# Limiting Spectra of inhomogeneous random graphs

Luca Avena  Leiden University, Mathematical Institute, Niels Bohrweg 1 2333
CA, Leiden. The Netherlands. DIMAI, Dipartimento di Matematica e Informatica
“Ulisse Dini”, Università degli studi di Firenze, Viale G.B. Morgagni 67A,
50134, Florence, Italy<EMAIL_ADDRESS>,  Rajat Subhra Hazra  University
of Leiden, Niels Bohrweg 1, 2333 CA, Leiden, The Netherlands
<EMAIL_ADDRESS>and  Nandan Malhotra  University of Leiden, Niels
Bohrweg 1, 2333 CA, Leiden, The Netherlands<EMAIL_ADDRESS>
###### Abstract.

We consider sparse inhomogeneous Erdős-Rényi random graph ensembles where
edges are connected independently with probability $p_{ij}$. We assume that
$p_{ij}=\varepsilon_{N}f(w_{i},w_{j})$ where $(w_{i})_{i\geq 1}$ is a sequence
of deterministic weights, $f$ is a bounded function and
$N\varepsilon_{N}\to\lambda\in(0,\infty)$. We characterise the limiting
moments in terms of graph homomorphisms and also classify the contributing
partitions. We present an analytic way to determine the Stieltjes transform of
the limiting measure. The convergence of the empirical distribution function
follows from the theory of local weak convergence in many examples but we do
not rely on this theory and exploit combinatorial and analytic techniques to
derive some interesting properties of the limit. We extend the methods of
Khorunzhy et al. (2004) and show that a fixed point equation determines the
limiting measure. The limiting measure crucially depends on $\lambda$ and it
is known that in the homogeneous case, if $\lambda\to\infty$, the measure
converges weakly to the semicircular law (Jung and Lee (2018)). We extend this
result of interpolating between the sparse and dense regimes to the
inhomogeneous setting and show that as $\lambda\to\infty$, the measure
converges weakly to a measure which is known as the operator-valued
semicircular law.

###### Key words and phrases:

generalized random graphs, inhomogeneous random graph, empirical spectral
distribution, spectrum, free probability

###### 2000 Mathematics Subject Classification:

05C80, 60B20, 60B10, 46L54

###### Contents

  1. 1 Introduction
  2. 2 Setting and Main results
    1. 2.1 Model
    2. 2.2 Main Results
      1. Method of moments: Combinatorial Approach
      2. Stieltjes transform: Analytic approach.
    3. 2.3 Examples
  3. 3 Method of Moments: Proof of Theorem 2.7
    1. 3.1 Expected Moments
    2. 3.2 Concentration and uniqueness.
  4. 4 Analytic approach: Proof of Theorem 2.9
    1. 4.1 Resolvent and Stieltjes Transform
    2. 4.2 Deriving the expression for the Stieltjes Transform
  5. 5 Appendix

##  1\. Introduction

Homogeneous Erdős-Rényi Random Graphs (ERRG) serve as the basis for many
mathematical theories in random graphs. Real-world networks are highly
inhomogeneous and have a far more complex structure. Various attempts have
been made to generalize this to other kinds of random graph models. One of the
successful extensions is the inhomogeneous Erdős-Rényi random graph model
introduced by Bollobás et al. (2007). This graph has $N$ vertices labeled by
$[N]={1,...,N}$, and edges are present independently with probability $p_{ij}$
given by $p_{ij}=\frac{f(x_{i},x_{j})}{N}\wedge 1$, where $f$ is a nice
symmetric kernel on a state space $S\times S$, and $x_{i}$ are certain
attributes associated with vertex $i$ belonging to $S$. If $f$ is bounded, the
graph is a sparse random graph. To introduce the non-sparse regime, in this
article, we consider a small variant of the above inhomogeneous random graph.
The vertex set remains the same, but the connection probabilities are given by

$p_{ij}=\varepsilon_{N}f(w_{i},w_{j})\wedge 1,$ (1.1)

where $\varepsilon_{N}$ is a tuning parameter, $(w_{i})$ is a sequence of
deterministic weights, and $f$ is a symmetric, bounded function on
$[0,\infty)^{2}$. The weights can signify a property of vertex $i$. They can
also be generally random, but we do not consider this case. Note that when
$N\varepsilon_{N}\to\infty$, the average degree is unbounded, and when
$N\varepsilon_{N}=\mathrm{O}(1)$, the average degree is bounded. We call the
former case _dense_ and the latter case _sparse_. In the sparse case, the
properties of the connected components were studied in Bollobás et al. (2007).
They studied the properties of the connected components and their relationship
with the branching process. It was shown that the largest component of the
graph has a size of order $N$ if the operator norm of the kernel operator
corresponding to $f$ is strictly greater than $1$ (see also (van der Hofstad,
2023, Theorem 3.9)). In the subcritical case, the sizes of the largest
connected components can exhibit different behaviour compared to the ERRG. The
study of the largest connected components in various inhomogeneous random
graphs has attracted a lot of attention (see, for example, Bhamidi et al.
(2010), Broutin et al. (2021), Bet et al. (2023), van der Hofstad (2013),
Devroye and Fraiman (2014)). In this article, we are interested in the
empirical distribution of the eigenvalues of the adjacency matrix of the graph
and how the transition occurs from the sparse to the dense case in terms of
the limiting spectral distribution. There hasn’t been much literature in this
area, even though various specific graphs have been studied. For example, the
largest eigenvalue of the sparse Chung-Lu random graph was studied in Chung et
al. (2003), and this was extended to an inhomogeneous setting by Benaych-
Georges et al. (2020, 2019). The bulk of the spectrum of sparse graphs is
mainly studied through local weak convergence. Here, we present a unifying
approach to understanding both the sparse and the dense cases, allowing us to
interpolate between the two regimes.

In the case of homogeneous ERRG, it is known that in the dense case, the
empirical distribution converges to the semicircle law after an appropriate
scaling (Tran et al. (2013)). In the sparse case, it converges to a measure
that depends on the parameter $N\varepsilon_{N}\to\lambda$. The behaviour is
much more complicated in the sparse case. Various interesting properties were
predicted by Bauer and Golinelli (2001). The existence of the limiting
distribution was proved by Khorunzhy et al. (2004), who also showed some
interesting properties of the moments and the limiting Stieltjes transform.
The local geometric behaviour of sparse random graphs can be well studied
using the theory of local weak convergence (LWC), which builds on the works
Aldous and Lyons (2007) and Benjamini and Schramm (2001). It roughly describes
how a graph looks like in the limit around a uniformly chosen vertex. For a
detailed review of LWC and various other applications, see van der Hofstad
(2023). In a remarkable work by Bordenave and Lelarge (2010), it was proved
that if a graph with $N$ vertices converges locally weakly to a Galton-Watson
tree, then the Stieltjes transform of the empirical spectral distribution
converges in $L^{1}$ to the Stieltjes transform of the spectral measure of the
tree, and it satisfies a recursive distributional equation. The example of
homogeneous ERRG was treated in (Bordenave and Lelarge, 2010, Example 2). The
limiting measure of sparse ERRG depends on $\lambda$ and is still very non-
explicit. It was proved by Bordenave et al. (2017), Arras and Bordenave (2021)
that the measure has an absolutely continuous component if and only if
$\lambda>1$. The size of the atom at the origin was shown by Bordenave et al.
(2011), and the nature of the atomic part of the measure was studied in the
same article. The study of so-called extended states at origin was initiated
in Coste and Salez (2021), and it was shown that for $\lambda<e$, there were
no extended states, and for $\lambda>e$, it has extended states. All these
results were conjectured in Bauer and Golinelli (2001). Most of these results
on local limits show that properties are generally true for unimodular Galton
Watson trees.

In the simulations of Bauer and Golinelli (2001), it is clear that when
$\lambda$ is slightly larger than 1, the limiting measure already starts
taking the shape of the semicircular law. It was shown in Jung and Lee (2018)
that indeed, if $\lambda\to\infty$, then the limiting measure converges to the
semicircular law. The main motivation of this work comes from the work of
(Jung and Lee, 2018, Theorem 1), and we extend the results from ERRG to
inhomogeneous models. We explicitly derive the moments of the limiting measure
for the inhomogeneous setting, extending the works of Khorunzhy et al. (2004),
albeit with a much simpler proof. The moments of the limiting measure depend
on certain kinds of graph homomorphism counts, which also appeared in the
works of Zhu (2020). Although the theory of local weak convergence is very
useful, we do not know if it can be used to derive the moments of the limiting
measure. We also study the Stieltjes transform of the limiting measure,
following the idea of Khorunzhy et al. (2004), and attempt an expansion of it
for $\lambda$ large enough. This has also gained attention in the physics
literature, see references in Akara-pipattana and Evnin (2023). We show that
when $\lambda\gg 1$, the limiting moments closely resemble those of the
inhomogeneous ERRG, as derived in Chakrabarty et al. (2021) and also implied
by the work of Zhu (2020). In Chakrabarty et al. (2021), they considered
inhomogeneous ERRG to have weights $w_{i}=i/N$, and
$N\varepsilon_{N}\to\infty$. This result can be extended to general
deterministic weights without significant effort, and we state this general
result in Section 2. The limiting measure is well-known in the free
probability literature and appears as a universal object in many inhomogeneous
systems, referred to as the _operator-valued semicircle law_ (Speicher, 2011,
Theorem 22.7.2). The Stieltjes transform satisfies a recursive analytic
equation. We derive the Stieltjes transform in the sparse setting using a
fixed-point equation. The fixed point is simpler in the case of homogeneous
ERRG, but in the inhomogeneous case, it becomes more complex. We explicitly
characterise this fixed-point equation. We believe that in the future, this
will aid in determining the rate of convergence of the empirical spectral
distribution, which can be precisely quantified in terms of $\lambda$ and $N$.
The rates of convergence in the free central limit theorem were recently
explored in Banna and Mai (2023), but these results are not directly
applicable to our setting. We leave this as an open problem. Obtaining an
explicit rate of convergence will provide an exact explanation of why the
limiting measure in the sparse setting is very close to the non-sparse setting
for relatively small $\lambda>1$.

Brief summary of the results. The two main results of this work aim to
characterise the limiting spectral measure of inhomogeneous Erdős-Rényi random
graphs. Our first result, Theorem 2.7, gives a characterisation of the moments
of this measure, where the $k^{\text{th}}$ moment for any $k\geq 0$ is
described in terms of homomorphism densities of the inhomogeneity function $f$
and special classes of partitions of the tuple $[k]$. We can recover the
moments of the dense regime asymptotically (as $\lambda\to\infty$) using this
result. The second result Theorem 2.9 provides an analytic characterisation of
the measure. In particular, we provide an analytic characterisation of a
functional of the resolvent of the adjacency matrix in terms of a fixed point
equation. As a consequence, in Corollaries 2.10 and 2.11, we obtain the
Stieltjes transform of the sparse and dense limiting measures. The form of the
limiting Stieltjes transform can be seen as an alternative description of the
form obtained through local weak convergence (whenever it applies).

Outline. We begin Section 2 by describing the model and stating the results of
the dense regime. We state the assumptions on the sparse setting more
explicitly and proceed by stating our main results for this setting. We then
describe a relationship with local weak convergence and also give some
examples of popular random graph models. We show that the sparse Chung-Lu type
model falls into our setting, and while the Norros-Riettu model and the
Generalised Random Graph models do not directly fall into our setting, we show
that asymptotically the three models have the same spectral distribution,
which has a free-multiplicative part that can be seen from our main results.

In Section 3, we prove our first main result which takes a combinatorial
approach, and we set up all the necessary tools used in proving the result. We
identify the moments of the limiting spectral measure in terms of partitions
of a tuple and graph-homomorphism densities. We provide a characterisation of
the partitions and explicit expressions for the moments that are given by
homomorphism densities defined based on these partitions. We further identify
a leading order of the moments and a polynomial in $\lambda^{-1}$, which was
also seen for the homogeneous setting in Jung and Lee (2018).

In Section 4, we prove our second main result which in contrast has an
analytic flavour. We set up the relevant analytic structures, and instead of
working directly with the Stieltjes Transform, we work with a functional of
the resolvent of the adjacency matrix, which was introduced in Khorunzhy et
al. (2004). We borrow both fundamental and advanced tools from analysis to
provide an exact analytic characterisation of the limiting spectral measure.
We conclude with the Appendix as Section 5 where we state the key analytic
tools we use in Section 4.

##  2\. Setting and Main results

###  2.1. Model

We consider the inhomogeneous Erdős-Rényi random graph (IER) $\mathbb{G}_{N}$
on the vertex set $[N]=\\{1,\ldots,N\\}$ where edges are independently added
with probability $p_{ij}$. As mentioned before we will assume that $p_{ij}$
has a special form as

$p_{ij}=\varepsilon_{N}f(w_{i},w_{j})\wedge 1,$

where $\varepsilon_{N}$ is a tuning parameter such that $\varepsilon_{N}\to
0$, $(w_{i})_{i\geq 1}$ is a sequence of deterministic non-negative weights
and $f:[0,\infty)^{2}\to[0,\infty)$ is bounded and continuous. We will use
$\mathbb{P}_{N}$ to denote the law of this random graph and we will drop the
subscript $N$ for notational convenience, and ${\mathbb{E}}$ will be the
expectation with respect to $\mathbb{P}$. We will always assume that $N$ is
large enough and hence $\varepsilon_{N}$ is small enough to make $p_{ij}\leq
1$ since $f$ is bounded.

Let $\mathbf{M}_{N}$ denote the adjacency matrix of the graph
$\mathbb{G}_{N}$, that is, the $(i,j)$-th entry is $1$ if $i$ shares an edge
with $j$, and $0$ otherwise. So $\mathbf{M}_{N}$ is a symmetric matrix, where
any entry $\mathbf{M}_{N}(i,j)$ is distributed as Bernoulli random variable
with parameter $p_{ij}$ as in (1.1) and $\\{\mathbf{M}_{N}(i,j),i\geq j\\}$ is
an independent collection. Instead of studying the adjacency matrix
$\mathbf{M}_{N}$ we will study the scaled adjacency matrix. In particular, we
do a CLT type scaling by the variance of the entries, that is, we study the
matrix

$\displaystyle\frac{1}{\sqrt{N\varepsilon_{N}(1-\varepsilon_{N})}}\mathbf{M}_{N}.$
(2.1)

The empirical measure which puts mass $1/N$ on each eigenvalue of an $N\times
N$ random matrix $\mathbf{A}_{N}$ is called the _Empirical Spectral
Distribution_ of $\mathbf{A}_{N}$, and is denoted by

$\operatorname{ESD}(\mathbf{A}_{N}):=\frac{1}{N}\sum_{i=1}^{N}\delta_{\lambda_{i}}.$
(2.2)

We are interested in studying the following object:

$\displaystyle\operatorname{ESD}\left(\frac{\mathbf{M}_{N}}{\sqrt{N\varepsilon_{N}(1-\varepsilon_{N})}}\right)=\frac{1}{N}\sum_{i=1}^{N}\delta_{\lambda_{i}},$

where $\lambda_{1},\ldots,\lambda_{N}$ are the eigenvalues of
$(N\varepsilon_{N}(1-\varepsilon_{N}))^{-1/2}\mathbf{M}_{N}$.

We are interested in the weak convergence (in probability) of the above
measure and the limiting measure is called the _Limiting Spectral
Distribution_ (LSD). The limiting measure depends on the following two
geometric regimes in random graphs and its properties differ in the two cases:

  * •

Dense Regime: $\varepsilon_{N}\to 0$ and $N\varepsilon_{N}\to\infty$. The
connectivity regime with $N\varepsilon_{N}\gg C\log N$ falls in this regime.

  * •

Sparse Regime : $\varepsilon_{N}\to 0$ and
$N\varepsilon_{N}\to\lambda\in(0,\infty)$.

##### Dense Regime:

In literature, the dense regime is characterised by
$\varepsilon_{N}\equiv\text{constant}$ but we will not use the features of
dense graphs in this article and hence by abuse of terminology we say that a
graph is dense when it is not sparse. Let us now recall briefly what happens
in the dense regime. The following result was proved in Chakrabarty et al.
(2021) and also can be obtained from Zhu (2020).

######  Theorem 2.1 ( ESD in the dense case).

Consider the inhomogeneous ER graph with $p_{ij}$ as in (1.1) with
$\varepsilon_{N}\to 0$ and $N\varepsilon_{N}\to\infty$ . Suppose the
deterministic weights satisfy the following assumption:

Let $o_{N}$ be an uniform random variable on $[N]$ and let $W_{N}=w_{o_{N}}$.
We assume that there exists a $W$ with law $\mu_{w}$ such that

$W_{N}\xrightarrow{d}W.$

Then there exists a measure $\mu_{f}$ which is compactly supported such that

$\lim_{N\to\infty}\operatorname{ESD}\left(\frac{\mathbf{M}_{N}}{\sqrt{N\varepsilon_{N}(1-\varepsilon_{N})}}\right)=\mu_{f}\text{
weakly in probability}.$

Many interesting properties of this limiting measure are known. To define the
moments we need a quantity which is similar to the homomorphism density of
graphons. Define

$t(H_{k},f,\mu_{w}):=\int_{[0,\infty)^{k}}\prod_{\\{a,b\\}\in
E(H_{k})}f(w_{a},w_{b})\mu^{\bigotimes k}_{w}(\operatorname{d}\mathbf{w}),$
(2.3)

where $H_{k}$ is a simple graph on $k$ vertices with the edge set $E(H_{k})$,
$\mu^{\bigotimes k}_{w}(\cdot)$ is the $k$-fold product measure of
$\mu_{w}(\cdot)$, and $\mathbf{w}=(w_{1},...,w_{k})$. If we restrict the range
of $f$ to $[0,1]$ and take $\mu_{w}(\cdot)$ as the Lebesgue measure on
$[0,1]$, then this quantity is the standard graph homomorphism density (see
Lovász and Szegedy (2006)).

The rooted planar tree is a planar graph with no cycles, with one
distinguished vertex as a root, and with a choice of ordering at each vertex.
The ordering defines a way to explore the tree starting at the root. One of
the algorithms used for traversing the rooted planar trees is depth-first
search. An enumeration of the vertices of a tree is said to have depth-first
search order if it is the output of the depth-first search.

We now recall the definition of a Stieltjes transform of a measure $\mu$ on
$\mathbb{R}$. For $z\in\mathbb{C}^{+}$, where $\mathbb{C}^{+}$ is the upper
half complex plane, the Stieltjes Transform of a measure $\mu$ is given by

$\operatorname{S}_{\mu}(z)=\int_{\mathbb{R}}\frac{1}{x-z}\mu(\operatorname{d}x).$

The following proposition gives the properties of the measure $\mu_{f}$ which
appear in Theorem 2.1.

######  Proposition 2.2.

  1. (a)

_[ Moments]_ The measure $\mu_{f}$ is the unique probability measure
identified by the following moments:

$\int
x^{2k}\mu_{f}(\operatorname{d}x)=\sum_{j=1}^{C_{k}}t(T_{j}^{k+1},f,\mu_{w}),\
\int x^{2k+1}\mu_{f}(\operatorname{d}x)=0,\,\,k\geq 0,$ (2.4)

where $T_{j}^{k+1}$ is the $j^{th}$ rooted planar tree with $k+1$ vertices and
$C_{k}$ is the $k^{th}$ Catalan number.

  2. (b)

_[ Stieltjes transform]_ There exists an unique analytic function
$\mathcal{H}$ defined on $\mathbb{C}^{+}\times[0,\infty)$ such that

$\operatorname{S}_{\mu_{f}}(z)=\int_{0}^{\infty}\mathcal{H}(z,x)\mu_{w}(\operatorname{d}x),$

and $\mathcal{H}(z,x)$ satisfies the integral equation

$z\mathcal{H}(z,x)=1+\mathcal{H}(z,x)\int_{0}^{\infty}\mathcal{H}(z,y)f(x,y)\mu_{w}(\operatorname{d}y),\quad
x\geq 0.$ (2.5)

######  Example 2.3 (Rank 1).

One special case which arises in many examples of random graphs, and will be
discussed later is when $f$ has a multiplicative structure, that is,
$f(x,y)=r(x)r(y)$, where $r:[0,\infty)\to[0,\infty)$ is a bounded continuous
function. In this case, the measure

$\mu_{f}=\mu_{s}\boxtimes\mu_{r(W)}$

where $\mu_{s}$ is the standard semicircle law and $\mu_{r(W)}$ is the law of
$r(W)$ and $\boxtimes$ is the free multiplicative convolution of the two
measures. When $r$ is identically equal to $1$ then $\mu_{f}=\mu_{s}$, the
standard semicircle law. We refer to Chakrabarty et al. (2021, Theorem 1.3)
for details.

##### Sparse regime.

The seminal work of Bordenave and Lelarge (2010) characterises the limiting
spectral distribution for locally tree-like graphs. In particular, if one
takes $\mathbf{A}_{N}$ to be the scaled adjacency matrix as given in (2.1),
they show that if the sequence of random graphs $\\{\mathbb{G}_{N}\\}_{N\geq
1}$ have a weak limit $\mathbb{G}$, and for any uniformly chosen root
$o_{N}\in\mathbb{G}_{N}$, its degree sequence
$(\mathrm{deg}(\mathbb{G}_{N},o_{N}))_{N}$ is uniformly integrable, then there
exists a unique probability measure $\mu_{\lambda}$ on $\mathbb{R}$ such that
$\lim_{N\to\infty}\operatorname{ESD}(\mathbf{A}_{N})=\mu_{\lambda}$ weakly in
probability. Furthermore, it is shown that when $f\equiv 1$, the measure
$\mu_{\lambda}$ represents the expected spectral measure associated with the
root of a Galton-Watson tree with an offspring distribution of
$\mathrm{Poi}(\lambda)$ and weights $1/\sqrt{\lambda}$. This result comes from
the theory of local weak convergence, also known as Benjamini-Schramm
convergence (see van der Hofstad (2023), Benjamini and Schramm (2001)), which
is a powerful tool to study spectral measures associated with many sparse
random graph models.

