Abstract
Consider an \(n \times n\) Hermitian random matrix with, above the diagonal, independent entries with \(\alpha \)-stable symmetric distribution and \(0 < \alpha < 2\). We establish new bounds on the rate of convergence of the empirical spectral distribution of this random matrix as \(n\) goes to infinity. When \(1 < \alpha < 2\) and \( p > 2\), we give vanishing bounds on the \(L^p\)-norm of the eigenvectors normalized to have unit \(L^2\)-norm. On the contrary, when \(0 < \alpha < 2/3\), we prove that these eigenvectors are localized.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
We consider an array \((X_{ij})_{1 \le i \le j}\) of i.i.d. real random variables and set, for \(i > j, X_{ij} = X_{ji}\). Then, for each integer \(n \ge 1\), we may define the random symmetric matrix:
The eigenvalues of the matrix \(X\) are real and are denoted by \(\lambda _n (X) \le \cdots \le \lambda _1(X)\). In the large \(n\) limit, the spectral properties of this matrix are now well understood as soon as \(X_{ij}\) has at least two finite moments see e.g. [3–5, 8, 15, 28] for reviews, or [20, 21, 23, 27, 29] for recent results on universality. The starting point of this analysis is the Wigner’s semi-circular law, which asserts that if the variance of \(X_{ij}\) is normalized to \(1\), then the empirical spectral measure
converges almost surely for the weak convergence topology to the semi-circular law \(\mu _2\) with support \([-2,2]\) and density \(f_2 ( x) = \frac{1}{2 \pi } \sqrt{ 4 - x ^2}\). As already advertised, many more properties of the spectrum are known. For example, if the entries are centered and have a subexponential tail, then, see [18, 19], for any \(p >2\) and \(\varepsilon > 0\),
is \(O ( n^{ 1 / p - 1/2 + \varepsilon })\), where \(\Vert v \Vert _p = \left( \sum _{i=1} ^n |v_i|^p \right)^{\frac{1}{p}}\). This implies that the eigenvectors are strongly delocalized.
When the second moment is no longer finite, much less is known and the picture is different. Let \(0 < \alpha < 2\) and assume for simplicity that
Then, we are not anymore in the basin of attraction of Wigner’s semi-circular law: now the empirical spectral measure
converges a.s. for the weak convergence topology to a new limit law \(\mu _\alpha \), see [7] and also [6, 10]. It is known that \(\mu _\alpha \) is symmetric, has full support, a bounded density \(f_\alpha \) which is analytic outside a finite set of points. Moreover, \(f_\alpha (0)\) has an explicit expression and as \(x\) goes to \(\pm \infty , f_\alpha ( x) \sim (\alpha / 2) |x|^{-\alpha -1}\). Finally, as \(\alpha \) goes to \(2, \mu _\alpha \) converges for the weak convergence topology to \(\mu _2\). One of the difficulty of this type of random matrices is the lack of an exactly solvable model as in the Gaussian Unitary Ensemble or the Gaussian Orthogonal Ensemble in the finite variance case.
In the present paper, we give a rate of local convergence to \(\mu _\alpha \) and investigate the behavior of the eigenvectors of \(X\). In a fascinating article [12], Bouchaud and Cizeau have made some prediction for the eigenvectors of \(X\). They argue that the situation is different for \(0< \alpha < 1\) and \(1 < \alpha < 2\). They quantify the localized nature of a vector \(v\) with \(\Vert v \Vert _2 =1\) by two scalars: \( \Vert v \Vert _4\) and \( \Vert v \Vert _1\). If \(\Vert v \Vert _4 = o (1)\) the vector is said to be delocalized, if \(\Vert v \Vert _4 \ne o (1)\) but \(\Vert v \Vert _1 \gg 1\) then \(v\) is weakly delocalized (we might also say weakly localized), while if \(\Vert v \Vert _1 = O(1)\) then the vector is localized. Now suppose that \(v\) is an eigenvector of \(n^{-1/\alpha } X\) associated to an eigenvalue \(\lambda \). For \(1 < \alpha < 2\), we have proved that all but \(o(n)\) of the eigenvectors are delocalized (this disproved the prediction of [12]). Our result is in fact stronger as we can show that the \(L^\infty \) norm of these vectors goes to zero, which insures that the \(L^p\) norm goes to zero for all \(p>2\) and to infinity for all \(p<2\) by duality.
For \(0 < \alpha < 1\), Bouchaud and Cizeau predict that with high probability, if \(|\lambda | < E_\alpha \) then \(v\) is weakly delocalized, while if \(|\lambda | > E_\alpha , v\) is localized. It is reasonable to predict that \(E_\alpha \) goes to \(0\) as \(\alpha \downarrow 0\) and goes to infinity as \(\alpha \uparrow 2\). It is not clear whether this threshold \(E_\alpha \) depends on the choices of the norms \(L^1\) and \(L^4\) to quantify localization and delocalization. We are far from proving the existence of such a threshold within the spectrum. Nevertheless, for \(0 < \alpha < 2/3\), we have proved that there exists \(E_\alpha >0\) such that if \(|\lambda | \ge E_\alpha \) then a localization occurs : the mass of \(v\) is carried by at most \( n^{1-\delta _\alpha }\) entries, for some \(\delta _\alpha >0\).
This heavy-tailed matrix model is in some sense similar to the adjacency matrix of Erdős-Rényi graphs with parameter \(p/n\) since its entries are of order one only with probability of order \(1/n\). In the regime where \(p\) is going to infinity faster than \(n^{2/3}\), this adjacency matrices were shown to belong to the university class of Wigner random matrices [16, 17]. If \(pn / ( \log n )^c\) goes to infinity for some constant \(c\), the delocalization of eigenvectors was also proved in these articles. In the related model of the adjacency matrix of uniformly sampled \(d\)-regular graphs on \(n\) vertices, the delocalization of eigenvectors has been studied in Dumitriu and Pal [14] and Tran et al. [30]. It was also conjectured by Sarnak that as soon as \(d \ge 3\), this model also belongs to the university class of Wigner random matrices.
