}$ defined in (5.3), tends to zero faster than the right-
hand side of (5.5). The existence of such $C$ is ensured by Lemma 5.1. Write

$\displaystyle{\mathbb{P}}\left(|E_{\rm high}|>n\rho\right)$
$\displaystyle={\mathbb{P}}\left(\sum_{x\in
V_{n}}\tilde{D}_{x}{\mathbbm{1}}_{\\{W_{x}\geq n^{a}\\}}>n\rho\right)$
$\displaystyle={\mathbb{P}}\left(\sum_{x\in
V_{n}}\tilde{D}_{x}{\mathbbm{1}}_{\\{W_{x}\geq
n^{a}\\}}>n\rho,\,\mathcal{F}_{n,C}\right)+O({\mathbb{P}}\left(\mathcal{F}_{n,C}^{\,c}\right)).$
(5.6)

By choice of $C$, the second term on the right-hand side tends to zero faster
than $n^{-k(\beta-2)}$. Using the definition of $\mathcal{F}_{n,C}$, the first
summand of (5.6) can be bounded from below by

$\displaystyle{\mathbb{P}}\left(\sum_{x\in
V_{n}}\tilde{D}_{x}{\mathbbm{1}}_{\\{W_{x}\geq
n^{a}\\}}>n\rho,\,\mathcal{F}_{n,C}\right)$
$\displaystyle\geq{\mathbb{P}}\left(\sum_{x\in
V_{n}}\big{(}\lambda_{n}(W_{x})-C\sqrt{\lambda_{n}(W_{x})\log{\lambda_{n}(W_{x})}}\big{)}{\mathbbm{1}}_{\\{W_{x}\geq
n^{a}\\}}>n\rho,\,\mathcal{F}_{n,C}\right)$
$\displaystyle\geq{\mathbb{P}}\left(\sum_{x\in
V_{n}}\big{(}\lambda_{n}(W_{x})-C\sqrt{\lambda_{n}(W_{x})\log{\lambda_{n}(W_{x})}}\big{)}{\mathbbm{1}}_{\\{W_{x}\geq
n^{a}\\}}>n\rho\right)-O({\mathbb{P}}\left(\mathcal{F}_{n,C}^{\,c}\right)),$
(5.7)

where as before, the term $O(\mathbb{P}(\mathcal{F}_{n,C}^{\,c}))$ tends to
zero faster than $n^{-k(\beta-2)}$. An upper bound follows similarly, i.e.,

$\displaystyle{\mathbb{P}}\left(\sum_{x\in
V_{n}}\tilde{D}_{x}{\mathbbm{1}}_{\\{W_{x}\geq
n^{a}\\}}>n\rho,\,\mathcal{F}_{n,C}\right)$
$\displaystyle\leq{\mathbb{P}}\left(\sum_{x\in
V_{n}}\left(\lambda_{n}(W_{x})+C\sqrt{\lambda_{n}(W_{x})\log{\lambda_{n}(W_{x})}}\right){\mathbbm{1}}_{\\{W_{x}\geq
n^{a}\\}}>n\rho,\,\mathcal{F}_{n,C}\right)$
$\displaystyle\leq{\mathbb{P}}\left(\sum_{x\in
V_{n}}\left(\lambda_{n}(W_{x})+C\sqrt{\lambda_{n}(W_{x})\log{\lambda_{n}(W_{x})}}\right){\mathbbm{1}}_{\\{W_{x}\geq
n^{a}\\}}>n\rho\right).$ (5.8)

Fix $\varepsilon>0$ such that
$\frac{\rho}{1-\varepsilon},\frac{\rho}{1+\varepsilon}\in(k-1,k)$. There
exists $L>0$ such that $h(x)=\varepsilon x-C\sqrt{x\log{x}}$ is increasing on
the interval $(L,\infty)$. In particular, when $W_{x}\geq n^{a}$ for $x\in
V_{n}$, we have $\lambda_{n}(W_{x})\geq\lambda_{n}(n^{a})\geq L$ for all $n$
sufficiently large, where the first inequality holds by monotonicity of
$\lambda_{n}$, and the second inequality follows from the fact that
$\lambda_{n}(n^{a})$ diverges by Lemma 5.2. Using the monotonicity of the
function $h$ on $(L,\infty)$, we can then conclude that when $W_{x}\geq n^{a}$
for $x\in V_{n}$ we have $h(\lambda_{n}(W_{x}))\geq h(\lambda_{n}(n^{a}))>0$
for all sufficiently large $n$. This implies that, for sufficiently large $n$
and all $x\in V_{n}$,

$\displaystyle(1+\varepsilon)\lambda_{n}(W_{x})$
$\displaystyle\geq\lambda_{n}(W_{x})+C\sqrt{\lambda_{n}(W_{x})\log{\lambda_{n}(W_{x})}},\quad\text{
and }$ (5.9) $\displaystyle(1-\varepsilon)\lambda_{n}(W_{x})$
$\displaystyle\leq\lambda_{n}(W_{x})-C\sqrt{\lambda_{n}(W_{x})\log{\lambda_{n}(W_{x})}}.$
(5.10)

Combining the above two inequalities together with (5.6), (5.7) and (5.8), it
follows that for sufficiently large $n$,

