Delocalization and Limiting Spectral Distribution of Erd\H{o}s-R\'{e}nyi Graphs with Constant Expected Degree

We consider Erd\H{o}s-R\'{e}nyi graphs $G(n,p_n)$ with large constant expected degree $\lambda$ and $p_n=\lambda/n$. Bordenave and Lelarge (2010) showed that the infinite-volume limit, in the Benjamini-Schramm topology, is a Galton-Watson tree with offspring distribution Pois($\lambda$) and the mean spectrum at the root of this tree has unbounded support and corresponds to the limiting spectral distribution of $G(n,p_n)$ as $n\to\infty$. We show that if one weights the edges by $1/\sqrt{\lambda}$ and sends $\lambda\to\infty$, then the support mostly vanishes and in fact, the limiting spectral distributions converge weakly to a semicircle distribution. We also find that for large $\lambda$, there is an orthonormal eigenvector basis of $G(n,p_n)$ such that most of the vectors delocalize with respect to the infinity norm, as $n\to\infty$. Our delocalization result provides a variant on a result of Tran, Vu and Wang (2013).


Introduction
The spectral theory of graphs is important since many principal invariants of graphs are essentially related with their spectra. On the other hand, powerful tools used to investigate the spectrum of random matrices have been developed following the seminal work by Wigner [19]. In this paper, we study a class of random matrices related to graphs, namely the adjacency matrices of Erdős-Rényi random graphs.
Let G(n, p) be the Erdős-Rényi random graph with n vertices and connection probability p. More precisely, letting M n,p denote the adjacency matrix of G(n, p), for i > j we independently set, 1 INTRODUCTION 2 in other words, under the condition that G(n, p) has an expected degree, np, diverging with n. Under this condition, the spectral distribution of the scaled Erdős-Rényi ensemble 1 np(1 − p) M n,p , n ∈ N weakly converges to the standard semicircle distribution [18]. Moreover, a local semicircle law holds [12]. Also, remarkably, all the l 2 -normalized eigenvectors "delocalize" in term of their l ∞ -norm [12,18]. The situation is different if the expected degree is fixed. If, for all n, we impose that p = λ/n for some fixed λ > 0, convergence to the semicircle law and delocalization do not hold [2,4,20]. Let ν n,λ be the empirical spectral distribution of the scaled random adjacency matrix As shown in [2,4,20], ν n,λ almost surely has a deterministic limiting distribution ν λ as λ goes to infinity; however, it is an open problem to find an explicit form for ν λ , or even to give a characterization of its decomposition into pure-point, absolutely-continuous, and singular-continuous parts [7].
In [2], Bauer and Golinelli analyzed ν λ using the moment method; we use the moment asymptotics given by their work as a starting point for this study. A numerical simulation is also given in [2], and one can see that the numerical approximation of ν λ there, simulates the semicircle distribution as λ increases.
Then, as λ goes to infinity, ν λ converges weakly to the standard semicircle distribution ρ sc where It was recently pointed out to us that the above result was proved in [11], nevertheless we provide two independent proofs of this fact since they are both different from the proof given in [11]. These proofs are provided also for the sake of completeness, since the above result will play a crucial role in the proof of our main result, Theorem 1.2.
Let us also remark that while the semicircle convergence results of [12,18] look similar to the above, there is a difference in the "order of limits": suppose {λ m } is an expected degree sequence such that lim m→∞ λ m = ∞. In [12,18], a limiting "diagonal" spectral distribution sequence is considered, whereas we are interested in the limit of limiting distributions {ν λm }, In addition to results about the spectral distribution, another natural question is whether the l 2normalized eigenvectors of M n,λ/n localize or delocalize. This question was raised, for example, by Dekel et al. [8]:

