Largest eigenvalues of sparse inhomogeneous Erdős-Rényi graphs

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1 Largest eigenvalues of sparse inhomogeneous Erős-Rényi graphs Florent Benaych-Georges Charles Borenave Antti Knowles April 26, 2017 We consier inhomogeneous Erős-Rényi graphs. We suppose that the maximal mean egree satisfies log n. We characterize the asymptotic behaviour of the n 1 o(1 largest eigenvalues of the ajacency matrix an its centre version. We prove that these extreme eigenvalues are governe at first orer by the largest egrees an, for the ajacency matrix, by the nonzero eigenvalues of the expectation matrix. Our results show that the extreme eigenvalues exhibit a novel behaviour which in particular rules out their convergence to a nonegenerate point process. Together with the companion paper [3], where we analyse the extreme eigenvalues in the complementary regime log n, this establishes a crossover in the behaviour of the extreme eigenvalues aroun log n. Our proof relies on a new tail estimate for the Poisson approximation of an inhomogeneous sum of inepenent Bernoulli ranom variables, as well as on an estimate on the operator norm of a prune graph ue to Le, Levina, an Vershynin from [12]. 1. Introuction The purpose of the present text is to unerstan the extreme eigenvalues of the ajacency matrix of an inhomogeneous Erős-Rényi ranom graph on n vertices in the regime where the maximal mean egree satisfies log n. Heuristically, such eigenvalues arise from three ifferent origins: (i the ege of the limiting bulk eigenvalue ensity, (ii vertices of large egrees, an (iii outliers associate with nonzero eigenvalues of the expectation matrix. One goal of this paper is a precise unerstaning of this interplay between ranom matrices on the one han an the geometry of ranom graphs on the other. Such questions have several motivations from applications, such as the estimation of the spectral gap an spectral clustering. The simplest ranom graph is the Erős-Rényi ranom graph G(n, /n, where each ege is present inepenently with probability /n. In this case it is rather well unerstoo that the behaviour of the extreme eigenvalues in the regime log n is governe by ranom matrix behaviour; see [3, 7 10, 14, 16]. In the complementary regime log n, the main result available up to now was ue to Suakov an Krivelevich [11], who showe that the largest eigenvalue of the ajacency matrix is asymptotically equivalent to the maximum of the maximal mean egree an the square root of the largest egree (their result hols in fact for all regimes of. Our main result is a escription of the behaviour of the n 1 o(1 largest an smallest eigenvalues of the ajacency matrix A an its centre version A.= A EA, for an inhomogeneous Erős-Rényi ranom graph whose mean egree is much smaller than log n. Informally, we prove that the k-th 1

2 largest eigenvalue eigenvalue of A satisfies (log(n/k λ k (A log((log n/, k n1 ε, ε (0, 1. (1.1 Uner mil aitional assumptions (satisfie for instance by stochastic block moels, we show that the same result hols for the eigenvalues of A, with the exception of some outlier eigenvalues whose locations we also characterize. A consequence of our results, combine with those from the companion paper [3], where we analyse the extreme eigenvalues in the complementary regime log n, is a crossover in the behaviour of the extreme eigenvalues aroun log n (the same threshol as for the graph connectivity. Inee, in [3] we prove that if log n then all eigenvalues are asymptotically containe within the support of the semicircle law escribing the macroscopic eigenvalue ensity, while in the current paper we establish for log n a novel behaviour of the extreme eigenvalues, which implies that n 1 o(1 eigenvalues escape the support of the semicircle law. Their locations are governe by (1.1 an efine a istribution that is illustrate in Figure 1 below. It is helpful to analyse the behaviour of the extreme eigenvalues for log n in the context of ranom matrix theory. Until now, in ranom matrix theory two ifferent types of universal behaviour at leaing orer of the extreme eigenvalues have been establishe, exhibite for instance by light- an heavy-taile Wigner matrices respectively. After a suitable eterministic rescaling of the matrix, these two classes may be characterize as follows. (a The extreme eigenvalues converge to the ege of the support of the asymptotic bulk spectrum. (b The extreme eigenvalues form asymptotically a Poisson point process. For example, it is known [1,13,15] that a Wigner matrix whose entries have tail ecay x α belongs to class (a if α > 4 an to class (b if α < 4. Moreover, as state above, in the companion paper [3] we prove that the Erős-Rényi graph belongs to class (a if log n. Also, sparse heavy-taile ranom matrices exhibit a transition between these classes epening on the sparsity an the tail ecay of the entries [4]. A consequence of our results is that, perhaps surprisingly, for log n, the (possibly inhomogeneous Erős-Rényi graph belongs to neither class (a nor class (b. Instea, the behaviour from (1.1 results in a sharp increase in the ensity of eigenvalues as one moves towars the centre of the spectrum, which implies that, no matter the rescaling of the spectrum, any nonegenerate limiting point process will be infinite on compact sets. The proof consists of two main steps. In a first step, we analyse the istribution of the n 1 o(1 largest egrees, an prove that the corresponing vertices are with high probability separate by istance at least 3 from each other. The key tool behin this step is a new sharp estimate (Theorem 3.1 below on the tail of a sum of inhomogeneous inepenent Bernoulli ranom variables. This estimate may be regare as an improvement for the tails of the well-known Poisson approximation provie by Le Cam s inequality [2]. It is of inepenent interest. In a secon step, we compare the n 1 o(1 largest eigenvalues of the graph with those of the graph obtaine by only keeping the eges incient to the n 1 o(1 vertices of largest egree. The latter correspons to a block-iagonal matrix whose blocks are associate with star graphs of high central egree. This comparison is base on a sharp estimate on the operator norm of the complementary graph ue to Le, Levina, an Vershynin [12, Theorem 2.1]. 2