$\displaystyle{\mathbb{P}}\Big{(}\sum_{x\in V_{n}}$
$\displaystyle\lambda_{n}(W_{x}){\mathbbm{1}}_{\\{W_{x}\geq
n^{a}\\}}>\tfrac{\rho}{1-\varepsilon}n\Big{)}+O({\mathbb{P}}\left(\mathcal{F}_{n,C}^{\,c}\right))$
$\displaystyle\leq{\mathbb{P}}\left(|E_{\rm
high}|>n\rho\right)\leq{\mathbb{P}}\Big{(}\sum_{x\in
V_{n}}\lambda_{n}(W_{x}){\mathbbm{1}}_{\\{W_{x}\geq
n^{a}\\}}>\tfrac{\rho}{1+\varepsilon}n\Big{)}+O({\mathbb{P}}\left(\mathcal{F}_{n,C}^{\,c}\right)).$
(5.11)

Proposition 2.3 follows by multiplying this series of inequalities by
$n^{k(\beta-2)}$, then letting $n\to\infty$, and finally $\varepsilon\searrow
0$, using (5.5) and the continuity of $F=F_{0}$ from Lemma 2.4.∎

We now give the proofs of Lemma 5.1 and Proposition 5.3 used in the proof of
Proposition 2.3.

###  5.2. Concentration of degrees: Proof of Lemma 5.1

We begin by introducing a lemma that states that the degree $\tilde{D}_{x}$ is
a Poisson random variable for the Poisson case:

######  Lemma 5.4.

In the Poisson case, given a vertex $x\in\mathcal{P}_{n}$ and its weight
$W_{x}$, the locations and the weights of the vertices contributing to
$\tilde{D}_{x}$ are distributed as a Poisson point process on
$\mathbb{T}_{n}^{d}\times(0,\infty)$ with intensity

$h(y,w)=f_{W}(w)\frac{1}{2}(1+{\mathbbm{1}}_{\\{w<n^{a}\\}})\varphi\left(\frac{d_{n}(x,y)^{d}}{\kappa(w,W_{x})}\right).$

Consequently, given the weight $W_{x}$, the degree $\tilde{D}_{x}$ is a
Poisson random variable with parameter
$\int_{1}^{\infty}\int_{\mathbb{T}_{n}^{d}}h(y,w)dydw$.

We omit the proof of this lemma, as it is a minor modification of [15, Lemma
2.1]. Next we state and prove a concentration result on $\tilde{D}_{x}$ given
its weight, for any fixed $x\in V_{n}$.

######  Lemma 5.5.

For $x\in\mathbb{T}_{n}^{d}$, any $w>1$ and $C>0$,

$\displaystyle\mathbb{P}^{\,x}\left(\left.|\tilde{D}_{x}-\lambda_{n}(w)|>C\sqrt{\lambda_{n}(w)\log{\lambda_{n}(w)}}\;\right|\;W_{x}=w\right)$
(5.12) $\displaystyle\leq
2\exp\left(-(\tfrac{C}{2})^{2}\tfrac{\log{\lambda_{n}(w)}}{4+\frac{2}{3}\sqrt{\log{\lambda_{n}(w)}/\lambda_{n}(w)}}\right)$

###### Proof of Lemma 5.5.

We can write $\tilde{D}_{x}=D_{x}-D^{\prime}_{x}$, where
$D^{\prime}_{x}=\frac{1}{2}\sum_{y\in V_{n}}A_{x,y}{\mathbbm{1}}_{\\{W_{y}\geq
n^{a}\\}}$ and $D_{x}$ given by (1.4). Recall that
$\pi_{n}(w)=\mathbb{E}^{x}\left[\left.D_{x}\;\right|\;W_{x}=w\right]$ for
$w>0$, and define
$\pi^{\prime}_{n}(w)=\mathbb{E}^{x}\left[\left.D^{\prime}_{x}\;\right|\;W_{x}=w\right]$.
For all $w>1$,

$\displaystyle\pi_{n}(w)\leq 2\lambda_{n}(w)$ $\displaystyle\phantom{XX}\text{
and }\phantom{XX}\pi^{\prime}_{n}(w)\leq\lambda_{n}(w),$ (5.13)

since we can write $\tilde{D}_{x}=D_{x}^{\prime}+\sum_{y\in
V_{n}}A_{x,y}{\mathbbm{1}}_{\\{W_{y}<n^{a}\\}}$ and
$2\tilde{D}_{x}=D_{x}+\sum_{y\in
V_{n}}A_{x,y}{\mathbbm{1}}_{\\{W_{y}<n^{a}\\}}$. We begin by proving the
statement concerning lattice models. Note that, conditionally on $W_{x}=w$,
both $D_{x}$ and $D_{x}^{\prime}$ are sums of independent Bernoulli random
variables. Hence, standard concentration arguments (see e.g. [17, Theorem
2.8]) imply that, for all $t>0$,

$\displaystyle\mathbb{P}^{\,x}\left(\left.|D_{x}-\pi_{n}(w)|>t\;\right|\;W_{x}=w\right)$
$\displaystyle\leq\exp{\left(-\tfrac{t^{2}}{2\pi_{n}(w)+2t/3}\right)},$ (5.14)
$\displaystyle\mathbb{P}^{\,x}\left(\left.|D^{\prime}_{x}-\pi^{\prime}_{n}(w)|>t\;\right|\;W_{x}=w\right)$
$\displaystyle\leq\exp{\left(-\tfrac{t^{2}}{2\pi^{\prime}_{n}(w)+2t/3}\right)}.$
(5.15)

