Central limit theorem for the principal eigenvalue and eigenvector of Chung–Lu random graphs

The spectral properties of adjacency matrices play an important role in various areas of network science. In the present paper we consider an inhomogeneous version of the Erdős–Rényi random graph called the Chung–Lu random graph and we derive a central limit theorem for the principal eigenvalue and eigenvector of its adjacency matrix.

1.1.1. Setting

1.1.2. Principal eigenvalue and eigenvector

The largest eigenvalue of the adjacency matrix A and its corresponding eigenvector, written as $(\lambda_1,v_1)$, contain important information about the random graph. Several community detection techniques depend on a proper understanding of these quantities [1, 25, 32], which in turn play an important role for various measures of network centrality [26, 27] and for the properties of dynamical processes (such as the spread of an epidemic) taking place on networks [12, 28]. For Erdős–Rényi random graphs, it was shown in [24] that with high probability (whp in the following) λ₁ scales like

$$\begin{align} \lambda_1 \sim \max\{\sqrt{D_\infty}, np\}, \qquad n \to \infty, \end{align} \tag{ 1.1 }$$

where $D_\infty$ is the maximum degree. This result was partially extended to $\mathcal{G}_n(\vec{d}_n)$ in [16], and more recently to a class of inhomogeneous Erdős–Rényi random graphs in [5, 6]. For a related discussion on the behaviour of $(\lambda_1,v_1)$ in real-world networks, see [12, 28]. In the present paper we analyse the fluctuations of $(\lambda_1,v_1)$. We will be interested specifically in the case where λ₁ is detached from the bulk, which for Erdős–Rényi random graphs occurs when $\lambda_1 \sim np$ whp, and for Chung–Lu random graphs when $\lambda_1 \sim m_2/m_1$, where $m_2 = \sum_{i\in [n]} d_i^2$. Note that the quotient $m_2/m_1$ arises from the fact that the average adjacency matrix is rank one and that its only non-zero eigenvalue is $m_2/m_1$. Such rank-one perturbations of a symmetric matrix with independent entries became prominent after the work in [4]. Later studies extended this work to finite-rank perturbations [3, 7, 10, 11, 20, 21]. Erdős–Rényi random graphs differ, in the sense that perturbations live on a scale different from $\sqrt{n}$. For Chung–Lu random graphs we assume that $m_2/m_1\to\infty$.

In the setting of inhomogeneous Erdős–Rényi random graphs, finite-rank perturbations were studied in [13]. In that paper the connection probability between between i and j is given by $p_{ij} = \varepsilon_n f(i/n, j/n)$, where $f\colon\,[0,1]^2\to [0,1]$ is almost everywhere continuous and of finite rank, $\varepsilon_n \in [0,1]$ and $n\varepsilon_n\gg (\log n)^{8}$. However, for a Chung–Lu random graph with a given degree sequence it is not always possible to construct an almost everywhere continuous function f independent of n such that $\varepsilon_n f(i/n, j/n) = d_i d_j/m_1$. In the present paper we extend the analysis in [13] to Chung–Lu random graphs by focussing on $(\lambda_1,v_1)$. For Erdős–Rényi random graphs it was shown in [18, 19] that λ₁ satisfies a central limit theorem (CLT) and that v₁ aligns with the unit vector. These papers extend the seminal work carried out in [22].

1.1.3. Chung–Lu random graphs

1.1.4. Outline

1.2.1. Set-up

Let $\mathbb{G}_n$ be the set of simple graphs with n vertices. Let $\vec{d}_n = (d_i)_{i \in [n]}$ be a sequence of degrees, such that $d_i\in \mathbb{N}$ for all $i\in [n]$ and abbreviate

$$\begin{align*} m_k = \sum_{i \in [n]} (d_i)^k, \qquad d_\uparrow = \max_{i \in [n]} d_i, \qquad d_\downarrow= \min_{i\in [n]} d_i. \end{align*}$$

Note that these numbers depend on n, but in the sequel we will suppress this dependence. For each pair of vertices $i,j$ (not necessarily distinct), we add an edge independently with probability

$$\begin{align} p_{ij}=\frac{d_id_j}{m_1}. \end{align} \tag{ 1.2 }$$

The resulting random graph, which we denote by $\mathcal{G}_n(\vec{d}_n)$, is referred to in the literature as the Chung–Lu random graph. In [15] it was assumed that $d_\uparrow^2 \leqslant m_1$ to ensure that $p_{ij} \leqslant 1$. In the present paper we need sharper restrictions.

Assumption 1.1. Throughout the paper we need two assumptions on $\vec{d}_n$ as $n\to\infty$:

• Connectivity and sparsity: There exists a ξ > 2 such that
  $$\begin{align*} (\log n)^{2\xi} \ll d_\uparrow \ll n^{1/2}. \end{align*}$$
• Bounded inhomogeneity: $d_\downarrow \asymp d_\uparrow$.

 $\spadesuit$

The lower bound in assumption 1.1(D1) guarantees that the random graph is connected whp and that it is not too sparse. The upper bound is needed in order to have $d_\uparrow = {{\mathrm{o}}}(\sqrt{m_1})$, which implies that (1.2) is well defined. Assumption 1.1(D2) is a restriction on the inhomogeneity of the model and requires that the smallest and the largest degree are comparable.

Remark 1.2. The lower bound on $d_\uparrow$ in assumption 1.1(D1) can be seen as an adaptation to our setting of the main condition in [16, theorem 2.1] for the asymptotics of λ₁. As mentioned in section 1.1, under the assumption

$$\begin{align*} \frac{m_2}{m_1}\gg \sqrt{d_\uparrow}\,(\log n)^\xi, \end{align*}$$

[16] shows that $\lambda_1 = [1+{{\mathrm{o}}}(1)]\,m_2/m_1$ whp. It is easy to see that the above condition together with assumption 1.1(D2) gives the lower bound in assumption 1.1(D1). $\spadesuit$