3 This text is organize as follows. In the remainer of the introuction, we state our main results, which are prove in Section 2. In Section 3 we state an prove the new tail estimate for Poisson approximation mentione above. Notation. The eigenvalues of a Hermitian n n matrix H are enote by λ 1 (H λ 2 (H λ n (H. Its operator norm is given by H = max(λ 1 (H, λ 1 ( H. For p [0, 1], we enote by B(p the Bernoulli law with parameter p, i.e. B(p = (1 pδ 0 + pδ 1. We enote by Bin(p 1,..., p n the law B(p 1 B(p n. In particular, Bin(p,..., p is the Binomial istribution with parameters (n, p. For x > 0 use the abbreviation [x].= {1, 2,..., x } Hypotheses an efinitions. Throughout this paper, A is the ajacency matrix of an inhomogeneous (unirecte Erős-Rényi ranom graph G with vertex set [n], where the ege {i, j} is inclue with probability p ij [0, 1] inepenently of the others. Note that we allow loops: there is a loop at vertex i with probability p ii. The maximal ege probability is p max.= max i j p ij. The mean egree of the vertex i [n] an the maximal mean egree are efine as i.= j p ij,.= max i i respectively. We always suppose that there are κ > 0 an η (0, 1 such that κ η log n an p max n 1+η. (1.2 As all of our error term controls will be uniform, with quantitative rates of convergence, in the parameters (p ij i,j [n] such that (1.2 hols, we introuce the following efinitions. Definition 1.1. (i An amissible error function is a function ψ(n, κ, η satisfying κ > 0, lim n η 0 ψ(n, κ, η = 0. (1.3 (ii Given an event E an a conition A on the parameters κ, η, (p ij i,j [n], we say that, uner A, E hols with high probability (w.h.p. if there is an amissible error function ψ(n, κ, η such that P(E 1 ψ(n, κ, η for all κ, η, (p ij i,j [n] satisfying A. (iii Given a conition A on the parameters κ, η, (p ij i,j [n], for two families of ranom variables (u t, (v t we say that uner A, for all t, u t v t if there is an amissible error function ψ(n, κ, η such that ( v t P 1 u t ψ(n, κ, η for all t 1 ψ(n, κ, η for all κ, η, (p ij i,j [n] satisfying A. 3

4 Let us emphasize that the point in this efinition is the uniformity of the error terms in the asymptotic regime where n, = o(log n, an p max = n 1+o(1. To simplify presentation, in the following we shall not ientify the error functions ψ(n, κ, η explicitly, although a careful look at our proofs will easily yiel explicit expressions for them. Finally, for k [n] we set L k.= log(n/k log((log n/. ( Relation between the centre ajacency matrix an the largest egrees. For i [n], let D i enote the egree of the vertex i in the graph G. Denote by D 1 D n the ecreasingly orere egrees D 1,..., D n. We also introuce the centre ajacency matrix A.= A EA. By efinition, (EA ij = p ij. The following theorem relates the largest eigenvalues of A to the largest egrees, whose behaviour is escribe in Propositions 1.11 an 1.12 below. Theorem 1.2. For any ε (0, 1, uner (1.2, w.h.p., max λ k (A D k [n 1 ε k C np max + ε L 1, (1.5 ] max λ n+1 k (A + C np max + ε L 1, (1.6 k [n 1 ε ] where C is a universal constant an L 1 is efine in (1.4. D k The proof of Theorem 1.2 is base on an analysis of the graph spanne by the largest egree vertices, an on [12, Theorem 2.1] ue to Le, Levina, an Vershynin on the operator norm of the centre ajacency matrix where all large egree vertices have been remove. The term ± arises as an eigenvalue of a star graph with central egree D k O(1 (see Definition 2.6 below. By Proposition 1.11 below, for any ε (0, 1, we have D 1 (1 + εl 1 w.h.p., which yiels the following corollary. Corollary 1.3. For any ε (0, 1, uner (1.2, w.h.p., A (1 + ε L 1 + C np max. As explaine, for example, in [12], Corollary 1.3 fins applications in the analysis of spectral clustering techniques on ranom graphs. Uner the aitional hypothesis that all vertices have the same mean egree, the behaviour of the largest egrees summarize in Corollary 1.13 below implies that D k L k, where L k was efine in (1.4. We euce the following result. Corollary 1.4. Let ε (0, 1. Then uner the conitions i = for all i, np max ηl 1, an (1.2, we have for all k [n 1 ε ] λ k (A L k an λ n+1 k (A L k. (1.7 D k 4