Now, we write

$\displaystyle\mathbb{P}^{\,x}\left(\left.|\tilde{D}_{x}-\lambda_{n}(w)|>t\;\right|\;W_{x}=w\right)$
$\displaystyle\leq\mathbb{P}^{\,x}\left(\left.|D_{x}-\pi_{n}(w)|>t/2\;\right|\;W_{x}=w\right)$
$\displaystyle\phantom{X}+\mathbb{P}^{\,x}\left(\left.|D^{\prime}_{x}-\pi^{\prime}_{n}(w)|>t/2\;\right|\;W_{x}=w\right).$

By (5.14) and (5.15) together with (5.13), and letting
$t=C\sqrt{\lambda_{n}(w)\log\lambda_{n}(w)}$, this in turn is bounded from
above by

$\displaystyle
2\exp\left(-(\tfrac{C}{2})^{2}\tfrac{\lambda_{n}(w)\log{\lambda_{n}(w)}}{4\lambda_{n}(w)+\frac{2}{3}C\sqrt{\lambda_{n}(w)\log{\lambda_{n}(w)}}}\right)=2\exp\left(-(\tfrac{C}{2})^{2}\tfrac{\log{\lambda_{n}(w)}}{4+\frac{2}{3}C\sqrt{\log{\lambda_{n}(w)}/\lambda_{n}(w)}}\right),$
(5.16)

which concludes the lemma for lattice models.

For Poisson models note that Lemma 5.4 states that $\tilde{D}_{x}$ is a
Poisson random variable. Standard Poisson concentration bounds [17, Remark
2.6] imply that, for $w>1$ and $t>0$,

$\displaystyle\mathbb{P}^{\,x}\left(\left.|\tilde{D}_{x}-\lambda_{n}(W_{x})|>t\;\right|\;W_{x}=w\right)$
$\displaystyle\leq 2\exp\left(-\frac{t^{2}}{2(\lambda_{n}(w)+t)}\right).$

Plugging in $t=C\sqrt{\lambda_{n}(w)\log\lambda_{n}(w)}$ and noting that the
upper bound for lattice models given by (5.16) is larger, we conclude the
proof. ∎

###### Proof of Lemma 5.1.

We begin by proving the lemma for lattice models. Note that a standard union
bound gives us that

$\displaystyle{\mathbb{P}}\left(\mathcal{F}_{n,C}^{\,c}\right)$
$\displaystyle={\mathbb{P}}\left(\exists\,x\in V_{n}\;\;\text{such
that}\;\;|\tilde{D}_{x}-\lambda_{n}(W_{x})|{\mathbbm{1}}_{\\{W_{x}\geq
n^{a}\\}}>C\sqrt{\lambda_{n}(W_{x})\log({\lambda}_{n}(W_{x}))}\right)$
$\displaystyle\leq
n{\mathbb{P}}^{\,x}\left(|\tilde{D}_{x}-{\lambda}_{n}(W_{x})|{\mathbbm{1}}_{\\{W_{x}\geq
n^{a}\\}}>C\sqrt{{\lambda}_{n}(W_{x})\log({\lambda}_{n}(W_{x}))}\right)$
$\displaystyle=n(\beta-1)\int_{n^{a}}^{\infty}w^{-\beta}\mathbb{P}^{\,x}\left(\left.|\tilde{D}_{x}-{\lambda}_{n}(w)|>C\sqrt{{\lambda}_{n}(w)\log({\lambda}_{n}(w))}\;\right|\;W_{x}=w\right)dw.$

The upper bound (5.12) from Lemma 5.5 gives us that

$\displaystyle{\mathbb{P}}\left(\mathcal{F}_{n,C}^{\,c}\right)$
$\displaystyle\leq
2(\beta-1)n\int_{n^{a}}^{\infty}w^{-\beta}\exp{\left(-\tfrac{(C/2)^{2}\log{{\lambda}_{n}(w)}}{4+\frac{2}{3}\sqrt{\log{{\lambda}_{n}(w)}/{\lambda}_{n}(w)}}\right)}dw.$

Remark that by Lemma 5.2, $\lambda_{n}(w)$ inside the integral above can be
bounded from below by ${\sf k}n^{a}$, which is at least ${\mathrm{e}}$ for all
large $n$. Together with the fact that the function $h(x)=(\log x)/x$ is
decreasing in $({\rm e},\infty)$ we get that $h(\lambda_{n}(w))\leq h({\sf
k}n^{a})\leq 1$ for large $n$ and $w>n^{a}$. Hence, for large $n$, the
probability of $\mathcal{F}_{n,C}^{\,c}$ can be bounded from above by

$\displaystyle 2(\beta-1)n$
$\displaystyle\int_{n^{a}}^{\infty}w^{-\beta}\exp{\left(-\tfrac{(C/2)^{2}\log{{\lambda}_{n}(w)}}{5}\right)}dw$
$\displaystyle\leq
2(\beta-1)n\int_{n^{a}}^{\infty}w^{-\beta}\exp{\left(-\tfrac{(C/2)^{2}\log{{\lambda}_{n}(n^{a})}}{5}\right)}dw=2n^{1-a(\beta-1)}{\lambda}_{n}(n^{a})^{-C^{2}/20}.$

By Lemma 5.2, we conclude that for any $\xi>0$ we can choose a constant
$C(\xi)$ large enough such that the last term tends to zero faster than
$n^{-\xi}$.