INTRODUCTION
3 If the answer to (i) is positive, we say that the unit eigenvectors delocalize. Tao and Vu [17] showed that (i) and (ii) hold when p = 1/2, which is of course independent of n. However, if p = λ/n, it is easy to see that G(n, p) almost surely has O(n) isolated vertices which persist in the limit. Thus, almost surely there exist at least O(n) eigenvectors such that their infinity norms are asymptotically 1, so delocalization fails.
One can, however, obtain a weak form of delocalization as follows. For any ǫ > 0, one can choose n and λ large enough so that most of the vectors in some l 2 -normalized orthonormal basis have an infinity norm smaller than ǫ. We need some notation in order to state this result more precisely. For any symmetric n × n matrix H, the eigenvalues of H are denoted by {Λ i (H)} n i=1 . Without loss of generality, we suppose throughout this paper. Since H is symmetric, H has an orthonormal basis {u i (H)} n i=1 such that u i (H) is a unit eigenvector corresponding to Λ i (H). Theorem 1.2. Let ǫ > 0. Using the above notation, define a subset U (n, λ, ǫ) of {1, 2, · · · , n} as follows, U (n, λ, ǫ) := {i ∈ {1, 2, · · · , n} : u i (M n,λ/n ) ∞ < ǫ}. The strategy and main tools for proving the above are provided by Theorem 1.16 in [18] which we restate here for the reader's convenience.
Then there exists, a.s., an orthonormal eigenvector basis {u i (M n ) : i = 1, 2, · · · , n} such that In fact, we also get a "diagonalized convergence" result as a corollary to Theorem 1.2. The corollary should be viewed as a variant of the above Theorem 1.3. While the conclusion of the corollary is weaker than that of Theorem 1.3, the assumptions also allow for a broader class of sequences {λ n }. This is one benefit of a priori considering the limiting behavior as two separate limits instead of one single diagonalized limit. Corollary 1.4. Let λ = λ n depend on n and set M n := M n,λn/n . Also, suppose lim n→∞ λ n = ∞. Let ǫ > 0, and using the above notation, define U ′ (n, ǫ) by Then, there exists a.s. an orthonormal eigenvector basis {u i (M n ) : i = 1, 2, · · · , n} such that The outline of the rest of this paper is as follows. In the next section (Section 2), we give two proofs of Theorem 1.1 using respectively the moment method and the Stieltjes transform method. Section 3 is devoted to the proofs of Theorem 1.2 and Corollary 1.4. 4 2 Convergence to the semicircle distribution As a preliminary to the two proofs, let us recall that the limiting distribution ν λ exists [2,4,20]. In particular, [4] argues this via showing that the sequence of random graphs {G(n, λ/n)} n∈N converges, in the Benjamini-Schramm topology on rooted graphs, to a Galton-Watson tree with offspring distribution Pois(λ) (Poisson with intensity λ). This fact will be useful to us in our second proof. Let us begin, however, with the classical moment method.

Moment method proof
Fix λ > 0 and suppose n ≥ λ. Let m ij be the (i, j) element of M n,λ/n . A standard calculation in random matrix theory gives We first obtain an asymptotic formula for E ν n,λ , x k using the method and terminology of [2]. If a k-tuple (i 1 , i 2 , · · · , i k ) satisfies i 1 = i 2 , i 2 = i 3 , · · · , i k−1 = i k and i k = i 1 , it is said to be admissible. Non-admissible k-tuples do not contribute to the sum (8) since M n,λ/n has vanishing diagonal entries. For each positive integer j ≤ k, define W j as the set of admissible k-tuple A k-tuple (i 1 , i 2 , · · · , i k ) is called normalized if it is admissible and i j > 1 implies that there exist j ′ < j such that i j ′ = i j − 1. Let N j be the set of normalized k-tuples (i 1 , i 2 , · · · , i k ) such that {i 1 , i 2 , · · · , i k } = {1, 2, · · · , j}. For j ≤ n, Per(j, n) is defined to be the set of injective maps from {1, 2, · · · , j} to {1, 2, · · · , n}. It is observed that, there is a one to one correspondence between W j and {(ω, σ)|ω ∈ N j and σ ∈ Per(j, n)}. The set N of all normalized k-tuples is expressed as In Eq. (8), m i1i2 m i2i3 · · · m i k i1 can be identified with a closed walk along the graph given by the adjacency matrix M n,λ/n . That is to say, m i1i2 m i2i3 · · · m i k i1 corresponds with the closed walk i 1 i 2 · · · i k i 1 ("closed" means that it ends where it started). Let the sets of distinct edges and distinct vertices in the closed walk i 1 i 2 · · · i k i 1 corresponding to k-tuple ω = (i 1 , i 2 , · · · , i k ) be denoted by E(ω) and V (ω), respectively. We denote an edge e connecting the vertices with indices i j and i j+1 by if and only if m e = 1 for all e ∈ E(ω), The moment method proof of Theorem 1.1 will follow from Lemma 2.1 and Lemma 2.2 below.
In particular this implies that G(ω) must be a tree (rooted at 1).
When k = 2m, it is clear that a m is precisely the Catalan number, C m , since the multiplicity of every edge in the closed walk i 1 i 2 · · · i k i 1 is exactly 2.