5 Remark 1.5. There is an equivalent way to state Corollary 1.4. Introuce the counting function of the renormalize eigenvalues of A = A EA, efine as { N A (x.= # k [n] : λ } k(a x. (1.8 The first estimate of (1.7 implies that for any x (0, 1, L1 log N A (x log n 1 x 2. (1.9 Inee, for any δ > 0 small enough, for k := n 1 x2 δ an k := n 1 x2 +δ. We have k N A (x < k λ k (A < x L 1 λ k, which happens w.h.p. by the first estimate of (1.7. Informally, (1.9 states that N A (x n 1 x2, from which we euce that the ensity of renormalize eigenvalues λ k(a L1 at x (0, 1 is asymptotically See Figure 1 below for an illustration. 2 log(n n 1 x2 x. (1.10 Figure 1: Histogram of the right ege of the spectrum of A in the case of a homogenous Erős-Rényi graph an ensity of (1.10. The eigenvalues are renormalize in such a way that λ 2 = 1 (λ 1 has been remove an the histogram as well as the ensity are normalize in such a way that the total area is 1. Here, n = an = 0.5 (left, = 1.5 (centre, = 2.5 (right. We see that as grows, the empirical ensity gets more convex at its ege, which agrees with the iea that the semicircle law approximation gets more accurate. Remark 1.6. The estimate (1.9 states there exists no eterministic sequence α = α n such that the point process Ξ.= {αλ k (A. k [n]} is asymptotically finite an nonzero on compact sets. In particular, Ξ cannot converge to a point process as n. Note, however, that our results o not rule out the existence of an affine transformation parametrize by α = α n an β = β n such that the point process {α(λ k (A β. k [n]} converges. 5

6 1.3. Consequences for the ajacency matrix. Gershgorin s Circle Theorem implies that EA. Then, writing A = A + EA, following corollary is an immeiate consequence of Theorem 1.2 an Weyl s inequality (see e.g. [5, Corollary III.2.6]. Corollary 1.7. (a Theorem 1.2 hols with A replace by A an the right-han sies of (1.5 (1.6 replace by C np max + ε L 1 +. (b Uner (1.2, w.h.p., for any k [n], for some universal constant C. λ k (A λ k (EA C ( L1 + np max, Remark 1.8. As L 1 D 1, for an homogenous Erős-Rényi ranom graph, Corollary 1.7 is consistent with [11, Theorem 1.1] which asserts that λ 1 (A = A max{ D 1, } in all regimes of Applications to stochastic block moels. In the stochastic block moel, EA has boune rank an all its nonzero eigenvalues are of orer. We enote by λ + 1 λ+ k + > 0 the positive eigenvalues of E an λ 1 λ k < 0 the negative eigenvalues of EA. For κ the constant of (1.2, we suppose that λ + 1 λ+ k + κ, λ 1 λ k κ, k + + k κ 1. (1.11 Then, there is a ichotomy in the behaviour of the k + + k largest (in absolute value eigenvalues of A, epening on whether L 1 or L 1. Uner mil conitions on, these conitions rea log n/ log log n or log n/ log log n. Proposition 1.9. Let ε (0, 1. (a Uner conitions (1.2 an η log n/ log log n, w.h.p. A (1 + ε L 1 + C np max. (1.12 Uner the aitional conitions i = for all i an np max ηl 1, we have for all k [n 1 ε ] λ k (A L k an λ n+1 k (A L k. (1.13 (b Uner conitions (1.2, (1.11, an η 1 log n/ log log n, an w.h.p., i = 1,..., k +, λ i (A λ + i, i = 1,..., k, λ n+1 i (A λ i (1.14 max{ λ k + +1(A, λ n k (A } (1 + ε L 1 + C np max. (1.15 Uner the aitional conitions i = for all i an np max ηl 1, we have for all k [n 1 ε ] λ k + +k(a L k an λ n+1 (k +k(a L k. (1.16 6

7 We remark that in the case (a of small egree, the nontrivial eigenvalues λ ± i of EA o not give rise to corresponing eigenvalues of A, an A may be regare as a perturbation of A. In contrast, in the case (b of large egree, the nontrivial eigenvalues λ ± i of EA gives rise to associate outlier eigenvalues of A, an A may be regare as a perturbation of EA. Hence, the spectrum of A retains some information about the spectrum of EA if an only if log n/ log log n. Remark Remarks 1.5 an 1.6 also hol for the eigenvalue counting measure of A. See Figure 1 for an illustration. Proof of Proposition 1.9. The proof is a simple consequence of the results of the preceing subsections an of Weyl s inequalities. More specifically, use [5, Corollary III.2.6] to prove (1.12, (1.13, (1.14 an (1.15 consiering A as a perturbation of A in (a an A as a perturbation of EA in (b. To prove the first part of (1.16 (the secon part can be prove in the same way, note that by [5, Corollary III.2.3 an Exercise III.2.4] we have, for any k, an use Corollary 1.4. λ k+2k + +k (A λ k + +k(a λ k (A 1.5. Behaviour of the largest egrees. In our regime of interest, the largest egrees of the graph play a key role in the analysis of the largest eigenvalues of the ajacency matrix. We now escribe their asymptotic behaviour. Proposition Let ε (0, 1. Uner (1.2, w.h.p., for any k [n 1 ε ] For any x > 0, we introuce the sets D k (1 + εl k. V x.= {i [n] : D i x}, V =x.= {i [n] : D i = x}. (1.17 These sets are relate to the orere egrees through D k x #V x k. (1.18 Let us introuce the function f on (, efine by ( x f(x = f (x.= x log (x log 2πx. (1.19 If Y is a Poisson ranom variable with mean, for a large integer x Stirling s approximation gives P(Y = x = e f(x+o(x 1. (1.20 We shall in fact prove that, roughly speaking, uner conition (1.2, we have #V x ne f(x, (1.21 which, uner the aitional assumption i = for all i, can be strengthene to #V x ne f(x. (1.22 7