In particular, consider the space $\mathbb{H}$ of holomorphic functions
$f:\mathbb{C}^{+}\to\mathbb{C}^{+}$, equipped with the topology induced by
uniform convergence on compact sets. Then, this is a complete separable
metrizable compact space. The _resolvent_ of the adjacency operator is given
as

$\displaystyle\operatorname{R}_{\mathbf{A}_{N}}(z)=(\mathbf{A}_{N}-zI)^{-1}$

for each $z\in\mathbb{C}^{+}$. The map
$z\mapsto\operatorname{R}_{\mathbf{A}_{N}}(z)(i,i)$ is in $\mathbb{H}$, and
the Stieltjes transform of $\operatorname{ESD}(\mathbf{A}_{N})$ is given by
$\operatorname{tr}\operatorname{R}_{\mathbf{A}_{N}}(z)$, where
$\operatorname{tr}=N^{-1}\operatorname{Tr}$ denotes the normalised trace
operator. Let $\mathcal{G}^{*}$ denote the set of rooted isomorphism classes
of rooted connected locally finite graphs. Assume that the random graph
sequence $(\mathbb{G}_{N})_{N}$ has the random local limit
$\mathbb{G}\in\mathcal{G^{*}}$, and further that $\mathbb{G}$ is a _Galton
Watson Tree_ with degree distribution $F_{*}$, that is, a rooted random tree
obtained from a Galton-Watson process with root having offspring distribution
$F_{*}$ and all children having a distribution $F$ (which may or may not be
the same as $F_{*}$).

Let $\operatorname{S}_{\mathbf{A}_{N}}(z)$ denote the Stieltjes transform of
the empirical measure $\operatorname{ESD}(\mathbf{A}_{N})$. It was shown in
Bordenave and Lelarge (2010, Theorem 2) that there exists a unique probability
measure $Q$ on $\mathbb{H}$, such that for each $z\in\mathbb{C}^{+}$

$Y(z)\overset{d}{=}\left(z+\sum_{i=1}^{P}Y_{i}(z)\right)^{-1}$

where $P$ has distribution $F$ and $Y,\\{Y_{i}\\}_{i\geq 1}$ are iid with law
$Q$ and independent of $P$. Moreover

$\displaystyle\lim_{N\to\infty}\operatorname{S}_{\mathbf{A}_{N}}(z)=\mbox{${\mathbb{E}}$}X(z)\text{
in $L^{1}$},$

where $X(z)$ is such that:

$X(z)\overset{d}{=}-\left(z+\sum_{i=1}^{P_{*}}Y_{i}(z)\right)^{-1},$ (2.6)

where $\\{Y_{i}\\}_{i\geq 1}$ are i.i.d. copies with law $Q$, and $P_{*}$ is a
random variable independent of $\\{Y_{i}\\}_{i\geq 1}$ having distribution
$F_{*}$.

In Bordenave and Lelarge (2010, Example 2), we see that the sparse Erdős-Rényi
random graph with $p=\frac{\lambda}{N}$ falls in their setup, and in
particular, $P$ is distributed as $\mathrm{Poi}(\lambda)$. For a general $f$,
Bordenave and Lelarge (2010, Theorem 1) still guarantees the existence of
$\mu_{\lambda}$, since the graphs we will consider will have a local weak
limit known as the _multitype branching process_ (see (van der Hofstad, 2023,
Chapter 3) for more details). As $f$ is bounded, we get that the degree
sequence will still remain uniformly integrable. As mentioned before we will
not follow this well-known route of local weak convergence. Instead, we show
the above convergence through albeit classical methods. We now introduce the
conditions under which we will work. We will have the following sparsity
assumption on $\varepsilon_{N}$ and a regularity assumption on the function
$f$ and the weights:

  1. A.1

Connectivity function: Let $f:[0,\infty)^{2}\to[0,\infty)$ be a bounded,
continuous function, with $|f|\leq C_{f}\in(0,\infty)$,

  2. A.2

Sparsity assumption : $N\varepsilon_{N}\to\lambda\in(0,\infty)$,

  3. A.3

Assumption on weights: Let $o_{N}$ be an uniform random variable on $[N]$ and
let $W_{N}=w_{o_{N}}$. We assume that there exists a $W$ with law $\mu_{w}$
such that

$W_{N}\xrightarrow{d}W.$

We make some preliminary remarks about the assumptions. Since $f$ is bounded,
we can easily see that $f$ is $\mu_{w}-$ integrable. In the sparse setting, in
most important examples, the graph is locally tree-like and this can be seen
from the theory of local weak convergence.

Note that the limit $\lambda\to\infty$ recovers the dense regime. By this
choice, we can see that $1-\varepsilon_{N}\approx 1$ as N becomes very large,
and $N\varepsilon_{N}(1-\varepsilon_{N})\to\lambda$. Thus, our matrix of
interest is a scaled adjacency matrix now defined as follows:

$\mathbf{A}_{N}=\frac{1}{\sqrt{\lambda}}\mathbf{M}_{N}.$ (2.7)

###  2.2. Main Results

In this subsection, we state the main results of this article. As mentioned
before in the introduction, we would like to understand first the limiting
empirical distribution of the sparse inhomogeneous Erdős Rényi (IER) Random
Graph and also study the behaviour of the measure when the sparsity parameter
increases. Recall that the adjacency matrix is defined in (2.7) and the
empirical spectral distribution is denoted by
$\operatorname{ESD}(\mathbf{A}_{N})$ (see (2.2)). In what follows, we will see
that

$\displaystyle\lim_{N\to\infty}\operatorname{ESD}(\mathbf{A}_{N})=\mu_{\lambda}\text{
weakly in probability}$ (2.8)

and $\mu_{\lambda}\Rightarrow\mu_{f}$ where $\mu_{f}$ is as in Theorem 2.1.
For the homogeneous case, where $f\equiv 1$, we get the final limit as the
classical Wigner’s semicircular law, that is, $\mu_{f}=\mu_{sc}$. These
iterated limits were studied in Jung and Lee (2018). An interesting open
question is how close is $\mu_{\lambda}$ to $\mu_{f}$. Although we do not
manage to give an explicit estimate, through the moment method we show that it
is very close and the structure of the moments of $\mu_{f}$ is hidden inside
the structure of moments of $\mu_{\lambda}$. This will be our first result. To
describe the moments we need to introduce some notation.

#### Method of moments: Combinatorial Approach

We first define the _Special Symmetric Partitions_ which was introduced in
Bose et al. (2022). Let $\mathcal{P}(k)$ denote the set of partitions of $k$
and $\mathcal{P}_{2}(k)$ be the set of pair partitions where each block has
size $2$. Let $NC(k)$ be the set of non-crossing partitions of $[k]$ and
$NC_{2}(k)$ be the set of non-crossing pair partitions of $[k]$. Note that
$|NC_{2}(2k)|=\frac{1}{1+k}{2k\choose k}$ and these are known as the Catalan
numbers and represent the even moments of the semicircle distribution.

##### Partition terminology.

Let $\pi$ be a partition of a tuple $[k]$. Let $\pi$ consist of disjoint
blocks $V_{1},V_{2},\ldots,V_{m}$, for some $1\leq m\leq k$. We arrange the
blocks in the ascending order of their smallest element. For any block
$V_{i}$, a _sub-block_ is defined to be a subset of _consecutive integers_ in
the block. Two elements $j$ and $k$ in a block $V_{i}$ are said to be
_successive_ if for all $a$ between $j$ and $k$, $a\notin V_{i}$.

######  Definition 2.1 (Special Symmetric Partition).

A partition $\pi$ of a tuple $[k]=\\{1,2,...,k\\}$ is said to be a Special
Symmetric partition if it satisfies the following:

  * •

All blocks of $\pi$ are of even size.

  * •

Let $V\in\pi$ be any arbitrary block, and let $a,b\in V$ be two successive
elements in $V$ with $b>a$. Then, either of the following is true:

    * 1.

$b=a+1$, or

    * 2.

between $a$ and $b$ there are sub-blocks of even size.
In other words, there exist elements
$\\{a_{i_{1}},a_{i_{1}+1},\ldots,a_{i_{1}+k_{1}}\\}\in V_{1}$,
$\\{a_{i_{2}},\ldots,a_{i_{2}+k_{2}}\\}\in V_{2}$,
$\ldots,\\{a_{i_{l}},\ldots,a_{i_{l}+k_{l}}\\}\in V_{l}$, with $a=a_{i_{1}}-1$
and $b={a_{i_{l}+k_{l}}}+1$, such that $k_{1},k_{2},\ldots,k_{l}$ are even.

We denote the class of Special Symmetric partitions as $SS(k)$. Note that for
$k$ odd, $SS(k)=\emptyset$. For example, take
$\pi=\\{\\{1,4,5,8\\},\\{2,3,6,7\\},\\{9,10\\}\\}\in SS(10)$. Note here that
between $4$ and $5$ in the first block, there are no elements from the other
blocks, and between $5$ and $8$, there is the sub-block $\\{6,7\\}$ that is of
even size.

In Bose et al. (2022) a more elaborate definition was given and this is useful
in computations. Later, it was shown by (Pernici, 2021, Section 3) that the
definition in Bose et al. (2022) is equivalent to the above one. In Pernici
(2021), the set $SS(2k)$ is denoted by $P_{2}^{(2)}(k)$, a special subclass of
$k$-divisible partitions.

######  Remark 2.2.

We note down some important properties of $SS(k)$:

  * 1.

If $k$ is even, then

$\\{\pi\in SS(k):|\pi|=k/2\\}=\\{\pi\in NC_{2}(k)\\}.$

  * 2.

$SS(2k)=NC(2k)$ for $1\leq k\leq 3$. When $k\geq 4$, there are partitions
$\pi\in SS(2k)$ that are either crossing or non-paired. For example, for
$k=8$, $\\{\\{1,2,5,6\\},\\{3,4,7,8\\}\\}$ is a Special Symmetric partition.
In particular, crossings start appearing when there are at least two or more
blocks in a partition having 4 or more elements.

  * 3.

The set of Special Symmetric partitions are in one-to-one correspondence with
coloured rooted trees (see (Bose et al., 2022, Lemma 5.1)) and these trees
appeared first in the analysis in the works of Bauer and Golinelli (2001).

Any partition $\pi\in\mathcal{P}(k)$ can be realized as a permutation of
$[k]$, that is, a mapping from $[k]\to[k]$. Let $S_{k}$ denote the set of
permutations on $k$ elements. Let $\gamma=(1,2,\ldots,k)\in S_{k}$ be the
shift by $1$ modulo $k$. We will be interested in the compositions of the two
permutations $\gamma$ and $\pi$, denoted by $\gamma\pi$, and this will be seen
below as a partition.

######  Definition 2.3 (Graph associated to a partition).

For a fixed $k\geq 1$, let $\gamma$ denote the cyclic permutation
$(1,2,\ldots,k)$. For a partition $\pi$, we define
$G_{\gamma\pi}=(V_{\gamma\pi},E_{\gamma\pi})$ as a rooted, labelled graph
associated with any partition $\pi$ of $[k]$, constructed as follows.

  * •

Initially consider the vertex set $V_{\gamma\pi}=[k]$ and perform a closed
walk on $[k]$ as $1\to 2\to 3\to\cdots\to k\to 1$ and with each step of the
walk, add an edge.

  * •

Evaluate $\gamma\pi$, which will be of the form
$\gamma\pi=\\{V_{1},V_{2},\ldots,V_{m}\\}$ for some $m\geq 1$ where
$\\{V_{i}\\}_{i}$ are disjoint blocks. Then, collapse vertices in
$V_{\gamma\pi}$ to a single vertex if they belong to the same block in
$\gamma\pi$, and collapse the corresponding edges. Thus,
$V_{\gamma\pi}=\\{V_{1},\ldots,V_{m}\\}$.

  * •

Finally root and label the graph as follows.

    * –

_Root:_ We always assume that the first element of the closed walk (in this
case ‘1’) is in $V_{1}$, and we fix the block $V_{1}$ as the root.

    * –

_Label:_ Each vertex $V_{i}$ gets labelled with the elements belonging to the
corresponding block in $\gamma\pi$.

######  Remark 2.4.

While $\pi$ is a _partition_ and $\gamma$ is a _permutation_ , we do a
composition in the permutation sense. We read the partition $\pi$ as a
permutation, compose it with the permutation $\gamma$, and finally read
$\gamma\pi$ as a partition. As an example, consider
$\pi=\\{\\{1,2\\},\\{3,4\\}\\}$ and $\gamma=(1,2,3,4)$. To compute
$\gamma\pi$, we read $\pi$ as $(1,2)(3,4)$, and compute
$\gamma\pi=(1,3)(2)(4)$. We finally read $\gamma\pi$ as
$\\{\\{1,3\\},\\{2\\},\\{4\\}\\}$.

######  Example 2.5.

Consider for example partitions of $k=6$ and reading the partitions as
permutations and evaluating their composition with $\gamma$ gives us:

  1. (1)

$\pi_{1}=\\{\\{1,2,5,6\\},\\{3,4\\}\\},$

  2. (2)

$\pi_{2}=\\{\\{1,2,3,4\\},\\{5,6\\}\\},$

  3. (3)

$\pi_{3}=\\{\\{1,6\\},\\{2,3,4,5\\}\\}.$

  1. (1)

$\gamma\pi_{1}=\\{\\{1,3,5\\},\\{2,6\\},\\{4\\}\\}$,

  2. (2)

$\gamma\pi_{2}=\\{\\{1,3,5\\},\\{2,4\\},\\{6\\}\\}$,

  3. (3)

$\gamma\pi_{3}=\\{\\{1\\},\\{2,4,6\\},\\{3,5\\}\\}$.

The corresponding graphs $G_{\gamma\pi_{1}},G_{\gamma\pi_{2}}$ and
$G_{\gamma\pi_{3}}$ are as follows:

2,61,3,541,3,52,462,4,613,5

One can see that structurally the three graphs are the same. However, if we
root them on $V_{1}$, then the first two graphs are different from the third.
Further, if we label the vertices as shown, all three graphs become distinct.

######  Example 2.6.

Here, we illustrate the type of graph structures that can occur for $\pi\in
SS(k)$. Consider $k=8$, and the following three partitions.

  1. (1)

$\pi_{1}=\\{\\{1,2,3,4\\},\\{5,6,7,8\\}\\}$.

  2. (2)

$\pi_{2}=\\{\\{1,4,5,8\\},\\{2,3,6,7\\}\\}$.

  3. (3)

$\pi_{3}=\\{\\{1,2,4,5\\},\\{3,6,7,8\\}\\}$.

  1. (1)

$\gamma\pi_{1}=\\{\\{1,3,5,7\\},\\{2,4\\},\\{6,8\\}\\}$,

  2. (2)

$\gamma\pi_{2}=\\{\\{(1,5\\},\\{2,4,6,8\\},\\{3,7\\}\\}$,

  3. (3)

$\gamma\pi_{3}=\\{\\{1,3,7\\},\\{2,5\\},\\{4,6,8\\}\\}$.

Then, $\pi_{1},\pi_{2}\in SS(8)$ but $\pi_{3}\notin SS(8)$. Moreover,
$\pi_{1}$ is non-crossing whereas $\pi_{2}$ has 2 crossings. The corresponding
graphs are as below.

2,41,3,5,76,82,4,6,81,53,74,6,81,3,72,5

The following result is the first main result of the article. This is an
extension of the results obtained recently in Bose et al. (2022) and the
homogeneous case obtained in Jung and Lee (2018).

######  Theorem 2.7 (Identification of moments).

(a) Let $\mathbf{A}_{N}$ be the adjacency matrix of the sparse IER random
graph as defined in (2.7) satisfying assumptions A.1–A.3. Then there exists a
deterministic measure $\mu_{\lambda}$ such that

$\lim_{N\to\infty}\operatorname{ESD}(\mathbf{A}_{N})=\mu_{\lambda}\text{
weakly in probability}.$

Moreover, $\mu_{\lambda}$ is uniquely determined by its moments, which are
given as follows:

$m_{k}(\mu_{\lambda})=\int
x^{k}\mu_{\lambda}(\operatorname{d}x)=\begin{cases}0,&\text{ when $k$ is
odd,}\\\ \sum\limits_{l=2}^{k/2+1}\sum\limits_{\begin{subarray}{c}\pi\in
SS(k):\\\ |\gamma\pi|=l\end{subarray}}\lambda^{l-1-\frac{k}{2}}\hskip
2.84544ptt(G_{\gamma\pi},f,\mu_{w}),&\text{ when $k$ is even},\end{cases}$
(2.9)

where $SS(k)$ is the set of all Special Symmetric partitions of $[k]$ as
defined in Definition 2.1, $G_{\gamma\pi}$ is the graph associated to a
partition $\pi$ as defined in Definition 2.3, and $t$ is the homomorphism
density as in (2.3).

(b) As $\lambda\to\infty$,

$\mu_{\lambda}\Rightarrow\mu_{f},$

where $\mu_{f}$ is the measure described in Theorem 2.1.

######  Remark 2.8.

Note that limiting second moment is given by
$m_{2}=t(G_{\gamma\pi},f,\mu_{w})$ where $\pi=\\{1,2\\}$ and
$\gamma\pi=\\{\\{1\\},\\{2\\}\\}$. Hence $G_{\gamma\pi}$ is the graph with $2$
vertices and $1$ edge. Therefore

$m_{2}(\mu_{\lambda})=\int_{(0,\infty]^{2}}f(x,y)\mu_{w}(\operatorname{d}x)\mu_{w}(\operatorname{d}y).$

#### Stieltjes transform: Analytic approach.

It is well-known that $\mu_{\lambda}$ can be characterised by its Stieltjes
transform, which, in turn, can be characterised by a random recursive
equation. Local weak convergence is a powerful tool for studying the Stieltjes
transform of spectral measures associated with sparse random graphs. However,
it becomes challenging to provide accurate estimates on the Stieltjes
transform to study local laws and extreme values. Therefore, we present an
alternative approach to studying the Stieltjes transform of the spectral
measure of IER graphs. The ideas used here originate from the works of
Khorunzhy et al. (2004).

We denote the upper half complex plane by

$\mathbb{C}^{+}=\\{z\in\mathbb{C}:\,z=\zeta+\iota\eta,\,\eta>0\\}.$

For an analytic approach to the problem, we analyse the resolvent of this
matrix, defined as

$\operatorname{R}_{\mathbf{A}_{N}}(z):=(\mathbf{A}_{N}-zI)^{-1},\,z\in\mathbb{C}^{+}.$

The Stieltjes transform of the empirical spectral distribution of
$\mathbf{A}_{N}$ is given by

$\operatorname{S}_{\mathbf{A}_{N}}(z)=\int_{\mathbb{R}}\frac{1}{x-z}\operatorname{ESD}(\mathbf{A}_{N})(\operatorname{d}x)=\operatorname{tr}(\operatorname{R}_{\mathbf{A}_{N}}(z)),$

where $\operatorname{tr}$ denotes the normalized trace. To get more refined
estimates we need an additional assumption on the connectivity function:

  1. A.4

$f:[0,\infty)^{2}\to[0,\infty)$ is symmetric and bounded by a constant
$C_{f}$. Moreover, $f$ is Lipschitz in one coordinate, that is, for all
$x_{1},x_{2},y\in[0,\infty)$,

$|f(x_{1},y)-f(x_{2},y)|\leq C_{L}|x_{1}-x_{2}|$

where $C_{L}$ is the Lipschitz constant for $f$.

To state the result we will need a Banach space of analytic functions.
Consider the space $\operatorname{\mathcal{B}}$ defined by

$\operatorname{\mathcal{B}}=\left\\{\phi:[0,\infty)\times[0,\infty)\to\mathbb{C}\hskip
5.69046pt\text{analytic }\middle|\hskip 5.69046pt\sup_{x,y\geq
0}\frac{|\phi(x,y)|}{\sqrt{1+y}}<\infty\right\\}$ (2.10)

and take the norm

$\|\phi\|_{\operatorname{\mathcal{B}}}=\sup_{x,y\geq
0}\frac{|\phi(x,y)|}{\sqrt{1+y}}.$

Then, $(\operatorname{\mathcal{B}},\|\cdot\|_{\operatorname{\mathcal{B}}})$ is
a Banach space. We defer the proof of this in Proposition 5.1 in the appendix.

Consider the function $G_{N}:[0,\infty)\times\mathbb{C}^{+}$ given by

$\displaystyle
G_{N}(u,z):=\frac{1}{N}\sum_{i=1}^{N}\mathrm{e}^{\dot{\iota}ur_{ii}^{N}(z)}$
(2.11)

where $r_{ii}^{N}(z)=\operatorname{R}_{\mathbf{A}_{N}}(z)(i,i)$, the
$i^{\text{th}}$ diagonal element of the resolvent of $\mathbf{A}_{N}$. It
turns out that

$\displaystyle\left.\frac{\partial G_{N}(u,z)}{\partial
u}\right|_{u=0}=\operatorname{S}_{\mathbf{A}_{N}}(z)$

and hence one can derive a form of the limiting Stieltjes transform.