6
Recall that ρ sc is the standard semicircle distribution. It is easy to see that Since ρ sc has bounded support, its moments characterize it uniquely, which implies that ν λ converges weakly to ρ sc (See Theorem 30.2 in [3]).

Stieltjes transform proof
For later use, recall from [14, pg. 225] the notion of a spectral measure ν φ , of a self-adjoint operator A, associated to a unit vector e φ . Such a probability measure, ν φ , can be defined by finding the unique measure satisfying for all bounded, continuous f . Using spectral theory and exchangeability, [4] argued that the mean of the random measure ν n,λ can be regarded as the expected spectral measure at vertex 1 (or any other fixed vertex) of the Erdős-Rényi graph G(n, λ/n) (with weights 1/ √ λ on the edges). Moreover, the limiting deterministic measure ν λ is the expected spectral measure associated to the root of a Galton-Watson tree with offspring distribution Pois(λ) and weights 1/ √ λ, which is the limit of {G(n, λ/n)} with weighted edges in the Benjamini-Schramm topology (see also [5,13]). The adjacency operator 1 ∞ of the limiting graph is self-adjoint ([13, Lemma 5.2]) and its resolvent R (λ) is well-defined. Letting φ denote the root of the tree and e φ denote the root vector, i.e. a Kronecker-delta function at the root, define the random variable where the domain of z is C\R. Let S λ be the Stieltjes transform of the limiting distribution ν λ . According to [4,Thm 2], where (R (λ) k,k (z)) k∈N is an i.i.d. sequence with the same distribution as R (λ) φ,φ (z) and Pois(λ) is a Poisson random variable independent from (R (λ) k,k (z)) k∈N . Thus, The strategy of the proof is to show that exists for all z ∈ C\R and satisfies the self-consistent equation, implying that S(z) = − 1 2 (z − √ z 2 − 4) by choosing the solution of (19) such that the imaginary parts of S(z) and z are the same. By the Stieltjes inversion formula, ν λ converges weakly to ρ sc , the standard semicircle law, as λ → ∞.
Let us now carry out the above strategy. Define Y λ and f λ as follows.
where o( 1 λ ) depends on θ. The last equality in (20) comes from the Taylor expansion of the characteristic function f which is possible since we have the a.s. bound Choose a subsequence λ n → ∞ such that a limit S(z) exists. Eq. (20) tells us that by convergence of the characteristic functions of {Y λn , λ n > 0}, and the fact that the limit is a constant. Next, suppose without loss of generality that z ∈ C + . Then ℑ(S(z) + z) ≥ ℑ(z) > 0 which implies S(z) = z. By the continuous mapping theorem, By (18), the left-hand side above has the same distribution as R (λn) φ,φ (z) which by (21) is bounded for any fixed z ∈ C\R. Thus Therefore S(z) satisfies (19) and must be the Stieltjes transform of the semicircle law. The proof follows since the measures {ν λ } are tight , while the above argument shows that there is a unique limit point.

Delocalization
Recall that {Λ i (H)} n i=1 and {u i (H)} n i=1 denote the eigenvalues and eigenvectors of a symmetric n × n matrix H, respectively. We begin with several lemmas, the first of which is Eq. (5.8) in [10]. We state the version from [ be an n × n symmetric matrix for some a ∈ R and X ∈ R n−1 , and let x v be the unit eigenvector with eigenvalue Λ i (H) where x ∈ R and v ∈ R n−1 . Assume that none of the eigenvalues ofH are equal to Λ i (H). Then, where ·, · denotes the inner product between vectors.
The second lemma is a consequence of Talagrand's inequality that was proved in Lemma 68 of [17]. We state the version from [18,Lemma 3.4