8 This leas us to introuce, for k [n], the solution k of the equation This solution is unique an satisfies f( k = log(n/k an k. k L k. (1.23 (See Lemma 2.1 below for the full etails. The combination of the characterization (1.18 of the largest egrees with estimates (1.21, (1.22 an (1.23 naturally leas to the estimates given in Propositions 1.11 an In the special case of a homogenous Erős-Rényi graph, the next proposition is essentially containe in [6, Chapter 3]. Hence, our next result may be viewe as a generalization of this result to the inhomogeneous case. It is a more precise version of Proposition 1.11 uner the aitional assumption that all vertices have the same mean egree. Proposition Let ε (0, 1. (a For any k [n 1 ε ] there exists a eterministic k { k, k } such that uner the conitions i = for all i an (1.2, w.h.p., for any k [n 1 ε ], we have (i D k { k, k 1}, (ii D k = k when ist( k, N ε. (b Uner the conitions i = for all i an (1.2, for all integers t [ε 1, 1 ε], #V t #V =t ne f(t. (1.24 Uner the same conitions, if t 1 + ε then, w.h.p., #V t = 0. An immeiate corollary of Proposition 1.12 an (1.23 is the following. Corollary Let ε (0, 1. Uner the conitions i = for all i an (1.2, for all k [n 1 ε ], D k L k. Remark 1.14 (Lack of limit point process of largest egrees. Proposition 1.12 (b shows that, perhaps surprisingly, there is no Poisson point process at the right ege of the multiset of egrees of G. There is instea a sharp transition at 1 : for any integer t 1 ε, w.h.p. the number of vertices with egree t is 1 an for any integer t 1 + ε, w.h.p. there is no vertex with egree t. 2. Estimation of the largest egrees an comparison with the eigenvalues The rest of this paper is evote to the proofs of our main results. Throughout this section we use the following conventions about convergence of eterministic quantities. Let u an v be eterministic quantities epening on n an (p ij i,j [n]. We write u = o(v, or, equivalently, u v, whenever u/v 0 as n an η 0, uniformly in (p ij i,j [n] satisfying (1.2 an all parameters except ε. We remark that such a convergence can always be upgrae to a quantitative convergence using some amissible error function from Definition 1.1, but for the sake of simplicity we shall not o this. 8

9 2.1. Largest egrees: proof of Proposition 1.11 an Proposition Recall that the function f was efine in (1.19. Lemma 2.1. Let ε (0, 1. For n large enough an η small enough, uner conition (1.2, for any k [n 1 ε ], there exists a unique solution k of the equation Moreover, uner conition (1.2, for any k [n 1 ε ], f( k = log(n/k an k. (2.1 k L k. (2.2 Proof. The function f is increasing on (, (inee, f (u = log(u/ + 1/(2u an satisfies f(u = u log(u/ + O(u as u, (2.3 so that for n large enough, k is well efine for any 1 k n 1 ε. Moreover, (unconitionally on, we have f(u + x = f(u + x log(u/ + O(x/u + x 2 /u. (2.4 Inee, ( f(u + x f(u = (u + x log 1 + x ( u + x log x + 1 ( u 2 log 1 + x. (2.5 u Let us now prove (2.2. As both k = k (n, an L k = L k (n, are eterministic an epen only on n an, by Definition 1.1, (2.2 reas lim sup sup k (n, L k (n, 1 = 0. (2.6 n η 0 κ η log n k [n 1 ε ] If it were not the case, there woul be an infinite set I of positive integers an some sequences (η n n I, ( n n I, (k n n I satisfying κ n η n log n, 1 k n n 1 ε an η n 0 as n I tens to infinity, such that kn (n, n L k (n, n 1 > c (2.7 for some positive constant c. Let us rop the inex n from the notation. One first verifies that k / as n I grows (by a simple argument by contraiction using (2.1. Then, introuce ϕ > 0 such that k = ϕl k. By (2.3, we have log(n/k = f ( k k log( k / = ϕ log(n/k ( log(log(n/k/ + log ϕ log log((log n/. log((log n/ By assumption, ε log n log(n/k log n so that 1 ϕ 1 + log ϕ + o(1. (2.8 log((log n/ On easily euces from (2.8 that ϕ is boune away from 0 an, an then that ϕ tens to one, which contraicts (2.7. Thus (2.6, hence also (2.2, are true. 9