1.1 Main results
Let us now be more precise. Throughout the paper, the array \((X_{ij})_{1 \le i \le j}\) will be real i.i.d. symmetric \(\alpha \)-stable random variable such that for all \(t \in \mathbb{R }\),
for some \(0 < \alpha < 2\) and \(w_\alpha = \pi / ( \sin (\pi \alpha / 2)\Gamma (\alpha )) \). With this choice, the random variables \((X_{ij})\) are normalized in the sense that (1) holds. The assumption that the random variables follow an \(\alpha \)-stable law should not be crucial for our results, it will however simplify substantially some proofs. We define the hermitian matrix
The eigenvalues of the matrix \(A\) are denoted by \(\lambda _n ( A) \le \cdots \le \lambda _1 (A)\). The empirical spectral measure of \(A\) is defined as
The resolvent of \(A\) will be denoted by
where \(z \in \mathbb{C }_+ = \{z \in \mathbb{C }: \mathrm{Im}(z) >0\}\). The Cauchy–Stieltjes transform of \(\mu _A\) is easily recovered from the resolvent:
From [7, 10], for any fixed interval \(I \subset \mathbb{R }\), a.s. as \(n \rightarrow \infty \),
where \(| I |\) denotes the length of the interval \(I\). As in [20, 21], the opening move for proving statements about the eigenvectors of \(A\) is to reinforce the convergence (3) for small intervals whose length vanishes with \(n\). We will express our main results in terms of a scalar \(\rho \) depending on \(\alpha \):
The scalar \(\rho \) depends continuously on \(\alpha \) and is non-decreasing. Roughly speaking we are able to prove that the convergence (3) holds for all intervals of size larger than \(n^{ - \rho + o(1)}\). A precise statement is the following.
Theorem 1.1
(Local convergence of the empirical spectral distribution) Let \(0 < \alpha < 2\). There exists a finite set \({\mathcal{E }}_\alpha \subset \mathbb{R }\) such that if \(K \subset \mathbb{R }\backslash {\mathcal{E }}_\alpha \) is a compact set and \(\delta >0\), the following holds. There are constants \(c_0, c_1 >0\) such that for all integers \(n \ge 1\), if \(I\subset K\) is an interval of length \( |I| \ge c_1 n^{-\rho } ( \log n)^2\), then
with probability at least \(1 - 2 \exp \left( - c_0 n \delta ^2 | I | ^2 \right)\).
In the forthcoming Theorem 3.5, we will give a slightly stronger form of Theorem 1.1: we will allow the parameter \(\delta \) to depend explicitly on \(n\) and \(| I |\) and the logarithmic correction in front of \(n^{-\rho }\) will be reinforced. The proof of Theorem 1.1 will be based on estimates of the diagonal coefficients of the resolvent matrix \(R(z)\) as \(z = E + i \eta \) gets close to the real axis with \(\eta = n^{ - \rho + o(1)}\). For technical reasons, we have only been able to establish (3) for intervals outside the finite set \({\mathcal{E }}_\alpha \) which contains \(0\). The same type of result should hold for all sufficiently large intervals. In Proposition 2.1, we will give an upper bound on \(\mu _{A} (I) \) (i.e. a Wegner’s estimate) which will be valid for all intervals of size larger than \(n^{- (\alpha + 2) / 4} \). The threshold \(\rho \le \frac{1}{2}\) may be optimal, even though for Wigner’s matrices it is simply one, since the spectral measure of heavy tails random matrices fluctuates like \(O(n^{-1/2})\) rather than like \(O(n^{-1})\) for Wigner’s matrices (see [9, 26]).
Theorem 1.1 will have the following corollary on the delocalization of the eigenvectors.
Theorem 1.2
(Delocalization of eigenvectors) Let \(1 < \alpha < 2\). There exist a finite set \({\mathcal{E }}_\alpha \subset \mathbb{R }\) and a constant \(c >0\) such that if \(K \subset \mathbb{R }\backslash {\mathcal{E }}_\alpha \) is a compact set, with probability tending to \(1\),
where \(v_1, \ldots , v_n\) is an orthogonal basis of eigenvectors of \(A\) associated to the eigenvalues \(\lambda _1 (A), \ldots , \lambda _n (A)\).
Notice that for \(p > 2, \Vert v \Vert _p \le \Vert v \Vert ^{2/p}_2 \Vert v \Vert ^{1 - 2/p}_\infty \). Hence, Theorem 1.2 implies that the \(L^p\)-norm of any eigenvector associated to an eigenvalue in \(K\) goes in probability to \(0\) as soon as \(p > 2\). Similarly, from \(\Vert v \Vert ^2_2 \le \Vert v \Vert _1 \Vert v \Vert _\infty \), we have a lower bound of order \(n^{ \rho (1 - \frac{1}{\alpha }) + o(1)}\) on the \(L^1\)-norm of the eigenvectors. Note that our estimate becomes trivial as \(\alpha \downarrow 1\) and give upper bound of order \(n ^ { - 1 / 4 + o (1)}\) as \(\alpha \uparrow 2\). For any \(\kappa > 0\), in the proof of Theorem 1.2, we will see that by increasing suitably \(c\), the probability that the event (4) holds is at least \(1 - n^{-\kappa }\).
We now present our result on localization of eigenvectors. We are not able to prove localization for all eigenvectors but only for ”typical” eigenvectors associated to an eigenvalue in a small interval. More precisely, we consider \(v_1, \ldots , v_n\) an orthogonal basis of eigenvectors of \(A\) associated to the eigenvalues \(\lambda _1 (A) , \ldots , \lambda _n (A)\). If \(I\) is an interval of \(\mathbb{R }\), we define \(\Lambda _I\) as the set of eigenvectors whose eigenvalues are in \(I\). Then, if \(\Lambda _I\) is not empty, for \( 1 \le k \le n\), set
where, throughout this paper,
is the cardinal of \(\Lambda _I. W_I(k) / n\) is the average amplitude of the \(k\)-th coordinate of eigenvectors in \(\Lambda _I\). By construction, the average amplitude of \(W_I\) is \(1\):
If the eigenvectors in \(\Lambda _I\) are localized and \(I\) contains few eigenvalues, then we might expect that for some \(i, W_I (k) \gg 1\), while for most of the others \(W_I(k) = o(1)\). More quantitatively, fix \(0 < \delta < 1\) and assume that for some \(0 < \kappa < 1\) and \(0 < \varepsilon < 1\)
then, setting \(J=\{ k : W_I (k) \ge (\delta ^{-1} \varepsilon )^{ - \frac{1}{ 1 - \kappa } } \}\), we find
In particular, all but a proportion \(\delta \) of the mass of \(W_I\) is carried by a set \(J\) of cardinal at most \(|J| \le n ( \delta ^{-1} \varepsilon )^{ \frac{1}{ 1 - \kappa } }\). If \(\varepsilon \) goes to \(0\) with \(n\), this indicates a localization phenomenon. With this in mind, we can state our result.