Next we prove the lemma for Poisson models. Similarly to the lattice case, and
using a union bound, we observe that

$\displaystyle{\mathbb{P}}\left(\mathcal{F}_{n,C}^{\,c}\right)$
$\displaystyle={\mathbb{P}}\left(\exists\,x\in\mathcal{P}_{n}\;\;\text{such
that}\;\;|\tilde{D}_{x}-\lambda_{n}(W_{x})|{\mathbbm{1}}_{\\{W_{x}\geq
n^{a}\\}}>C\sqrt{\lambda_{n}(W_{x})\log({\lambda}_{n}(W_{x}))}\right)$
$\displaystyle\leq{\mathbb{E}}\left[\,\sum_{x\in\mathcal{P}_{n}}\mathbb{P}\left(\left.|\tilde{D}_{x}-\lambda_{n}(W_{x})|{\mathbbm{1}}_{\\{W_{x}\geq
n^{a}\\}}>C\sqrt{\lambda_{n}(W_{x})\log({\lambda}_{n}(W_{x}))}\;\right|\;\mathcal{P}_{n}^{x}\right)\right]$
$\displaystyle=\int_{\mathbb{T}_{n}^{d}}{\mathbb{E}}\left[\mathbb{P}\left(\left.|\tilde{D}_{x}-\lambda_{n}(W_{x})|{\mathbbm{1}}_{\\{W_{x}\geq
n^{a}\\}}>C\sqrt{\lambda_{n}(W_{x})\log({\lambda}_{n}(W_{x}))}\;\right|\;\mathcal{P}_{n}^{x}\right)\right]dx$
$\displaystyle=\int_{\mathbb{T}_{n}^{d}}{\mathbb{P}}^{\,x}\left(|\tilde{D}_{x}-\lambda_{n}(W_{x})|{\mathbbm{1}}_{\\{W_{x}\geq
n^{a}\\}}>C\sqrt{\lambda_{n}(W_{x})\log({\lambda}_{n}(W_{x}))}\right)dx,$

where the second equality holds by Mecke’s formula [20, Theorem 4.1]. The
remainder of the proof is identical to that of lattice models and is therefore
omitted. ∎

###  5.3. The few-jumps phenomenon: Proof of Proposition 5.3

It remains to prove Proposition 5.3. We first state and prove a result that
shows that it is unlikely to have many small macroscopic contributions.

######  Lemma 5.6.

Let $k$ be a positive integer. For all $\delta>0$, there exists
$\varepsilon>0$ such that

$\displaystyle{\mathbb{P}}\left(\sum_{x\in
V_{n}}\lambda_{n}(W_{x}){\mathbbm{1}}_{\\{n^{a}\leq W_{x}\leq\varepsilon
n\\}}>\delta n\right)=o(n^{-k(\beta-2)}).$ (5.17)

Lemma 5.6 tells us that the sum $\sum_{x\in
V_{n}}\lambda_{n}(W_{x}){\mathbbm{1}}_{\\{n^{a}\leq W_{x}\leq\varepsilon
n\\}}$ cannot contribute significantly at a linear scale. Let us discuss the
proof strategy. First, we use Lemma 5.2 to reduce the question to
understanding the probability

$\displaystyle{\mathbb{P}}\left(\sum_{i=1}^{N}W_{i}{\mathbbm{1}}_{\\{n^{a}\leq
W_{i}\leq\varepsilon n\\}}>K\delta n\right),$

for some fixed constant $K>0$. The sum inside the last probability cannot
contribute at least $K\delta n$ with good probability when we choose
$\varepsilon$ arbitrarily small. To show this, we use a Chernoff inequality to
reduce the problem to understanding exponential moments of the form
$\mathbb{E}[\exp{(s_{n}\hat{W}^{\scriptscriptstyle(n)})}]$, where
$\hat{W}^{\scriptscriptstyle(n)}$ is the truncated random variable
$W{\mathbbm{1}}_{\\{n^{a}<W<\varepsilon n\\}}$ and where $W$ has density
(1.1), and we take $s_{n}=b\log{n}/n$ for a suitable $b>0$. This we achieve by
a second order Taylor expansion and choosing the scale $b$ appropriately.

###### Proof.

Let $(W_{i})_{i\geq 1}$ be an i.i.d. collection with law (1.1). For both
lattice and Poisson models, we write

$\displaystyle{\mathbb{P}}\bigg{(}\sum_{x\in
V_{n}}\lambda_{n}(W_{x}){\mathbbm{1}}_{\\{n^{a}\leq W_{x}\leq\varepsilon
n\\}}>\delta n\bigg{)}$
$\displaystyle\leq{\mathbb{P}}\bigg{(}\sum_{i=1}^{N}W_{i}{\mathbbm{1}}_{\\{n^{a}\leq
W_{i}\leq\varepsilon n\\}}>\delta n/{\sf K}\bigg{)},$

where the inequality holds by Lemma 5.2, and we use that the $W_{x}$ are
independent of the location of $x$. Recall that for lattice models, $N$ equals
the number $n$ of points on the integer lattice of $\mathbb{T}^{d}_{n}$ and
for Poisson models, $N$ is a Poi($n$)-distributed random variable. Recalling
$I_{M}(n)$ from (3.1) and using (3.3) we deduce that, for Poisson models,