Remark 1.3. When $d_\uparrow\ll n^{1/6}$, [33, theorem 6.19] implies that our results also hold for the Generalized Random Graph (GRG) model with the same average degrees. This model is defined by choosing connection probabilities of the form

$$\begin{align*} p_{ij}=\frac{d_id_j}{m_1+d_id_j}, \end{align*}$$

and arises in statistical physics as the canonical ensemble constrained on the expected degrees, which is also called the canonical configuration model. Note that in the above connection probability, d_i plays the role of a hidden variable, or a Lagrange multiplier controlling the expected degree of vertex i, but does not in general coincide with the expected degree itself. However, under the assumptions considered here, d_i does coincide with the expected degree asymptotically. The reader can find more about GRG and their use in [33, Chapter 6], and about their role in statistical physics in [31]. In the corresponding microcanonical ensemble the degrees are not only fixed in their expectation but they take a precise deterministic value, which corresponds to the microcanonical configuration model. The two ensembles were found to be nonequivalent in the limit as $n\to\infty$ [30]. This result was shown to imply a finite difference between the expected values of the largest eigenvalue λ₁ in the two models [17] when the degree sequence was chosen to be constant ($d_i = d$ for all $i \in [n]$). In this latter case the canonical ensemble reduces to the Erdős–Rényi random graph with $p = d/n$, while the microcanonical ensemble reduces to the d-regular random graph model. Although ensemble nonequivalence is not our main focus here, we will briefly relate some of our results to this phenomenon. $\spadesuit$

1.2.2. Notation

Let A be the adjacency matrix of $\mathcal{G}_n(\vec{d}_n)$ and $\mathbb{E}[A]$ its expectation. The (i, j)th entry of $\mathbb{E}[A]$ equals to p_ij in (1.2). The (i, j)th entry of $A-\mathbb{E}[A]$ is an independent centered Bernoulli random variable with parameter p_ij . Let $\lambda_1\geqslant \ldots\geqslant \lambda_n$ be the eigenvalues of A and let $v_1, \ldots, v_n$ be the corresponding eigenvectors. The vector e will be the n-dimensional column vector

$$\begin{align} e=\frac{1}{\sqrt{m_1}}( d_1, \ldots, d_n)^t, \end{align} \tag{ 1.3 }$$

where t stands for transpose. It is easy to see that $\mathbb{E}[A] = ee^t$.

Definition 1.4. Following [19], we say that an event $\mathcal{E}$ holds with $(\xi,\nu)$-high probability (written $(\xi,\nu)$-hp) when there exist ξ > 2 and ν > 0 such that

$$\begin{align} \mathbb{P}(\mathcal{E}^c) \leqslant \mathrm{e}^{-\nu (\log n)^\xi}. \end{align} \tag{ 1.4 }$$

   $\spadesuit$

Note that this is different from the classical notion of whp, because it comes with a specific rate.

Remark 1.5. Our results hold for any ν > 0 as soon as ξ > 2 (think of ν = 1). The role of ν becomes important when we consider specific subsets $\mathcal{S}$ of the event space and split into $\mathcal{S}\cap\mathcal{E}$ and $\mathcal{S}\cap\mathcal{E}^c$ (see e.g. [19]).$\spadesuit$

We write $\stackrel{w}{\longrightarrow}$ to denote weak convergence as $n\to\infty$, and use the symbols ${{\mathrm{o}}},\mathrm{O}$ to denote asymptotic order for sequences of real numbers.

1.2.3. CLT for the principal eigenvalue

1.2.4. CLT for the principal eigenvector

We place the theorems in their proper context.

• (a)  
  Theorems 1.6 and 1.7 provide a CLT for $\lambda_1,v_1$. We note that $m_2/m_1$ is the leading order term in the expansion of λ₁, while $m_1m_3/m_2^2$ is a correction term. We observe that theorem 1.6(I) does not follow from the results in [16], because the largest eigenvalue need not be uniformly integrable and also the second order expansion is not considered there. We also note that in theorem 1.6(II) the centering of the largest eigenvalue, $\mathbb{E}[\lambda_1]$, cannot be replaced by its asymptotic value as the error term is not compatible with the required variance.
• (b)  
  The lower bound in assumption 1.1(D1) is needed to ensure that the random graph is connected, and is crucial because the largest eigenvalue is very sensitive to connectivity properties. Assumption 1.1(D2) is needed to control the inhomogeneity of the random graph. It plays a crucial role in deriving concentration bounds on the central moments $\langle e, (A-\mathbb{E}[A])^k e\rangle$, $k \in \mathbb{N}$, with the help of a result from [19]. Further refinements may come from different tools, such as the non-backtracking matrices used in [5, 6]. While assumption 1.1(D1) appears to be close to optimal, assumption 1.1(D2) is far from optimal. It would be interesting to allow for empirical degree distributions that converge to a limiting degree distribution with a power law tail.
• (c)  
  As already noted, if the expected degrees are all equal to each other, i.e. $d_i = d$ for all $i \in [n]$, then the Chung–Lu random graph, or canonical configuration model, reduces to the homogeneous Erdős–Rényi random graph with $p = d/n$, while the corresponding microcanonical configuration model reduces to the homogeneous d-regular random graph model (here, all models allow for self-loops). This implies that, for the homogeneous Erdős–Rényi random graph with connection probability $p \gg (\log n)^{2\xi}/n$, ξ > 2, theorem 1.6(I) reduces to

  $$\begin{align*} \mathbb{E} [\lambda_1] = np+1+{{\mathrm{o}}}(1), \qquad n\to\infty, \end{align*}$$

  while theorem 1.6(II) reduces to

  $$\begin{align*} \frac{1}{\sqrt{p}} \left(\lambda_1- \mathbb{E}[\lambda_1]\right) \stackrel{w}{\longrightarrow} \mathcal{N}(0, 2), \qquad n \to \infty. \end{align*}$$

  Both these properties were derived in [18] for homogeneous Erdős–Rényi random graphs and also for rank-1 perturbations of Wigner matrices. In [17], the fact that $\mathbb{E}[\lambda_1]$ in the canonical ensemble differs by a finite amount from the corresponding expected value (here, d = np) in the microcanonical ensemble (d-regular random graph) was shown to be a signature of ensemble nonequivalence.
• (d)  
  In case $d_i = d$ for all $i \in [n]$, theorem 1.7(III) reduces to the following CLT, which was not covered by [18] and [17].

Corollary 1.8. For the Erdős–Rényi random graph with $(\log n)^{4\xi}/n \ll p \ll n^{-1/2}$ for some ξ > 2,

$$\begin{align*} n\sqrt{\frac{p}{1-p}}\left( v_1(i)-\frac{1}{\sqrt{n}}\right) \stackrel{w}{\longrightarrow} \mathcal{N}(0,1), \qquad n \to \infty. \end{align*}$$

Note that, in the corresponding microcanonical ensemble (d-regular random graph), v₁ coincides with the constant vector where $v_1(i) = 1/\sqrt{n}$ for all $i \in [n]$. Therefore in the canonical ensemble each coordinate $v_1(i)$ has Gaussian fluctuations around the corresponding deterministic value for the microcanonical ensemble. This behaviour is similar to the degrees having, in the canonical configuration model, either Gaussian (in the dense setting) or Poisson (in the sparse setting) fluctuations around the corresponding deterministic degrees for the microcanonical configuration model [23].