Theorem 1.3
(Localization of eigenvectors) Let \(0 < \alpha < 2/3, 0 < \kappa < \alpha /2\) and \(\rho \) be as above. There exists \(E_{\alpha ,\kappa }\) such that for any compact \(K \subset [-E_{\alpha ,\kappa },E_{\alpha ,\kappa }]^c\), there are constants \(c_0,c_1 >0\) and for all integers \(n \ge 1\), if \(I\subset K\) is an interval of length \( |I| \ge n^{-\rho } ( \log n)^2\),
with probability at least \(1 - 2 \exp \left( - c_0 n | I | ^4 \right)\).
This result is interesting when \(I = [ E - n^{-\rho + o(1)}, E + n^{-\rho + o(1)} ]\) is a small neighborhood around some large \(E\). Then it shows that for any \(0 < \kappa < \alpha /2\), the mass of the eigenvectors corresponding to eigenvalues around \(E\) is concentrated around order \(n^{ 1 - 2 \rho \kappa / ( 2 - \alpha ) }\) entries as long as \(|E|\) is large enough. The proof of Theorem 1.3 will be done by showing that
vanishes if \(\eta =n^{-\rho + o(1)} \) even though that
stays bounded away from 0. This phenomenon will have an interpretation in terms of a random self-adjoint operator introduced in [10] which is, in some sense, the limit of the matrices \(A\). We will prove that the imaginary part of its resolvent vanishes at \(E + i \eta \), with \(\eta = o(1)\) and \(|E|\) large enough, while its expectation does not, see Theorem 5.1. Note that if \(0 < \eta \ll n^{-1}\), then we necessarily have that for almost all \(E, \mathrm{Im}R(E + i \eta )_{kk}\) converges to \(0\). The fact that our estimate \( |I| \ge n^{-\rho }\) gets worse as \(\alpha \) goes to \(0\) is an artifact of the proof : our rate of convergence of \(R(E + i \eta )_{kk}\) to its limit gets worse as \(\alpha \) gets small. It is however intuitively clear that the localization should be stronger when \(\alpha \) is smaller. However, in the forthcoming Theorem 5.9, we will prove that, for any \(0 < \alpha < 2/3\), the expression (6) goes to \(0\) if \(\eta =n^{-\frac{1}{6}} \). Finally, it is worth to notice that computing the fractional moments of the resolvent matrix as a way to prove localization is already present in the literature on random Schrödinger operators, see e.g. [2].
The remainder of the paper is organized as follows. In Sect. 2 we establish general upper bounds on \(N_I\) defined by (5). Section 3 contains the proof of Theorem 1.1. Section 4 is devoted to the proof of Theorem 1.2. The arguments developed in these two sections are based on ideas from the seminal work of Erdős, Schlein, Yau (see e.g. [18–20]) concerning Wigner’s matrices with enough moments, as well as on the analytic approach of heavy tailed matrices as initiated in [6, 7]. However, there was a technical gap due to the lack of concentration inequalities, as well as of simple loop equations, that hold for finite second moment Wigner matrices. A few of the required new estimates due to the specific nature of heavy tailed matrices are contained in the appendix on concentration inequalities and stable laws. In Sect. 5 we prove Theorem 1.3, which is based on the representation of the asymptotic spectral measure given in [10] and a new fixed point argument which allows to prove the vanishing of the imaginary part of the resolvent in the regime \(\alpha \in (0,2/3)\).
The whole article is quite technical, but hopefully shall be useful for further local study of the spectrum of random matrices which do not belong to the universality class of Wigner’s semi-circle law.
2 Upper bound on the spectral counting measure
For \(\eta > 0\) and \( E \in \mathbb{R }\), we set \(I = [E - \eta , E + \eta ]\). The goal of this section is to provide a rough upper bound on \(N_I\) when \(\eta \) is large enough, where \(N_I\) was defined by (5). Let
Proposition 2.1
(Upper bound on counting measure) Let \(0 < \alpha < 2\). There exist \(c,c^{\prime }>0\) depending only on \(\alpha \) such that if \(\eta \ge n^{- \frac{\alpha + 2}{ 4}}\), then, for all integers \(n\), for all \(t\ge c\),
This bound will later be refined in the forthcoming Proposition 3.6. The upper bound on the eigenvalues counting measure implies an upper bound for the trace of the resolvent.
Corollary 2.2
(Trace of resolvent) Let \(0 < \alpha < 2\) and \(z = E + i \eta \in \mathbb{C }_{+}\). There exists \(c >0\) depending only on \(\alpha \) such that if \(\eta \ge n^{- \frac{\alpha + 2}{ 4}}\), then, for all integers \(n\),
Proof
By the spectral theorem
Let \(\eta \) be the imaginary part of \(z\). Define \(I_0 = [E - \eta , E + \eta ]\) and for integer \(k \ge 0, I_{k+1} = [E - 2^{k+1} \eta , E + 2^{k+1} \eta ] \backslash [E - 2^{k} \eta , E + 2^{k} \eta ] \). By construction, if \(\lambda _j (A) \in I_k\) then \(|\lambda _j (A)- z|^{-2} \le 2^{- 2k + 1} \eta ^{-2}\). Therefore, if \(N_{I_k} = n \mu _{A} (I_k)\) is the number of eigenvalues in \(I_k\), then
We write \(I_k = I^+_ k \cup I^-_k\), where \(I^\pm _k = \mathbb{R }_{\pm } \cap I_k\). To estimate \(\mathbb{E }[ N_{I_k^\pm }]\) we apply Proposition 2.1. Namely it yields that for each interval \(I\) of length \(\eta \ge n^{-\frac{\alpha +2}{4}}\), for any \(\tau \ge c\)
for some finite constant \(c_0 > 0 \) and \(\tau \ge 1\). Therefore, taking \(\tau \) of order \((\log n)^{\frac{2+\alpha }{4}}\), we deduce that there exists some finite constant \(c_1 > 0 \) such that
Therefore, we deduce from (8) that
\(\square \)
The rest of this section is devoted to the proof of Proposition 2.1.