$\displaystyle{\mathbb{P}}\left(\sum_{i=1}^{N}W_{i}{\mathbbm{1}}_{\\{n^{a}\leq
W_{i}\leq\varepsilon n\\}}>\delta n/{\sf K}\right)$
$\displaystyle={\mathbb{P}}\left(\sum_{i=1}^{N}W_{i}{\mathbbm{1}}_{\\{n^{a}\leq
W_{i}\leq\varepsilon n\\}}>\delta n/{\sf K},\,N\in
I_{M}(n)\right)+o(n^{-k(\beta-2)})\phantom{wefillthisspacetoalign}$
$\displaystyle={\mathbb{E}}\left[{\mathbbm{1}}_{\\{N\in
I_{M}(n)\\}}\mathbb{P}\left(\left.\sum_{i=1}^{N}W_{i}{\mathbbm{1}}_{\\{n^{a}\leq
W_{i}\leq\varepsilon n\\}}>\delta n/{\sf
K}\;\right|\;N\right)\right]+o(n^{-k(\beta-2)}).$

From this it is clear that to prove the lemma for both lattice and Poisson
models, it suffices to show that, for all $\delta>0$,

$\displaystyle{\mathbb{P}}\left(\sum_{i=1}^{m_{n}}W_{i}{\mathbbm{1}}_{\\{n^{a}\leq
W_{i}\leq\varepsilon n\\}}>\delta n\right)=o(n^{-k(\beta-2)}),$ (5.18)

where $(m_{n})_{n}$ is any deterministic sequence satisfying
$m_{n}=(1+o(1))n$. To prove (5.18), define the truncated random variable
$\hat{W}^{\scriptscriptstyle(n)}:=W{\mathbbm{1}}_{\\{n^{a}\leq W<\varepsilon
n\\}}$. The Chernoff bound gives us that, for all $s_{n}>0$,

$\displaystyle{\mathbb{P}}\left(\sum_{i=1}^{m_{n}}\hat{W}^{\scriptscriptstyle(n)}_{i}>m\right)\leq{\rm
e}^{-s_{n}m_{n}}{\mathbb{E}}\left[\exp(s_{n}\hat{W}^{\scriptscriptstyle(n)})\right]^{m_{n}}.$

Let $s_{n}:=b\log n/n$, for a constant $b>0$ to be chosen later. By the Taylor
expansion of ${\rm e}^{x}$,

$\displaystyle{\mathbb{E}}\left[\exp(s_{n}\hat{W}^{\scriptscriptstyle(n)})\right]$
$\displaystyle\leq 1+s_{n}\hat{\mu}_{n}+{\mathbb{E}}\left[\sum_{k\geq
2}\frac{(s_{n}\hat{W}^{\scriptscriptstyle(n)})^{k}}{k!}\right],$ (5.19)

where $\hat{\mu}_{n}:=\mathbb{E}[\hat{W}^{\scriptscriptstyle(n)}]$ converges
to zero as $n$ tends to infinity. We start by investigating the third term in
(5.19). We choose constants $c,C>0$ such that ${\rm e}^{x}\leq 1+x+Cx^{2}$ for
all $x\in(0,c)$. Thus, for sufficiently large $n$,

$\displaystyle{\mathbb{E}}\left[\sum_{k\geq
2}\frac{(s_{n}\hat{W}^{\scriptscriptstyle(n)})^{k}}{k!}\right]$
$\displaystyle\leq
Cs_{n}^{2}{\mathbb{E}}\left[(\hat{W}^{\scriptscriptstyle(n)})^{2}{\mathbbm{1}}_{\\{\hat{W}^{\scriptscriptstyle(n)}<c/s_{n}\\}}\right]+{\mathbb{E}}\left[\exp{(s_{n}\hat{W}^{\scriptscriptstyle(n)})}{\mathbbm{1}}_{\\{\hat{W}^{\scriptscriptstyle(n)}\geq
c/s_{n}\\}}\right]$ $\displaystyle\leq
Cs_{n}^{2}{\mathbb{E}}\left[W^{2}{\mathbbm{1}}_{\\{W<c/s_{n}\\}}\right]+{\mathbb{E}}\left[\exp{(s_{n}W)}{\mathbbm{1}}_{\\{W\in[c/s_{n},\varepsilon
n)\\}}\right],$ (5.20)

where the second inequality holds since

$(\hat{W}^{\scriptscriptstyle(n)})^{2}{\mathbbm{1}}_{\\{\hat{W}^{\scriptscriptstyle(n)}<c/s_{n}\\}}=W^{2}{\mathbbm{1}}_{\\{n^{a}<W<\varepsilon
n\\}}{\mathbbm{1}}_{\\{W<c/s_{n}\\}}$

and
${\mathbbm{1}}_{\\{\hat{W}^{\scriptscriptstyle(n)}>c/s_{n}\\}}={\mathbbm{1}}_{\\{n^{a}<W<\varepsilon
n\\}}{\mathbbm{1}}_{\\{W>c/s_{n}\\}}$. Note that $W$ is $L^{2-\ell}$-bounded
for $3-\beta<\ell<2$. Thus we can choose $\ell>0$ such that $1<2-\ell<\beta-1$
and