• (e)  
  One way to satisfy assumption 1.1 is to specify functions $\omega,c_1,\ldots,c_n$, satisfying $(\log n)^{2\xi}\ll\omega(n)\ll \sqrt{n}$ and $c \leqslant c_1(n) \leqslant \ldots \leqslant c_n(n)\leqslant C$ with $c,C\geqslant0$, such that

  $$\begin{align*} d_i(n)=c_i(n)\omega(n), \qquad p_{ij}=\frac{c_ic_j}{\frac{1}{n}\sum_k c_k}\frac{\omega}{n}. \end{align*}$$

  The reason why we avoid such a description is that our setting is potentially broader. The concentration estimate in lemma 3.4 requires us to assume homogeneous degree sequences as above, while theorem 1.6(I) holds for much more general degree sequences. A further refinement of lemma 3.4 may be possible. The advantage of the above description is that it makes the scale $\omega(n)$ on which the degrees live explicit. However, most of the bounds in our proofs depend on some power of $d_\uparrow$, up to some multiplicative constant. This means that, in the bounded inhomogeneity setting, expressing the asymptotics through $\omega(n)$ or $d_\uparrow$ are equivalent. Bounds expressed through $\omega(n)$ would cease to be meaningful as soon as we manage to push beyond the bounded inhomogeneity setting of our model, while the skeleton of our proof would still hold.
• (f)  
  In [14] the empirical spectral distribution of A was considered under the assumption that

  $$\begin{align*} (d_\uparrow)^2/m_1 \ll 1 \ll n(d_\uparrow)^2/m_1, \end{align*}$$

  which is weaker than assumption 1.1. It was shown that if $\mu_n \stackrel{w}{\longrightarrow} \mu$ with $\mu_n = n^{-1} \sum_{i = 1}^n \delta_{d_i/d_\uparrow}$ and µ some probability distribution on $\mathbb{R}$, then

  $$\begin{align*} \mathrm{ESD}\left(\frac{A}{\sqrt{n(d_\uparrow)^2/m_1}}\right) \stackrel{w}{\longrightarrow} \mu \boxtimes \mu_\mathrm{sc} \end{align*}$$

  with $\mu_{\mathrm{sc}}$ the Wigner semicircle law and $\boxtimes$ the free multiplicative convolution. Since $\mu\boxtimes \mu_{\mathrm{sc}}$ is compactly supported, this shows that the scaling for the largest eigenvalue and the spectral distribution are different.

In figure 1 we show histograms for the quantity

$$\begin{align*} \bar{\lambda}_1=\frac{m_2}{m_1\sigma_1} (\lambda_1 - \mathbb{E}[\lambda_1]), \end{align*}$$

which should be close to normal with mean 0 and variance 2 (for $\mathbb{E}[\lambda_1]$ the correction term ${{\mathrm{o}}}(1)$ is neglected). The convergence is fast: already for n = 500 the Gaussian shape emerges and represents an excellent fit: the sample mean µ is close to 0 and the sample standard deviation σ is close to $\sqrt{2}$.

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In figure 2 we show histograms for the quantity

$$\begin{align*} \bar{v}_1(i)=\frac{m^{3/2}_2}{m_1s_1(i)}\left(v_1(i)-d_i/\sqrt{m_2}\,\right), \end{align*}$$

which should be close to normal with mean 0 and variance 1. The fit is again excellent.

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What happens when the degrees are of order $\log n$? As can be seen in figure 3, in that range the Gaussian approximation for the largest eigenvalue is visibly worse, especially for the centering. The same happens for the components of the largest eigenvector, as can be seen in figure 4, where the Gaussian shape is lost and two peaks appear.

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Zoom In Zoom Out Reset image size

In order to study λ₁, we need good bounds on the spectral norm of H. The spectral norm of matrices with inhomogeneous entries has been studied in a series of papers [2, 5, 6] for different density regimes.

An important role is played by $\lambda_1(\mathbb{E}[A])$. In recent literature this quantity has been shown to play a prominent role in the so-called BBP-transition [4]. Given our setting (1.2), it is easy to see that

$$\begin{align} \lambda_1(\mathbb{E}[A])=\frac{m_2}{m_1}, \end{align} \tag{ 3.3 }$$

while all other eigenvalues of $\mathbb{E}[A]$ are zero.

Remark 3.1. Since $d_\downarrow\leqslant \frac{m_2}{m_1}\leqslant d_\uparrow$, assumption 1.1(D2) implies that

$$\begin{align} \frac{m_2}{m_1}\asymp d_\uparrow. \end{align} \tag{ 3.4 }$$

   $\spadesuit$

We start with the following lemma, which ensures concentration of λ₁ and is a direct consequence of the results in [6] (which matches assumption 1.1). In particular, we use [6, theorem 3.2] to check that the boundaries of the bulk of the spectral distribution live on a scale smaller than the scale of λ₁.

Lemma 3.2. Under assumption 1.1, with $(\xi,\nu)$-hp

$$\begin{align*} \left|\frac{\lambda_1(A)-\lambda_1(\mathbb{E}[A])}{\lambda_1(\mathbb{E}[A])}\right| = \mathrm{O}\left(\frac{1}{\sqrt{d_\uparrow}}\right), \qquad n \to \infty, \end{align*}$$

and consequently

$$\begin{align*} \frac{\lambda_1(A)}{\lambda_1\left(\mathbb{E}[A]\right)} \overset{\mathbb{P}}\to 1, \qquad n \to \infty. \end{align*}$$

Proof. In the proof it is understood that all statements hold with $(\xi,\nu)$-hp in the sense of (1.4). Let $A = H+\mathbb{E}[A]$. Due to Weyl's inequality, we have that

$$\begin{align*} \lambda_1(\mathbb{E}[A])-\|H\|\leqslant\lambda_1(A)\leqslant\lambda_1(\mathbb{E}[A])+\|H\|. \end{align*}$$

From [6, theorem 3.2] we know that there is a universal constant C > 0 such that

$$\begin{align*} \mathbb{E}\left[\|A-\mathbb{E} [A]\|\right]=\mathbb{E}\left[\|H\|\right]\leqslant \sqrt{d_\uparrow}\left(2+\frac{C}{q}\sqrt{\frac{\log n}{1\vee\log\left(\frac{\sqrt{\log n}}{q}\right)}}\right), \end{align*}$$

where

$$\begin{align*} q=\sqrt{d_\uparrow}\wedge n^{1/10} \kappa^{-1/9} \end{align*}$$

with κ defined by

$$\begin{align*} \kappa = \max_{ij} \frac{p_{ij}}{d_\uparrow/n}=\frac{n d_\uparrow}{m_1}. \end{align*}$$