######  Theorem 2.9 (Analytic functional of the resolvent).

Let $\mathbf{A}_{N}$ be the adjacency of the IER random graph as defined in
(2.7) and satisfying assumptions (A.2)–(A.4). Further, consider $G_{N}$ as
defined in (2.11). Define the function $d_{f}(x)$ as

$d_{f}(y)=\int_{0}^{\infty}f(x,y)\mu_{w}(\operatorname{d}x).$ (2.12)

Then, for $z\in\mathbb{C}^{+}$ there exists a function
$\phi^{*}(x,u):=\phi^{*}_{z}(x,u)\in\operatorname{\mathcal{B}}$ such that for
each $z\in\mathbb{C}^{+}$ and uniformly in $u\in(0,1]$ we have

$\lim_{N\to\infty}\mbox{${\mathbb{E}}$}[G_{N}(u,z)]=1-\sqrt{u}\int_{0}^{\infty}\mathrm{e}^{-\lambda
d_{f}(y)}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{\dot{\iota}vz}\mathrm{e}^{\lambda\phi^{*}(y,v/\lambda)}\operatorname{d}v\hskip
5.69046pt\mu_{w}(\operatorname{d}y)$ (2.13)

and

$\operatorname{Var}[G_{N}(u,z)]\to 0.$

Here, $\phi^{*}:=\phi^{*}_{z}$ is a unique analytic solution (in the space
$\operatorname{\mathcal{B}}$) for the fixed point equation:

$\begin{split}\phi^{*}(x,u)&=F_{z}(\phi^{*})(x,u)\\\
&=d_{f}(x)-\int_{0}^{\infty}f(x,y)\mathrm{e}^{-\lambda
d_{f}(y)}\left(\sqrt{u}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{\dot{\iota}vz}\mathrm{e}^{\lambda\phi^{*}\left(y,\frac{v}{\lambda}\right)}\operatorname{d}{v}\right)\mu_{w}(\operatorname{d}{y}),\end{split}$
(2.14)

where $\operatorname{J_{1}}$ is the Bessel function of the first order of the
first kind defined as

$\operatorname{J_{1}}(x)=\frac{x}{2}\sum_{k=0}^{\infty}\frac{(-1)^{k}(x^{2}/4)^{k}}{k!(k+1)!}.$
(2.15)

Observe that there is a slight difference in the right-hand sides of (2.13)
and (2.14) but in the case $f=1$ both are the same. The next corollary
describes the convergence of the Stieltjes transform.

######  Corollary 2.10 (Identification of the Stieltjes Transform).

Under the assumptions of the above theorem, we have that any
$z\in\mathbb{C}^{+}$,

$\operatorname{S}_{\mathbf{A}_{N}}(z)\to\operatorname{S}_{\mu_{\lambda}}(z)\text{
in probability},$

where $\mu_{\lambda}$ is as in Theorem 2.7. The
$\operatorname{S}_{\mu_{\lambda}}(\cdot)$ satisfies the following equation:

$\operatorname{S}_{\mu_{\lambda}}(z)=\dot{\iota}\int_{0}^{\infty}\mathrm{e}^{-\lambda
d_{f}(y)}\int_{0}^{\infty}\mathrm{e}^{\dot{\iota}vz}\mathrm{e}^{\lambda\phi_{z}^{*}(y,\frac{v}{\lambda})}\operatorname{d}v\hskip
5.69046pt\mu_{w}(\operatorname{d}y),\,\,z\in\mathbb{C}^{+}.$ (2.16)

To recover the dense regime, we study the asymptotic $\lambda\to\infty$ as in
the next corollary.

######  Corollary 2.11 (Stieltjes Transform as $\lambda\to\infty$).

For $\lambda\to\infty$, we have that

$\displaystyle\lim_{\lambda\to\infty}\operatorname{S}_{\mu_{\lambda}}(z)=\operatorname{S}_{\mu_{f}}(z)$
(2.17)

for each $z\in\mathbb{C}^{+}$, where $\operatorname{S}_{\mu_{f}}(z)$ satisfies
an integral equation given by

$\displaystyle\operatorname{S}_{\mu_{f}}(z):=\int_{0}^{\infty}\mathcal{H}(z,x)\mu_{w}(\operatorname{d}{x}),$
(2.18)

where $\mathcal{H}(z,x)$ satisfies the $f$ dependent fixed point equation
(2.5).

######  Remark 2.12 ($\phi^{*}_{z}$ and the Stieltjes Transform in the
homogeneous setting).

In the case when $f\equiv 1$, we recover the homogeneous setting. We know
$\phi^{*}_{z}$ satisfies the fixed point equation (2.14). If we substitute
$f=1$ in (2.14) we get

$\phi^{*}(x,u)=1-\sqrt{u}\int_{0}^{\infty}\mathrm{e}^{-\lambda}\left(\sqrt{u}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{\dot{\iota}vz}\mathrm{e}^{\lambda\phi^{*}\left(y,\frac{v}{\lambda}\right)}\operatorname{d}{v}\right)\mu_{w}(\operatorname{d}y).$

We see that the right-hand side has no dependency on the parameter $x$, and
so, we have a unique analytical functional
$\widetilde{\phi^{*}}(u)=\phi^{*}(x,u)$ that satisfies the fixed point
equation

$\widetilde{\phi^{*}}(u)=1-\mathrm{e}^{-\lambda}\sqrt{u}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{\dot{\iota}vz}\mathrm{e}^{\lambda\widetilde{\phi^{*}}(v/\lambda)}\operatorname{d}{v}.$
(2.19)

This matches the result of Khorunzhy et al. (2004).

From Example 2 of Bordenave and Lelarge (2010), we have that
$\widetilde{\phi^{*}_{z}}$ has the form
$\widetilde{\phi^{*}_{z}}(u)=\mbox{${\mathbb{E}}$}[\mathrm{e}^{\dot{\iota}uX(z)}]$
for each $z\in\mathbb{C}^{+}$, where $X(z)$ has the law $Q$ as described in
(2.6). So,

$\displaystyle\operatorname{S}_{\mu_{\lambda}}(z)=\dot{\iota}\int_{0}^{\infty}\mathrm{e}^{\dot{\iota}vz}e^{-\lambda+\lambda\mbox{${\mathbb{E}}$}\left[\mathrm{e}^{\dot{\iota}\frac{v}{\lambda}X(z)}\right])}\operatorname{d}{v}=\dot{\iota}\int_{0}^{\infty}\mathrm{e}^{\dot{\iota}vz}\varphi_{P}\left(\mbox{${\mathbb{E}}$}\left[\mathrm{e}^{\dot{\iota}\frac{v}{\lambda}X(z)}\right]\right)\operatorname{d}v\hskip
5.69046pt,\,\,z\in\mathbb{C}^{+},$

where

$\displaystyle\varphi_{P}(z)=\mbox{${\mathbb{E}}$}[z^{P}]=\mathrm{e}^{\lambda(z-1)},\,\,,P\sim\mathrm{Poi}(\lambda).$

###  2.3. Examples

We now list out a few examples of the model that can be approached by our
methods.

##### Example 1: Homogeneous Erdős-Rényi Random Graph.

When we have $f\equiv 1$, the model reduces to the standard homogeneous Erdős-
Rényi graph with edge probability $p=\lambda/N$. As discussed, in this case
the moments of $\mu_{\lambda}$ can be computed. In particular, we have
$t(G_{\gamma\pi},f,\mu_{w})=1$ for all $\pi$. Hence we have

$m_{2k}(\mu_{\lambda})=\sum_{l=1}^{k}\lambda^{l-k}|\\{\pi\in
SS(2k):|\pi|=l\\}|=|NC_{2}(2k)|+\sum_{l=1}^{k-1}\lambda^{l-k}|\\{\pi\in
SS(2k):|\pi|=l\\}|$

Since the (even) moments of the semicircle law are given by the Catalan
numbers, it is immediate that

$\lim_{\lambda\to\infty}m_{2k}(\mu_{\lambda})=m_{2k}(\mu_{s}).$

Hence Theorem 2.7(b) is true in this special case. It is known that
$\mu_{\lambda}$ has an absolutely continuous spectrum when $\lambda>1$ (see
Bordenave et al. (2017), Arras and Bordenave (2021)). In this case, the
Stieltjes transform is given by

$\operatorname{S}_{\mu_{\lambda}}(z)=-\dot{\iota}\int_{0}^{\infty}\mathrm{e}^{\dot{\iota}vz}\mathrm{e}^{-\lambda+\lambda\widetilde{\phi^{*}}(v/\lambda)}\operatorname{d}{v},$

and $\widetilde{\phi^{*}}(v/\lambda)$ satisfies the equation (2.19). What is
interesting and cannot be immediately derived from our results is the rate of
convergence of the measure $\mu_{\lambda}$ to $\mu_{s}$ as $\lambda$ becomes
large. In the simulation below we consider the $\lambda=10$ and the simulation
already suggests the appearance of semicircle law. We believe the
representation above of the Stieltjes transform as in Corollary 2.10 can be
used to prove the rate of convergence as done in the classical Wigner case in
Bai (2008).

(a) $\lambda=5$.

(b) $\lambda=10$.

Figure 1. The homogeneous Erdős-Rényi Random Graph on 10,000 vertices.

##### Example 2: Chung-Lu Random Graph:

Let $(d_{i})_{i\in[n]}$ be a graphical sequence and denote by
$m_{1}=\sum_{i}d_{i}$ and $m_{\infty}=\max_{i}d_{i}$, the total and the
maximum degree, respectively. Let $f$ be defined on $[0,1]^{2}$ as

$f(x,y)=xy\wedge 1$

and

$w_{i}=\frac{d_{i}}{m_{\infty}},\hskip
8.5359pt\varepsilon_{N}=\frac{m_{\infty}^{2}}{m_{1}}.$

We can choose an appropriate degree sequence $(d_{i})_{i\geq 1}$ such that
$m_{\infty}=\mbox{${\mathrm{o}}$}(\sqrt{m_{1}})$ and
$N\varepsilon_{N}\to\lambda$. The connection probabilities will be given by

$p_{ij}^{\mathrm{cl}}=\varepsilon_{N}\left(\frac{d_{i}d_{j}}{m_{\infty}^{2}}\wedge
1\right)=\frac{d_{i}d_{j}}{m_{1}}.$

Let $o_{N}$ be a uniformly chosen vertex and $d_{o_{N}}$ be the degree of this
vertex. We assume that

$\frac{d_{o_{N}}}{m_{\infty}}\overset{d}{\to}W$

_where $W$ has law $\mu_{w}$ which is compactly supported._ Then the
conditions of Theorem 2.7 are satisfied. Hence there exists a limiting
spectral distribution which we call $\mu_{CL,\lambda}$ and the even moments
can identified in the following way.

Let $SS_{\ell}(2k)$ be the set of Special Symmetric partitions with $l$
blocks. Then,

$\displaystyle\int_{\mathbb{R}}x^{2k}\mu_{CL,\lambda}(\operatorname{d}{x})$
$\displaystyle=\sum_{\ell=1}^{k}\sum_{\pi\in
SS_{\ell}(2k)}\lambda^{\ell-k}t(G_{\gamma\pi},f,\mu_{w})$
$\displaystyle=\sum_{\ell=1}^{2k}\sum_{\pi\in
SS_{\ell}(2k)}\lambda^{\ell-k}\prod_{j=1}^{\\#(\gamma\pi)}\int_{\mathbb{R}}x^{b_{j}(\gamma\pi)}\mu_{w}(\operatorname{d}{x}),$

where $b_{1}(\sigma),\cdots,b_{\\#\sigma}(\sigma)$ denotes the size of the
blocks of a partition $\sigma$. For $\sigma\in NC_{2}(k)$, its _Kreweras
complement_ $K(\sigma)$ is the maximal non-crossing partition $\bar{\sigma}$
of $\\{\bar{1},\ldots,\bar{k}\\}$, such that $\sigma\cup\bar{\sigma}$ is a
non-crossing partition of $\\{\bar{1},1,\ldots,\bar{k},k\\}$. For example,

$\displaystyle K\left(\\{\\{1,2\\},\\{3,4\\},\\{5,6\\}\\}\right)$
$\displaystyle=$ $\displaystyle\\{\\{1\\},\\{2,4,6\\},\\{3\\},\\{5\\}\\},$
$\displaystyle K\left(\\{(\\{1,2\\},\\{3,6\\},\\{4,5\\},\\{7,8\\}\\}\right)$
$\displaystyle=$
$\displaystyle\\{\\{1,3,7\\},\\{4,6\\},\\{2\\},\\{5\\},\\{8\\}\\}.$

Note that this slightly differs from the standard notation of Kreweras
complement in Nica and Speicher (2006) but for pairings, the $\pi$ and
$\pi^{-1}$ coincide. It follows easily that when $\pi\in NC_{2}(2k)$,
$\gamma\pi$ can be replaced by $K(\pi)$. The benefit of this representation is
the following. It follows from (Nica and Speicher, 2006, Page 228) that

$\int_{\mathbb{R}}x^{2k}(\mu_{w}\boxtimes\mu_{s})(\operatorname{d}{x})=\sum_{\pi\in
NC_{2}(2k)}\prod_{j=1}^{k+1}\int_{\mathbb{R}}x^{b_{j}(K(\pi))}\mu_{w}(\operatorname{d}{x}),$

where $\mu_{w}\boxtimes\mu_{s}$ is the free multiplicative convolution of the
measures $\mu_{w}$ and semicircle law $\mu_{s}$. Hence the moments of
$\mu_{CL,\lambda}$ can be written as

$\int_{\mathbb{R}}x^{2k}\mu_{CL,\lambda}(\operatorname{d}{x})=\int_{\mathbb{R}}x^{2k}(\mu_{w}\boxtimes\mu_{s})(\operatorname{d}{x})+\sum_{\ell=1}^{k-1}\sum_{\pi\in
SS_{\ell}(2k)}\lambda^{\ell-k}\prod_{j=1}^{\\#(\gamma\pi)}\int_{\mathbb{R}}x^{b_{j}(\gamma\pi)}\mu_{w}(\operatorname{d}{x}).$

This also shows that

$\lim_{\lambda\to\infty}\int_{\mathbb{R}}x^{2k}\mu_{CL,\lambda}(\operatorname{d}{x})=\int_{\mathbb{R}}x^{2k}(\mu_{w}\boxtimes\mu_{s})(\operatorname{d}{x}),$

and consequently, $\mu_{f}$ is of the form $\mu_{w}\boxtimes\mu_{s}$.

######  Remark 2.1.

We want to add a remark about heavy-tailed degrees. Our conditions are not
satisfied when the degree sequence follows a power-law distribution. In that
case, the $w_{i}$ need to be scaled differently, and the limiting $W$ will not
have a compact support. For further discussion on inhomogeneous random graphs
with heavy tails, we refer to (van der Hofstad, 2017, Chapter 6).

##### Example 3: Generalized random graph:

Again let $(d_{i})$ be as above. Let $f(x,y)=\frac{xy}{1+xy}$ and
$w_{i}=d_{1}/\sqrt{m_{1}}$. Then,

$p_{ij}^{\mathrm{grg}}=\frac{d_{i}d_{j}}{m_{1}+d_{i}d_{j}}.$

Although the above example does not directly fall in our set-up (due to lack
of $\varepsilon_{N}$), one can still derive the limiting spectral distribution
using the Chung-Lu model. We will use the following two facts. The first is
(Bai, 2008, Corollary A.41), and is also a corollary of the Hoffman-Wielandt
inequality.

######  Fact 2.2.

If $d_{L}$ denotes the Lévy distance between two probability measures, then
for $N\times N$ symmetric matrices $A$ and $B$,

$d_{L}^{3}\left(\operatorname{ESD}(A),\operatorname{ESD}(B)\right)\leq\frac{1}{N}\operatorname{Tr}\left((A-B)^{2}\right)\,.$

The following is a fact about the coupling of two Bernoulli random variables
with parameters $p$ and $q$ (see van der Hofstad (2023, Theorem 2.9))

######  Fact 2.3.

There exits a coupling between $X\sim\mathrm{Ber}(p)$ and
$Y\sim\mathrm{Ber}(q)$ such that

$\mathbb{P}(X\neq Y)=|p-q|.$

Using the above coupling, we can construct a sequence of independent Bernoulli
random variables $(b_{ij})$ and $(c_{ij})$ with parameters
$p_{ij}^{\mathrm{cl}}$ and $q_{ij}^{\mathrm{grg}}$, respectively. Let
$\mathbf{M}_{N}^{\mathrm{cl}}$ and $\mathbf{M}_{N}^{\mathrm{grg}}$ be the
adjacency matrix of Chung-Lu and Generalized random graph models respectively,
with the above coupled Bernoulli random variables. Suppose the sequence
$(d_{i})_{i\in[n]}$ satisfies the assumptions described in Example 2 and let
$N\varepsilon_{N}\to\lambda$ and
$\mathbf{A}_{N}^{\mathrm{cl}}=\lambda^{-1/2}\mathbf{M}_{N}^{\mathrm{cl}}$ and
$\mathbf{A}_{N}^{\mathrm{grg}}=\lambda^{-1/2}\mathbf{M}_{N}^{\mathrm{grg}}$.
Then,

$\begin{split}\mbox{${\mathbb{E}}$}\left[d_{L}^{3}\left(\operatorname{ESD}(\mathbf{A}_{N}^{\mathrm{cl}}),\operatorname{ESD}(\mathbf{A}_{N}^{\mathrm{grg}})\right)\right]&\leq\frac{1}{N}\mbox{${\mathbb{E}}$}\left[\operatorname{Tr}(\mathbf{A}_{N}^{\mathrm{cl}}-\mathbf{A}_{N}^{\mathrm{grg}})^{2}\right]=\frac{1}{N\lambda}\mbox{${\mathbb{E}}$}\left[\sum_{i,j=1}^{N}(b_{ij}-c_{ij})^{2}\right]\\\
&=\frac{1}{\lambda
N}\mbox{${\mathbb{E}}$}\left[\sum_{i,j=1}^{N}(b_{ij}-c_{ij})^{2}\mathbbm{1}_{\\{b_{ij}\neq
c_{ij}\\}}\right]\\\ &\leq\frac{1}{\lambda
N}\sum_{i,j=1}^{N}\mathbb{P}(b_{ij}\neq c_{ij})\leq\frac{1}{\lambda
N}\sum_{i,j=1}^{N}\left|p_{ij}^{\mathrm{cl}}-p_{ij}^{\mathrm{grg}}\right|.\end{split}$

since $(b_{ij}-c_{ij})^{2}$ can be trivially bounded by 1. Using
$x-\frac{x}{1+x}\leq\frac{x^{2}}{1+x}\leq x^{2}$ for any $x>0$, we have

$p_{ij}^{\mathrm{cl}}-p_{ij}^{\mathrm{grg}}=\frac{d_{i}d_{j}}{m_{1}}-\frac{d_{i}d_{j}}{m_{1}+d_{i}d_{j}}\leq\frac{d_{i}^{2}d_{j}^{2}}{m_{1}^{2}}\leq\frac{m_{\infty}^{4}}{m_{1}^{2}}.$

Therefore

$\displaystyle\mbox{${\mathbb{E}}$}\left[d_{L}^{3}\left(\operatorname{ESD}(\mathbf{A}_{N}^{\mathrm{cl}}),\operatorname{ESD}(\mathbf{A}_{N}^{\mathrm{grg}})\right)\right]$
$\displaystyle\leq\frac{C}{\lambda
N}\sum_{i,j=1}^{N}\frac{m_{\infty}^{4}}{m_{1}^{2}}$
$\displaystyle=\frac{C}{\lambda
N}N^{2}\frac{m_{\infty}^{4}}{m_{1}^{2}}\leq\mbox{${\mathrm{O}}$}\left(\frac{Nm_{\infty}^{4}}{m_{1}^{2}}\right).$

If we consider $m_{\infty}=o(m_{1}^{1/4})$, then the empirical distribution
functions are close. Now using Markov inequality and the fact that
$\operatorname{ESD}(\mathbf{A}_{N}^{\mathrm{cl}})$ converges weakly in
probability to $\mu_{CL,\lambda}$ it follows that

$\lim_{N\to\infty}\operatorname{ESD}(\mathbf{A}_{N}^{\mathrm{grg}})=\mu_{CL,\lambda}\,\text{
weakly in probability}.$

##### Example 4: Norros-Riettu.

Let $(d_{i})_{i}$ be a given sequence and $w_{i}=\frac{d_{i}}{\sqrt{m_{1}}}$.
Take $f(x,y)=1-\exp(-xy)$. Then,

$p_{ij}^{\mathrm{nr}}=1-\exp\left(-\frac{d_{i}d_{j}}{m_{1}}\right).$

Again the form of the above connection probability does not fall directly in
our set-up but we can show that Norros-Riettu model is close to the
generalized random graph models. Let
$\mathbf{A}_{N}^{\mathrm{nr}}=\lambda^{-1/2}\mathbf{M}_{N}^{\mathrm{nr}}$
where $\mathbf{M}_{N}^{\mathrm{nr}}$ is the adjacency of the Norros-Riettu
model. Without loss of generality, we assume that we can couple Bernoulli
random variable $c_{ij}$ and $d_{ij}$ with parameters $p_{ij}^{\mathrm{grg}}$
and $p_{ij}^{\mathrm{nr}}$ using Fact 2.3. Just as in the previous example, it
follows using Fact 2.2 that

$\displaystyle\mbox{${\mathbb{E}}$}\left[d_{L}^{3}\left(\operatorname{ESD}(\mathbf{A}_{N}^{\mathrm{grg}}),\operatorname{ESD}(\mathbf{A}_{N}^{\mathrm{nr}})\right)\right]\leq\frac{1}{\lambda
N}\sum_{i,j=1}^{N}\mbox{${\mathbb{E}}$}\left[(c_{ij}-d_{ij})^{2}\operatorname{\mathbbm{1}}_{\\{c_{ij}\neq
d_{ij}\\}}\right].$