$\displaystyle
s_{n}^{2}{\mathbb{E}}\left[W^{2}{\mathbbm{1}}_{\\{W<c/s_{n}\\}}\right]$
$\displaystyle=s_{n}^{2}{\mathbb{E}}\left[W^{\ell}W^{2-\ell}{\mathbbm{1}}_{\\{W<c/s_{n}\\}}\right]\leq
c^{\ell}s_{n}^{2-\ell}{\mathbb{E}}\left[W^{2-\ell}\right]=o(s_{n}),$

where the last step holds since $2-\ell>1$ and $s_{n}$ converges to $0$ as $n$
increases, so $s_{n}^{2-\ell}=o(s_{n})$. Next, note that

$\displaystyle{\mathbb{E}}\left[\exp{(s_{n}W}){\mathbbm{1}}_{\\{W\in[c/s_{n},\varepsilon
n)\\}}\right]$ $\displaystyle\leq{\mathrm{e}}^{s_{n}\varepsilon
n}{\mathbb{P}}\left(W\in[c/s_{n},\varepsilon
n)\right)\leq{\mathrm{e}}^{s_{n}\varepsilon n}{\mathbb{P}}\left(W\geq
c/s_{n}\right)$ $\displaystyle={\mathrm{e}}^{s_{n}\varepsilon
n}(s_{n}/c)^{\beta-1}=s_{n}c^{1-\beta}{\mathrm{e}}^{s_{n}\varepsilon
n}s_{n}^{\beta-2}.$

Plugging in $s_{n}=b\log n/n$, we get that, for some constant $C^{\prime}$,

$\displaystyle{\mathbb{E}}\left[\exp{(s_{n}W^{\scriptscriptstyle(n)}}){\mathbbm{1}}_{\\{W\in[c/s_{n},\varepsilon
n)\\}}\right]$ $\displaystyle\leq s_{n}C^{\prime}n^{b\varepsilon+2-\beta}(\log
n)^{\beta-2}.$

For $b<(\beta-2)\varepsilon^{-1}$, we get $n^{b\varepsilon+2-\beta}(\log
n)^{\beta-2}=o(1)$ and so, by (5.20),

$\displaystyle{\mathbb{E}}\left[\sum_{k\geq
2}\frac{(s_{n}\hat{W}^{\scriptscriptstyle(n)})^{k}}{k!}\right]$
$\displaystyle=o(s_{n}),$ (5.21)

as $n\rightarrow\infty$. From (5.21) it follows that

$\displaystyle{\mathbb{E}}\left[\exp{(s_{n}\hat{W}^{\scriptscriptstyle(n)})}\right]$
$\displaystyle\leq
1+s_{n}\hat{\mu}_{n}+o(s_{n})=1+o(s_{n})\leq\exp{(o(s_{n}))}.$

Putting everything together, it follows that

$\displaystyle{\mathbb{P}}\left(\sum_{i\leq
m_{n}}W_{i}{\mathbbm{1}}_{\\{n^{a}\leq W_{i}<\varepsilon n\\}}>\delta
n\right)$ $\displaystyle\leq\exp{(-s_{n}\delta
n+m_{n}o(s_{n}))}=\exp(s_{n}n(-\delta+o(1)))$
$\displaystyle=o(n^{1-b\delta}),$

using that $s_{n}=b\log n/n$, $m_{n}=(1+o(1))n$ and $n^{o(1)}=o(n)$. Choosing
$b>0$ and $\varepsilon>0$ such that
$(1-k(\beta-2))/\delta<b<(\beta-2)\varepsilon^{-1}$ is satisfied, we obtain
that $1-b\delta<k(\beta-2)$ which concludes the proof of (5.18). ∎

Next, we investigate the asymptotic behaviour of $\lambda_{n}(bn)$.

######  Lemma 5.7.

As $n\to\infty$, for $w\in(0,\infty)$,

$\displaystyle\frac{1}{n}{\lambda}_{n}(wn)\longrightarrow{\Lambda}(w),$

where we recall that
${\Lambda}(w)=\int_{[-\frac{1}{2},\frac{1}{2}]^{d}}{\mathbb{E}}\left[\varphi\left(\frac{\|z\|^{d}}{\mathcal{K}(w,W)}\right)\right]dz$
in (1.10).

###### Proof.

We prove the lemma first for lattice models. Recall that
$\mathbb{Q}_{n}\subset\mathbb{R}^{d}$ is the cube of volume $n$ centred at
$0$. Embedding the torus around $x$ into the integer lattice and recalling the
definition of $\tilde{D}_{x}$, we write

$\displaystyle\frac{{\lambda}_{n}^{x}(wn)}{n}$
$\displaystyle=\frac{1}{n}\int_{1}^{\infty}f_{W}(t)\sum_{y\in(\mathbb{Q}_{n}\cap\mathbb{Z}^{d})\setminus\\{0\\}}\varphi\left(\frac{\|y\|^{d}}{\kappa(wn,t)}\right)dt$
$\displaystyle\phantom{X}-\frac{1}{2n}\int_{n^{a}}^{\infty}f_{W}(t)\sum_{y\in(\mathbb{Q}_{n}\cap\mathbb{Z}^{d})\setminus\\{0\\}}\varphi\left(\frac{\|y\|^{d}}{\kappa(wn,t)}\right)dt.$
(5.22)