Thanks to assumption 1.1(D2), we have $\kappa = \mathrm{O}(1)$. By remark 3.1 of [6, Remark 3.1] (which gives us that $q = \sqrt{d_\uparrow}$ for n large enough) and assumption 1.1, we get that

$$\begin{align} \mathbb{E}\left[\|H\|\right] \leqslant \begin{cases} \sqrt{d_\uparrow}\left(2+\frac{C\sqrt{\log n}}{\sqrt{d_\uparrow}}\right), \qquad (\log n)^{2\xi}\leqslant\sqrt{d_\uparrow} \leqslant n^{1/10} \kappa^{-1/9},\\[9pt] \sqrt{d_\uparrow}\left(2+\frac{C^{^{\prime}}\sqrt{\log n}}{n^{1/10}}\right), \qquad \sqrt{d_\uparrow}\geqslant n^{1/10} \kappa^{-1/9}. \end{cases} \end{align} \tag{ 3.5 }$$

Using [8, example 8.7] or [6, equation 2.4] (the Talagrand inequality), we know that there exists a universal constant c > 0 such that

$$\begin{align*} \mathbb{P}\left( \left|\|H\|-\mathbb{E}[\|H\|]\right|>t\right)\leqslant 2\mathrm{e}^{-ct^2}. \end{align*}$$

For $t = \sqrt{\nu (\log n)^\xi}$,

$$\begin{align} \mathbb{E}[\|H\|]-\sqrt{\nu} (\log n)^{\xi/2}\leqslant\|H\|\leqslant \mathbb{E}[\|H\|]+\sqrt{\nu} (\log n)^{\xi/2}. \end{align} \tag{ 3.6 }$$

Thus, we have

$$\begin{align} \left|\lambda_1(A)-\lambda_1(\mathbb{E}[A])\right|\leqslant \|H\|\leqslant \sqrt{d_\uparrow}(2+{{\mathrm{o}}}(1))+ \sqrt{\nu} (\log n)^{\xi/2}. \end{align} \tag{ 3.7 }$$

Using that $\lambda_1(\mathbb{E}[A]) = m_2/m_1$, we have that with $(\xi,\nu)$-hp the following bound holds:

$$\begin{align*} \left|\frac{\lambda_1(A)-\lambda_1(\mathbb{E}[A])}{\lambda_1(\mathrm{E}[A])}\right| \leqslant \frac{\sqrt{d_\uparrow}}{m_2/m_1}\left( 2+{{\mathrm{o}}}(1)\right)+\frac{\sqrt{\nu} (\log n)^{\xi/2}}{m_2/m_1} = \mathrm{O}\left(\frac{\sqrt{d_\uparrow}}{m_2/m_1}\right). \end{align*}$$

Via assumption 1.1 and (3.4) the claim follows.

Remark 3.3. 

• (a)  
  The proof of lemma 3.2 works well if we replace assumption 1.1(D2) by a milder condition. Indeed, the former is directly linked to the parameter κ that appears in the proof of lemma 3.2 and in the proof of [6, theorem 3.2], which contains a more general condition on the inhomogeneity of the degrees.
• (b)  
  Note that a consequence of proof of lemma 3.2 is that with $(\xi,\nu)$-hp

  $$\begin{align} \frac{\|H\|}{\lambda_1(A)}\leqslant 1-C_0 \end{align} \tag{ 3.8 }$$

  for some $C_0\in (0,1)$. This allows us to claim that with $(\xi,\nu)$-hp the inverse

  $$\begin{align} \left(I- \frac{H}{\lambda_1(A)}\right)^{-1} \end{align} \tag{ 3.9 }$$

  exists.

   $\spadesuit$

Lemma 3.4. Let $1\leqslant k\leqslant L$. Then, under assumption 1.1, with $(\xi,\nu)$-hp

$$\begin{align*} \left| \left\langle e, H^k e\right\rangle - {\mathbb E} \left[\left\langle e, H^k e\right\rangle \right]\right| \leqslant C \frac{m_2}{m_1}\frac{d_\uparrow^{\frac{k}{2}}(\log n)^{k\xi}}{\sqrt{n}}, \end{align*}$$

i.e.

$$\begin{align*} \max_{1 \leqslant k \leqslant L} \mathbb{P}\left( \left|\left\langle e, H^k e\right\rangle - {\mathbb E} \left[\left\langle e, H^k e\right\rangle \right]\right| > \frac{C (\log n)^{k\xi}d_\uparrow^{\frac{k}{2}}}{\sqrt{n}} \frac{m_2}{m_1}\right) \leqslant \mathrm{e}^{-\nu(\log n)^\xi}, \qquad n\geqslant n_1(\nu,\xi). \end{align*}$$

Lemma 3.4 is a generalization to the inhomogeneous setting of [19, lemma 6.5]. We skip the proof because it requires a straightforward modification of the arguments in [19].

Lemma 3.5. Under assumption 1.1, for $2\leqslant k\leqslant L$, there exists a constant C > 0 such that

$$\begin{align} \mathbb{E}\left[ \left\langle e, H^k e\right\rangle\right] \leqslant \frac{m_2}{m_1} (Cd_\uparrow)^{k/2}. \end{align} \tag{ 3.10 }$$

Proof. Let $\mathcal E$ be the high probability event defined by (3.6), i.e.

$$\begin{align*} \| H\| \leqslant \mathrm{E}[\|H\|] + \sqrt{\nu}(\log n)^{\xi/2}\leqslant d_\uparrow\left( 1+ \mathrm{O}\left( \frac{(\log n)^{\xi/2}}{d_\uparrow}\right)\right). \end{align*}$$

Due to assumption 1.1(D1) we can bound the right-hand side by $Cd_\uparrow$. Since $\|e\|_2^2 = m_2/m_1$, on this event we have

$$\begin{align*} \mathbb{E}\left[ \left( \left\langle e, H^k e\right\rangle\right){\mathbf{1}}_{\mathcal E}\right] \leqslant \|e\|_2^2\, \mathbb{E}[\|H\|^k{\mathbf{1}}_{\mathcal E}]\leqslant \frac{m_2}{m_1} (Cd_\uparrow)^{k/2} . \end{align*}$$

We show that the expectation when evaluated on the complementary event is negligible. Indeed, observe that

$$\begin{align*} \mathbb{E}\left[ \left\langle e, H^k e\right\rangle\right] &= \mathbb{E}\left(\sum_{i_1, \ldots, i_{k+1}=1}^n e_{i_1}e_{i_{k+1}} \prod_{j=1}^k H(i_j, i_{j+1})\right)^2\\ &\leqslant \left( \frac{n^{k+1} d_\uparrow^2}{m_1}\right)^2\leqslant C\mathrm{e}^{(2k+2)\log n} \leqslant \mathrm{e}^{2(\log n)^2}, \end{align*}$$

where in the last inequality we use that $d_\uparrow = {{\mathrm{o}}}(\sqrt{m_1})$. This, combined with the exponential decay of the event $\mathcal E^c$, gives

$$\begin{align*} \mathbb{E}\left[ \left\langle e, H^k e\right\rangle{\mathbf{1}}_{A^c}\right] \leqslant C\mathrm{e}^{- \nu(\log n)^{\xi}}, \end{align*}$$

and so the claim follows.