We bound trivially $(c_{ij}-d_{ij})^{2}$ by a constant $C_{1}>0$ and hence we
get that

$\displaystyle\mbox{${\mathbb{E}}$}\left[d_{L}^{3}\left(\operatorname{ESD}(\mathbf{A}_{N}^{\mathrm{grg}}),\operatorname{ESD}(\mathbf{A}_{N}^{\mathrm{nr}})\right)\right]\leq\frac{C_{1}}{\lambda
N}\sum_{i,j=1}^{N}\mathbb{P}\left(c_{ij}\neq
d_{ij}\right)=\frac{C_{1}}{\lambda
N}\sum_{i,j=1}^{N}(p_{ij}^{\mathrm{nr}}-p_{ij}^{\mathrm{grg}})$

Now,

$\begin{split}p_{ij}^{\mathrm{nr}}-p_{ij}^{\mathrm{grg}}&=\left(1-\exp\left(-\frac{d_{i}d_{j}}{m_{1}}\right)-\frac{d_{i}d_{j}}{m_{1}+d_{i}d_{j}}\right)\\\
&=\left(\frac{d_{i}^{2}d_{j}^{2}}{m_{1}^{2}+m_{1}d_{i}d_{j}}\right)+\frac{\lambda}{N}\mbox{${\mathrm{O}}$}\left(\frac{d_{i}^{2}d_{j}^{2}}{m_{1}^{2}}\right)\\\
&\leq C^{\prime}\frac{d_{i}^{2}d_{j}^{2}}{m_{1}^{2}},\end{split}$

for some constant $C^{\prime}>0$. Therefore, for some new constant
$C_{1}^{\prime}>0$,

$\displaystyle\mbox{${\mathbb{E}}$}\left[d_{L}^{3}\left(\operatorname{ESD}(\mathbf{A}_{N}^{\mathrm{grg}}),\operatorname{ESD}(\mathbf{A}_{N}^{\mathrm{nr}})\right)\right]\leq\frac{C^{\prime}_{1}}{\lambda
N}\frac{m_{2}^{2}}{m_{1}^{2}}$ (2.20)

where $m_{2}=\sum_{i=1}^{N}d_{i}^{2}$. Since $W$ has compact support, we have
that $\frac{m_{2}}{Nm_{\infty}}\to\mbox{${\mathbb{E}}$}[W^{2}]$ and
$\frac{m_{1}}{Nm_{\infty}}\to\mbox{${\mathbb{E}}$}[W]$. So
$\frac{m_{2}^{2}}{m_{1}^{2}}$ is bounded for large $N$ and hence the right
hand side of (2.20) goes to $0$. This shows that

$\lim_{N\to\infty}\operatorname{ESD}(\mathbf{A}_{N}^{\mathrm{nr}})=\mu_{CL,\lambda}\,\text{
weakly in probability}.$

Figure 2. Spectral distributions for the Chung-Lu random graph, the
generalised random graph, and the Norros-Riettu random graph on 10,000
vertices with $\\{d_{i}\\}_{i}$ uniformly generated integers in $[1,5]$

##### Example 5: Inhomogeneous Random Graphs:

Let $w_{i}=\frac{i}{N}$ and $f:[0,1]^{2}\to[0,1]$ be any continuous function.
Then,

$p_{ij}=\varepsilon_{N}f\left(\frac{i}{N},\frac{j}{N}\right).$

This is a case which falls directly into our set-up if we assume
$N\varepsilon_{N}\to\lambda$ and the measure $\mu_{w}$ is the Lebesgue
measure. The other examples considered in this section are mostly of the
rank-1 type but through this example, one can achieve limiting measures which
are of wide variety.

(a) $\lambda=5$.

(b) $\lambda=10$.

Figure 3. The Inhomogeneous Random Graph on 10,000 vertices, with the
inhomogeneity function $f(x,y)=r_{1}(x)r_{1}(y)+r_{2}(x)r_{2}(y)$, where
$r_{1}(x)=\frac{x}{1+x}$ and $r_{2}(x)=x$.

We note that in van der Hofstad (2023), inhomogeneous random graphs are
introduced in a much more abstract setting, following the works of Bollobás et
al. (2007). The connectivity function $f$ is generally continuous and also
satisfies reducibility properties. The above examples also fall under the
setup described there.

##  3\. Method of Moments: Proof of Theorem 2.7

In this section, we will prove the main result Theorem 2.7 using the method of
moments.

We begin with a small observation. Recall from Assumption A.3 that if $o_{N}$
is an uniformly chosen vertex and $W_{N}=w_{o_{N}}$ and we assume
$W_{N}\xrightarrow{\operatorname{d}}W$. This means that $W_{N}$ has a
distribution function $F_{N}(x)$ given by

$F_{N}(x)=\frac{1}{N}\sum_{i=1}^{N}\operatorname{\mathbbm{1}}_{\\{w_{i}\leq
x\\}}$

and if we denote by $F$ the distribution of $W$ then for any continuity point
$x$ of $F$ we have

$F_{N}(x)\to F(x).$

Also for any bounded continuous function $g$, we have
$\mbox{${\mathbb{E}}$}[g(W_{N})]\to\mbox{${\mathbb{E}}$}[g(W)]$. Let
$o_{1},\ldots,o_{k}$ be i.i.d. Uniform random variables on $[N]$. Let
$W_{N,i}=w_{o_{i}}$ for $i=1,\ldots,k$. Then

$\left(W_{N,1},...,W_{N,k}\right)\xrightarrow{\operatorname{d}}(W_{1},W_{2},...,W_{k})$

where $W_{1},\ldots,W_{k}$ are $k$ independent copies of the limiting variable
$W$. Hence for any bounded continuous $g$ in $k$-variables we have

$\mbox{${\mathbb{E}}$}\left[g(W_{N,1},\ldots,W_{N,k})\right]\to\mbox{${\mathbb{E}}$}\left[g(W_{1},\dots,W_{k})\right].$
(3.1)

In our model, we can allow self-loops as we are not imposing that $f(x,x)=0$
but the presence of self-loops does not affect the $\operatorname{ESD}$. The
following lemma shows that we can remove the self-loops.

######  Lemma 3.1 (Diagonal contribution).

Let $\widetilde{\mathbf{A}}_{N}$ be the matrix $\mathbf{A}_{N}$ with zero on
the diagonal, and let $d_{L}$ denote the Lévy distance. Then,

$d_{L}\left(\operatorname{ESD}(\widetilde{\mathbf{A}}_{N}),\operatorname{ESD}(\mathbf{A}_{N})\right)\xrightarrow{\mathbb{P}}0.$

In particular, if $\operatorname{ESD}(\mathbf{A}_{N})$ converges weakly in
probability to $\mu_{\lambda}$, then so will
$\operatorname{ESD}(\widetilde{\mathbf{A}}_{N})$ and visa-versa.

###### Proof.

Let $D_{N}$ denote the diagonal of $\mathbf{A}_{N}$. Then,
$D_{N}=\mathbf{A}_{N}-\tilde{\mathbf{A}}_{N}$. Using Fact 2.2 we have

$d_{L}^{3}\left(\operatorname{ESD}(\widetilde{\mathbf{A}}_{N}),\operatorname{ESD}(\mathbf{A}_{N})\right)\leq\frac{1}{N}\operatorname{Tr}(D_{N}^{2})=\frac{1}{N\lambda}\sum_{1\leq
i\leq N}a_{ii}^{2}.$

Hence we have

$\displaystyle\mbox{${\mathbb{E}}$}\left[d_{L}^{3}\left(\operatorname{ESD}(\widetilde{\mathbf{A}}_{N}),\operatorname{ESD}(\mathbf{A}_{N})\right)\right]$
$\displaystyle\leq\frac{\sqrt{\lambda}}{N^{2}}\sum_{1\leq i\leq
N}f\left(w_{i},w_{i}\right)$ $\displaystyle\leq
C_{f}\frac{\sqrt{\lambda}}{N},$

for some constant $C_{f}$, which comes from the fact that $f$ is bounded. The
result follows using Markov’s inequality. ∎

We are now ready to begin with the proofs of the main results.

###  3.1. Expected Moments

We split up the proof into three parts. To ease the notation we abbreviate the
empirical spectral distribution and its expectation as

$\mu_{N,\lambda}(\cdot)=\operatorname{ESD}(\mathbf{A}_{N})(\cdot)\qquad\text{
and
}\qquad\bar{\mu}_{N,\lambda}(\cdot)=\mbox{${\mathbb{E}}$}[\operatorname{ESD}[\mathbf{A}_{N}]](\cdot)=\frac{1}{N}\sum_{i=1}^{N}\mathbb{P}(\lambda_{i}\in\cdot).$
(3.2)

Note that $\bar{\mu}_{N,\lambda}$ is now a deterministic measure, for which we
compute the moments as

$\begin{split}\int
x^{k}\bar{\mu}_{N,\lambda}(\operatorname{d}x)&=\frac{1}{N}\sum_{i=1}^{N}\int_{\mathbb{R}}x^{k}\mathbb{P}(\lambda_{i}\in\operatorname{d}{x})=\frac{1}{N}\sum_{i=1}^{N}\mbox{${\mathbb{E}}$}[\lambda_{i}^{k}]=\mbox{${\mathbb{E}}$}\operatorname{tr}\mathbf{A}_{N}^{k},\end{split}$

where $\operatorname{tr}$ denotes the normalized trace. Using the trace
formula it follows that

$\mbox{${\mathbb{E}}$}[\operatorname{tr}\mathbf{A}_{N}^{k}]=\frac{1}{N}\mathbb{E}[\operatorname{Tr}\mathbf{A}_{N}^{k}]=\frac{1}{N\lambda^{k/2}}\sum_{1\leq
i_{1},i_{2},...,i_{k}\leq
N}\mathbb{E}[a_{i_{1}i_{2}}a_{i_{2}i_{3}}...a_{i_{k}i_{1}}],$ (3.3)

where $a_{ij}$ are entries of the adjacency matrix $\mathbf{M}$. We compute
the expected moments and demonstrate that they are finite. Subsequently, we
establish a concentration result to show that the moments of the empirical
measure converge to $m_{k}$ in probability. Next, we prove that the sequence
$m_{k}$ satisfies Carleman’s condition, thereby uniquely determining the
limiting measure.

Let $SS(k)$ be the set of Special Symmetric partitions, and
$\gamma=(1,2,\ldots,k)$ be the cyclic permutation. For the following
computations, one has to read the partition $\pi$ as a permutation, with
elements of a block in the partition set in an ascending manner in the
permutation. That is, if $\pi=\\{\\{1,2,5,6\\},\\{3,4\\}\\}$, then the
corresponding permutation is $(1,2,5,6)(3,4)$.

######  Lemma 3.1 (Expected moments).

Let $\mu_{N,\lambda}$ be the $\operatorname{ESD}$ of $\mathbf{A}_{N}$ and
$\bar{\mu}_{N,\lambda}=\mbox{${\mathbb{E}}$}\mu_{N,\lambda}$. Let $\gamma\pi$
be decomposed into blocks of the form

$\gamma\pi=\\{V_{1},V_{2},\ldots,V_{m}\\}.$

where $m=|\gamma\pi|$ be the number of blocks. Define
$\mathcal{F}_{\gamma\pi}$ as

$\mathcal{F}_{\gamma\pi}:=\\{\mathbf{i}\in\mathbb{N}^{k}\mid
i_{j}=i_{j^{\prime}}\text{ if and only if there exists }l\in[m]\text{ s.t.
}j,j^{\prime}\in V_{l}\\}.$ (3.4)

Then,

$\displaystyle\int
x^{k}\bar{\mu}_{N,\lambda}(\operatorname{d}x)=\begin{cases}\mbox{${\mathrm{O}}$}(\lambda^{k/2}N^{-1}),&\text{$k$
odd}\\\ \sum\limits_{\pi\in
SS(k)}\lambda^{|\gamma\pi|-1-k/2}\sum\limits_{\mathbf{i}\in\mathcal{F}_{\gamma\pi}}\frac{1}{N^{|\gamma\pi|}}\prod\limits_{(a,b)\in
E_{\gamma\pi}}f\left(w_{i_{a}},w_{i_{b}}\right)+\mbox{${\mathrm{O}}$}(\lambda^{k/2}N^{-1}),&\text{$k$
even}\end{cases}.$ (3.5)

######  Example 3.2.

For $k=4$, take $\pi=\\{\\{1,2\\},\\{3,4\\}\\}$. Then,
$\gamma\pi=\\{\\{1,3\\},\\{2\\},\\{4\\}\\}$. We see that tuples of the form
$(1,2,1,3)$ and $(2,3,2,4)$ belong in $\mathcal{F}_{\gamma\pi}$.

###### Proof of Lemma 3.1.

Recall from (3.3) that

$\frac{1}{N\lambda^{k/2}}\mathbb{E}[\operatorname{Tr}\mathbf{A}_{N}^{k}]=\frac{1}{N\lambda^{k/2}}\sum_{\mathbf{i}\in
N^{k}}\mathbb{E}[a_{i_{1}i_{2}}a_{i_{2}i_{3}}...a_{i_{k}i_{1}}],$

where $\mathbf{i}=(i_{1},\ldots,i_{k})$. The term
$a_{i_{1}i_{2}}a_{i_{2}i_{3}}...a_{i_{k}i_{1}}$ is associated with the closed
walk $i_{1}i_{2}\ldots i_{k}i_{1}$. Let the set of distinct vertices and edges
along a closed walk correspond to a $k$-tuple $\mathbf{i}$ be denoted by
$V(\mathbf{i})$ and $E(\mathbf{i})$, respectively. An edge that connects
vertices $i_{j}$ and $i_{j+1}$, will be denoted by $e=(i_{j},i_{j+1})$.
Without loss of generality, we assume that in $V(\mathbf{i})$ we assign the
positions where the first of distinct indices appear in $\mathbf{i}$.

For example, for the 4-tuple $\mathbf{i}=(1,2,1,3)$, we have
$V(\mathbf{i})=\\{1,2,4\\}$. So, $E(\mathbf{i})=\\{(1,2),(1,4)\\}$. Since

$a_{i_{1}i_{2}}a_{i_{2}i_{3}}...a_{i_{k}i_{1}}=1\text{ if and only
$a_{l}a_{l+1}=1$ for all $(l,l+1)\in E(\mathbf{i})$}$

we can rewrite (3.3) as

$\begin{split}\frac{1}{N\lambda^{k/2}}\mathbb{E}[\operatorname{Tr}\mathbf{A}_{N}^{k}]&=\frac{1}{N\lambda^{k/2}}\sum_{1\leq
i_{j}\leq N:j\in
V(\mathbf{i})}\left(\frac{\lambda}{N}\right)^{|E(\mathbf{i})|}\prod_{(a,b)\in
E(\mathbf{i})}f(w_{i_{a}},w_{i_{b}}).\end{split}$ (3.6)

Let $\pi$ be a partition of $[k]:=\\{1,2,\ldots,k\\}$ and
$\gamma\pi=\\{V_{1},V_{2},\ldots,V_{m}\\}$, where $m=|\gamma\pi|$. Recall the
definition of $\mathcal{F}_{\gamma\pi}$ as in (3.4) and also the graph
$G_{\gamma\pi}$ corresponding to $\gamma\pi$ as in Definition 2.3. Note that
for a fixed $\mathbf{i}\in\mathcal{F}_{\gamma\pi}$,
$V(\mathbf{i})=V_{\gamma\pi}$ and $E(\mathbf{i})=E_{\gamma\pi}$. Moreover, if
$\mathbf{i},\mathbf{i}^{\prime}\in\mathcal{F}_{\gamma\pi}$, then
$V(\mathbf{i})=V(\mathbf{i}^{\prime})$ and
$E(\mathbf{i})=E(\mathbf{i}^{\prime})$. Using this formulation, we can rewrite
our summation in (3.6) once again as

$\frac{1}{N\lambda^{k/2}}\mathbb{E}[\operatorname{Tr}\mathbf{A}_{N}^{k}]=\frac{1}{N\lambda^{k/2}}\sum_{\pi\in\mathcal{P}(k)}\sum_{\mathbf{i}\in\mathcal{F}_{\gamma\pi}}\left(\frac{\lambda}{N}\right)^{|E_{\gamma\pi}|}\prod_{(a,b)\in
E_{\gamma\pi}}f\left(w_{i_{a}},w_{i_{b}}\right).$

Since $|\gamma\pi|=|V(\mathbf{i})|$, we can multiply and divide by
$N^{|\gamma\pi|}$ to get

$\frac{1}{N\lambda^{k/2}}\mathbb{E}[\operatorname{Tr}\mathbf{A}_{N}^{k}]=\sum_{\pi\in\mathcal{P}(k)}\frac{1}{N^{|\gamma\pi|}}\sum_{\mathbf{i}\in\mathcal{F}_{\gamma\pi}}\lambda^{|E_{\gamma\pi}|-k/2}N^{|\gamma\pi|-|E_{\gamma\pi}|-1}\prod_{(a,b)\in
E_{\gamma\pi}}f\left(w_{i_{a}},w_{i_{b}}\right).$

Note that since $f$ is bounded, then the product is bounded. For a fixed $k$
and a partition $\pi$ of $[k]$, $|E_{\gamma\pi}|\leq k$. One can also see that
$|\mathcal{F}_{\gamma\pi}|\sim N^{|\gamma\pi|}$. We thus focus only on
$\lambda^{|E_{\gamma\pi}|-k/2}N^{|\gamma\pi|-|E_{\gamma\pi}|-1}$. For this to
contribute, a tuple $\mathbf{i}$ must yield a tree structure in
$G_{\gamma\pi}$, this will give us $|V(\mathbf{i})|=|E(\mathbf{i})|+1$, which
would imply $|\gamma\pi|=|E_{\gamma\pi}|+1$. In particular, all tuples
$\bf{i}\in\mathcal{F}_{\gamma\pi}$ such that $G_{\gamma\pi}$ is a coloured
rooted tree as defined in Definition 2.3 contribute to the summation.

For other graphs with $|V(\mathbf{i})|<|E(\mathbf{i})|+1$, the leading error
would be of the order $\mbox{${\mathrm{O}}$}(N^{-1})$. The leading order error
is given when $G_{\gamma\pi}$ is a $k$-cycle and hence the error is of the
order of $\lambda^{k/2}N^{-1}$. Thus our sum reduces to

$\frac{1}{N\lambda^{k/2}}\mathbb{E}[\operatorname{Tr}\mathbf{A}_{N}^{k}]=\sum_{\begin{subarray}{c}\pi\in\mathcal{P}(k):\\\
G_{\gamma\pi}\text{ is a rooted labelled
tree}\end{subarray}}\sum_{\mathbf{i}\in\mathcal{F}_{\gamma\pi}}\lambda^{|E_{\gamma\pi}|-k/2}\frac{1}{N^{|\gamma\pi|}}\prod_{(a,b)\in
E_{\gamma\pi}}f\left(w_{i_{a}},w_{i_{b}}\right)\hskip
5.69046pt+\mbox{${\mathrm{O}}$}(\lambda^{k/2}N^{-1}).$

Thus rewriting the expression with $|E_{\gamma\pi}|=|\gamma\pi|+1$ we get,

$\frac{1}{N\lambda^{k/2}}\mathbb{E}[\operatorname{Tr}\mathbf{A}_{N}^{k}]=\sum_{\begin{subarray}{c}\pi\in\mathcal{P}(k):\\\
G_{\gamma\pi}\text{ is a rooted labelled
tree}\end{subarray}}\lambda^{|\gamma\pi|+1-k/2}\sum_{\mathbf{i}\in\mathcal{F}_{\gamma\pi}}\frac{1}{N^{|\gamma\pi|}}\prod_{(a,b)\in
E_{\gamma\pi}}f\left(w_{i_{a}},w_{i_{b}}\right)\hskip
5.69046pt+\mbox{${\mathrm{O}}$}(\lambda^{k/2}N^{-1}).$ (3.7)

######  Remark 3.3.

We would like to remark here that if there exists an edge $e$, such that it is
traversed only once in the closed walk, then the graph cannot be a tree.
Consider, without loss of generality, that this edge $e$ is $(1,2)$, with
$1\in V_{1}$ and $2\in V_{2}$, as in figure 4, where
$V_{1},V_{2}\in\gamma\pi$. Here $C_{1}$ and $C_{2}$ are the remaining
components of the graph $G_{\gamma\pi}$.

$V_{2}$$V_{1}$$C_{1}$$C_{2}$ Figure 4. Graph associated to $\gamma\pi$ having
blocks $V_{1}$ and $V_{2}$ with the edge between them traversed only once.

Thus, since the closed walk $1\to 2,2\to 3,\ldots k\to 1$ has to return back
to $V_{1}$, it has to do so via $C_{1}$ since the edge $e$ cannot be traversed
again. Clearly, this will form a cycle in the graph. Thus, every edge must be
traversed at least twice.

It is well-known (see Nica and Speicher (2006)) that for $\pi\in NC_{2}(k)$ if
and only if $|\gamma\pi|=1+k/2$, but in the above setting we shall see that
other partitions will also contribute as $|\mathcal{F}_{\gamma\pi}|\sim
N^{|\gamma\pi|}$. In particular, we need to sum over only those $\pi$ that
give rise to a tree structure. We show in a series of characterizations that
the resulting partitions are $SS(k)$.

##### Characterising partitions

Recall from Definition 2.3 that to construct a graph $G_{\gamma\pi}$
associated with a partition $\pi$ of $[k]$, we need to evaluate $\gamma\pi$ to
construct the vertex set and then perform a closed walk. We prove a property
that will play a key role in characterising partitions in the proof of Theorem
2.7.