10 Lemma 2.2. Suppose that n is large enough an η small enough, an that satisfies conition (1.2, so that 1 is well efine (see Lemma 2.1. Let q 1,..., q n > 0 satisfy = i q i, an let X be a ranom variable with law Bin(q 1,..., q n. Suppose that q max.= max i q i /(log n 5/2. Then for any u 1 an x such that x 2 u an u + x 2 is integer, where δ = O ( ( + x 2 /u + q max u 5/2 /. ( u xe P(X u + x = (1 + δ f(u. (2.9 Proof. By Theorem 3.1 below, P(X u + x = (1 + δp(y = u + x where Y has Poisson istribution with mean an δ = O ( /u + q max u 5/2 /. Now, by (1.20, we have Then, the estimate (2.4 allows to conclue. We are now reay to prove Proposition P(Y = u + x = e f(u+x+o(1/u. (2.10 Proof of Proposition Let k [n 1 ε ]. It is sufficient to prove that w.h.p. we have D k (1 + εl k. Inee, the inversion of w.h.p. an for all k [n 1 ε ] is straightforwar using k = n 1 lε for each l [1/ε], as all error terms o(1 in what follows are terms tening to zero uniformly in k [n 1 ε ] as n an η 0. We note that D k t is equivalent to #V t k (the set V t has been efine in (1.17. By (2.4, we have ne f( k+x = k(1 + o(1( k / x. (2.11 Thus (2.11 applie to x = k +2 k implies that ne f( k +2 = o(k. By (2.9 in Lemma 2.2, it follows that E#V t = o(k if t = k + 2. It remains to use (2.2 an Markov s inequality. Our proof of Proposition 1.12 will require a sharp boun on the variance of #V t. Lemma 2.3. Let D i enote the egree of the vertex i in the graph G. Then any integer t 0, Var(#V t E#V t + 3n max P(D i t 1 2. i Proof. For ease of notation, we set q i := P(D i t. Since #V t = n i=1 1 D i t, we have E#V t = n i=1 q i an Var(#V t = i,j = i Cov(1 Di t, 1 Dj t q i (1 q i + i j Cov(1 Di t, 1 Dj t. Hence it suffices to prove that for i j, Cov(1 Di t, 1 Dj t 3p ij max P(D i t 1 2. i 10

11 Let us fix i j. We have D i = k A ik an D j = k A jk. We introuce the events { } E.= A ik t, A ij = 0, E.= A ik t, k j k { } F.= A jk t, A ij = 0, F.= A jk t. k i Then E E, F F an (E \E F = (F \F E =, the latter follows from E \E = { k j A ik t 1, A ij = 1} (an similarly for F \F. Thus by Lemma A.1 an the inepenence of the events { k j A ik s} an { k i A jk s } Cov(1 Di t, 1 Dj t = Cov(1 E, 1 F which allows to conclue. P ( (E \E (F \F + P(F \F P(E + P(E \EP(F ( ( 3p ij P A ik t 1 P A jk t 1, k j We are reay to prove Proposition Proof of Proposition First we remark that the inversion of w.h.p. an for all for all integers t [ε 1, 1 ε] for (b or for all k [n 1 ε ] for (a can be treate as in the proof of Proposition (b By (2.9 in Lemma 2.2, if t 1 + ε, P(#V t 1 E#V t (1 + ε(/ 1 ε tens to 0. Moreover, in the regime ε 1 t 1 ε, as ne f(t goes to infinity, to prove the left-han sie of (1.24, by Markov s inequality it suffices to prove that E#V t = ne f(t (1 + o(1 an Var(#V t = ne f(t (1 + o(1, which follows irectly from (2.9 in Lemma 2.1 an Lemma 2.3. It remains to prove that #V =t ne f(t when ε 1 t 1 ε. We note that k i #V =t = #V t #V t+1. From what precees, it suffices to check that #V t+1 = o(ne f(t. The latter is a consequence of (2.9 in Lemma 2.1 which implies that E#V t+1 (1 + o(1(/tne f(t. (a We note that for any k, t, the claim D k t is equivalent to #V t k. Thus (2.11 applie to x = k 1 k an (b imply that with high probability D k k 1. Similarly, (2.11 applie to x = k + 2 k an (b imply with high probability D k k + 1. Moreover, if k + 1 k ε, then with high probability D k k, while if k k ε then with high probability D k k. Note that either k k ε or k + 1 k ε hols when ε = 1/2. k 11

12 2.2. Proof of Theorem 1.2. First it is easy to see, by Weyl s inequality, that we may assume without loss of generality that p ii = 0 for all i [n]. As pointe in introuction, our strategy is to escribe the graph spanne by the vertices of high egree. We start with a eviation inequality on the egrees. Define h(x.= (1 + x log(1 + x x. (2.12 Lemma 2.4. For istinct i 1,..., i k [n] an t 0 we have ( k ( ( ( t t 2 P D il k( + t exp kh + k 2 p max + 1. (2.13 l=1 In particular, if t an kt/ log(t/ 2 /p max, we have Proof. We have Now, ( k P D il kt l=1 ( exp (1 + o(1kt log ( k ( k P D il k( + t P D il ED il kt. l=1 k D il = 2 l=1 k k l=1 j=l+1 A il i j + k l=1 l=1 j / {i 1,...,i k } ( t. (2.14 A il j =. 2S 1 + S 2, where S 1 an S 2 are inepenent. From Chernov s boun, for any λ 0, ( k P D il ED il kt exp( λkt + αφ(λ + βφ(2λ, where φ(λ = e λ λ 1, α = k l=1 l=1 j / {i 1,...,i k } p il j an β = 2 k k l=1 j=l+1 p il i j = k l=1 j=1 k p il i j. By hypothesis, α + β k an β k 2 p max. We take λ = log(t/ + 1 an use that φ(2λ e 2λ, we arrive at (2.13. The secon claim (2.14 is an immeiate consequence of the fact that the function h from (2.12 satisfies h(x x log x when x goes to infinity. Lemma 2.5. There exists a constant c > 0 such that the following hols. Let ε (0, 1 an t = εl 1. For S {1,..., n}, let N (S = {j : A ij = 1 for some i S} enote the set of neighbours of elements in S. Then, uner (1.2, w.h.p. for any i V t we have # [ N (i (V t N (V t \{i} ] c/ε. 12