We denote the event in lemma 3.2 by $\mathcal E$, which has high probability. As noted in remark 3.3(b), $I-\frac{H}{\lambda_1}$ is invertible on $\mathcal E$. Hence, expanding on $\mathcal E$, we get

$$\begin{align*} \lambda_1= \sum_{k=0}^\infty \left\langle e, \frac{H^k}{\lambda_1^k} e\right\rangle . \end{align*}$$

We split the sum into two parts:

$$\begin{align} \lambda_1 = \sum_{k=0}^{L} \frac{\left\langle e, H^k e\right\rangle}{\lambda_1^{k}} + \sum_{k=L+1}^\infty \frac{\left\langle e, H^k e\right\rangle}{\lambda_1^{k}}. \end{align} \tag{ 3.11 }$$

First we show that we may ignore the second sum. To that end we observe that, by assumption 1.1 (D1), on the event $\mathcal E$ we can estimate

$$\begin{align} \left| \sum_{k= L+1}^\infty \frac{\left\langle e, H^k e\right\rangle}{\lambda_1^{k}}\right| &\leqslant \sum_{k=L+1}^\infty \frac{\|e\|_2^2 \|H\|^k}{\lambda_1^k} \leqslant \sum_{k=L+1}^\infty \frac{m_2}{m_1} \frac{d_\uparrow^{k/2}}{(Cm_2/m_1)^k}\nonumber\\ &\leqslant \sum_{k=L+1}^\infty \frac{C^{^{\prime}}}{d_\uparrow^{k/2-1}}=\mathrm{O}\left(\mathrm{e}^{-c\log \sqrt{n}}\right). \end{align} \tag{ 3.12 }$$

Because of (3.12) and the fact that $\mathbb{E}(\left\langle e,H e\right\rangle) = 0$, (3.11) reduces to

$$\begin{align*} \lambda_1&=\sum_{k=3}^{L} \frac{\mathbb{E} \left[\left\langle e, H^k e\right\rangle\right]}{\lambda_1^{k}} + \sum_{k=3}^{L} \frac{\left\langle e, H^k e\right\rangle - \mathbb{E}\left[\left\langle e, H^k e\right\rangle\right]}{\lambda_1^{k}}\\ &\quad +\left\langle e, e\right\rangle + \frac{1}{\lambda_1} \left\langle e, H e\right\rangle + \frac{1}{\lambda_1^2} \left\langle e, H^2 e\right\rangle+{{\mathrm{o}}}(1). \end{align*}$$

Next, we estimate the second sum in the above equation. Using lemma 3.2, we get

$$\begin{align*} \left|\sum_{k=3}^{L} \frac{\left\langle e, H^k e\right\rangle - \mathbb{E} \left[\left\langle e, H^k e\right\rangle \right]}{\lambda_1^{k}}\right| \leqslant \sum_{k=3}^{L} \frac{C d_\uparrow^{\frac{k}{2}}(\log n)^{k\xi}}{\sqrt{n}(m_2/m_1)^{k-1}} \leqslant \sum_{k=3}^{L} \frac{C(\log n)^{k\xi}}{\sqrt{n}d_\uparrow^{k/2-1}} \leqslant\mathrm{O}\left( \frac{C(\log n)^{\xi+1}}{\sqrt{nd_\uparrow}}\right) = {{\mathrm{o}}}(1). \end{align*}$$

From lemma 3.5 we have

$$\begin{align*} \sum_{k=3}^L \frac{\mathbb{E} \left\langle e, H^k e\right\rangle }{\lambda_1^{k}} \leqslant \sum_{k=3}^L \frac{\frac{m_2}{m_1} (Cd_\uparrow)^{k/2} }{\left(m_2/m_1\right)^k} =\mathrm{O}\left( \frac{1}{\sqrt{d_\uparrow}}\right)= {{\mathrm{o}}}(1), \end{align*}$$

where the last estimate follows from assumption 1.1(D1). Hence, on $\mathcal E$,

$$\begin{align*} \lambda_1= \left\langle e, e\right\rangle +\frac{1}{\lambda_1} \left\langle e, H e\right\rangle+ \frac{\left\langle e, H^2 e\right\rangle}{\lambda_1^2} + {{\mathrm{o}}}(1). \end{align*}$$

Iterating the expression for λ₁ in the right-hand side, we get

$$\begin{align*} \lambda_1 &= \left\langle e, e\right\rangle + \left\langle e, H e\right\rangle \left( \left\langle e, e\right\rangle + \frac{1}{\lambda_1}\left\langle e, He\right\rangle +\frac{1}{\lambda_1^2} \left\langle e, H^2 e\right\rangle + {{\mathrm{o}}} (1) \right)^{-1} \\ &\quad + \left\langle e, H^2 e\right\rangle \left( \left\langle e, e\right\rangle + \frac{1}{\lambda_1}\left\langle e, H e\right\rangle +\frac{1}{\lambda_1^2} \left\langle e, H^2 e\right\rangle + {{\mathrm{o}}} (1) \right)^{-2} + {{\mathrm{o}}} (1). \end{align*}$$