]:
Lemma 3.2 (Lemma 3.4 in [18]). Let Y = (y 1 , · · · , y n ) ∈ R n be a random vector whose coordinates are i.i.d. centered random variables which are a.s. bounded in absolute value by 1 and have variance σ 2 . Let H be a subspace of dimension k and π H the orthogonal projection onto H. Then, where · is the Euclidean norm.
Let N n be a symmetric n × n matrix whose upper triangular elements are independent standard normal variables N (i, j). Note that even though the perturbed matrix elements are unbounded, we have that As n n 2 e − n 2 < ∞, by Borel-Cantelli we have that |N (i, j)| ≤ √ n a.s. for all 1 ≤ i, j, ≤ n and all n large enough. This will allow us to use Lemma 3.2 later on.
Assume that N n is also independent from M n,λ/n . Let {δ(n)} n∈N be a sequence of positive numbers satisfying Denote the scaled adjacency matrix and a perturbed version of it as follows: B n,λ := A n,λ + δ(n)N n .
The reason for introducing the perturbed matrix is that it almost surely has a simple spectrum (see [16, Exercise 1.3.10]): Write B n,λ in the following matrix form: where a ∈ R and X ∈ R n−1 . Then, {Λ i (B n,λ ) : i = 1, 2, · · · , n} ∩ {Λ i (B n,λ ) : i = 1, 2, · · · , n − 1} = ∅ almost surely (33) by (31) and the Cauchy interlacing principle. Note that (33) allows us to use Lemma 3.1. Our third preliminary lemma bounds the effect of the above perturbation on infinity norms of eigenvectors: Recall that B n,λ is defined as the perturbation A n,λ + δ(n)N n . There exists an orthonormal basis of eigenvectors {u i (A n,λ )} n i=1 such that, for every 1 ≤ i ≤ n, where α(n) → 0 as n → ∞, and α(n) can be chosen to be arbitrarily small depending only on δ(n).
Henceforth assume u i (A n,λ ) = u i (M n,λ/n ) for all i and n and that the orthonormal basis The above lemma follows simply from Theorem 1.1 and Weyl's inequality; however, for completeness, we provide an explicit proof in Appendix A.1.
By (33), we can apply Lemma 3.1 to get where x = u i (1) is the first coordinate of u i (B n,λ ). A similar bound holds for any other coordinate u i (k) of u i (B n,λ ) by replacingB n,λ with an appropriate submatrix. Thus, we will see that it suffices to find an upper bound of |x| 2 , with high enough probability, in order to get an upper bound for u i (B n,λ ) ∞ , uniformly in i with high probability. Let Q be a positive integer and set l := 4/Q.
Choose Q large enough so that Q ≥ 5 and We fix this value of Q (thus fixing l) henceforth and note that they only depend on ǫ.
For q i as above Therefore for H i = H(n, q i ), we get the inequality Now, define a random vector Y from the vector X which is as in (32) Let H ′ be the orthogonal complement of span{1(n − 1)}. Then, for generic q and H = H(n, q), Observe in particular, that dim(H ∩ H ′ ) ≥ dim(H) − 1. SinceB n,λ is independent of Y , Lemma 3.2 can be applied with after conditioning onB n,λ , and also after normalizing Y so that σ = 1. Thus with probability at least 1 − 10 exp(−( √ n · log n)/4), The Borel-Cantelli lemma implies that the inequality (48) holds almost surely for large n and for every subspace H(n, q) with q ∈ [1, Q], so in particular it holds for H(n, q i ). Plugging (48) into (44), and recalling that δ(n) = o(n −1/2 ), we have, almost surely, Recall thatμ n,λ is the empirical spectral distribution ofB n,λ . Fix q ∈ {1, 2, · · · , Q} and note that Applying Theorem 1.1 and Lemma 3.4 tõ Combining (39), (41), (49) and (52) we get that |x| < ǫ/2 almost surely for large n and large λ under the assumption Λ i (B n,λ ) ∈ [−2, 2]. Finally, recall that a relation similar to (39) holds for any other coordinate u i (k) of u i (B n,λ ) and so using a union bound over k ∈ {1, . . . , n}, and noting that n 10n exp(−( √ n·log n)/4) < ∞ (in order to invoke the Borel-Cantelli lemma for a union of probabilities), we obtain (37). This completes the proof.

Proof of Corollary 1.4
From now on, we let the expected degree depend on n, i.e., λ = λ n . Recall that in contrast to Theorem 1.3 where the growth condition (6) is required, we consider the more general case where Recall that M n := M n,λn/n . Also, let ν n := ν n,λn . According to Theorem 1.3 in [18], the empirical spectral measure ν n weakly converges to the standard semicircle distribution ρ sc as n goes to infinity. We can use the same argument as in the proof of Theorem 1.2 up until (50). After that, set µ n := µ n,λn andμ n :=μ n,λn and use the following inequality instead of (51): Consequently, lim inf n→∞ min q∈{1,2,··· ,Q} |J(n, q)| n · l 2 ≥ min q∈{1,2,··· ,Q} ρ sc [a q , a q+1 ] Since lim inf n→∞ µ n ([−2, 2]) = 1, the result follows. While Corollary 1.4 has the advantage of holding without any growth rate condition on λ n , it has the drawback that it give no information about the infinity norms of eigenvectors corresponding to the eigenvalues outside of [−2, 2]. Note that [−2, 2] corresponds to the support of the standard semicircle law.