##### Property 1.

[Block characterisation] For $\pi\in\mathcal{P}(k)$ with
$\gamma\pi=\\{V_{1},\ldots,V_{l}\\}$, if $G_{\gamma\pi}$ has a tree structure,
then all elements of a block $V_{j}$, $\forall\,1\leq j\leq l$, have either
all odd elements or all even elements.

###### Proof of Property 1.

For simplicity, we show that the first block has this property. Assume that
$V_{1}$ has all odd elements except one special element $a\in[k]$. We assume
that element ‘1’ belongs to $V_{1}$.

Recall from the definition of $G_{\gamma\pi}$ that we first perform a closed
walk on $[k]$ as $1\to 2\to 3\to\ldots\to k\to 1$, and then collapse elements
of the same block of $\gamma\pi$ into a single vertex. Thus, if $a-1$ (or
$a+1$) belong to $V_{1}$, then, we get a self-loop since $a-1$ and $a$
collapse to the same vertex and the edge $a-1\to a$ (or $a\to a+1$) forms a
loop, which does not give a tree structure. Hence $a-1$ (respectively $a+1$)
is not in $V_{1}$.

Now, suppose $a-1\in V_{j}$ for some $j\neq 1$. Then, there exists a path from
$V_{1}$ to $V_{j}$ of length $t>1$, since if $t=1$, the closed walk $1\to
2\to\ldots$ would imply that $a-2\in V_{1}$, which contradicts our claim. Now,
if $t>1$, the next edge $\\{a-1\to a\\}$ from the closed walk will be from
$V_{j}$ to $V_{1}$, leading to a cycle in the graph. Thus, violating property
1 yields a graph that is not a tree.

∎

##### Property 2.

[Initial characterisation of $\pi$] If $\pi\in\mathcal{P}(k)$ then in any
block of $\pi$, no two consecutive elements can either be both odd or both
even.

###### Proof of Property 2.

Suppose $a_{1}$ and $a_{2}$ belong in the same block of $\pi$ with no elements
between them, and $a_{1}<a_{2}$, either both even or odd. Then in $\gamma\pi$,
$a_{1}$ and $a_{2}+1$ belong in the same block, which contradicts Property 1.
∎

##### Property 3.

[Diagonal terms] If $\pi$ is a contributing partition, then for any
$\mathbf{i}=(i_{1},\ldots,i_{k})$ in $\mathcal{F}_{\gamma\pi}$, each element
of $\mathbf{i}$ must be pairwise distinct, that is, $i_{1}\neq i_{2},i_{2}\neq
i_{3},\ldots,i_{k-1}\neq i_{k}$.

###### Proof of Property 3.

Suppose not, and assume $i_{a}=i_{a+1}$ for some $1\leq a\leq k-1$. Then, in
$\gamma\pi$, ‘a’ and ‘a+1’ belong to the same block. This contradicts Property
1. ∎

We now use the above properties for further characterisation of the
partitions.

######  Lemma 3.4.

Every block in $\pi$ must be of even size.

###### Proof of Lemma 3.4.

We prove this by contradiction. Consider an odd-sized block
$V=\\{l_{1},\ldots,l_{r}\\}\in\pi$ with $l_{1}<l_{2}<\cdots<l_{r}$. Assume
that $l_{1}$ is odd. By Property 2, $l_{2}$ must be even, and by continuing
the argument, we have that at every even position, the element is even, and at
odd positions, it is odd. Since $r$ is odd, and $l_{r}$ is on the
$r^{\text{th}}$ position, which is an odd position, $l_{r}$ must be odd. Then,
in $\gamma\pi$, the element $l_{r}$ will map to the element $l_{1}+1$ which is
even, which contradicts Property 1. A similar argument holds when $l_{1}$ is
taken to be even. This proves the result. ∎

######  Corollary 3.5 (Vanishing odd moments).

The odd moments vanish as $N\rightarrow\infty$.

###### Proof of Corollary 3.5.

Recall that partitions whose graphs do not yield a tree structure contribute
to the error term with leading order $\mbox{${\mathrm{O}}$}(N^{-1})$. For $k$
odd, every $\pi\in SS(k)$ must have at least one block of odd size. Therefore,
Lemma 3.4 is violated, and consequently, the odd moments asymptotically. ∎

######  Proposition 3.6.

Let $\pi\in\mathcal{P}(k)$ such that $G_{\gamma\pi}$ is a rooted labelled
tree. Then $\pi$ must satisfy the following properties.

  * •

All blocks of the partition must be of even size.

  * •

Between any two successive elements of a block, there lie sub-blocks of even
sizes.

###### Proof of Proposition 3.6.

The first condition is already proved using Lemma 3.4. For the second
condition, begin by considering a block $B$ that is of the form

$B=\\{\ldots,a_{1},a_{1}+1,\ldots,a_{1}+e,a_{2},\ldots\\}$

with $a_{1}-1\notin B$, and there doesn’t exist any element $a^{\prime}$ such
that $a_{1}+e<a^{\prime}<a_{2}$ and $a^{\prime}\in B$. The sub-block here of
interest is $\\{a_{1},a_{1}+1,\ldots,a_{1}+e\\}$. We claim that this sub-block
has an odd number of elements, or equivalently, $e$ is an even number. We can
also assume, without loss of generality, that $a_{1}$ is an odd number. As a
consequence of Property 2, $a_{2}$ must be even. If we now evaluate
$\gamma\pi$ using the above information, we have that $\gamma\pi$ contains the
following three (and possibly more) blocks.

$\begin{split}V_{1}&=\\{\ldots,a_{1},a_{1}+2,\ldots,a_{1}+e,a_{2}+1,\ldots\\},\\\
V_{2}&=\\{\ldots,a_{1}+1,a_{1}+3,\ldots,a_{1}+e-1,a_{1}+e+1,\ldots\\},\\\
V_{3}&=\\{\ldots,a_{2},\ldots\\}.\end{split}$

Thus, the graph associated with $\gamma\pi$ will be as shown in figure 5,
where $C_{1}$, $C_{2}$, and $C_{3}$ are the remaining components of the graph.

$V_{2}$$V_{1}$$V_{3}$$C_{1}$$C_{2}$$C_{3}$ Figure 5. Graph associated to
$\gamma\pi$ having blocks $V_{1}$, $V_{2}$ and $V_{3}$.

We now focus on the closed walk that occurs on the tuple $[k]$. Since this is
a closed walk, it does not matter if instead of beginning at $1$, we begin at
an arbitrary element $k_{1}\in[k]$ and perform $\\{k_{1}\to
k_{1}+1,\ldots,k\to 1,1\to 2,\ldots k_{1}-1\to k_{1}\\}$. So, we pick $a_{1}$
as the starting point and consequently, without loss of generality, we assume
the walk begins at $V_{1}$.

The walk will immediately proceed to move back and forth between $V_{1}$ and
$V_{2}$ due to the path $\\{a_{1}\to a_{1}+1,a_{1}+1\to
a_{1}+2,\ldots,a_{1}+e\to a_{1}+e+1\\}$, and will eventually end at $V_{2}$.

Now, the walk will jump from $V_{2}$ into the component $C_{2}$. On the other
hand, when the walk eventually enters $V_{3}$, it will move at least once to
$V_{1}$, due to the path $\\{a_{2}\to a_{2}+1\\}$. So, to preserve the tree
structure, the walk must first come back to $V_{2}$ and then proceed to
$V_{3}$ via $V_{1}$. Thus, there is an element $a^{\prime}$ such that
$a^{\prime}\in V_{2}$ and $a^{\prime}+1\in V_{1}$, where $a^{\prime}>a_{1}+e$
and $a^{\prime}<a_{2}$. Therefore, in $\gamma\pi$, $a_{1}+e$ maps to
$a^{\prime}+1$. This implies that $a_{1}+e$ and $a^{\prime}$ belong to the
same block in $\pi$, and thus, $a^{\prime}\in B$. This contradicts our
construction, and therefore, the walk must form a cycle from $V_{2}$ or
$C_{2}$ to either $C_{1}$, $C_{3}$ or $V_{3}$. ∎

Recall the definition of Special Symmetric Partitions as provided in
Definition 2.1, where the two properties outlined in Proposition 3.6 are the
main characteristics. As a result, we have demonstrated (3.5), leading us to
the conclusion of the proof of Lemma 3.1. ∎

We would now like to take limits in (3.5) and finally get the expression for
the moments. The following lemma is an easy consequence of Lemma 3.1 and the
fact that $|\mathcal{F}_{\gamma\pi}|\sim N^{|\gamma\pi|}$.

######  Lemma 3.7.

Let $\pi\in SS(k)$ and $\mathcal{F}_{\gamma\pi}$ be as in Lemma 3.1. Also,
$G_{\gamma\pi}=(V_{\gamma\pi},E_{\gamma\pi})$ be the graph as in Definition
2.3.

$\lim_{N\to\infty}\sum_{\mathbf{i}\in\mathcal{F}_{\gamma\pi}}\frac{1}{N^{|\gamma\pi|}}\prod_{(a,b)\in
E_{\gamma\pi}}f\left(w_{i_{a}},w_{i_{b}}\right)=\int_{[0,\infty)^{|\gamma\pi|}}\prod_{(a,b)\in
E_{\gamma\pi}}f(w_{a},w_{b})\mu_{w}^{\bigotimes|\gamma\pi|}(d\mathbf{w}).$
(3.8)

Now, going back to equation (3.7) and taking limits gives us

$\displaystyle\lim_{N\to\infty}\mbox{${\mathbb{E}}$}[\operatorname{tr}\mathbf{A}_{N}^{k}]=\begin{cases}0,&\text{$k$
odd}\\\ \sum\limits_{\begin{subarray}{c}\pi\in
SS(k)\end{subarray}}\lambda^{|\gamma\pi|-1-k/2}t(G_{\gamma\pi},f,\mu_{w}),&\text{$k$
even}\end{cases}.$ (3.9)

Now, the sum over $SS(k)$ can be further split up as the sum over $NC_{2}(k)$
and the remaining partitions. Moreover, for $\pi\in SS(k)$, we have
$|V_{\gamma\pi}|=|\gamma\pi|\in\\{2,3,\ldots,k/2+1\\}$. In particular, for
$\pi\in NC_{2}(k)$, $|\gamma\pi|=k/2+1$, and when $\pi$ is the full partition
$\\{\\{1,2,\ldots,k\\}\\}$, $|\gamma\pi|=2$. So, we can write

$\displaystyle\lim_{N\to\infty}\mbox{${\mathbb{E}}$}[\operatorname{tr}\mathbf{A}_{N}^{k}]=\begin{cases}0,&\text{$k$
odd}\\\ \sum\limits_{\pi\in
NC_{2}(k)}t(G_{\gamma\pi},f,\mu_{w})+\sum\limits_{l=2}^{k/2}\sum\limits_{\begin{subarray}{c}\pi\in
SS(k)\setminus NC_{2}(k):\\\
|\gamma\pi|=l\end{subarray}}\lambda^{l-1-k/2}t(G_{\gamma\pi},f,\mu_{w}),&\text{$k$
even}\end{cases}.$ (3.10)

###  3.2. Concentration and uniqueness.

We now show a concentration result to obtain convergence in probability.

######  Lemma 3.1 (Concentration of trace).

For all $k\geq 0$, we have that

$\operatorname{Var}\left[\operatorname{tr}\mathbf{A}_{N}^{k}\right]=\mathrm{O}((\lambda
N)^{-1}).$

###### Proof.

We shall proceed to compute the variance

$\operatorname{Var}\left[\operatorname{tr}\mathbf{A}_{N}^{k}\right].$

Let $\mathbf{i}$ and $\mathbf{i^{\prime}}$ denote the tuples

$\mathbf{i}=\\{i_{1},\ldots,i_{k}\\},\hskip
5.69046pt\mathbf{i^{\prime}}=\\{i_{k+1},\ldots,i_{2k}\\}$

and denote by $P(\mathbf{i})$ the expectation

$P(\mathbf{i})=\mbox{${\mathbb{E}}$}[a_{i_{1}i_{2}}a_{i_{2}i_{3}}\ldots
a_{i_{k}i_{1}}].$

Similarly, we have

$P(\mathbf{i^{\prime}})=\mbox{${\mathbb{E}}$}[a_{i_{k+1}i_{k+2}}a_{i_{k+2}i_{k+3}}\ldots
a_{i_{2k}i_{k+1}}].$

For the tuple $\mathbf{i}$, we can define a closed walk as in the proof of
Lemma 3.1 to get a graph $G(\mathbf{i}):=(V(\mathbf{i}),\,E(\mathbf{i}))$. In
the same spirit, one can define
$G(\mathbf{i},\mathbf{i^{\prime}})=(V(\mathbf{i},\mathbf{i^{\prime}}),E(\mathbf{i},\mathbf{i^{\prime}}))$,
with the closed walk now performed as

$1\to 2\to\ldots k\to 1,k+1\to k+2\to\ldots 2k\to k+1,$

where the jump from 1 to $k+1$ is without an edge. Then, we can define

$P(\mathbf{i},\mathbf{i^{\prime}})=\mbox{${\mathbb{E}}$}[a_{i_{1}i_{2}}a_{i_{2}i_{3}}\ldots
a_{i_{k}i_{1}}a_{i_{k+1}i_{k+2}}\ldots a_{i_{2k}i_{k+1}}].$

With this notation set up, one can see that

$\begin{split}\operatorname{Var}\left[\operatorname{tr}\mathbf{A}_{N}^{k}\right]&=\frac{1}{N^{2}}\left[\mbox{${\mathbb{E}}$}[(\operatorname{Tr}(\mathbf{A}_{N}^{k})^{2}]-(\mbox{${\mathbb{E}}$}[\operatorname{Tr}(\mathbf{A}_{N}^{k})])^{2}\right]\\\
&=\frac{1}{N^{2}\lambda^{k}}\sum_{i_{1},i_{2},\ldots,i_{k},i_{k+1},\ldots,i_{2k}=1}^{N}P(\mathbf{i},\mathbf{i^{\prime}})-P(\mathbf{i})P(\mathbf{i^{\prime}}).\end{split}$
(3.11)

We remark here that the construction of the graph
$G(\mathbf{i},\mathbf{i^{\prime}})$ is similar to how we did in Lemma 3.1,
with the essential difference being the closed walk structure over two
separate $k-$tuples.

Suppose that $E(\mathbf{i})\cap E(\mathbf{i^{\prime}})=\phi$. Then by
independence, (3.11) becomes 0. Thus, we must have $E(\mathbf{i})\cap
E(\mathbf{i^{\prime}})\neq\phi$. Moreover, due to remark 3.3, each term must
appear at least twice in $P(\mathbf{i},\mathbf{i^{\prime}})$, that is, each
edge in $E(\mathbf{i},\mathbf{i^{\prime}})$ is traversed at least twice. This
implies that the maximum number of edges our graph can have is $k$.

Next, note that the only way the graph $G(\mathbf{i},\mathbf{i^{\prime}})$
will be disconnected is when the closed walk over the two $k-$ tuples yields
two disjoint graphs, and thus we once again obtain
$P(\mathbf{i},\mathbf{i^{\prime}})=P(\mathbf{i})P(\mathbf{i^{\prime}})$.

Thus, our computation boils down to the case where
$G(\mathbf{i},\mathbf{i^{\prime}})$ is a connected graph, with each edge
appearing at least twice, and $E(\mathbf{i})\cap
E(\mathbf{i^{\prime}})\neq\phi$. Note that one can have
$G(\mathbf{i},\mathbf{i^{\prime}})$ to be connected and still have
$E(\mathbf{i})\cap E(\mathbf{i^{\prime}})=\phi$, for example when $i_{1}$ and
$i_{k+1}$ are collapsed into the same vertex. This gives us that
$|V(\mathbf{i},\mathbf{i^{\prime}})|\leq|E(\mathbf{i},\mathbf{i^{\prime}})|+1\leq
k+1$. Using $|f|\leq C_{f}$ gives us that

$\operatorname{Var}\left[\operatorname{tr}\mathbf{A}_{N}^{k}\right]\leq
C_{f}\frac{1}{N^{2}\lambda^{k}}N^{|V|}\left(\frac{\lambda}{N}\right)^{|E|}=C_{f}\lambda^{|E|-k}N^{|V|-|E|-2}=\mbox{${\mathrm{O}}$}(N^{-1}).$

This completes the proof. ∎

An immediate consequence using Chebychev’s inequality is that the moments
concentrate around their mean as $N\to\infty$. In other words, for all $k\geq
1$,

$\lim_{N\to\infty}\operatorname{tr}\mathbf{A}_{N}^{k}=m_{k}(\mu_{\lambda})\hskip
5.69046pt\text{in probability},$

where $m_{k}(\mu_{\lambda})$ are as in (2.9). To conclude Theorem 2.7, we now
further analyse the sequence $\\{m_{k}\\}_{k\geq 0}$, and show that it is
unique for the measure $\mu_{\lambda}$. A measure $\mu$ is said to be uniquely
determined by its moment sequence $\\{m_{k}\\}_{k\geq 0}$ if the following
holds (Carleman’s condition):

$\displaystyle\sum_{k\geq 0}m_{2k}^{-1/2k}=\infty.$ (3.12)

######  Lemma 3.2 (Uniqueness of moments).

For $\lambda$ bounded away from 0, that is, $\lambda>0$, the moments uniquely
determine the limiting spectral measure.

###### Proof.

Let $m_{k}$ denote the $k^{\text{th}}$ moment. Since $f$ is bounded, we have

$\begin{split}m_{2k}&=\sum_{\pi\in
SS(2k)}\lambda^{|\gamma\pi|-1-k}\int_{[0,1]^{|\gamma\pi|}}\prod_{(ab)\in
E_{\gamma\pi}}f(x_{a},x_{b})\prod_{i=1}^{|\gamma\pi|}\mu_{w}(\operatorname{d}x_{i})\\\
&\leq\sum_{\pi\in SS(2k)}C_{f}^{|\gamma\pi|}\lambda^{|\gamma\pi|-1-k}\\\
&=\sum_{l=2}^{k+1}\sum_{\pi\in
SS(2k):|\gamma\pi|=l}C_{f}^{l}\lambda^{l-1-k},\end{split}$

Let $A_{k}$ be defined as

$A_{k}=\begin{cases}1,&\hskip 8.5359pt\text{if $\lambda\geq 1$},\\\
\lambda^{-k},&\hskip 8.5359pt\text{if $1>\lambda>0$}.\end{cases}$

Then,

$\begin{split}m_{2k}&\leq C_{f}^{k+1}A_{k}\sum_{l=2}^{k+1}|\\{\pi\in
SS(2k):|\gamma\pi|=l\\}|\\\ &\leq A_{k}C_{f}^{k+1}|\\{SS(2k)\\}|\\\ &\leq
A_{k}C_{f}^{k+1}(2k)^{2k},\end{split}$

where the last inequality follows since $SS(2k)\subset P(2k)$ and $|P(2k)|$ is
bounded by $2k^{2k}$. Thus,

$m_{2k}^{-1/2k}\geq\frac{1}{2k\sqrt{C_{f}}}.\frac{1}{(A_{k}C_{f})^{\frac{1}{2k}}}$

So, we have the series $\sum_{k\geq 1}m_{2k}^{-1/2k}$ to be lower bounded by
$\sum_{k\geq 1}a_{k}$, where

$a_{k}=\frac{1}{2k\sqrt{C_{f}}}.\frac{1}{(A_{k}C_{f})^{\frac{1}{2k}}}=\frac{1}{C_{1}k\mathrm{e}^{\frac{1}{2k}\log(A_{k}C_{f})}}.$

Thus,

$\displaystyle
a_{k}=\begin{cases}\frac{\mathrm{e}^{-C_{2}/2k}}{C_{1}k},&\text{for
$\lambda\geq 1$},\\\
\frac{\sqrt{\lambda}\mathrm{e}^{-C_{2}/2k}}{C_{1}k},&\text{for
$\lambda<1$}.\end{cases}$

Since $\mathrm{e}^{-x}>1-x$, we see that the series $\sum_{k\geq 1}a_{k}$
diverges, and consequently,

$\sum_{k\geq 0}m_{2k}^{-1/2k}=\infty.$

∎

##  4\. Analytic approach: Proof of Theorem 2.9

###  4.1. Resolvent and Stieltjes Transform

We fix a $z\in\mathbb{C}^{+}$ throughout this argument, with $\Im(z)=\eta>0$.
Recall that resolvent is given by

$\operatorname{R}_{\mathbf{A}_{N}}(z):=(\mathbf{A}_{N}-zI)^{-1},\,z\in\mathbb{C}^{+}.$

The Stieltjes transform of the empirical spectral distribution of
$\mathbf{A}_{N}$ is given by

$\operatorname{S}_{\mathbf{A}_{N}}(z)=\int_{\mathbb{R}}\frac{1}{x-z}\operatorname{ESD}(\mathbf{A}_{N})(\operatorname{d}x)=\operatorname{tr}(\operatorname{R}_{\mathbf{A}_{N}}(z)),$
(4.1)

where $\operatorname{tr}$ denotes the normalized trace.

######  Lemma 4.1 (Resolvent Properties).

For any $z\in\mathbb{C}^{+},1\leq i,j\leq N$, the following properties are
well-known for the resolvent $\operatorname{R}_{\mathbf{A}}$ of an $N\times N$
matrix $\mathbf{A}$.

  * (i)

Analytic: $z\mapsto\operatorname{R}_{\mathbf{A}}(z)(i,j)$ is an analytic
function on $\mathbb{C}^{+}\rightarrow\mathbb{C}^{+}$.