2.1 A geometric upper bound
In this paragraph, we recall a general upper bound for \(N_I\) that is due to Erdős–Schlein–Yau [20], namely if we let \(A^{(k)}\) be the principal minor matrix of \(A\) where the \(k\)-th row and column have been removed, \(W^{(k)}\) be the vector space generated by the eigenvectors of \(A^{(k)}\) corresponding to eigenvalues at distance greater than \(\eta \) from \(E\), we have
Let us prove (9). We start with the resolvent formula,
where \(X_k = ( X_{k1}, \ldots ,X_{k k-1}, X_{k k+1}, \ldots , X_{kn}) \in \mathbb{R }^{n-1}\) and \(R^{(k)} = ( A^{(k)} - z ) ^{-1}\).
Identifying the real and imaginary part, we get, using the fact that \(z\) and the eigenvalues of \(R^{(k)}\) are in \(\mathbb{C }_+\),
Let \((\lambda _{i}^{(k)})_{1 \le i \le n-1}\) and \((u_{i}^{(k)})_{1 \le i \le n-1}\) be the eigenvalues and eigenvectors of \(A^{(k)}\). Choosing \(z = E + i \eta \), we have the spectral decomposition
If \( |\lambda _i^{(k)} - E| \le \eta \), then \( \frac{\eta }{ ( \lambda _i - E)^2 + \eta ^2 } \ge 1 / ( 2 \eta )\). Therefore, we deduce
We rewrite the above expression as
where
Since \(I = [E - \eta , E + \eta ]\), the inequalities (11) and
give (9). We set
From Weyl interlacement theorem, we have
2.2 Proof of Proposition 2.1
We note that up to increasing the constant \(c\) and \(1/c^{\prime }\), it is sufficient to prove the proposition only for all \(\eta \ge c_1 n^{- \frac{\alpha + 2}{ 4}}\) for some \(c_1\) (indeed, \(N_{I^{\prime }} \le N_I\) if \(I^{\prime } \subset I\) and if \(N_I \le t n | I |^\gamma \) then \(N_{I^{\prime }} \le t n |I^{\prime }|^\gamma (|I | / |I^{\prime } | )^\gamma \)).
In the sequel we denote in short \({\mathrm{dist }}_k = {\mathrm{dist }}( X_k , W^{(k)})\). From (9)–(12), we write
We have \( {\mathrm{dist }}^2 _k = \langle X_k , P_k X_k \rangle \), where \(P_k\) is the orthogonal projection onto \(W^{(k)}\). We note that \(X_k\) is independent of \(W^{(k)}\). From Lemma 7.1, there exists a positive \(\alpha /2\)-stable random variable \(S_k\) and a standard Gaussian vector \(G_k\), independent from \(S_k\), such that
Note that \(\eta \ge c n^{- \frac{\alpha + 2}{ 4}}\) is equivalent to \(n \eta ^{\frac{2\gamma }{\alpha }} \ge c^{\frac{2\gamma }{\alpha }}\). By Corollary 6.2 (applied to \(A=P_k\)), there exists universal constants \(C,\delta \) so that if \(N^{(k)}_I\ge t c^{-\gamma } n \eta ^{\gamma } \ge t n^{1-\frac{\alpha }{2}}\) for \(t \ge C c^{\gamma }\), with probability at least
we have,
Hence, if \(F_n\) denotes the event that \(N_I > \lfloor t n \eta ^{\gamma } \rfloor \) and for all \(k, \Vert P_k G_k \Vert _\alpha \ge \delta \left( t n \eta ^\gamma \right)^{\frac{1}{\alpha }}\), we have from (9) and (13)
With our choice of \(\gamma , 2 - 2 \gamma / \alpha = \gamma \), hence, with \(c_1 = 4 \delta ^{-2}\), we deduce
The variables \((S_k)_{1 \le k \le n}\) have the same distribution but are correlated. Nevertheless, note that the function \(x \mapsto \exp ( x^\delta )\) is convex on \([b_\delta , +\infty ) \) with \(b_\delta = 0\) if \(\delta \ge 1\), and \(b_\delta = (1/\delta -1)^{1/\delta }\) if \(0 \le \delta \le 1\). Hence from the Jensen inequality, we deduce
In particular for every \(c_2 > 0\),
where \(c_3 = \exp ( c_2 b_{\alpha / (2 - \alpha ) } ^{\alpha / (2 - \alpha )} )\). By Lemma 7.3, for \(c_2\) small enough, the above is finite. Thus, from the Markov inequality, for some constants \(c_4,c_5 >0\),
Therefore, by (14), we deduce
which completes the proof of the proposition.
3 Local convergence of the spectral measure
To prove the local convergence of the spectral measure, we shall prove that an observable of the resolvent satisfies nearly a fixed point equation, which also entails an approximate equation for the resolvent. Such an equation was already derived in [6, 7] but the error terms are here carefully estimated. This step will be crucial to obtain, in the second part of this section, a rate of convergence of the Stieltjes transform of the spectral measure toward its limit. The range of convergence will be first derived roughly, and then improved for \(\alpha >1\) thanks to bootstraps arguments.
3.1 Approximate fixed point equation
The observables we shall be interested in will be
(For \(1 \le k , \ell \le n\), we will write indifferently \(R_{ k \ell } (z)\) or \(R (z)_{ k \ell }\)). For \(\beta \in [0,2]\), we define \({\mathcal{K }}_\beta = \{z \in \mathbb{C }: | \arg (z) | \le \frac{\pi \beta }{2} \}\). By construction \( - i R_{11} ( z) \in {\mathcal{K }}_{1}\) for \(z\in \mathbb C _+\), so that \( Y(z) \in {\mathcal{K }}_{\alpha / 2}\) and \( X(z) \in {\mathcal{K }}_{1}. \) On \({\mathcal{K }}_{\alpha / 2}\), we may define the entire functions
and
For further use, we define, with the notation of (10),
Note that, writing explicitly the dependence in \(n, R_n^{(1)} = ( A_n^{(1)} - z ) ^{-1}\) and \(\frac{a_n}{a_{n-1}} A_n^{(1)}\) has the same distribution than \(A_{n-1}\). We may thus apply Corollary 2.2 to \(R^{(1)}\) (it can be checked the difference between \(a_{n}\) and \(a_{n-1}\) is harmless). For some constant \(c>0\), we therefore have the upper bound for all \(z = E + i \eta \) and \(\eta \ge n^{ -\frac{\alpha +2}{4}}\),
The main result of this paragraph is the following approximate fixed point equations.