By (1.6), for every $w\in(0,\infty)$,

$\displaystyle\frac{1}{n}\sum_{y\in(\mathbb{Q}_{n}\cap\mathbb{Z}^{d})\setminus\\{0\\}}\varphi\left(\frac{\|y\|^{d}}{\kappa(wn,t)}\right)$
$\displaystyle=\frac{1}{n}\sum_{y\in(\mathbb{Q}_{n}\cap\mathbb{Z}^{d})\setminus\\{0\\}}\varphi\left(\frac{\|y\|^{d}}{(1+o(1))n\,\mathcal{K}(w,t)}\right),$

where the $o(1)$ is uniform in $y\in\mathbb{Z}^{d}$. This term therefore
converges to

$\int_{[-\frac{1}{2},\frac{1}{2}]^{d}}\varphi\left(\frac{\|z\|^{d}}{\mathcal{K}(w,t)}\right)dz.$

The fact that $|\mathbb{Q}_{n}\cap\mathbb{Z}^{d}|=n$ together with the
dominated convergence theorem implies that, for every $w\in(0,\infty)$,

$\displaystyle\frac{1}{n}\int_{1}^{\infty}f_{W}(t)\sum_{y\in(\mathbb{Q}_{n}\cap\mathbb{Z}^{d})\setminus\\{0\\}}\varphi\left(\frac{\|y\|^{d}}{\kappa(wn,t)}\right)dt\to\int_{1}^{\infty}\int_{[-\tfrac{1}{2},\tfrac{1}{2}]^{d}}f_{W}(t)\varphi\left(\tfrac{\|z\|^{d}}{\mathcal{K}(w,t)}\right)\,dzdt,$

as $n\to\infty$. From this it also follows that the second summand in (5.3)
converges to zero, concluding the proof for lattice models. The proof for
Poisson models follows similarly and is therefore omitted. ∎

In the next lemma, we investigate the contribution of the macroscopic weights.

######  Lemma 5.8.

For all non-integers $\rho>0$, with $k=\lceil\rho\rceil$ and for all $b>0$,

$\displaystyle{\mathbb{P}}\left(\sum_{x\in
V_{n}}{\lambda}_{n}(W_{x}){\mathbbm{1}}_{\\{W_{x}>bn\\}}>\rho
n\right)=(F_{b}(\rho)+o(1))n^{-k(\beta-2)}.$

###### Proof.

For Poisson models, we observe that

$\displaystyle{\mathbb{P}}\left(\sum_{x\in
V_{n}}\lambda_{n}(W_{x}){\mathbbm{1}}_{\\{W_{x}>bn\\}}>\rho n,\,N\in
I_{M}(n)\right)$ $\displaystyle\leq{\mathbb{P}}\left(\sum_{x\in
V_{n}}\lambda_{n}(W_{x}){\mathbbm{1}}_{\\{W_{x}>bn\\}}>\rho n\right)$
$\displaystyle\leq{\mathbb{P}}\left(\sum_{x\in
V_{n}}\lambda_{n}(W_{x}){\mathbbm{1}}_{\\{W_{x}>bn\\}}>\rho n,\,N\in
I_{M}(n)\right)+o(n^{-k(\beta-2)}),$ (5.23)

recalling $I_{M}(n)$ from (3.1) and using the bound (3.3). To prove the lemma
for Poisson models it therefore suffices to show that for $b>0$ and
sufficiently large $M>0$,

$\displaystyle{\mathbb{E}}\left[{\mathbbm{1}}_{\\{N\in
I_{M}(n)\\}}\mathbb{P}\left(\left.\sum_{x\in\mathcal{P}_{n}}\lambda_{n}(W_{x}){\mathbbm{1}}_{\\{W_{x}>bn\\}}>\rho
n\;\right|\;N\right)\right]=(1+o(1))n^{-k(\beta-2)}F_{b}(\rho).$

Note that, for $x\in V_{n}$, $\lambda_{n}(W_{x})$ is independent of the
location of $x$ for lattice and Poisson models. Therefore in order to prove
Lemma 5.8 for both models, it suffices to show that

$\displaystyle{\mathbb{P}}\left(\sum_{i=1}^{m_{n}}\lambda_{n}(W_{i}){\mathbbm{1}}_{\\{W_{i}>bn\\}}>\rho
n\right)=(1+o(1))n^{-k(\beta-{2})}F_{b}(\rho),$ (5.24)

for an i.i.d. collection $(W_{i})_{i=1}^{m_{n}}$ having law (1.1), where
$m_{n}$ is any deterministic sequence satisfying $m_{n}=(1+o(1))n$. Thus for
the rest of the proof, we work with such a sequence.