Expanding the second and third term we get,

$$\begin{align*} \lambda_1&= \left\langle e, e\right\rangle + \frac{\left\langle e, H e\right\rangle}{\left\langle e, e\right\rangle }\left( 1 - \frac{\left\langle e, H e\right\rangle }{\lambda_1 \left\langle e, e\right\rangle } -\frac{\left\langle e, H^2 e\right\rangle }{\lambda_1^2 \left\langle e, e\right\rangle } + {{\mathrm{o}}} (1) \right) \\ &\quad + \frac{\left\langle e, H^2 e\right\rangle }{(\left\langle e, e\right\rangle )^2}\left( 1 - \frac{2\left\langle e, H e\right\rangle }{\lambda_1 \left\langle e, e\right\rangle } -\frac{2\left\langle e, H^2 e\right\rangle }{\lambda_1^2 \left\langle e, e\right\rangle} + {{\mathrm{o}}} (1) \right) + {{\mathrm{o}}} (1),\\ &= \left\langle e, e \right\rangle + \frac{\left\langle e, H e\right\rangle}{\left\langle e, e\right\rangle } - \frac{\left\langle e, H e\right\rangle^2}{\lambda_1 \left\langle e, e\right\rangle^2} + \frac{\left\langle e, H^2 e\right\rangle }{\left\langle e, e\right\rangle^2} + {{\mathrm{o}}} (1). \end{align*}$$

Here we use that $\left\langle e, e\right\rangle = m_2/m_1\to \infty$, and we ignore several other terms because they are small with $(\xi,\nu)$-hp , for example,

$$\begin{align*} \frac{\left\langle e, H e\right\rangle \left\langle e, H^2 e\right\rangle}{\lambda_1^2\left\langle e, e\right\rangle^2} = \mathrm{O}\left( \frac{d_\uparrow^{3/2}}{(m_2/m_1)^4}\right)={{\mathrm{o}}}(1). \end{align*}$$

One more iteration gives

$$\begin{align*} \lambda_1 &= \left\langle e, e \right\rangle + \frac{\left\langle e, H e\right\rangle}{\left\langle e, e \right\rangle} + \frac{\left\langle e, H^2 e\right\rangle}{\left\langle e, e \right\rangle^2} \\ &\quad - \frac{\left\langle e, H e\right\rangle^2}{\left\langle e, e \right\rangle^2}\left( \left\langle e, e \right\rangle + \frac{1}{\lambda_1}\left\langle e, H e\right\rangle +\frac{1}{\lambda_1^2} \left\langle e, H^2 e\right\rangle + {{\mathrm{o}}} (1) \right)^{-1} + {{\mathrm{o}}} (1) \\ &= \left\langle e, e \right\rangle + \frac{\left\langle e, H e\right\rangle}{\left\langle e, e \right\rangle} + \frac{\left\langle e, H^2 e\right\rangle}{\left\langle e, e \right\rangle^2} - \frac{\left\langle e, He\right\rangle^2}{\left\langle e, e \right\rangle^3} + \frac{\left\langle e, H^2 e\right\rangle^2 \left\langle e, H e\right\rangle}{\lambda_1 \left\langle e, e \right\rangle^3} + \frac{\left\langle e, H^2 e\right\rangle^3}{\lambda_1^2 \left\langle e, e \right\rangle^3} + {{\mathrm{o}}} (1). \end{align*}$$

Proof of theorem 1.6 (I) Since the probability of $\mathcal E^c$ decays exponentially with n, taking the expectation of the above term and using that $\mathbb{E}[\left\langle e, H e\right\rangle] = 0$, we obtain

$$\begin{align*} \mathbb{E} [\lambda_1] = \left\langle e, e \right\rangle + \frac{\mathbb{E}[\left\langle e, H^2 e\right\rangle ]}{\left\langle e, e \right\rangle^2} - \frac{\mathbb{E}[\left\langle e, He\right\rangle ^2]}{\left\langle e, e \right\rangle^3} + {{\mathrm{o}}} (1) = \frac{m_2}{m_1}+ \frac{m_1m_3}{m_2^2}- \frac{m_3^2}{m_2^3} + {{\mathrm{o}}}(1). \end{align*}$$

Note that

$$\begin{align*} \frac{m_3^2}{m_2^2}\leqslant \frac{d_\uparrow^2}{n}={{\mathrm{o}}}(1), \qquad \frac{m_1 m_3}{m_2^2}\leqslant \left(\frac{d_\uparrow}{d_\downarrow}\right)^4=\mathrm{O}(1), \end{align*}$$

and so we can write

$$\begin{align} \mathbb{E} [\lambda_1] = \frac{m_2}{m_1}+\frac{m_1m_3}{m_2^2}+ {{\mathrm{o}}}(1). \end{align} \tag{ 3.13 }$$

Again consider the high probability event on which (3.9) holds. Recall that from the series decomposition in (3.11) we have

$$\begin{align} \lambda_1&= \frac{\left\langle e, He\right\rangle}{\lambda_1} + \sum_{k=0}^{L} \frac{\mathbb{E}\left\langle e, H^ke\right\rangle}{\lambda_1^k} + \sum_{k=2}^{L} \frac{\left\langle e, H^ke\right\rangle- \mathbb{E} \left\langle e, H^k e\right\rangle}{\lambda_1^k} + \sum_{k>L} \frac{\left\langle e, H^k e\right\rangle}{\lambda_1^k}. \end{align} \tag{ 3.14 }$$

Lemma 3.6. The equation

$$\begin{align} x= \sum_{k=0}^{L} \frac{\mathbb{E}\left\langle e, H^ke\right\rangle}{x^k} \end{align} \tag{ 3.15 }$$

has a solution x₀ satisfying

$$\begin{align*} \lim_{n\to\infty} \frac{x_0}{m_2/m_1} = 1. \end{align*}$$

Proof. Define the function $h\colon\, (0,\infty)\to\mathbb R$ by

$$\begin{align*} h(x)= \sum_{k=0}^{\log n} \frac{\mathbb{E}\left\langle e, H^ke\right\rangle}{x^k}. \end{align*}$$

Since $\mathbb{E}[e^{\prime} He] = 0$, we have

$$\begin{align*} h\left(\frac{xm_2}{m_1}\right)= \frac{m_2}{m_1} +\sum_{k=2}^{\log n} \frac{\mathbb{E}\left\langle e, H^ke\right\rangle}{(xm_2/m_1)^k}. \end{align*}$$

For x > 0,

$$\begin{align*} \left|\sum_{k=2}^{\log n} \frac{\mathbb{E}[\left\langle e, H^ke\right\rangle]}{(xm_2/m_1)^k}\right| &\leqslant \sum_{k=2}^\infty \frac{1}{(xm_2/m_1)^k}\frac{m_2}{m_1} (Cd_\uparrow)^{k/2} \\ &= {{\mathrm{o}}}\left( \frac{m_2}{m_1} \sum_{k=2}^\infty\frac{1}{x^k (\log n)^{k\xi}}\right) ={{\mathrm{o}}}\left( \frac{m_2}{m_1} x^{-2}\right). \end{align*}$$

This shows that

$$\begin{align*} \lim_{n\to\infty} \frac{1}{m_2/m_1} \sum_{k=0}^{\log n} \frac{\mathbb{E}\left\langle e, H^ke\right\rangle}{(xm_2/m_1)^k} = 1. \end{align*}$$