Proposition 3.1
(Approximate fixed point equation) Let \(0 < \alpha < 2\) and \(z = E + i \eta \in \mathbb{C }_+, E \ne 0\). There exists \(c =c(E)>0, c(E)\le \text{ const.} |E|^{-\alpha }\vee |E|^{-\alpha /2}\), such that if \(n^{ -\frac{\alpha +2}{4}} \le \eta \le 1\) and
then, for any integer \(n \ge 1\),
and
We note that we could use the bound (19) to get an explicit upper bound on \(\varepsilon \). In the forthcoming Proposition 3.6 we will however improve this bound for some range of \(\eta \).
In the first step of the proof, we compare \(Y(z)\) with an expression which gets rid of the off-diagonal terms of \(R^{(1)}\) in (10). More precisely, with the notation of (10), we define
and similarly,
We shall prove that
Lemma 3.2
(Diagonal approximation) Let \(0 < \alpha < 2\) and \(z = E + i \eta \in \mathbb{C }_+\). There exists \(c >0\) such that if \(\varepsilon \) is given by (20) then
We start with a technical lemma.
Lemma 3.3
(Off-diagonal terms) Let \(B\) be an hermitian matrix with resolvent \(G = ( B - z)^{-1}\). For any \(0 < \alpha < 2\), there exists a constant \(c = c(\alpha ) >0\) such that for \(n \ge 2\),
and if \(1 < \alpha < 2\),
Proof
Let \( 0 < \alpha \le 1\). We use a decoupling technique: from [13, Theorem 3.4.1] there exists a universal constant \(c > 0\) such that
where \(X^{\prime }_1\) is an independent copy of \(X_1\). From the stable property of \(X^{\prime }_1\), we deduce that
and similarly for the imaginary part. From the stable property of \(X_1\),
where \((\hat{X}_\ell )_\ell \) is a random vector whose marginal distribution is again the law of \(X_{11}\) (note however that the entries of \((\hat{X}_\ell )_\ell \) are correlated). Let \(\rho _\ell = \sum _{k\ne \ell } | \mathrm{Re}( G_{k\ell } ) | ^ \alpha \) and \(\rho = \sum _\ell \rho _\ell \). For \( s \ge 2\) to be chosen later, we define \(Y_\ell = \hat{X}_{\ell } {\mathbf{1 }}( |\hat{X}_{\ell }| \le s a_n )\) and \(Y^{\prime }_1 = X^{\prime }_{11} {\mathbf{1 }}( | X^{\prime }_{11}| \le s)\). It is straightforward to check that
where we have used the Markov inequality and the union bound
We choose \(s = 2 \vee ( t / \rho ^{1 / \alpha })\), we find that
The same statement holds for \(\mathrm{Im}(G)\). To sum up, we deduce from (23) that
Then, the first statement of the lemma follows Hölder’s inequality which asserts that
where we used that \(G_{k\ell } =\bar{G}_{\ell k}\).
Now, assume \(\alpha > 1\). By integrating the above bound, we find easily
(In fact, with slightly more care, we may replace \(\log n\) by \((\log n)^{\frac{1}{\alpha }} \log (\log n)\)). It remains to check that we can remove the term \(\log n\) for \(\alpha > 4/3\). It will come easily from the bound
Let \(s \ge 1\) to be chosen later. We now set \(Y_k = X_{1k} {\mathbf{1 }}( |X_{1k}| \le s a_n ) / a_n \). We write
The variables \(Y_k\) are iid and by symmetry \(\mathbb{E }Y_1 = \mathbb{E }Y_1^3 = 0\). Hence, since \(G_{k\ell } = G_{\ell k}\), taking expectation we obtain
It is routine to check that, for \(p > \alpha \), \( \mathbb{E }[ |Y _{1} |^p ] \le c(p) s^{p - \alpha } n^{-1}\). Hence, arguing as above, we find
We conclude by choosing \(s = 1 \vee ( n^2 t^2 / {\mathrm{tr }}GG^* ) ^{1 / (4 - \alpha )}\). \(\square \)
We now can turn to the proof of Lemma 3.2
(Proof of Lemma 3.2)
Define
We notice that for any, \(z, z^{\prime } \in \mathbb{C }_+\) and \(\alpha > 0\),
With the notation of (10), we also note that \(\mathrm{Im}(\sum _{k=1}^n X_{1k}^2 R^{(1)}_{kk} ) \ge 0\) and \(\mathrm{Im}( - a_n^{-1} X_{11}+ \langle X_1 , R^{(1)}X_1 \rangle ) \ge 0\). Hence, from (10), for any event \(\Omega \),
Applying the same argument with \(X(z),J(z)\), we get
with
We may bound \(D(z)\) by using that by Lemma 3.3. Indeed, since \(R^{(1)} \) is independent of \(X^{(1)}\), it gives
and for \( 1 < \alpha < 2\),
-
Let us first assume that \(1 < \alpha < 2\), then taking \(\Omega ^c = \emptyset \) in (29) and using (32), it shows that for some constant \(c>0\),
$$\begin{aligned} D(z) \le c \eta ^{ - 1} \left( n^{-\frac{1}{\alpha }} + \sqrt{\frac{M_n(z)}{n}} \left(1 + {\mathbf{1 }}_{1< \alpha \le 4/3} \log n \right) \right). \end{aligned}$$ -
Assume that \(0< \alpha \le 1\), we take in (30)
$$\begin{aligned} \Omega = \left\{ a_n^{-1} |X_{11}| \le t \, ; a_n ^{-2} \left| \sum _{1 \le k \ne \ell \le n } X_{1k} X_{1\ell } R_{k\ell }^{(1)}(z)\right| \le t \right\} . \end{aligned}$$Then, we have
$$\begin{aligned} \mathbb{E }[|T(z)|\!{\mathbf{1 }}_{\Omega }]&\!\le \!&\!\int _0^t \mathbb{P }\left(\! a_n^{-1} |X_{11}|\!\ge \!y\!\right)dy \!+\! \int _0^t \mathbb{P }\left(\!\left| a_n ^{-2}\!\sum _{1 \le k \ne \ell \le n } X_{1k} X_{1\ell } R_{k\ell }^{(1)}(z)\!\right| \!\ge \! y\!\right)dy. \end{aligned}$$Assume that \(1 / n \le t \le 1\). Then, using (31), we find that \( \mathbb{E }[|T(z)|{\mathbf{1 }}_{\Omega }] \) is bounded up to multiplicative constant (depending on \(\alpha \)) by