13 Proof. Fix an integer k. Let P (1 be the probability that there exists a vertex of V t which is neighbour to at least k other elements of V t. We have P (1 P(D i0 t an l {0,..., k} : D il t, A i0 i l = 1 i 0,...,i k istinct i 0,...,i k istinct Since for any fixe i 0 we have we euce that P (1 n k max i n k exp P(D i0 t kp( l {1,..., k} : D il t 1 i 1,...,i k l=1 istinct P(D i t k max i 1,...,i k k l=1 p i0 i l k p i0 i l k (2.15 istinct ( (1 + o(1(k + 1t log t P( l {1,..., k} : D il t 1 if t an tp max 2 log(t/ (from (2.14. For t = εl 1 an k fixe such that k + 1 > a/ε with a > 1. We fin P (1 n 1 a(1+o(1 = o(1. Similarly, let P (2 be the probability that there exists a vertex i V t which is neighbour to at least k elements of N (V t \{i}. Then P (2 k s=1 τ i 0,...,i s,j 1,...,j k istinct ( P D i0,..., D is t an l = 1,..., k : A i0 j l = A jl i τ(l = 1, where the secon sum is over all surjective maps τ : {1,..., k} {1,..., s}. We euce that P (2 k s=1 τ i 0,...,i s,j 1,...,j k istinct P(D i0 t sp ( r {1,..., s} : D ir t #τ 1 ({r} k p i0 j l p jl i τ(l. Now, note that for any fixe surjective map τ : [k] [s], for any i 1,..., i s, we have { r {1,..., s} : Dir t #τ 1 ({r} } { s } D ir st k ; for any fixe j 1,..., j k, we have, as in (2.15, k i 1,...,i s l=1 istinct p jl i τ(l s p k s max. r=1 l=1 13

14 We euce, using (2.15 an (2.14 again, that P (2 k n k+s p k s s=1 n k τ k s=1 max max i The o(1 is uniform over 1 s k. Hence, P(D i t k max P i 1,...,i s istinct ( k k k s s p k s s maxe (1+o(1(s+1t log( t. ( s D il st k l=1 P (2 n k e (1+o(1t log( t ( e (1+o(1t log( t + kpmax k. As above for t = εl 1 an k a fixe integer such that k + 1 > a/ε with a > 1. P (2 = o(1. This conclues the proof of the first claim of the lemma. We fin Definition 2.6. A star graph with central egree D is the graph with vertex set {0, 1,..., D} an ege set {{0, 1}, {0, 2},..., {0, D}}. We may now prove Theorem 1.2. Proof of Theorem 1.2. Let 0 < δ < ε an set t := δl 1. By Proposition 1.11, w.h.p. for any k [n 1 ε ], D k t. Let G be the graph obtaine from G as follows. The vertex set of G is [n]. The ege set of G is the set of eges {i, j} of G (i.e. A ij = 1 such that i V t an j / V t N (V t \{i} (where the notation N ( was introuce in Lemma 2.5. By construction, G is a isjoint union of isolate vertices an of star graphs with central egrees Di, i V t. By Lemma 2.5, w.h.p., the central egrees of the stars satisfy D i c/δ Di D i. Let A be the ajacency matrix of G an let A = A A be the ajacency matrix of G\G. By Lemma A.2, the nonzero eigenvalues of A are ± Di, i V t. From what precees, w.h.p. for all i V t, D i D i = D i D i Di + D i c δ t. (2.16 for c the universal constant of Lemma 2.5. Besies, w.h.p. the maximal egree in G\G is boune by max(t, c/δ. Inee, let i [n] be a vertex. If the egree of i in G is < t, then there is nothing to prove (as the egree of i in G\G is boune by its egree in G, whereas if D i t, then the egree of i in G\G is D i Di c/δ. By Proposition 1.12, we know that w.h.p., the carinal number of V t is at most n 1 δ+o(1 10/p max, hence by Theorem A.3 of Le, Levina, an Vershynin, w.h.p. A EA C((np max 1/2 + (δl 1 1/2. (2.17 Then, one conclues thanks to Weyl s perturbation inequality (see e.g. [5, Corollary III.2.6], noticing that the constants from (2.16 an (2.17 o not epen on the choice of δ > 0. 14