Hence, for any $0\lt\delta \lt1$,

$$\begin{align*} \lim_{n\to\infty} \frac{1}{m_2/m_1} \left[ \frac{m_2}{m_1}(1+\delta)- h\left((1+\delta) \frac{m_2}{m_1}\right)\right] = \delta. \end{align*}$$

So, for large enough n,

$$\begin{align*} h\left((1+\delta) \frac{m_2}{m_1}\right) < \frac{m_2}{m_1} (1+\delta). \end{align*}$$

Similarly, for any $0\lt\delta\lt1$,

$$\begin{align*} h\left((1-\delta) \frac{m_2}{m_1}\right) > \frac{m_2}{m_1} (1-\delta). \end{align*}$$

This shows that there is a solution for (3.15), which lies in the interval $[ \frac{m_2}{m_1}(1-\delta), \frac{m_2}{m_1}(1-\delta)]$.

Lemma 3.7. Let x₀ be a solution for (3.15). Define

$$\begin{align*} R_n=\lambda_1 -x_0 -\frac{\left\langle e, H e\right\rangle}{m_2/m_1}. \end{align*}$$

Then

$$\begin{align*} R_n={{\mathrm{o}}}_{\mathbb{P}}\left( \frac{m_3}{m_2\sqrt{m_1}}\right), \qquad \mathbb{E}\left[|R_n|\right]={{\mathrm{o}}}\left( \frac{m_3}{m_2\sqrt{m_1}}\right). \end{align*}$$

Proof of theorem 1.6 (II) From the previous lemmas we have

$$\begin{align*} \lambda_1= x_0+ \frac{\left\langle e, H e\right\rangle}{m_2/m_1}+ R_n. \end{align*}$$

Therefore

$$\begin{align*} \mathbb{E}[\lambda_1]= x_0+ \mathbb{E}[R_n] \end{align*}$$

and

$$\begin{align*} \lambda_1- \mathbb{E}[\lambda_1]= \frac{\left\langle e, H e\right\rangle}{m_2/m_1}+ {{\mathrm{o}}}\left(\frac{m_3}{m_2\sqrt{m_1}}\right). \end{align*}$$

Hence

$$\begin{align} \frac{m_2}{m_1}\left(\lambda_1- \mathbb{E}[\lambda_1]\right)= \left\langle e, H, e\right\rangle + {{\mathrm{o}}}\left( \frac{m_3}{m_1^{3/2}}\right). \end{align} \tag{ 3.16 }$$

Observe that

$$\begin{align*} \left\langle e, H e\right\rangle = \sum_{i,j=1}^N h_{i,j} \frac{d_i d_j}{m_1}= 2\sum_{i\leqslant j}h_{i,j} \frac{d_i d_j}{m_1}. \end{align*}$$

Let

$$\begin{align*} \sigma_1^2= \sum_{i\leqslant j} \mathrm{Var}\left( \frac{2}{m_1}h_{i,j} d_i d_j\right)=\sum_{i\leqslant j} \frac{4d_i^3d_j^3}{m_1^3}\left(1- \frac{d_id_j}{m_1}\right)\sim 2 \frac{m_3^2}{m_1^3} \left( 1+\mathrm{O}\left(\frac{d_\uparrow^2}{n}\right)\right), \end{align*}$$

where we use the symmetry of the expression in the last equality. We can apply Lyapunov's central limit theorem, because $\{h_{i,j}\colon\, i\leqslant j\}$ is an independent collection of random variables and Lyapunov's condition is satisfied, i.e.

$$\begin{align*} \lim_{n \to \infty} \frac{1}{\sigma_n^3} \sum_{i > j} {\mathbb E} \left[ \left| H(i,j) d_i d_j \right|^3\right] \leqslant K\lim_{n \to \infty} \frac{m_1^{3/2}}{m_3^3}\frac{m_4^2}{m_1} =0, \end{align*}$$

where K is a constant that does not depend on n. Hence

$$\begin{align*} \frac{m_1^{3/2}\left\langle e, H e\right\rangle}{\sqrt{2} m_3}\overset{w}\longrightarrow N(0,1). \end{align*}$$

Returning to the eigenvalue equation in (3.16) and dividing by σ₁, we have

$$\begin{align*} \frac{\sqrt{m_1}{m_2}}{m_3} \left( \lambda_1- \mathrm{E}[ \lambda_1]\right) = \frac{m_1^{3/2}\left\langle e, H e\right\rangle }{m_3}+ {{\mathrm{o}}}(1)\overset{w}\longrightarrow N(0, 2). \end{align*}$$

We next prove lemma 3.7, on which the proof of the central limit theorem relied.

Proof. Note that by (3.14) and (3.15) we can write

$$\begin{align} \lambda_1-x_0= \frac{\left\langle e, H e\right\rangle}{\lambda_1}+\sum_{k=2}^{L} \mathbb{E}\left\langle e, H^k e\right\rangle \left( \frac{1}{\lambda_1^k}- \frac{1}{x_0^k}\right)+L_n, \end{align} \tag{ 3.17 }$$

where

$$\begin{align*} L_n=\sum_{k=2}^L \frac{\left\langle e, H^k e\right\rangle- \mathbb{E} \left\langle e, H^k e\right\rangle}{\lambda_1^k}+\sum_{k>L} \frac{\left\langle e, H^k e\right\rangle}{\lambda_1^k}. \end{align*}$$

Thanks to lemma 3.2, 3.4 and (3.12) we have

$$\begin{align*} L_n = \mathrm{O}\left(\frac{d_\uparrow(\log n)^{2\xi}}{\sqrt{n}m_2/m_1}\right). \end{align*}$$

Note that $L_n = {{\mathrm{o}}}( \frac{m_3}{m_2 \sqrt{m_1}})$. Indeed, using $m_3\geqslant n d_\downarrow^3$ and assumption 1.1(D1), we get

$$\begin{align*} \frac{d_\uparrow(\log n)^{2\xi} m_2\sqrt{m_1}}{\sqrt{n}(m_2/m_1) m_3} \leqslant \frac{d_\uparrow^{5/2} n^{3/2} (\log n)^{2\xi}}{\sqrt{n} d_\downarrow n d_\downarrow^3 (\log n)^\xi}=\frac{d_\uparrow^{5/2}(\log n)^{\xi}}{d_\downarrow^4 } =\mathrm{O}\left( \frac{(\log n)^{\xi}}{d_\downarrow^{3/2}}\right). \end{align*}$$