$$\begin{aligned} \frac{t^{1 - \alpha }}{n} + \frac{\log (n)}{n} {\mathbf{1 }}_{\alpha =1} + \left(\frac{M_n(z)}{n}\right)^{\frac{\alpha }{2} } \left( \log n \right)^2 t^{1 - \alpha }. \end{aligned}$$(using (20) we may safely bound the terms \( \log ( t n / M_n )\) by \(\log n\)). With \(t = \eta \) we find
$$\begin{aligned} D(z) \le c \eta ^{-\alpha } n ^{-1} +c \frac{1_{\alpha =1} ( \log n) }{\eta n} + c \left(\frac{M_n(z)}{n\eta ^2}\right)^{\frac{\alpha }{2}} \left( \log n \right)^2 \!. \end{aligned}$$
This yields the claimed bounds. \(\square \)
We next relate \(I(z)\) and \(J(z)\) with the functions \(\varphi _{\alpha ,z}\) and \(\psi _{\alpha ,z}\) defined in (16)–(17) by using the well known identities
valid for \(x\) in the interior of \( {\mathcal{K }}_1, \delta > 0\) and \(0 <\delta < 1\) respectively. We get
We may apply Corollary 7.2 to take the expectation over \(X_1\) and get
where \(w_\alpha > 0\) was defined in the introduction and \((g_i)_{i \ge 1}\) are iid standard gaussian variables. Similarly, we find that
The next lemma due to Belinschi et al. [6] will be crucial in the sequel. Recall that the functions \(\varphi _{\alpha ,z}\) and \(\psi _{\alpha ,z}\) were defined in (16)–(17).
Lemma 3.4
([6, Lemma 3.6]) For any \(z \in \mathbb{C }_+\), the functions \(\varphi _{\alpha ,z}\) and \(\psi _{\alpha ,z}\) are Lipschitz with constant \(c = c(\alpha ) |z|^{ - \alpha } \) and \(c = c(\alpha ) |z|^{ - \alpha /2} \) on \({\mathcal{K }}_{\alpha /2}\). Moreover \(\varphi _{\alpha ,z}\) maps \({\mathcal{K }}_{\alpha /2}\) into \({\mathcal{K }}_{\alpha /2}\) and \(\psi _{\alpha ,z}\) maps \({\mathcal{K }}_{\alpha /2}\) into \({\mathcal{K }}_{1}\).
Proof
The first statement follows from [6, Lemma 3.6] by a change of variable. For the second, we note that if \(x \in {\mathcal{K }}_{\alpha /2}\) then \(x = ( - i w ) ^{\alpha /2}\) with \(w \in \mathbb{C }_+\) and from (33) \(\varphi _{\alpha ,z} ( x) = \mathbb{E }( i z + i w S )^{- \alpha /2}\) where \(S\) is a non-negative \(\alpha /2\) stable law with Laplace transform, for \(y > 0, \mathbb{E }\exp ( - y S) = \exp ( - \Gamma ( 1- \alpha /2) y^{\alpha /2} )\). Similarly, \(\psi _{\alpha ,z} (x) = \mathbb{E }( i z + i w S )^{- 1}\). In particular, \(\varphi _{\alpha ,z} (x) \in {\mathcal{K }}_{\alpha /2}\) and \(\psi _{\alpha ,z} (x) \in {\mathcal{K }}_{\alpha /2}\). \(\square \)
We are now able to prove Proposition 3.1.
Proof of Proposition 3.1
We recall that \(I(z)\) and \(J(z)\) were defined by (21)–(22). The point is that the Lipschitz constant in Lemma 3.4 depends on \(|z|\) and not only \(\mathrm{Im}(z)\). Hence since \( \rho _k := \left( - i R^{(1)}_{kk}(z)\right)^{\frac{\alpha }{2}} \in {\mathcal{K }}_{\alpha /2}\), using exchangeability, we deduce that with \(c=c(\alpha ) |z|^{ - \alpha /2} \) as in Lemma 3.4,
By the Cauchy–Schwarz inequality and Lemma 8.4, we obtain
By applying the Jensen inequality, we also notice that since \(0 < \alpha \le 2\),
Hence we obtain an error of
In the forthcoming computations, we shall always consider \(\eta \) so that \(\varepsilon \) of (20) is smaller than one so that \( n^{-1} M_n(z)^{\frac{\alpha }{2}}\) vanishes and is neglectable compared to \( n^{-\frac{1}{2}} M_n(z)^{\frac{\alpha }{4}}\). However \(\mathbb{E }\rho _2\) and \(Y(z)\) are close. More precisely, by equation (97) (in Appendix) applied with \(f(x)=(-ix)^{\frac{\alpha }{2}}\) we find that
Taking the expectation, we get
By Lemma 3.4, the function \(\varphi _{\alpha ,z}\) is Lipschitz for some constant \(c = c(\alpha ,|z|)\) on \({\mathcal{K }}_{\alpha /2}\). We deduce from (36) that
where we used the upper bound on \(M_n(z)\) given by (19). We note finally that the first term is always smaller for \(n\) large enough and \(\eta \le 1\) than the second term. We have thus proved that there exists \(c >0\), such that for all \(n^{- \frac{\alpha + 2}{ 4}} \le \eta \le 1\) and all integers,
The statement of Proposition 3.1 on \(Y(z)\) follows by applying Lemma 3.2: we find
Finally, we observe that the bound on \(X(z)\) follows similarly from (35):
We now use Lemma 3.2 and Proposition 3.1 is proved.n\(\square \)
3.2 Rate of convergence of the resolvent
We will now use Proposition 3.1 as the stepping stone to obtain a quantitative rate of convergence of the spectral measure \(\mu _A\) toward its limit.
By Lemma 3.4, if \(z = E + i \eta \) and \(| z|\) is large enough, say \(E_0\), then \(\varphi _{\alpha ,z}\) and \(\psi _{\alpha ,z}\) are Lipschitz with constant \(L < 1\) and in particular, it has a unique fixed point
Existence and uniqueness of such a fixed point extends to all \(z\) except on a finite set by the implicit function theorem (see [6]).