15 3. Poisson tail aproximation The following sharp tail asymptotic of Bin(p 1,..., p n is of inepenent interest. It is stronger than what can be euce from Le Cam s inequality (see [2]. Theorem 3.1. Let X with istribution Bin(p 1,..., p n, = p 1 + +p n > 0 an p max = max i p i /n. Let Y be a Poisson variable with mean. There exists a universal constant C > 0 such that for any integer k satisfying 2 k (/p max 2/5 /C, we have P(X = k P(Y = k 1 C p maxk 5 2, (3.1 an P(X > k C ( k + p maxk 5 2 P(X = k. (3.2 We first check that Theorem 3.1 hols for stanar binomial variable. Lemma 3.2 (Tails of binomial laws. Let Z be istribute accoring to the binomial istribution with parameters (n, /n. There exists a universal constant C > 0 such that for any integer k with 2 k n/c, P(Z = k P(Y = k 1 Ck2 n, (3.3 an Proof. We have using (n k k n! (n k! nk, we get that P(Z > k C P(Z = k. (3.4 k ( P(Z = k P(Y = k = n! (n k!(n k 1 n e n ( 1 k k ( 1 n ( e P(Z = k n k n n P(Y = k e k/(n. (3.5 n ( k Using log(1 + x x, we get n n e k/(n. Then, it is easy to see that there is C such that as soon as, k n an n 2, we have e k/(n 1 + C k n, so that P(Z = k k 1 + C P(Y = k n. (3.6 For the lower boun, first note that there is C such that for any 1 k n, 1 Ck2 n ( 1 k n k (3.7 15

16 (inee, it comes own to prove that there is a constant C such that for any n 1, for x (0, (Cn 1/2, log(1 Cnx 2 nx log(1 x, which is easily obtaine thanks to the series expansion. Then note that there are C, C such that as soon as n an n 2, ( log 1 + ( 2 n n C an e nc(/n2 1 C 2 n n. (3.8 From (3.5, (3.7 an (3.8, we euce that P(Z = k P(Y = k (1 C k2 n The claim (3.3 then follows from (3.6 an (3.9. Note that for any integer j, We euce that for ε.= k, This conclues the proof. P(Z = j + 1 = P(Z = j (n j + n P(Z > k P(Z = k l 1 (1 C 2. (3.9 n ε l = j + 1 P(Z = j. j + 1 ε P(Z = k. 1 ε The classical Bennett s inequality gives a first tail boun for the istribution Bin(p 1,..., p n. Lemma 3.3 (Half of Bennett s inequality. Let X with istribution Bin(p 1,..., p n an set = p p n. Then for any t 0, we have where h was efine in (2.12. P(X (1 + t exp{ h(t} (3.10 The next crucial lemma will use the Lineberg s replacement principle to compare the istributions Bin(p 1,..., p n an Bin(p,..., p with p = 1 n i p i. Lemma 3.4 (Comparison principle. Let P = Bin(p 1,..., p n an Q = Bin(/n,..., /n with.= n i=1 p i. For any integer k, we have k 2 h(( P (k Q(k 2p max e + an k 2 h(( P (k, Q(k, p max e +. Proof. Fix the integers 1 i, j n. We write f(t = P (t an f ij (t is the probability of t for the istribution Bin((p l l {i,j}. We have f(t = (1 p i (1 p j f ij (t + p i (1 p i f ij (t 1 + p j (1 p j f ij (t 1 + p i p j f ij (t 2. (3.11 As (1 p i (1 p j + p i (1 p i + p j (1 p j + p i p j = 1, we first euce from (3.11 that for any k 0, P (k, = f(t f ij (t. (3.12 t k t k 16

17 We also euce from (3.11 that f(t is a polynomial of orer 2 in (p i, p j. Interestingly, the term of orer 1 is symmetric an equal to The secon orer term is equal to (p i + p j (f ij (t 1 f ij (t. p i p j (f ij (t 2f ij (t 1 + f ij (t 2. Now, let (q 1,..., q n be such that p l = q l for k {i, j}, q i + q j = p i + p j. Hence, if g(t is the probability of t for Bin(q 1,..., q n, we fin an f(t g(t = (p i p j q i q j (f ij (t 2f ij (t 1 + f ij (t 2. f(t g(t = (p i p j q i q j (f ij (k 2 f ij (k 1. t k Assume that p i p j an q i q j are both boune by p max /n. Then from the previous equations an from (3.12, we euce that f(k g(k 2 p max P (k 2, an n f(t g(t p max P (k 2,. n By Lemma 3.3, we fin that f(k g(k 2 p max n k 2 e h(( + an t k f(t g(t p max n t k e h(( k 2 +. (3.13 Now, if (p 1,..., p n (/n,..., /n then there exists (i, j such that p i < /n < p j (since the average /n is in the convex hull of (p 1,..., p n. We consier (q 1,..., q n as above such that q i = /n an p i < q j = p i + p j /n < p j. Then the boun (3.13 applies here an q max = max k q k p max. We may thus repeat the same operation to p 1 = (q 1,..., q n an get p 2 an so on. After m n iterations, we arrive at p m = (/n,..., /n. Summing the m times (3.13 gives P (k Q(k 2m p max k 2 e h(( n + k 2 h(( 2p max e +. This gives the first claim. The same argument applie to the right-han sie of (3.13 gives the secon claim. We are finally reay for the proof of Theorem 3.1 Proof of Theorem 3.1. Let Z be a ranom variable with istribution Bin(/n,..., /n. In view of Lemma 3.2, it sufficient to prove (up to ajusting the universal constant C > 0 that P(X > k P(Z > k + P(X = k P(Z = k C p maxk 5 2 P(Y = k. Then, from Lemma 3.4, we euce that it is sufficient to check that for k, k 2 h(( e + P(Y = k C k 5 2. (