  * (ii)

Bounded : $\|\operatorname{R}_{\mathbf{A}}(z)\|_{\text{op}}\leq\Im(z)^{-1}$,
where $\|\cdot\|_{\text{op}}$ denotes the operator norm.

  * (iii)

Normal :
$\operatorname{R}_{\mathbf{A}}(z)\operatorname{R}_{\mathbf{A}}(z)^{*}=\operatorname{R}_{A}(z)^{*}\operatorname{R}_{A}(z)$.

  * (iv)

Diagonals are bounded:
$|\operatorname{R}_{\mathbf{A}}(z)(i,j)|\leq\Im(z)^{-1}.$

  * (v)

Trace bounded:
$|\operatorname{tr}(\operatorname{R}_{\mathbf{A}}(z))|\leq\Im(z)^{-1}$. In
particular,

$\left|\operatorname{tr}(\operatorname{R}^{p}_{\mathbf{A}}(z))\right|\leq\Im(z)^{-p},\text{
for any $p\geq 1$.}$

For the first three properties see (Bordenave, 2019, Chapter 3). Note that the
property (iv) follows from (iii) by the following argument:

$|\operatorname{R}_{\mathbf{A}}(z)(i,j)|\leq|\langle\delta_{i},\,\operatorname{R}_{\mathbf{A}}(z)\delta_{j}\rangle|\leq\sup_{v:\|v\|=1}|\langle\delta_{i},\operatorname{R}_{\mathbf{A}}(z)\delta_{j}\rangle|=\|\operatorname{R}_{\mathbf{A}}(z)\|_{\text{op}}.$

The last property (v) follows from (iv). We now state the Ward’s identity, for
which we refer the reader to Erdős and Yau (2017, Lemma 8.3).

######  Lemma 4.2 (Ward’s identity).

Let $\mathbf{A}$ be a Hermitian matrix and $\operatorname{R}_{\mathbf{A}}$ be
the resolvent. Let $z\in\mathbb{C}^{+}$. Then for any fixed $k$, we have

$\sum_{l\neq
k}|R_{\mathbf{A}}(l,k)|^{2}=\frac{1}{\eta}\Im(R_{\mathbf{A}}(k,k)).$

Since we have already shown in the previous section
$\lim_{n\to\infty}\operatorname{ESD}(\mathbf{A}_{N})=\mu_{\lambda}$ weakly in
probability and hence it follows that for any $z\in\mathbb{C}^{+}$

$\lim_{N\to\infty}\operatorname{S}_{\mathbf{A}_{N}}(z)\to
S_{\mu_{\lambda}}(z).$

Due to the involved structure of the moments, it is not immediately evident
what the limiting Stieltjes transform looks like.

Recall the notation of _expected_ empirical spectral distribution of
$\mathbf{A}_{N}$ from (3.2). Let $\bar{\operatorname{S}}_{\mathbf{A}_{N}}(z)$
denote the Stieltjes transform of $\bar{\mu}_{N,\lambda}$. Notice that
$\bar{\operatorname{S}}_{{\mathbf{A}}_{N}}(z)=\mbox{${\mathbb{E}}$}[\operatorname{S}_{\mathbf{A}_{N}}(z)]$.
It is known that if a measure $\mu_{N}$ converges weakly in probability to a
measure $\mu$, then the corresponding Stieltjes transforms converge. In
particular, we have the following lemma.

######  Lemma 4.3.

(Anderson et al., 2010, Theorem 2.4.4) A sequence of measures $\mu_{N}$
converge weakly in probability to a measure $\mu$ if and only if
$\operatorname{S}_{\mu_{N}}(z)$ converges in probability to
$\operatorname{S}_{\mu}(z)$ for each $z\in\mathbb{C}^{+}$.

Thus, we compute an expression for the expected Stieltjes transform
$\operatorname{S}_{\bar{\mathbf{A}}_{N}}$, and using convergence in
probability from Theorem 2.7, we can claim that the Stieltjes transform
$\operatorname{S}_{\mathbf{A}_{N}}(z)$ converges in probability to the same
expression. For ease of notation we shall denote by
$r_{kk}^{N}(z):=\operatorname{R}_{\mathbf{A}_{N}}(z)(k,k)$ for $1\leq k\leq
N$.

The following identity can be found in Abramowitz and Stegun (1964). For any
complex number $z\in\mathbb{C}^{+}$, we have for all $u\geq 0$,

$\mathrm{e}^{\dot{\iota}uz}=1-\sqrt{u}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{-\dot{\iota}vz^{-1}}\operatorname{d}{v},$
(4.2)

where $\operatorname{J_{1}}(x)$ is the first-order Bessel function of the
first kind given by (2.15). Note that for all $x\geq 0$,
$|\operatorname{J_{1}}(x)|\leq 1$ (see (Abramowitz and Stegun, 1964, Chapter
9)). We know that resolvent maps the upper half complex plane to the upper
half complex plane. Thus, we begin by fixing $r^{N}_{jj}(z)$, the
$j^{\text{th}}$ diagonal entry of the $N\times N$ resolvent matrix, as our
complex variable in $\mathbb{C}^{+}$. So we can get

$\mathrm{e}^{\dot{\iota}ur_{jj}^{N}(z)}=1-\sqrt{u}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{-\dot{\iota}v(r_{jj}^{N})^{-1}}\operatorname{d}{v}.$
(4.3)

If we look at $\sum_{j=1}^{N}\mathrm{e}^{\dot{\iota}ur_{jj}^{N}(z)}$ then the
relation between the Stieltjes transform and the above equation becomes
apparent. It turns out that

$\operatorname{S}_{\mathbf{A}_{N}}(z)=\left.\frac{\partial}{\partial
u}\frac{1}{N}\sum_{j=1}^{N}\mathrm{e}^{\dot{\iota}ur_{jj}^{N}(z)}\right|_{u=0}.$
(4.4)

To understand the Stieltjes transform we will first try to understand the
behaviour of (4.3). We will adapt the approach of Khorunzhy et al. (2004). For
ease of notation, for what follows, $\|\cdot\|$ will denote the norm
$\|\cdot\|_{\operatorname{\mathcal{B}}}$ as defined in (2.10), unless stated
otherwise.

######  Proposition 4.4.

Let $r^{N}_{jj}:=r^{N}_{jj}(z)$ denote the $j^{\text{th}}$ diagonal entry of
the resolvent $\operatorname{R}_{\mathbf{A}_{N}}(z)$. Let

$d_{j}=\frac{1}{N}\sum\limits_{k=1}^{N}f(w_{j},w_{k})$ (4.5)

and for any $b>0$ define the function
$g_{N}:(0,\infty)\times(0,\infty)\times\mathbb{C}^{+}\to\mathbb{C}$ as follows

$g_{N}(x,b,z):=\frac{1}{N}\sum\limits_{k=1}^{N}f(x,w_{k})\mathrm{e}^{\dot{\iota}br^{N}_{kk}(z)}.$
(4.6)

Then, for any $z\in\mathbb{C}^{+}$,

$\mbox{${\mathbb{E}}$}[\mathrm{e}^{r^{N}_{jj}}]=1-\mathrm{e}^{-\lambda
d_{j}}\sqrt{u}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{\dot{\iota}vz}\mbox{${\mathbb{E}}$}\left[\mathrm{e}^{\lambda
g_{N}\left(w_{j},\frac{v}{\lambda},z\right)}\right]\operatorname{d}{v}+q_{N,\lambda}(u,z),$
(4.7)

where
$q_{N,\lambda}(u,z)=\mbox{${\mathrm{O}}$}\left(\frac{\lambda\sqrt{u}}{\eta^{5/2}\sqrt{N}}\right)$.

We begin by stating two results we use in this proof. Note that we
conveniently drop the dependence on $z$ for $r_{jj}^{N}(z)$, since we fix
$z\in\mathbb{C}^{+}$ throughout and hence just use the notation $r_{jj}^{N}$.

######  Fact 4.5 (Exponential Inequalities).

The following holds true for any real numbers $a,b\in\mathbb{R}$ and complex
numbers $z_{1},z_{2}\in\mathbb{C}^{+}$.

$\displaystyle|e^{a\dot{\iota}z_{1}}-\mathrm{e}^{a\dot{\iota}z_{2}}|\leq|a\|z_{1}-z_{2}|$
(4.8) $\displaystyle|e^{a}-\mathrm{e}^{b}|\leq|a-b|e^{|a|+|b|}$ (4.9)

###### Proof of Proposition 4.4.

For the resolvent of a matrix with zero diagonal, we have the relation

$r^{N}_{jj}=-\left(z+\sum_{k,l\neq
j}\tilde{r}^{N-1}_{kl}a_{kj}a_{lj}\right)^{-1},$

for any diagonal element $r_{jj}^{N}$ of the resolvent
$\operatorname{R}_{\mathbf{A}_{N}}(z)$, where
$\tilde{r}^{N-1}_{kl}:=\tilde{r}^{N-1}_{kl}(z)$ are the entries of the
resolvent of $\mathbf{A}_{N-1}^{(j)}$ in $z\in\mathbb{C}^{+}$, which is the
adjacency matrix with deleted $j^{\text{th}}$ row and column. Plugging into
(4.2) yields

$\displaystyle\mathrm{e}^{\dot{\iota}ur^{N}_{jj}}=1-\sqrt{u}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{\dot{\iota}vz}\exp\left(\dot{\iota}v\sum_{k,l\neq
j}\tilde{r}^{N-1}_{kl}a_{kj}a_{lj}\right)\operatorname{d}{v}.$ (4.10)

Adding and subtracting the appropriate exponential to (4.10) yields

$\displaystyle\mathrm{e}^{\dot{\iota}ur^{N}_{jj}}=1-\sqrt{u}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{\dot{\iota}vz}\exp\left(\dot{\iota}v\sum_{k\neq
j}\tilde{r}^{N-1}_{kk}a_{kj}^{2}\right)\operatorname{d}{v}+E_{1},$ (4.11)

where $E_{1}$ is an error term given by

$\displaystyle
E_{1}=\sqrt{u}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{\dot{\iota}vz}\left(\exp\left(\dot{\iota}v\sum_{k,l\neq
j}\tilde{r}^{N-1}_{kl}a_{kj}a_{lj}\right)-\exp\left(\dot{\iota}v\sum_{k\neq
j}\tilde{r}^{N-1}_{kk}a_{kj}^{2}\right)\right)\operatorname{d}{v}.$

It is easy to see that for $z\in\mathbb{C}^{+}$ with
$\Re(z)=\zeta\in\mathbb{R}$ and $\Im(z)=\eta>0$, we have
$|\mathrm{e}^{\dot{\iota}vz}|=|\mathrm{e}^{\dot{\iota}\zeta
v}\mathrm{e}^{-\eta v}|\leq\mathrm{e}^{-\eta v}$. Thus,

$\displaystyle\begin{split}|E_{1}|&=\left|\sqrt{u}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{\dot{\iota}vz}\left(\exp\left(\dot{\iota}v\sum_{k,l\neq
j}\tilde{r}^{N-1}_{kl}a_{kj}a_{lj}\right)-\exp\left(\dot{\iota}v\sum_{k\neq
j}\tilde{r}^{N-1}_{kk}a_{kj}^{2}\right)\right)\operatorname{d}{v}\right|\\\
&\leq\sqrt{u}\int_{0}^{\infty}\frac{v\mathrm{e}^{-\eta
v}}{\sqrt{v}}\sum_{k\leq N}\sum_{l\neq
k}|\tilde{r}^{N-1}_{kl}|a_{kj}a_{lj}\operatorname{d}{v}=\left(\sqrt{u}\int_{0}^{\infty}\sqrt{v}\mathrm{e}^{-\eta
v}\operatorname{d}{v}\right)\sum_{k\leq N}\sum_{l\neq
k}|\tilde{r}^{N-1}_{kl}|a_{kj}a_{lj}\end{split}$ (4.12)

where in the last step, we use inequality (4.8) and the bound
$|\operatorname{J_{1}}(x)|\leq 1$ for $x\geq 0$. Note that in the last sum in
(4.12), the entries $a_{kj}$ and $a_{lj}$ are independent of one another, and
of $\tilde{r}_{kl}^{N-1}$. Thus, since $f$ is bounded by a constant $C_{f}$,
taking expectation on the summation gives us

$\displaystyle\mbox{${\mathbb{E}}$}\left[\sum_{l\neq
k}|\tilde{r}^{N-1}_{kl}|a_{kj}a_{lj}\right]\leq\frac{\lambda
C_{f}^{2}}{N^{2}}\sum_{l\neq k}|\tilde{r}^{N-1}_{kl}|$ (4.13)

since $a_{ij}$ are distributed as Bernoulli random variables with parameter
$p_{ij}$, and are scaled by a factor $\lambda^{-1/2}$. Using (4.13) and taking
expectation in (4.12) gives us

$\displaystyle\begin{split}\mbox{${\mathbb{E}}$}\left[|E_{1}|\right]&\leq
C_{f}^{2}\sqrt{u}\int_{0}^{\infty}\frac{\sqrt{v}\mathrm{e}^{-\eta
v}\lambda}{N^{2}}\sum_{k\leq N}\sum_{l\neq
k}|\tilde{r}^{N-1}_{kl}|\operatorname{d}{v}\\\ &\leq
C_{f}^{2}\sqrt{u}\int_{0}^{\infty}\frac{\sqrt{v}\mathrm{e}^{-\eta
v}\lambda}{N\sqrt{N}}\sum_{k\leq N}\left(\sum_{l\neq
k}|\tilde{r}^{N-1}_{kl}|^{2}\right)^{\frac{1}{2}}\operatorname{d}{v}\hskip
8.5359pt\text{(Cauchy-Schwarz)}\\\ &\leq
C_{f}^{2}\sqrt{u}\int_{0}^{\infty}\frac{\sqrt{v}\mathrm{e}^{-\eta
v}\lambda}{N\sqrt{N\eta}}\sum_{k\leq
N}(\Im(\tilde{r}^{N-1}_{kk}))^{\frac{1}{2}}\operatorname{d}{v}\hskip
5.69046pt\text{(using Lemma \ref{pap1-Wald})}\\\ &\leq
C_{f}^{2}\sqrt{u}\int_{0}^{\infty}\frac{\sqrt{v}\mathrm{e}^{-\eta
v}\lambda}{\sqrt{N}\eta}\operatorname{d}{v}\hskip 8.5359pt\text{(using
property (iv) from Lemma \ref{resolventproperties})}\\\
&=C_{f}^{2}\frac{\sqrt{u}\lambda}{\eta^{5/2}\sqrt{N}}\int_{0}^{\infty}\sqrt{\eta
v}\mathrm{e}^{-\eta v}\operatorname{d}{(\eta
v)}=\mbox{${\mathrm{O}}$}\left(\frac{\lambda\sqrt{u}}{\eta^{5/2}\sqrt{N}}\right),\end{split}$

where in the last step we do a change of variable $\eta v=v^{\prime}$ to show
the integral is finite. So, if we now take an expectation in (4.11), we get

$\mbox{${\mathbb{E}}$}[\mathrm{e}^{\dot{\iota}ur^{N}_{jj}}]=1-\sqrt{u}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{\dot{\iota}vz}\mbox{${\mathbb{E}}$}\left[\exp\left(\dot{\iota}v\sum_{k\neq
j}\tilde{r}^{N-1}_{kk}a_{kj}^{2}\right)\right]\operatorname{d}{v}+q_{N,\lambda}(u,z),$
(4.14)

where
$q_{N,\lambda}(u,z)=\mbox{${\mathrm{O}}$}\left(\frac{\lambda\sqrt{u}}{\eta^{5/2}\sqrt{N}}\right)$.
Note that the expectation could be pulled inside the integral in (4.11) using
Fubini’s Theorem since the integral is bounded above by a constant. To
evaluate the expectation inside (4.14), we use a conditioning argument as
follows. We have

$\displaystyle\mbox{${\mathbb{E}}$}\left[\exp\left(\dot{\iota}v\sum_{k\neq
j}\tilde{r}^{N-1}_{kk}a_{kj}^{2}\right)\right]$
$\displaystyle=\mbox{${\mathbb{E}}$}\left[\mbox{${\mathbb{E}}$}\left[\exp\left(\left.\dot{\iota}v\sum_{k\neq
j}\tilde{r}^{N-1}_{kk}a_{kj}^{2}\right)\right|\mathbf{A}_{N-1}^{(j)}\right]\right]$
$\displaystyle=\mbox{${\mathbb{E}}$}\left[\prod_{k=1}^{N}\left(1-\frac{\lambda}{N}f(w_{k},w_{j})+\frac{\lambda}{N}f(w_{k},w_{j})\mathrm{e}^{\dot{\iota}v\tilde{r}^{N-1}_{kk}/\lambda}\right)\right]$
$\displaystyle=\mbox{${\mathbb{E}}$}\left[\prod_{k=1}^{N}\left(1+\frac{\lambda}{N}f(w_{k},w_{j})\left(\mathrm{e}^{\dot{\iota}v\tilde{r}^{N-1}_{kk}/\lambda}-1\right)\right)\right]$
$\displaystyle=\mbox{${\mathbb{E}}$}\left[\prod_{k=1}^{N}\left(\exp\left(\frac{\lambda}{N}f(w_{k},w_{j})\left(\mathrm{e}^{\dot{\iota}v\tilde{r}^{N-1}_{kk}/\lambda}-1\right)\right)+q^{\prime}_{k}(N,\lambda)\right)\right],$
(4.15)

where $q^{\prime}_{k}(N,\lambda)$ is an error given by

$\displaystyle q^{\prime}_{k}(N,\lambda)$
$\displaystyle=1+\frac{\lambda}{N}f(w_{k},w_{j})\left(\mathrm{e}^{\dot{\iota}v\tilde{r}^{N-1}_{kk}/\lambda}-1\right)-\exp\left(\frac{\lambda}{N}f(w_{k},w_{j})\left(\mathrm{e}^{\dot{\iota}v\tilde{r}^{N-1}_{kk}/\lambda}-1\right)\right).$

Since $|\mathrm{e}^{\dot{\iota}v\tilde{r}^{N-1}_{kk}/\lambda}-1|\leq 2$, doing
a Taylor expansion for the exponential term in $q_{k}^{\prime}(N,\lambda)$
gives us

$\displaystyle|q^{\prime}_{k}(N,\lambda)|\leq\frac{4C_{f}^{2}\lambda^{2}}{N^{2}}=\mbox{${\mathrm{O}}$}\left(\frac{\lambda^{2}}{N^{2}}\right).$
(4.16)

We can write

$\displaystyle\mbox{${\mathbb{E}}$}\left[\exp\left(\dot{\iota}v\sum_{k\neq
j}\tilde{r}^{N-1}_{kk}a_{kj}^{2}\right)\right]=\mbox{${\mathbb{E}}$}\left[\prod_{k=1}^{N}\left(\exp\left(\frac{\lambda}{N}f(w_{k},w_{j})\left(\mathrm{e}^{\dot{\iota}v\tilde{r}^{N-1}_{kk}/\lambda}-1\right)\right)\right)\right]+\mbox{${\mathbb{E}}$}[E_{2}],$
(4.17)

where $E_{2}$ is an expression involving all the other terms of the product in
(4.15). To get the order of $E_{2}$, we take a supremum over $k$ in (4.15) and
compute the binomial expansion of the form $(a+b)^{N}$ modulo the leading term
$a^{N}$. In particular, since
$|\mathrm{e}^{\dot{\iota}v\tilde{r}^{N-1}_{kk}/\lambda}-1|\leq 2$, and again
using (4.16), we have

$\displaystyle|E_{2}|\leq\sum_{j=1}^{N}{N\choose
j}\left(\mathrm{e}^{\frac{2\lambda
C_{f}}{N}}\right)^{N-j}\left(\frac{4C_{f}^{2}\lambda^{2}}{N^{2}}\right)^{j},$

which for some constant $C_{a}>0$ and $N$ large enough further simplifies to

$\displaystyle|E_{2}|\leq
C_{a}\sum_{j=1}^{N}(2C_{f}\lambda)^{2j}N^{j}\mathrm{e}^{-\frac{2j\lambda
C_{f}}{N}}N^{-2j}=C_{a}\sum_{j=1}^{N}(2C_{f}\lambda)^{2j}N^{-j}\mathrm{e}^{-\frac{2j\lambda
C_{f}}{N}}=C_{a}\frac{4C_{f}\lambda^{2}N^{-1}\mathrm{e}^{-\frac{2\lambda
C_{f}}{N}}}{1-4C_{f}\lambda^{2}N^{-1}\mathrm{e}^{-\frac{2\lambda C_{f}}{N}}}.$

where the last equality is due to the sum being a geometric series. Thus,

$\displaystyle|E_{2}|=\mbox{${\mathrm{O}}$}\left(\frac{\lambda^{2}}{N}\right),$
(4.18)

which is a faster error than $q_{N,\lambda}(u,z)$ so we can later absorb it
into the existing error of (4.14). Thus, using (4.18), we can rewrite (4.17)
as

$\displaystyle\mbox{${\mathbb{E}}$}\left[\exp\left(\dot{\iota}v\sum_{k\neq
j}\tilde{r}^{N-1}_{kk}a_{kj}^{2}\right)\right]=\mbox{${\mathbb{E}}$}\left[\mathrm{e}^{-\lambda{d_{j}}}\exp\left(\lambda\tilde{g}_{N-1}\left(w_{j},\frac{v}{\lambda},z\right)\right)\right]+\mbox{${\mathrm{O}}$}\left(\frac{\lambda^{2}}{N}\right)$
(4.19)

where

$\displaystyle d_{j}=\frac{1}{N}\sum\limits_{k=1}^{N}f(w_{j},w_{k})\hskip
8.5359pt\text{and}\hskip
8.5359pt\tilde{g}_{N-1}(w_{j},b,z)=\sum\limits_{k=1}^{N}f(w_{j},w_{k})\mathrm{e}^{\dot{\iota}b\tilde{r}^{N-1}_{kk}}.$
(4.20)

Note that $\tilde{g}_{N}$ is a bounded function and is bounded above by
$C_{f}$. To get the error down from the exponent, we again use inequality
(4.9).