From [7, Theorem 1.4], the empirical measure \(\mu _A\) converges a.s. to a probability measure \(\mu _\alpha \) (for the topology of weak convergence). The Cauchy–Stieltjes transform of the limit measure \(\mu _\alpha \) is equal to
The above identity characterizes the probability measure \(\mu _\alpha \).
Theorem 3.5
(Convergence of Stieltjes transform) For all \(0 < \alpha < 2\), there exists a finite set \({\mathcal{E }}_\alpha \subset \mathbb{R }\) such that if \(K\) is a compact set with \(K \cap {\mathcal{E }}_\alpha = \emptyset \) the following holds for some constant \(c = c ( \alpha , K)\).
-
(i)
If \(1 < \alpha < 2\) : for any integer \(n \ge 1, z = E + i \eta \) with \(E \in I, c \sqrt{ \frac{ \log n}{ n }} \vee \left( n^{ - \frac{\alpha }{ 8 - 3 \alpha } } ( 1 + {\mathbf{1 }}_{ 1 < \alpha < 4/3} ( \log n ) ^{ \frac{2 \alpha }{ 8 - 3 \alpha } })\right) \le \eta \le 1\),
$$\begin{aligned} \left|\mathbb{E }g_{\mu _A} (z) - g_{\mu _\alpha } ( z) \right| \le c \delta , \end{aligned}$$(38)where \( \delta = \eta ^{-\frac{\alpha }{2}} n^{-\frac{\alpha }{4}} + \eta ^{- \frac{8 - 3 \alpha }{2\alpha } } n^{-\frac{1}{2}} ( 1 + {\mathbf{1 }}_{ 1 < \alpha < 4/3} \log n ) + \eta ^{-1} \exp ( - \delta n \eta ^2 )\).
-
(ii)
If \(0 < \alpha \le 1\), the same statement holds with \(c n^{-\frac{\alpha }{2 +3 \alpha }} ( \log n )^{\frac{4}{2 + 3 \alpha } } \le \eta \le 1\) and \(\delta = \eta ^{-\frac{\alpha }{2}} n^{-\frac{\alpha }{4}} + \eta ^{ - \frac{2 + 3 \alpha }{2} } n^{-\frac{\alpha }{2}}( \log n ) ^2\).
Moreover for any interval \(I \subset K\) of length \(|I |\ge \eta \,( 1 \vee \delta | \log ( \delta ) |^{-1}) \),
This result implies Theorem 1.1. Indeed the presence of the expectation of \(\mu _A(I)\) instead of \(\mu _A(I)\) does not pose a problem due to Lemma 8.1 in Appendix. We start the proof of Theorem 3.5 with a weaker statement.
Proposition 3.6
(Convergence of Stieltjes transform : weak form) Statement (ii) of Theorem 3.5 holds and
-
(i’)
If \(1 < \alpha < 2\) and \( c n^{-1/5} \left( 1 + {\mathbf{1 }}_{ 1 < \alpha \le \frac{4}{3}} ( \log n )^{\frac{2 }{5}} \right) \le \eta \le 1 \) then (38) holds with \(\delta = \eta ^{-\frac{\alpha }{2}} n^{-\frac{\alpha }{4}} + \eta ^{- 5/2 } n^{-1/2} \left( 1 + {\mathbf{1 }}_{ 1 < \alpha \le \frac{4}{3}} \log n \right)\).
Proof
Assume first that \(z = E + i \eta \) with \(| E | \ge E_0\) and \(\eta \le 1\). If we apply Lemma 3.4 to \(\varphi _{\alpha ,z}\) we find
Also, by exchangeability \(\mathbb{E }g_{\mu _A} (z) = \mathbb{E }G_{11} (z) = i X(z)\). Hence applying Lemma 3.4 to \(\psi _{\alpha ,z}\), we deduce
Also, the Cauchy–Weyl interlacing theorem implies that the same type of bounds holds for the minor \(A^{(1)}\) instead of \(A\). Indeed applying Lemma 8.2 to \(f (x) = ( x - z)^{-1}\), we have
We recall that \(\mu _{\alpha }\) has a bounded density (see [6, 7, 10]). Hence \(\mathrm{Im}( g_{\mu _\alpha } (z))\) is uniformly bounded. We get for any \(z = E + i \eta , | E | \ge E_0, \eta \ge n ^{-\frac{\alpha +2}{4}}\),
On the other hand, the spectral theorem implies the important identity
We first consider the case \(1 < \alpha < 2\). Then, by Proposition 3.1, we obtain for \(\eta \ge n^{-1/{2\alpha }} \ge n^{-1/2}\),
By monotonicity, we find that \(\eta M_n(z) \) is upper bounded by \(x^*\) where \(x^*\) is the unique fixed point of
It is easy to check that the unique fixed point of \(x = a + b x^\beta \), with \(a,b >0\) and \(0 < \beta < 1\) is upper bounded by \(\kappa (\beta ) a\) if \(a \ge b^{\frac{1}{1 - \beta }}\). We deduce that, for some constant \(c_1 > 0\) and all \(n^{-1/5} \left( 1 + {\mathbf{1 }}_{ 1 < \alpha \le \frac{4}{3}} ( \log n )^{\frac{2 }{5}} \right) \le \eta \le 1 \),
So finally, from Proposition 3.1, we find that for all \(n^{-1/5} \left( 1 + {\mathbf{1 }}_{ 1 < \alpha \le \frac{4}{3}} ( \log n )^{\frac{2 }{5}} \right) \le \eta \le 1 \),
We notice that the middle term is negligible compared to the last for our range of \(\eta \).
Assume finally that \(0 < \alpha \le 1\), then arguing as above for \(\eta \ge n^{-1/2} \ge n^{-1/{2\alpha }} \),
We deduce that, for some \(c_1 >0\) and all \(n^{-\frac{\alpha }{2 +3 \alpha }} ( \log n )^{\frac{4}{2 + 3 \alpha } } \le \eta \le 1, \eta M_n \le c_1\). We find that for all \(n^{-\frac{\alpha }{2 +3 \alpha }} ( \log n )^{\frac{4}{2 + 3 \alpha } } \le \eta \le 1\),
Again the middle term is negligible compared to the first for our range of \(\eta \).