Observe that (3.17) can be rearranged as

$$\begin{align*} (\lambda_1-x_0)= \frac{\left\langle e, He\right\rangle}{\lambda_1}-\sum_{k=2}^{L} (\lambda_1-x_0) \mathbb{E}\left\langle e, H^ke\right\rangle\lambda_1^{-k} x_0^{-k} \sum_{j=0}^{k-1} x_0^{k-1-j}+ L_n. \end{align*}$$

Hence, bringing the second term from the right to the left, we have

$$\begin{align*} (\lambda_1-x_0)\left[ 1+ \sum_{k=2}^{L} \mathbb{E}\left\langle e, H^ke\right\rangle\lambda_1^{-k} x_0^{-k} \sum_{j=0}^{k-1} x_0^{k-1-j}\right]= \frac{\left\langle e, H e\right\rangle}{\lambda_1} + L_n. \end{align*}$$

Using the bounds on λ₁ and x₀, we get

$$\begin{align*} \left|\sum_{k=2}^{L} \mathbb{E}\left\langle e, H^ke\right\rangle\lambda_1^{-k} x_0^{-k} \sum_{j=0}^{k-1} x_0^{k-1-j}\right| &\leqslant \sum_{k=2}^{L} \frac{k}{(m_2/m_1)^{k+1}} \mathbb{E}\left\langle e, H^ke\right\rangle\\ &\leqslant \sum_{k=2}^{L} \frac{k}{(m_2/m_1)^{k+1}}\frac{m_2}{m_1} (Cd_\uparrow)^{k/2} = \mathrm{O}\left(\frac{d_\uparrow}{(m_2/m_1)^2(\log n)^{2\xi-1}}\right)={{\mathrm{o}}}(1). \end{align*}$$

We can therefore write

$$\begin{align*} \lambda_1-x_0= \frac{\left\langle e,H e\right\rangle}{\lambda_1}+L_n, \end{align*}$$

where $L_n = {{\mathrm{o}}}_P( \frac{m_3}{m_2 \sqrt{m_1}})$. Finally, to go to R_n , note that

$$\begin{align} R_n= \lambda_1-x_0- \frac{\left\langle e, H e\right\rangle} {m_2/m_1} =\left\langle e, H e\right\rangle \left( \frac{1}{\lambda_1}- \frac{1}{m_2/m_1}\right)+ L_n. \end{align} \tag{ 3.18 }$$

To bound R_n , it is enough to show that the first term on the right-hand side is with $(\xi,\nu)$-hp bounded by $\frac{m_3}{m_2 \sqrt{m_1}}$. Using lemma 3.4 (for k = 1) and (3.7), we have with $(\xi,\nu)$-hp

$$\begin{align} \frac{\left| \left\langle e, H e\right\rangle \right| |\lambda_1- m_2/m_1|}{\lambda_1 m_2/m_1} \leqslant \frac{\sqrt{d_\uparrow}(\log n)^{\xi}}{\sqrt{n}}\frac{\sqrt{d_\uparrow}}{(m_2/m_1)}. \end{align} \tag{ 3.19 }$$

Using again assumption 1.1(D1), $m_3\geqslant n d_\downarrow^3$, $m_1\leqslant n d_\uparrow$ and $m_2\leqslant n d_\uparrow^2$, we get that

$$\begin{align*} \frac{d_\uparrow (\log n)^{\xi}}{\sqrt{n}(m_2/m_1)}\frac{m_2\sqrt{m_1}}{m_3}\leqslant \left(\frac{d_\uparrow}{d_\downarrow}\right)^3 \frac{c}{\sqrt{d_\uparrow}}={{\mathrm{o}}}(1). \end{align*}$$

This controls the right-hand side of (3.19), and hence $R_n = {{\mathrm{o}}}( \frac{m_3}{m_2\sqrt{m_1}})$ with $(\xi,\nu)$-hp .

We want to show that the latter is negligible both pointwise and in expectation. We already have that this is so with $(\xi,\nu)$-hp on R_n . We want to show that the same bound holds in expectation. Let $\mathcal{A}$ be the high probability event of lemmas 3.2 and 3.4, and write

$$\begin{align*} \mathbb{E}[|R_n|] = \mathbb{E}[|R_n| \textbf{1}_{\mathcal{A}^c}] + \mathbb{E}[|R_n|\textbf{1}_{\mathcal{A}}], \end{align*}$$

where $\textbf{1}_{\mathcal{A}}$ is the indicator function of the event $\mathcal{A}$. Since all the bounds hold on the high probability event $\mathcal A$, it is immediate that

$$\begin{align*} \mathbb{E}[|R_n|\textbf{1}_{\mathcal{A}}] ={{\mathrm{o}}}\left(\frac{m_3}{\sqrt{m_1} m_2}\right). \end{align*}$$

The remainder can be bounded via the Cauchy–Schwarz inequality, namely,

$$\begin{align*} \mathbb{E}[|R_n| \textbf{1}_{\mathcal{A}^c}] \leqslant \left(\mathbb{E}[|R_n|^2]\mathbb{E}[\textbf{1}_{\mathcal{A}^c}]\right)^{\frac{1}{2}}\leqslant\left(\mathbb{E}\left[|R_n|^2\right]e^{-\nu(\log n)^\xi}\right)^{\frac{1}{2}}. \end{align*}$$

We see that if $\mathbb{E}[|R_n|^2] = {{\mathrm{o}}}(\mathrm{e}^{-\nu(\log n)^\xi})$, then we are done. Expanding, we see that

$$\begin{align*} \mathbb{E}[|R_n|^2] = \mathbb{E}\left[\left|\lambda_1 - x_0 - \frac{\left\langle e, H e\right\rangle}{m_2/m_1} \right|^2\right] \leqslant n^C \end{align*}$$

for some C > 0, where we use that

$$\begin{align*} \mathbb{E}[(\lambda_1^2)] \leqslant \mathbb{E}[\mathrm{Tr\,} A^2] = \sum_{i,j=1}^N \mathbb{E}[(A(i,j))^2] \leqslant d_\uparrow n \end{align*}$$

and the trivial bound $|\left\langle e, He\right\rangle|\leqslant n^{C_*}$ for some $C_*\lt C$. Hence we have $\left(\mathbb{E}[|R_n|^2]\mathbb{E}[\textbf{1}_{A^c}]\right)^{\frac{1}{2}} \leqslant \mathrm{e}^{-\nu(\log n)^{\xi}}$ and

$$\begin{align*} \mathbb{E}[|R_n|] ={{\mathrm{o}}}\left(\frac{m_3}{\sqrt{m_1} m_2}\right). \end{align*}$$