18 First, from Stirling s formula, we fin P(Y = k = k k! e e k log(k/+k C k Seconly, from the convexity of x h(x +, we fin (( (( k 2 k h = h 2 It follows that This conclues the proof of ( k 2 h(( e + P(Y = k + = e h( k C k ( k h C ( k 2 k.. 2 log ( k. A. Auxiliary results Lemma A.1. Let, in a probability space, E E an F F be some events. (E \E F = (F \F E =. Then Cov(1 E, 1 F = Cov(1 E, 1 F + P(E \EP(F + P(F \F P(E. Proof. Set E.= E \E an F.= F \F. We have As we conclue easily. Cov(1 E, 1 F Cov(1 E, 1 F = P(E F P(E F (P(E P(F P(EP(F = P((E F \(E F [(P(E + P(E (P(F + P(F P(EP(F ] (E F \(E F = (E F (E F (E F, Assume that Lemma A.2. The ajacency matrix A of a star graph with central egree D 1 (see Definition 2.6 is a (D + 1 (D + 1 real symmetric matrix with nonzero eigenvalues ± D an associate eigenvectors (± D, 1,..., 1 (the first coorinate correspons to the centre of the star. Proof. One notices that the matrix has rank 2 an that the vectors given here are actually eigenvectors for ± D. The following result is [12, Theorem 2.1]. It concerns general inhomogeneous Erős-Rényi graphs with mean ajacency matrix (p ij i,j [n]. We recall that a weighte graph has ajacency matrix A whose entries are nonnegative real numbers, with the entry A ij 0 enoting the weight of the ege {i, j}. Theorem A.3. Set p max.= max i,j p ij an choose r 1. Then the following hols with probability at least 1 n r. Consier a subset of at most 10p 1 max vertices an reuce the weights of the eges incient to those vertices in an arbitrary way. Then the ajacency matrix A of the new (weighte graph satisfies A EA Cr 3/2 ( np max +, where C is a constant inepenent of r, = max 1 i n R i l 1 with R 1,..., R n the rows of A. 18

19 References [1] A. Auffinger, G. Ben Arous, an S. Péché. Poisson convergence for the largest eigenvalues of heavy taile ranom matrices. Ann. Inst. Henri Poincaré (B, 45: , [2] A. D. Barbour, L. Holst, an S. Janson. Poisson approximation, volume 2 of Oxfor Stuies in Probability. The Clarenon Press, Oxfor University Press, New York, Oxfor Science Publications. [3] F. Benaych-Georges, C. Borenave, an A. Knowles. Spectral raii of sparse ranom matrices. Preprint arxiv: , [4] F. Benaych-Georges an S. Péché. Localization an elocalization for heavy taile ban matrices. Ann. Inst. Henri Poincaré Probab. Stat., 50(4: , [5] R. Bhatia. Matrix analysis, volume 169 of Grauate Texts in Mathematics. Springer-Verlag, New York, [6] B. Bollobás. Ranom graphs, volume 73 of Cambrige Stuies in Avance Mathematics. Cambrige University Press, Cambrige, secon eition, [7] L. Erős, A. Knowles, H.-T. Yau, an J. Yin, Spectral Statistics of Erős-Rényi Graphs II: Eigenvalue Spacing an the Extreme Eigenvalues, Comm. Math. Phys., 314: , [8] L. Erős, A. Knowles, H.-T. Yau, an J. Yin, Spectral Statistics of Erős-Rényi Graphs I: Local Semicircle Law, Ann. Prob., 41: , [9] U. Feige an E. Ofek. Spectral techniques applie to sparse ranom graphs. Ranom Structures Algorithms, 27(2: , [10] Z. Fürei an J. Komlós. The eigenvalues of ranom symmetric matrices. Combinatorica, 1(3: , [11] M. Krivelevich an B. Suakov. The largest eigenvalue of sparse ranom graphs. Combin. Probab. Comput., 12(1:61 72, [12] C. M. L. Le, E. Levina, an R. Vershynin. Concentration an regularization of ranom graphs. arxiv: , [13] J. O. Lee an J. Yin. A necessary an sufficient conition for ege universality of Wigner matrices. Preprint arxiv: [14] J. O. Lee an K. Schnelli, Local law an Tracy-Wiom limit for sparse ranom matrices, Preprint arxiv: [15] A. Soshnikov. Poisson statistics for the largest eigenvalues of Wigner ranom matrices with heavy tails. Electr. Comm. Prob., 9:82 91, [16] V. H. Vu. Spectral norm of ranom matrices. Combinatorica, 27(6: ,

20 Florent Benaych-Georges, MAP 5 (CNRS UMR Université Paris Descartes, 45 rue es Saints-Pères Paris ceex 6, France. florent.benaych-georges@parisescartes.fr. Charles Borenave, Institut e Mathématiques (CNRS UMR Université Paul Sabatier Toulouse ceex 09. France. borenave@math.univ-toulouse.fr. Antti Knowles, University of Geneva, Section of Mathematics, 2-4 Rue u Lièvre, 1211 Genève 4, Switzerlan. antti.knowles@unige.ch. Acknowlegements. C. B. is supporte by grants ANR-14-CE an ANR-16-CE A. K. is supporte by the Swiss National Science Founation an the European Research Council. 20