Let $b>0$ and define the events $\mathcal{A}_{m}=\mathcal{A}_{m}(b)$ for
$1\leq m\leq m_{n}$ as

$\displaystyle\mathcal{A}_{m}(b):=\\{\text{there exist
exactly}\;m\;\text{random variables with}\;W_{i}\geq bn\\}.$ (5.25)

We can write

${\mathbb{P}}\left(\sum_{i=1}^{m_{n}}{\lambda}_{n}(W_{i}){\mathbbm{1}}_{\\{W_{i}>bn\\}}>\rho
n\right)=\sum_{m=1}^{m_{n}}{\mathbb{P}}\left(\sum_{i=1}^{m_{n}}{\lambda}_{n}(W_{i}){\mathbbm{1}}_{\\{W_{i}>bn\\}}>\rho
n,\,\mathcal{A}_{m}\right).$

Let $m<k$. Then, using the fact that $(W_{i})_{i}$ are i.i.d., we remark that

$\displaystyle{\mathbb{P}}\left(\sum_{i=1}^{m_{n}}{\lambda}_{n}(W_{i}){\mathbbm{1}}_{\\{W_{i}>bn\\}}>\rho
n,\mathcal{A}_{m}\right)$
$\displaystyle=\binom{m_{n}}{m}{\mathbb{P}}\left(\sum_{j=1}^{m}{\lambda}_{n}(W_{i_{j}})>\rho
n,W_{i_{1}},\ldots,W_{i_{m}}>bn,W_{i_{m+1}},\ldots,W_{i_{m_{n}}}\leq
bn\right)=0,$

for sufficiently large $n$, because ${\lambda}_{n}(W)\leq N$ deterministically
and so $\sum_{j=1}^{m}{\lambda}_{n}(W_{i_{j}})\leq mN\leq(1+o(1))(k-1)n<\rho
n$. Next, we note that, by a union bound of the at least $k+1$ vertices with
weight larger than $bn$,

$\displaystyle\sum_{m=k+1}^{m_{n}}{\mathbb{P}}\left(\sum_{i=1}^{m_{n}}{\lambda}_{n}(W_{i}){\mathbbm{1}}_{\\{W_{i}>bn\\}}>\rho
n,\,\mathcal{A}_{m}\right)$
$\displaystyle\leq\sum_{m=k+1}^{m_{n}}{\mathbb{P}}\left(\mathcal{A}_{m}\right)$
$\displaystyle\leq(1+o(1))n^{k+1}{\mathbb{P}}\left(W_{1},\ldots,W_{{k+1}}>bn\right).$

Since $(W_{i})_{i}$ have law (1.1), this equals $o(n^{k(2-\beta)})$.
Therefore,

$\displaystyle\mathbb{P}\Big{(}\sum_{i=1}^{m_{n}}{\lambda}_{n}(W_{i}){\mathbbm{1}}_{\\{W_{i}>bn\\}}>\rho
n\Big{)}\leq
o(n^{k(2-\beta)})+\mathbb{P}\Big{(}\sum_{i=1}^{m_{n}}{\lambda}_{n}(W_{i}){\mathbbm{1}}_{\\{W_{i}>bn\\}}>\rho
n,\mathcal{A}_{k}\Big{)}.$ (5.26)

It remains to study

$\displaystyle\mathbb{P}\Big{(}\sum_{i=1}^{m_{n}}{\lambda}_{n}(W_{i}){\mathbbm{1}}_{\\{W_{i}>bn\\}}>\rho
n,\mathcal{A}_{k}\Big{)}$
$\displaystyle=\binom{m_{n}}{k}\mathbb{P}\Big{(}\sum_{i=1}^{m_{n}}{\lambda}_{n}(W_{i}){\mathbbm{1}}_{\\{W_{i}>bn\\}}>\rho
n,W_{1},\dots,W_{k}>bn,W_{{k+1}},\dots,W_{{m_{n}}}\leq bn\Big{)}$
$\displaystyle=\binom{m_{n}}{k}(1-(\beta-1)(bn)^{1-\beta})^{m_{n}-k}(\beta-1)^{k}\int_{bn}^{\infty}\cdots\int_{bn}^{\infty}{\mathbbm{1}}_{\\{\frac{1}{n}({\lambda}_{n}(t_{i})+\ldots+{\lambda}_{n}(t_{k}))>\rho\\}}\prod_{i=1}^{k}t_{i}^{-\beta}dt_{i}.$

Observe that $(1-(\beta-1)(bn)^{1-\beta})^{m_{n}-k}=1+o(1)$ since
$n^{-(\beta-1)}=o(n^{-1})$, using $(m_{n}-k)/n\to 1$. By a change of variables
$y_{i}=t_{i}/n$, we can write the last expression as

$\displaystyle(1+o(1))\binom{n}{k}n^{-k(\beta-1)}(\beta-1)^{k}\int_{b}^{\infty}\cdots\int_{b}^{\infty}{\mathbbm{1}}_{\\{\frac{1}{n}({\lambda}_{n}(ny_{1})+\cdots+{\lambda}_{n}(ny_{k}))>\rho\\}}\prod_{i=1}^{k}y_{i}^{-\beta}dy_{i},$

since $m_{n}=(1+o(1))n$. By Lemma 5.7 and dominated convergence, the last
expression is asymptotically equivalent to $n^{-k(\beta-2)}F_{b}(\rho)$. ∎

Now we are ready to prove Proposition 5.3.

###### Proof of Proposition 5.3.

The following proof holds for both lattice and Poisson models. Recall the
definition of $F_{b}(\rho)$ given in Lemma 5.8 and recall that we write
$F(\rho)$ for $F_{0}(\rho)$. Let $k=\lceil\rho\rceil$. Fix $0<\delta<1$ and