To conclude the proof of the proposition, we need to return back to an
expression involving terms of the form $r_{kk}^{N}$ of the original resolvent.
To do so, we do an interpolation argument. Let $0\leq t\leq 1$ and define
$\mathbf{A}_{N}^{t}=(1-t)\mathbf{A}_{N}+t\mathbf{A}_{N-1}^{(j)}$ with the
resolvent $\operatorname{R}_{\mathbf{A}_{N}^{t}}(z)$, whose entries we denote
by $r^{N}_{kl}(t)$, that also implicitly depends on $z$ but we drop that for
convenience of notation. Also, define

$\mathbf{g}_{N}^{t}(w_{j},b,z)=\frac{1}{N}\sum_{i=1}^{N}f(w_{i},w_{j})\mathrm{e}^{\dot{\iota}br^{N}_{kk}(t)}.$

We remark using property (i) from Lemma 4.1 that $\mathbf{g}_{N}^{t}$ is also
bounded above by $C_{f}$ for all values of $t$, since the complex exponential
$\mathrm{e}^{\dot{\iota}br_{kk}^{N}(t)}$ is bounded by 1 for any $b\geq 0$ and
$1\leq k\leq N$. In particular, we have that $|g_{N}(x,b,z)|\leq C_{f}$ for
all $x,b\geq 0$.

Our target function is
$g_{N}(w_{j},b,z)=\frac{1}{N}\sum\limits_{i=1}^{N}f(w_{i},w_{j})\mathrm{e}^{\dot{\iota}br^{N}_{kk}}$.
By the fundamental theorem of calculus,

$\begin{split}|g_{N}(w_{j},b,z)-\tilde{g}_{N-1}(w_{j},b,z)|&=\left|\mathbf{g}_{N}^{0}(w_{j},b,z)-\mathbf{g}_{N}^{1}(w_{j},b,z)\right|\\\
&=\left|\int_{0}^{1}\frac{\partial}{\partial
t}\mathbf{g}_{N}^{t}\operatorname{d}{t}\right|=\left|\int_{0}^{1}\frac{b}{N}\sum_{k=1}^{N}\mathrm{e}^{\dot{\iota}br^{N}_{kk}(t)}\frac{\partial}{\partial
t}r^{N}_{kk}(t)\right|.\end{split}$

Now, $\operatorname{R}_{\mathbf{A}_{N}^{t}}(z)=(\mathbf{A}_{N}^{t}-zI)^{-1}$
and thus,
$\frac{\operatorname{d}}{\operatorname{d}{t}}\operatorname{R}_{\mathbf{A}_{N}^{t}}(z)=-\operatorname{R}_{\mathbf{A}_{N}^{t}}(z)\frac{\operatorname{d}{\mathbf{A}_{N}^{t}}}{\operatorname{d}{t}}\operatorname{R}_{\mathbf{A}_{N}^{t}}(z)$.

Note that
$\frac{\operatorname{d}{\mathbf{A}_{N}^{t}}}{\operatorname{d}{t}}=-J_{N}$,
where $J_{N}$ is given by

$\displaystyle J_{N}(k,l)=\begin{cases}0,&\text{if $k,l\neq j$}\\\
a_{kl},&\text{if $k=j$ or $l=j$}.\end{cases}$

Thus,

$\displaystyle|g_{N}(w_{j},b,z)-\tilde{g}_{N-1}(w_{j},b,z)|$
$\displaystyle=\left|\int_{0}^{1}\frac{b}{N}\sum_{k=1}^{N}\mathrm{e}^{\dot{\iota}br^{N}_{kk}(t)}\sum_{m,n=1}^{N}r^{N}_{km}(t)\frac{\partial
a^{t}_{mn}}{\partial t}r^{N}_{nk}(t)\right|$
$\displaystyle=\left|\int_{0}^{1}\frac{b}{N}\sum_{k=1}^{N}\mathrm{e}^{\dot{\iota}br^{N}_{kk}(t)}\sum_{m=1}^{N}r^{N}_{km}(t)a_{mj}r^{N}_{jk}(t)\operatorname{d}{t}\right|$
$\displaystyle\leq\int_{0}^{1}\frac{b}{N}\sum_{k=1}^{N}\sum_{m=1}^{N}|r^{N}_{km}(t)a_{mj}r^{N}_{jk}(t)|\operatorname{d}{t}$
(4.21)

since the complex exponential $\mathrm{e}^{\dot{\iota}br_{kk}^{N}(t)}$ is
trivially bounded by 1 as $r_{kk}^{N}(t)\in\mathbb{C}^{+}$. Then, using
Cauchy-Schwarz and Lemma 4.2 in (4.21), we have

$\displaystyle|g_{N}(w_{j},b,z)-\tilde{g}_{N-1}(w_{j},b,z)|\leq\int_{0}^{1}\frac{b}{N}\sum_{k=1}^{N}|r_{jk}^{N}(t)|\left(\frac{\Im(r^{N}_{kk}(t))}{\eta}\right)^{1/2}\left(\sum_{m=1}^{N}a_{mj}^{2}\right)^{1/2}\operatorname{d}{t}.$

Bounding $\Im(r_{kk}^{N}(t))$ by $1/\eta$ (property (iv) of Lemma 4.1) and
taking expectation, we get

$\displaystyle\mbox{${\mathbb{E}}$}[|g_{N}(w_{j},b,z)-\tilde{g}_{N-1}(w_{j},b,z)|]\leq\int_{0}^{1}\frac{b}{N\eta}\mbox{${\mathbb{E}}$}\left[\sum_{k=1}^{N}|r_{jk}^{N}(t)|\left(\sum_{m=1}^{N}a_{mj}^{2}\right)^{1/2}\right]\operatorname{d}{t}.$
(4.22)

Now, again using Cauchy-Schwarz and Lemma 4.2, we have for some constant
$C^{\prime}$ that

$\displaystyle\sum_{k=1}^{N}|r_{jk}^{N}(t)|\leq\sqrt{N}\left(\sum_{k=1}^{N}|r_{jk}^{N}(t)|^{2}\right)^{1/2}\leq
C^{\prime}\frac{\sqrt{N}}{\sqrt{\eta}}.$ (4.23)

Thus, using (4.23) and Jensen’s inequality on the function $\sqrt{X}$ in
(4.22), we get

$\displaystyle\mbox{${\mathbb{E}}$}[|g_{N}(w_{j},b,z)-\tilde{g}_{N-1}(w_{j},b,z)|]$
$\displaystyle\leq\int_{0}^{1}\frac{b}{N\eta}\mbox{${\mathbb{E}}$}\left[\frac{C^{\prime}\sqrt{N}}{\sqrt{\eta}}\left(\sum_{m=1}^{N}a_{mj}^{2}\right)^{1/2}\right]\operatorname{d}{t}$
$\displaystyle
C^{\prime}\int_{0}^{1}\frac{b}{\sqrt{N}\eta^{3/2}}\left(\mbox{${\mathbb{E}}$}\left[\sum_{m=1}^{N}a_{mj}^{2}\right]\right)^{1/2}\operatorname{d}{t}.$

Since $f$ is bounded, we have for some new constant $C_{f}^{\prime}$ that

$\displaystyle\mbox{${\mathbb{E}}$}[|g_{N}(w_{j},b,z)-\tilde{g}_{N-1}(w_{j},b,z)|]\leq\frac{C_{f}^{\prime}b\sqrt{\lambda}}{\eta^{3/2}\sqrt{N}}.$

Using the fact that $\mathbf{g}_{N}^{t}$ is bounded by $C_{f}$ for all $t$, we
get

$\mbox{${\mathbb{E}}$}[|\mathrm{e}^{\lambda\tilde{g}_{N-1}}-\mathrm{e}^{\lambda
g_{N}}|]\leq\mbox{${\mathbb{E}}$}[|\tilde{g}_{N-1}-g_{N}|]\mathrm{e}^{2C_{f}\lambda}=\mbox{${\mathrm{O}}$}\left(\frac{\sqrt{\lambda}}{\eta^{3/2}\sqrt{N}}\right).$

Since this is an error of the same order as $q_{N,\lambda}(u,z)$, we can
absorb it into the existing error $q_{N,\lambda}$. Finally, using (4.19) and
the interpolation argument allows us to write (4.14) as

$\mbox{${\mathbb{E}}$}[\mathrm{e}^{\dot{\iota}ur^{N}_{jj}}]=1-\mathrm{e}^{-\lambda
d_{j}}\sqrt{u}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{\dot{\iota}vz}\mbox{${\mathbb{E}}$}\left[\mathrm{e}^{\lambda
g_{N}\left(w_{j},\frac{v}{\lambda},z\right)}\right]\operatorname{d}{v}+q_{N,\lambda}(u,z),$

which proves the proposition. ∎

Now, consider the expression (4.7) from the Proposition 4.4. If we multiply
throughout by $f(x,w_{j})$ and then sum over $j$, and finally scale by $N$, we
get

$\displaystyle\begin{split}&\mbox{${\mathbb{E}}$}[g_{N}(x,u,z)]\\\
&=\frac{1}{N}\sum_{j=1}^{N}f(x,w_{j})-\frac{1}{N}\sum_{j=1}^{N}f(x,w_{j})\mathrm{e}^{-\lambda
d_{j}}\sqrt{u}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{\dot{\iota}vz}\mbox{${\mathbb{E}}$}\left[\mathrm{e}^{\lambda
g_{N}\left(w_{j},\frac{v}{\lambda},z\right)}\right]\operatorname{d}{v}+q_{N,\lambda}(u,z).\end{split}$
(4.24)

Consider the space of Lipschitz functions $Lip(\mathbb{R})$ defined as

$\displaystyle\mathrm{Lip}(\mathbb{R})=\left\\{h\in
C_{b}(\mathbb{R}):\sup_{x}|h(x)|\leq 1,\sup_{x\neq
y}\frac{|h(x)-h(y)|}{|x-y|}\leq C_{L},0<C_{L}<\infty\right\\}.$

Now, under the bounded Lipshitz metric $d_{BL}(\cdot,\cdot)$ given by

$d_{BL}(\mu,\nu)=\sup\limits_{h\in\mathrm{Lip}(\mathbb{R})}\left\\{\left|\int
h\operatorname{d}{\mu}-\int h\operatorname{d}{\nu}\right|\right\\},$

we have

$\mu_{W_{N}}\implies\mu_{w}\hskip 8.5359pt\text{if and only if}\hskip
8.5359ptd_{BL}(\mu_{W_{N}},\mu_{w})\to 0,$

where $W_{N}=w_{o_{N}}$ for a uniformly chosen vertex $o_{N}$. So, taking $f$
to be Lipschitz in one coordinate (and since we already have that $f$ is
bounded), the first term in the RHS of (4.24) becomes

$\displaystyle\frac{1}{N}\sum_{j=1}^{N}f(x,w_{j})=\int
f(x,y)\mu_{W_{N}}(\operatorname{d}{y})\leq d_{f}(x)+E_{N},$ (4.25)

where $E_{N}=d_{BL}(\mu_{W_{N}},\mu_{w})$.

Recall from (2.12) that we have

$d_{f}(w_{j}):=\int f(x,w_{j})\mu_{w}(\operatorname{d}{x}).$

Then, one simply gets

$|\mathrm{e}^{-\lambda d_{j}}-\mathrm{e}^{-\lambda d_{f}(w_{j})}|\leq\lambda
E_{N}\mathrm{e}^{2\lambda}.$ (4.26)

Thus, using (4.25) and (4.26) in (4.7) gives us

$\displaystyle\begin{split}&\mbox{${\mathbb{E}}$}[g_{N}(x,u,z)]\\\
&=d_{f}(x)-\frac{1}{N}\sum_{j=1}^{N}f(x,w_{j})\mathrm{e}^{-\lambda
d_{f}(w_{j})}\left(\sqrt{u}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{\dot{\iota}vz}\mbox{${\mathbb{E}}$}\left[\mathrm{e}^{\lambda
g_{N}\left(w_{j},\frac{v}{\lambda},z\right)}\right]\operatorname{d}{v}\right)\hskip
2.84544pt+\tilde{q}_{N,\lambda}(u,z),\end{split}$ (4.27)

where

$\tilde{q}_{N,\lambda}(u,z)=q_{N,\lambda}(u,z)+\mbox{${\mathrm{O}}$}(E_{N}).$

Finally, for a fixed $x\in[0,\infty)$, define

$\mathbf{I}_{g}(y)=f(x,y)\mathrm{e}^{-\lambda
d_{f}(y)}\left(\sqrt{u}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{\dot{\iota}vz}\mbox{${\mathbb{E}}$}\left[\mathrm{e}^{\lambda
g_{N}\left(y,\frac{v}{\lambda},z\right)}\right]\operatorname{d}{v}\right).$

Then, we have the following lemma.

######  Lemma 4.6.

$\mathbf{I}_{g}(y)$ is Lipschitz.

###### Proof.

Consider $\mathbf{I}_{g}(y)$ as defined. Then,

$\begin{split}&\left|\partial_{y}\mathbf{I}_{g}(y)\right|\\\
&\leq\left|\partial_{y}f(x,y)\mathrm{e}^{-\lambda
d_{f}(y)}\left(\sqrt{u}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{\dot{\iota}vz}\mbox{${\mathbb{E}}$}\left[\mathrm{e}^{\lambda
g_{N}\left(y,\frac{v}{\lambda},z\right)}\right]\operatorname{d}{v}\right)\right|\\\
&+\left|f(x,y)\mathrm{e}^{-\lambda
d_{f}(y)}\partial_{y}d_{f}(y)\left(\sqrt{u}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{\dot{\iota}vz}\mbox{${\mathbb{E}}$}\left[\mathrm{e}^{\lambda
g_{N}\left(y,\frac{v}{\lambda},z\right)}\right]\operatorname{d}{v}\right)\right|\\\
&+\left|f(x,y)\mathrm{e}^{-\lambda
d_{f}(y)}\left(\sqrt{u}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{\dot{\iota}vz}\mbox{${\mathbb{E}}$}\left[\mathrm{e}^{\lambda
g_{N}\left(y,\frac{v}{\lambda},z\right)}\right]\partial_{y}g_{N}(y,v/\lambda,z)\operatorname{d}{v}\right)\right|.\end{split}$
(4.28)

Recall that a function is Lipschitz if and only if it has a bounded
derivative. Thus, if $f$ is Lipschitz in $y$, the first term in (4.28)
uniformly bounded in $y$. Moreover, this makes the second term in (4.28)
bounded as well since

$\displaystyle\left|\partial_{y}d_{f}(y)\right|\leq\int_{0}^{\infty}|\partial_{y}f(x,y)|\mu_{w}(\operatorname{d}{x})$
(4.29)

is bounded. To justify interchanging the derivative and the integral in
(4.29), we have to utilise Theorem 5.2 for which we need to verify the
following conditions.

  * •

$f(x,y)$ is $\mu_{w}-$integrable for each $y$ and the map $y\mapsto f(x,y)$ is
continuous for each $x$.

  * •

For each $x$, the derivative $\partial_{y}f(x,y)$ exists.

  * •

For each $y$, there is a $\mu_{w}-$integrable function $\Psi_{y}(x)$ and a
neighbourhood $U_{y}$ containing $y$, such that for all $y^{\prime}\in U_{y}$,
$|\partial_{y^{\prime}}f(x,y^{\prime})|\leq\Psi_{y}(x)$.

The first and second are trivial to check, and by Lipschitz property, since
$\partial_{y}f(x,y)\equiv const.$, we have $\Psi_{y}(x)\equiv const$, which is
integrable on $[0,\infty)$ since $\mu_{w}$ is a probability measure.

Finally, for notational convenience, let $h(y,v)$ be denote

$h(y,v)=\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{\dot{\iota}vz}\mbox{${\mathbb{E}}$}\left[\mathrm{e}^{\lambda
g_{N}(y,v,z)}\right].$

Once again, we need to verify the three conditions as above so as to apply
Theorem 5.2. Note that $h(y,v)$ is integrable with respect to $v$. Moreover,

$\partial_{y}h(y,v)=h(y,v)\partial_{y}g_{N}(y,v,z)$

where one can compute

$\partial_{y}g_{N}(y,v,z)=\frac{1}{N}\sum_{k=1}^{N}\partial_{y}f(w_{k},y)\mathrm{e}^{\dot{\iota}vr_{kk}},$

which again is bounded. Thus, $\partial_{y}h(y,v)$ exists, and is bounded
above by $C_{0}v^{-\frac{1}{2}}\mathrm{e}^{-\eta v}$, which is integrable with
respect to $v$. This verifies the three conditions and allows us to pull the
derivative inside the third term in (4.28), and also makes that term bounded.
Thus, $\mathbf{I}_{g}(y)$ is Lipschitz. ∎

Since $\mathbf{I}_{g}(y)$ is Lipschitz, we can exploit the weak convergence of
$\mu_{w}$ under the Lipschitz metric $d_{BL}$ in (4.27) to give us

$\begin{split}&\mbox{${\mathbb{E}}$}[g_{N}(x,u,z)]\\\
&=d_{f}(x)-\int_{0}^{\infty}f(x,y)\mathrm{e}^{-\lambda
d_{f}(y)}\left(\sqrt{u}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{\dot{\iota}vz}\mbox{${\mathbb{E}}$}\left[\mathrm{e}^{\lambda
g_{N}\left(y,\frac{v}{\lambda},z\right)}\right]\operatorname{d}{v}\right)\mu_{w}(\operatorname{d}{y})\hskip
2.84544pt+\tilde{q}_{N,\lambda}(u,z).\end{split}$ (4.30)

Recall the Banach space as defined in (2.10), and consider
$\phi\in(\operatorname{\mathcal{B}},\|\cdot\|)$. In this space, consider the
map

$F_{z}(\phi)(x,u)=d_{f}(x)-\sqrt{u}\int_{0}^{\infty}f(x,y)\mathrm{e}^{-\lambda
d_{f}(y)}\left(\sqrt{u}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{\dot{\iota}vz}\mathrm{e}^{\lambda\phi\left(y,\frac{v}{\lambda},z\right)}\operatorname{d}{v}\right)\mu_{w}(\operatorname{d}{y}).$
(4.31)

Note that $\phi$ also implicitly depends on $z$ but we drop that for
notational purposes since we fix $z$ throughout.

Take $\phi_{1},\phi_{2}\in(\mathcal{B},\|\cdot\|)$ such that
$\|\phi_{1}\|,\|\phi_{2}\|\leq C_{f}$. Then, using the norm we defined in
(2.10) and inequality 4.9, from (4.31) we get

$\begin{split}&\|F_{z}(\phi_{1})-F_{z}(\phi_{2})\|\\\ &\leq\sup_{x,u\geq
0}\sqrt{\frac{1}{1+u}}\left|\int_{0}^{\infty}f(x,y)\mathrm{e}^{-\lambda
d_{f}(y)}\left(\sqrt{u}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{\dot{\iota}vz}\left(\mathrm{e}^{\lambda\phi_{1}\left(y,\frac{v}{\lambda}\right)}-\mathrm{e}^{\lambda\phi_{2}\left(y,\frac{v}{\lambda}\right)}\right)\operatorname{d}{v}\right)\mu_{w}(\operatorname{d}{y})\right|\\\
&\leq\sup_{u\geq
0}\sqrt{\frac{1}{1+u}}\int_{0}^{\infty}\int_{0}^{\infty}\frac{\lambda}{\sqrt{v}}\mathrm{e}^{-\eta
v}\left|\phi_{1}\left(y,\frac{v}{\lambda}\right)-\phi_{2}\left(y,\frac{v}{\lambda}\right)\right|\mathrm{e}^{\lambda\left|\phi_{1}\left(y,\frac{v}{\lambda}\right)\right|+\lambda\left|\phi_{2}\left(y,\frac{v}{\lambda}\right)\right|}\operatorname{d}{v}\hskip
5.69046pt\mu_{w}(\operatorname{d}{y})\\\
&\leq\lambda\|\phi_{1}-\phi_{2}\|\int_{0}^{\infty}\int_{0}^{\infty}\frac{\lambda}{\sqrt{v}}\mathrm{e}^{-\eta
v}\sup_{y,v\geq
0}\frac{\sqrt{1+v/\lambda}}{\sqrt{1+v/\lambda}}\mathrm{e}^{\lambda\left|\phi_{1}\left(y,\frac{v}{\lambda}\right)\right|+\lambda\left|\phi_{2}\left(y,\frac{v}{\lambda}\right)\right|}\operatorname{d}{v}\hskip
5.69046pt\mu_{w}(\operatorname{d}{y})\\\
&\leq\lambda\|\phi_{1}-\phi_{2}\|\int_{0}^{\infty}\int_{0}^{\infty}\frac{\sqrt{1+v/\lambda}}{\sqrt{v}}\mathrm{e}^{-\eta
v}\exp\left(\lambda\sqrt{1+v/\lambda}(\|\phi_{1}\|+\|\phi_{2}\|)\right)\operatorname{d}{v}\hskip
5.69046pt\mu_{w}(\operatorname{d}{y})\\\
&\leq\|\phi_{1}-\phi_{2}\|\int_{0}^{\infty}\left(\frac{\mathrm{e}^{-\eta
v}}{\sqrt{v}}+\frac{\mathrm{e}^{-\eta
v}}{\sqrt{\lambda}}\right)\mathrm{e}^{2C_{f}\sqrt{\lambda
v}}\operatorname{d}{v}\leq\frac{C_{1}}{\eta^{5/2}}\|\phi_{1}-\phi_{2}\|,\end{split}$

where $C_{1}$ is the constant upper bound to the integral of the form

$\displaystyle\int_{0}^{\infty}c_{1}\mathrm{e}^{-c_{2}x+c_{3}\sqrt{x}}\operatorname{d}{x},$

which is finite. Taking $\eta>0$ sufficiently large, we get that $F_{z}$ is a
contraction in an open ball $B\subset\mathcal{B}$ of radius $C_{f}<\infty$,
and thus, by the Banach Fixed Point Theorem, there exists a unique $\phi^{*}$
such that $\phi^{*}=F_{z}(\phi^{*})$ for $F_{z}:B\to B$.

We are now ready to prove a concentration result. Recall the function
$G_{N}(u)$ defined in (2.11) as

$\displaystyle
G_{N}(u)=\frac{1}{N}\sum_{i=1}^{N}\mathrm{e}^{\dot{\iota}ur_{ii}^{N}}.$

If we now define a new function $\tilde{G}_{N}(x,u)$ that acts identically on
the first coordinate as

$\tilde{G}_{N}(x,u):=G_{N}(u),$

then one can see that
$\sup_{x,u}\frac{1}{\sqrt{1+u}}\tilde{G}_{N}(x,u)<\infty$, and so
$\tilde{G}_{N}(x,u)\in\operatorname{\mathcal{B}}$, and consequently, a
concentration result for $\tilde{G}_{N}$ would imply concentration for
$G_{N}$.

######  Proposition 4.7 (Concentration and convergence).

For any $z\in\mathbb{C}^{+}$ and $x\in[0,\infty)$, and uniformly over $u$ in
$[0,1]$, we have
$\mbox{${\mathbb{E}}$}[g_{N}(x,u,z)]\xrightarrow{N\to\infty}\phi^{*}(x,u)$.
Further, we have

$\displaystyle\mbox{${\mathbb{E}}$}\left[\left\|g_{N}-\mbox{${\mathbb{E}}$}[g_{N}]\right\|^{2}\right]$
$\displaystyle=o(1),\hskip 8.5359pt\text{and}$
$\displaystyle\mbox{${\mathbb{E}}$}\left[\left\|\tilde{G}_{N}-\mbox{${\mathbb{E}}$}[\tilde{G}_{N}]\right\|^{2}\right]$
$\displaystyle=o(1).$

###### Proof of Proposition 4.7.

Let $\delta_{N}(x,u,z)$ denote the error

$\delta_{N}(x,u,z):=\mathrm{e}^{\lambda
g_{N}(x,u,z)}-\mathrm{e}^{\lambda\mbox{${\mathbb{E}}$}{[g_{N}(x,u,z)]}}.$

Let $1\leq k\neq l\leq N$ and consider the covariance

$A_{k,l}:=\mbox{${\mathbb{E}}$}[\mathrm{e}^{\dot{\iota}ur_{kk}^{N}}\mathrm{e}^{\dot{\iota}ur_{ll}^{N}}]-\mbox{${\mathbb{E}}$}[\mathrm{e}^{\dot{\iota}ur_{kk}^{N}}]\mbox{${\mathbb{E}}$}[\mathrm{e}^{\dot{\iota}ur_{kk}^{N}}].$

Using (4.11) for the first term and Proposition 4.4 for the second term, we
get

$\displaystyle\begin{split}&A_{k,l}=\\\
&-\mbox{${\mathbb{E}}$}[T_{j}]-\mbox{${\mathbb{E}}$}[T_{k}]+u\int\int\frac{\operatorname{J_{1}}(2\sqrt{uv_{1}})}{\sqrt{v_{1}}}\frac{\operatorname{J_{1}}(2\sqrt{uv_{2}})}{\sqrt{v_{2}}}\mathrm{e}^{\dot{\iota}(v_{1}+v_{2})z}\mbox{${\mathbb{E}}$}\left[\mathrm{e}^{\dot{\iota}v_{1}\sum\limits_{l\neq
j}\tilde{r}_{ll}^{N-1}a_{jl}^{2}+\dot{\iota}v_{2}\sum\limits_{l\neq
k}\tilde{r}_{ll}^{N-1}a_{kl}^{2}}\right]\operatorname{d}{v_{1}}\operatorname{d}{v_{2}}\\\
&+\mbox{${\mathbb{E}}$}[\tilde{T}_{j}]+\mbox{${\mathbb{E}}$}[\tilde{T}_{k}]-u\int\int\frac{\operatorname{J_{1}}(2\sqrt{uv_{1}})}{\sqrt{v_{1}}}\frac{\operatorname{J_{1}}(2\sqrt{uv_{2}})}{\sqrt{v_{2}}}\mathrm{e}^{\dot{\iota}(v_{1}+v_{2})z}\mbox{${\mathbb{E}}$}\left[\mathrm{e}^{\lambda
g_{N}(w_{j},\frac{v_{1}}{\lambda},z)+\lambda
g_{N}(w_{k},\frac{v_{2}}{\lambda},z)}\right]{}\operatorname{d}{v_{1}}\operatorname{d}{v_{2}},\end{split}$
(4.32)

where $T_{i}$ and $\tilde{T}_{i}$ are the RHS of equations (4.11) and (4.7)
respectively, and differ by the error $q_{N,\lambda}(u,z)$ in expectation. In
the first double integral of (4.32), one can do the interpolation argument
term-wise, and obtain the error
$C_{I}q_{N,\lambda}^{2}(u,z)+q_{N,\lambda}^{2}(u,z)$ by making a difference
with the second double integral in (4.32), where $C_{I}$ is the constant upper
bound to $\tilde{T}_{k}$ for any $k$. Thus, we have that

$\displaystyle|A_{k,l}|\leq
C_{I}^{\prime}q_{N,\lambda}(u,z)+q_{N,\lambda}^{2}(u,z).$ (4.33)

Using inequality 4.9 on $\delta_{N}(x,u,z)$ gives us

$\displaystyle\mbox{${\mathbb{E}}$}[|\delta_{N}(x,u,z)|^{2}]=\mbox{${\mathbb{E}}$}\left[\left|\mathrm{e}^{\lambda
g_{N}(x,u,z)}-\mathrm{e}^{\lambda\mbox{${\mathbb{E}}$}\left[g_{N}(x,u,z)\right]}\right|^{2}\right]\leq
C_{1}\mbox{${\mathbb{E}}$}\left[|g_{N}(x,u,z)-\mbox{${\mathbb{E}}$}[g_{N}(x,u,z)]|^{2}\right].$

since $|g_{N}(x,v,z)|\leq C_{f}$ and $C_{1}=\mathrm{e}^{2\lambda C_{f}}$. We
can now bound this by using the definition of $g_{N}$ to get

$\displaystyle\mbox{${\mathbb{E}}$}[|\delta_{N}(x,u,z)|^{2}]$
$\displaystyle\leq\frac{C_{1}}{N^{2}}\left|\sum_{k,l=1}^{N}\mbox{${\mathbb{E}}$}[f(x,w_{k})\mathrm{e}^{\dot{\iota}ur^{N}_{kk}}f(x,w_{l})\mathrm{e}^{\dot{\iota}ur^{N}_{ll}}]-\mbox{${\mathbb{E}}$}[f(x,w_{k})\mathrm{e}^{\dot{\iota}ur^{N}_{kk}}]\mbox{${\mathbb{E}}$}[f(x,w_{l})\mathrm{e}^{\dot{\iota}ur^{N}_{ll}}]\right|.$
(4.34)

Since $f$ is deterministic, we can pull it out of the expectation and take it
common, giving us

$\displaystyle\mbox{${\mathbb{E}}$}[|\delta_{N}(x,u,z)|^{2}]$
$\displaystyle\leq\frac{C_{1}}{N^{2}}\left|\sum_{k,l=1}^{N}f(x,w_{k})f(x,w_{l})A_{k,l}\right|.$

We can conclude using triangle inequality that

$\displaystyle\mbox{${\mathbb{E}}$}[|\delta_{N}(x,u,z)|^{2}]\leq
C_{1}C_{f}^{2}\sup_{k,l}|A_{k,l}|=\mbox{${\mathrm{O}}$}\left(\frac{\lambda\sqrt{u}}{\eta^{5/2}\sqrt{N}}\right).$
(4.35)

For $\eta>0$ sufficiently large, taking the norm, we get

$\mbox{${\mathbb{E}}$}\left[\left\|\mathrm{e}^{\lambda
g_{N}}-\mathrm{e}^{\lambda\mbox{${\mathbb{E}}$}[g_{N}]}\right\|^{2}\right]=o(1).$
(4.36)

However, $\delta_{N}$ is a bounded analytic function in
$[0,\infty)^{2}\times\mathbb{C}^{+}$. Using the identity theorem from complex
analysis , which states that if two holomorphic functions agree in an open set
of the domain then they must agree everywhere on the domain, we have that
since $\delta_{N}\to 0$ on an open set of the upper-half complex plane, it
must approach 0 everywhere on the upper-half plane. Since the error in (4.35)
can be absorbed in $\tilde{q}_{N,\lambda}(u,z)$, using 4.9 gives us

$\begin{split}&\mbox{${\mathbb{E}}$}[g_{N}(x,u,z)]\\\
&=d_{f}(x)-\int_{0}^{\infty}f(x,y)\mathrm{e}^{-\lambda
d_{f}(y)}\left(\sqrt{u}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{\dot{\iota}vz}\mathrm{e}^{\lambda\mbox{${\mathbb{E}}$}\left[g_{N}\left(y,\frac{v}{\lambda},z\right)\right]}\operatorname{d}{v}\right)\mu_{w}(\operatorname{d}{y})\hskip
2.84544pt+\tilde{q}_{N,\lambda}(u,z),\end{split}$ (4.37)

where the error vanishes in the norm as

$\|\tilde{q}_{N,\lambda}\|=\|q_{N,\lambda}(u,z)+\mbox{${\mathrm{O}}$}(E_{N})\|\leq\sup_{x,u\geq
0}\left|C\frac{\lambda\sqrt{u}}{\eta^{5/2}\sqrt{N}}\right|+E_{N}=o(1).$

Now, consider the function $\tilde{G}_{N}(x,u)$ and the error

$\Delta_{N}(u):=\tilde{G}_{N}(x,u)-\mbox{${\mathbb{E}}$}[\tilde{G}_{N}(x,u)].$

By definition of $\tilde{G}_{N}$, one can see that expanding $\Delta_{N}(u)$
will yield an expression similar to (4.34) modulo $f$, and so, using (4.33)
again, we get that

$\mbox{${\mathbb{E}}$}[|\Delta_{N}|^{2}]\leq
C_{1}C_{f}^{2}\sup_{k,l}|A_{k,l}|=\mbox{${\mathrm{O}}$}\left(\frac{\lambda\sqrt{u}}{\eta^{5/2}\sqrt{N}}\right).$

By taking the norm and again using identity theorem, we get that $\Delta_{N}$
vanishes in $[0,\infty)^{2}\times\mathbb{C}^{+}$ and thus

$\mbox{${\mathbb{E}}$}\left[\left\|\tilde{G}_{N}-\mbox{${\mathbb{E}}$}[\tilde{G}_{N}]\right\|^{2}\right]=o(1).$
(4.38)

A quick inspection of (4.34) shows that in fact we also have the concentration
for $g_{N}$, since the RHS is precisely the upper bound on

$\displaystyle\mbox{${\mathbb{E}}$}[|g_{N}(x,u,z)-\mbox{${\mathbb{E}}$}[g_{N}(x,u,z)]|^{2}],$

and so,

$\mbox{${\mathbb{E}}$}\left[\left\|g_{N}-\mbox{${\mathbb{E}}$}[g_{N}]\right\|^{2}\right]=o(1).$
(4.39)

Finally, comparing (4.37) with the contraction mapping (4.31), we have the
following:

$\begin{split}\mbox{${\mathbb{E}}$}[g_{N}(x,u,z)]&=F_{z}(\mbox{${\mathbb{E}}$}[g_{N}(x,u,z)])+\tilde{q}_{N,\lambda}(u,z),\\\
\phi^{*}(x,u)&=F_{z}(\phi^{*}(x,u)).\end{split}$

So, with $\eta>0$ large enough and $F_{z}$ being a contraction on
$B\subset\operatorname{\mathcal{B}}$ of radius $C_{f}$, we have

$\displaystyle\|\mbox{${\mathbb{E}}$}[g_{N}]-\phi^{*}\|\leq\|F_{z}(\mbox{${\mathbb{E}}$}[g_{N}])-F_{z}(\phi^{*})\|+\|\tilde{q}_{N}\|,$

and consequently,

$\displaystyle\frac{1}{2}\|\mbox{${\mathbb{E}}$}[g_{N}]-\phi^{*}\|\leq\|\tilde{q}_{N}\|.$

Thus, since $\|\mbox{${\mathbb{E}}$}g_{N}\|\leq C_{f}$,

$\|\mbox{${\mathbb{E}}$}[g_{N}]-\phi^{*}\|\xrightarrow{N\to\infty}0.$

As a quick remark, notice that

$\|\phi^{*}\|\leq C_{f},$ (4.40)

since $g_{N}$ is bounded.

Now, since $\mbox{${\mathbb{E}}$}[g_{N}(x,u,z)]$ is an analytic function on
$[0,\infty)^{2}\times\mathbb{C}^{+}$, we have
$\lim_{N\to\infty}\mbox{${\mathbb{E}}$}[g_{N}(x,u,z)]$ is an analytic
function. Again from the identity theorem of complex analysis, since
$\lim_{N\to\infty}\mbox{${\mathbb{E}}$}[g_{N}]$ and $\phi^{*}$ are analytic
and agree on an open set of $[0,\infty)^{2}\times\mathbb{C}^{+}$, they agree
everywhere in the complex domain $[0,\infty)^{2}\times\mathbb{C}^{+}$, and
thus the convergence holds for any $z\in\mathbb{C}^{+}$. Note that _for a
fixed $z\in\mathbb{C}^{+}$_, although both the functionals
$\mbox{${\mathbb{E}}$}[g_{N}]$ and $\phi^{*}$ live in
$(\operatorname{\mathcal{B}},\|\cdot\|_{\operatorname{\mathcal{B}}})$, the
domain of $\phi^{*}$ is $[0,\infty)^{2}\times\mathbb{C}^{+}$ since
$\mbox{${\mathbb{E}}$}g_{N}$ has the domain
$[0,\infty)^{2}\times\mathbb{C}^{+}$. Now, for each $z\in\mathbb{C}^{+}$,
fixing $u$ in the compact set $[0,1]$ gives us that for each $x\in[0,\infty)$
and uniformly over $u\in[0,1]$,

$\mbox{${\mathbb{E}}$}[g_{N}(x,u,z)]\xrightarrow{N\to\infty}\phi^{*}(x,u)$
(4.41)

∎

We can now prove Theorem 2.9.

###### Proof of Theorem 2.9.

Equation (4.38) proves the concentration statement of Theorem 2.9. Recall that
we had shown that

$\mbox{${\mathbb{E}}$}\left[\mathrm{e}^{\dot{\iota}ur^{N}_{jj}}\right]=1-\mathrm{e}^{-\lambda
d_{j}}\sqrt{u}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{\dot{\iota}vz}\mbox{${\mathbb{E}}$}\left[\mathrm{e}^{\lambda
g_{N}\left(w_{j},\frac{v}{\lambda},z\right)}\right]\operatorname{d}{v}+q_{N,\lambda}(u,z),$

and so,

$\displaystyle\mbox{${\mathbb{E}}$}[G_{N}(u,z)]$
$\displaystyle=\frac{1}{N}\sum_{j=1}^{N}\mbox{${\mathbb{E}}$}[\mathrm{e}^{\dot{\iota}ur^{N}_{jj}}]$
$\displaystyle=1-\frac{1}{N}\sum_{j=1}^{N}\mathrm{e}^{-\lambda
d_{j}}\sqrt{u}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{\dot{\iota}vz}\mbox{${\mathbb{E}}$}\left[\mathrm{e}^{\lambda
g_{N}\left(w_{j},\frac{v}{\lambda},z\right)}\right]\operatorname{d}{v}+q_{N,\lambda}(u,z).$
(4.42)

Next, we see that the function

$\displaystyle\tilde{I}_{g}(y)=\mathrm{e}^{-\lambda
d_{f}(y)}\sqrt{u}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{\dot{\iota}vz}\mathrm{e}^{\lambda\phi^{*}(y,v/\lambda)}\operatorname{d}{v}$

is Lipschitz by using an argument similar to Lemma 4.6. Thus, we get

$\displaystyle\mbox{${\mathbb{E}}$}[G_{N}(u,z)]=1-\int_{0}^{\infty}\mathrm{e}^{-\lambda
d_{f}(y)}\left(\sqrt{u}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{\dot{\iota}vz}\mbox{${\mathbb{E}}$}\left[\mathrm{e}^{\lambda
g_{N}\left(y,\frac{v}{\lambda},z\right)}\right]\operatorname{d}{v}\right)\mu_{w}(\operatorname{d}{y})\hskip
2.84544pt+\tilde{q}_{N,\lambda}(u,z).$

Since from Proposition 4.7 we have concentration for $g_{N}$, using inequality
(4.9) we have that

$\mbox{${\mathbb{E}}$}[G_{N}(u,z)]=1-\int_{0}^{\infty}\mathrm{e}^{-\lambda
d_{f}(y)}\sqrt{u}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{\dot{\iota}vz}\mathrm{e}^{\lambda\mbox{${\mathbb{E}}$}\left[g_{N}\left(y,\frac{v}{\lambda},z\right)\right]}\operatorname{d}{v}\hskip
5.69046pt\mu_{w}(\operatorname{d}{y})+\tilde{q}_{N,\lambda}(u,z).$

Finally, taking the limit $N\to\infty$ gives us

$\lim_{N\to\infty}\mbox{${\mathbb{E}}$}[G_{N}(u,z)]=1-\int_{0}^{\infty}\mathrm{e}^{-\lambda
d_{f}(y)}\sqrt{u}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{\dot{\iota}vz}\mathrm{e}^{\lambda\phi^{*}(y,v/\lambda)}\operatorname{d}{v}\hskip
5.69046pt\mu_{w}(\operatorname{d}{y}),$ (4.43)

completing the proof of Theorem 2.9. ∎

###  4.2. Deriving the expression for the Stieltjes Transform

Since we took $u$ to be in $[0,1]$, we can take a derivative with respect to
$u$ and evaluate it at $u=0$. Recall from equation (4.42) that we have

$\displaystyle\begin{split}\mbox{${\mathbb{E}}$}[G_{N}(u,z)]&=\frac{1}{N}\mbox{${\mathbb{E}}$}\sum_{j=1}^{N}\mathrm{e}^{\dot{\iota}ur^{N}_{jj}}\\\
&=1-\frac{1}{N}\sum_{j=1}^{N}\mathrm{e}^{-\lambda
d_{j}}\sqrt{u}\int_{0}^{\infty}\frac{\operatorname{J_{1}}(2\sqrt{uv})}{\sqrt{v}}\mathrm{e}^{\dot{\iota}vz}\mbox{${\mathbb{E}}$}\left[\mathrm{e}^{\lambda
g_{N}\left(w_{j},\frac{v}{\lambda},z\right)}\right]\operatorname{d}{v}+q_{N,\lambda}(u,z).\end{split}$

Note that by definition, $G_{N}(u,z)$ is a bounded function, and thus by DCT,
limit operations can be interchanged with expectation. We would like to take a
derivative with respect to $u$ and evaluate at $u=0$ to extract out
$\operatorname{tr}(\operatorname{R}_{\mathbf{A}_{N}}(z))$ from the LHS of
(4.42). On the other hand, we would first like to take $N\to\infty$ for the
RHS to remove the error term. To interchange these operations, we have the
following result.

######  Proposition 4.1.

Both the limits $\lim_{N\to\infty}\left.\frac{\partial}{\partial
u}\mbox{${\mathbb{E}}$}[G_{N}(u,z)]\right|_{u=0}$ and
$\left.\frac{\partial}{\partial
u}\lim_{N\to\infty}\mbox{${\mathbb{E}}$}[G_{N}(u,z)]\right|_{u=0}$ exist and
are equal.

###### Proof.

We fix a $z\in\mathbb{C}^{+}$. Now,
$\lim_{N\to\infty}\mbox{${\mathbb{E}}$}[G_{N}(u,z)]$ exists due to the RHS of
(4.42), which we denote by $G(u,z)$. If we define $H_{N}(u,z)$ and $H(u,z)$ as

$\displaystyle H_{N}(u,z)$
$\displaystyle=\frac{\mbox{${\mathbb{E}}$}[G_{N}(u,z)]-\mbox{${\mathbb{E}}$}[G_{N}(0,z)]}{u},$
$\displaystyle H(u,z)$ $\displaystyle=\frac{G(u,z)-G(0,z)}{u}.$

Then,

$\displaystyle\lim_{u\to 0}H_{N}(u,z)$
$\displaystyle=\left.\frac{\partial}{\partial
u}\mbox{${\mathbb{E}}$}[G_{N}(u,z)]\right|_{u=0},$ $\displaystyle\lim_{u\to
0}H(u,z)$ $\displaystyle=\left.\frac{\partial}{\partial
u}G(u,z)\right|_{u=0}.$

We would like to claim

$\lim_{N\to\infty}\left.\frac{\partial}{\partial
u}\mbox{${\mathbb{E}}$}[G_{N}(u,z)]\right|_{u=0}=\left.\frac{\partial}{\partial
u}G(u,z)\right|_{u=0}.$