We have proven the proposition if \(z = E + i \eta \) and \(|E| \ge E_0\) is large enough. It remains to prove the statement for all \(E\) outside of a finite set. It is proven in [6] that \(\varphi _{\alpha ,z} ( y ) = z^{-\alpha } g (y)\) where \(g\) is an analytic function on \({\mathcal{K }}_{\alpha /2} \backslash \{0\}\) (which depends on \(\alpha \)). Let \(y(z)\) be the fixed point defined in (37). It follows that the set of \(z \in \mathbb{C }\) such that \(\varphi ^{\prime }_{\alpha , z} (y (z) ) = 1\) is finite (for details see [6, §5.3]). We define \({\mathcal{E }}^{\prime }_\alpha \) as the set of real \(E\) such that there exists \(0 \le \eta \le E_0 + 1\) with \(\varphi _{\alpha , E + i \eta } ^{\prime } ( y ( E + i \eta )) = 1\). We set finally \({\mathcal{E }}_\alpha = \{ 0 \} \cup {\mathcal{E }}^{\prime }_\alpha \). This set is finite. Let \(K\) be a compact interval which does not intersect \({\mathcal{E }}_\alpha \). From the inverse function theorem, \(\varphi _{\alpha ,z}\) is invertible in a neighborhood of \(y\) with inverse with locally bounded derivative. Hence, there exist \(\tau , c_0 > 0\) such that for \(t \ge 0 , 0 \le \eta \le E_0 + 1, E \in K\),
Therefore, we may use Lemma 3.4 and an alternative version of (39) : for any \(z = E + i \eta \) with \(E \in K \) and \(0 \le \eta \le E_0 + 1\), if \(| Y(z) - y (z) | \le \tau \) then
To apply Proposition 3.1, we shall use an inductive argument to insure that the hypothesis \(| Y(z) - y (z) | \le \tau \) is satisfied. We set for integer \(\ell , \eta _0 = \eta , \eta _{\ell +1} = \eta _\ell + \frac{\tau }{3} ( 1 \wedge \eta _\ell ) ^2\) and \(z_\ell = E + i \eta _\ell \). As \(\eta _k\) goes to infinity as \(k\) goes to infinity when \(\eta _0\ne 0\), there exists \(k\) such that \(E_0 \le \eta _k \le E_0 + \tau \). Then \( \varphi _{\alpha , z_k}\) is a contraction and the above argument proves that
(note that \(\delta \) is a pessimistic bound since \(\mathrm{Im}( z_k ) \) is bounded away from \(0\)). We notice that it is sufficient to prove the statement of the proposition in the range, for \(1 < \alpha < 2, \kappa n^{-1/5} \left( 1 + {\mathbf{1 }}_{ 1 < \alpha \le \frac{4}{3}} ( \log n )^{\frac{2 }{5\alpha }} \right) \le \eta \le 1 \) and for \(0 < \alpha \le 1, \kappa n^{-\frac{\alpha }{2 +3 \alpha }} ( \log n )^{\frac{4}{2 + 3 \alpha } }\), where \( \kappa > 0\) is any fixed constant. Hence, up to increasing \( \kappa \), we may assume that \(s c \delta \le \tau /3\), where \(s \ge 1\) is large number that will be chosen later on.
To obtain a priori bounds for \(Y(z)-y(z)\) and \(|\mathbb{E }g_{\mu _A} (z) - g_{\mu _\alpha } ( z) |\) at \(z=z_{k-1}\) from those at \(z=z_{k}\) observe that for any probability measure \(\mu \) on \(\mathbb{R }\) and \(0 \le \beta \le 1\),
Using the above control with \(\mu =\sum _{k=1}^n \langle v_k, e_1\rangle ^2 \delta _{\lambda _k}\) so that \(g_\mu (z)=R_{11}(z)\) we deduce by applying (44) with \(\beta =\alpha /2\) that
and thus \(| Y (z_{k-1}) - y ( z_{k-1}) | \le \tau \). We get a similar control for \(\mathbb{E }g_{\mu _A} (z_{k-1})\) by applying (44) with \(\beta =1\) so that
Therefore, using Proposition 3.1, we find for some constant \(c^{\prime } >0\).
From what precedes, it implies that \(| Y (z_{k-1}) - y ( z_{k-1}) | \le c_0 c^{\prime } \delta \). We choose \(s\) large enough so that \( c^{\prime }c_0 \le s c\), so that we have \( c^{\prime } c_0 \delta \le \tau /3\). Also, we may use (43). We find for some new constant \(c^{\prime \prime }\),
Finally, if \(s\) was also chosen large enough so that \( c^{\prime \prime } \le s c\), then \(c^{\prime \prime } \delta \le \tau /3\), and we may repeat the above argument down to \(\ell = 0\). \(\square \)
When \(\alpha \in (1,2)\), a bootstrap argument allows to improve significantly Proposition 3.6. The idea is, in the spirit of [20], that if the imaginary part of \(\left\langle X, R^{(1)} X\right\rangle \) is a priori bounded below by something going to zero more slowly than \(\eta \), then we can improve the result of the key Lemma 3.2. Before moving to the proof, we state a classical deconvolution lemma.
Lemma 3.7
(From Stieltjes transform to counting measure) Let \(L > 0, 0 < \varepsilon < 1, K\) be an interval of \(\mathbb{R }\) and \(\mu \) be a probability measure on \(\mathbb{R }\). We assume that for some \(\eta > 0 \) and all \(E \in K\), either
Then, there exists a universal constant \(c\) such that for any interval \( I \subset K\) of size at least \(\eta \) and such that \({\mathrm{dist }}( I , K^c ) \ge \varepsilon \), we have
Proof
Let us prove the first statement. We observe that
where \(P_\eta \) is the Cauchy law with parameter \(\eta \). We thus need to perform a classical deconvolution. We may for example adapt Tao and Vu [29, Lemma 64] (see also e.g. [22, p. 15]). Define
In particular
Now, the Cauchy law has density, \( P_{\eta } ( t ) = \frac{1}{\pi } \frac{\eta }{ \eta ^2+ t^2 }.\) It follows that for \(\{ y \in I\}, \{ y \in I^c, {\mathrm{dist }}( y , I) \le |I | \}\) and \(\{ y \in I^c, {\mathrm{dist }}( y , I) \ge |I | \}\) we may use respectively the bounds
We write if \(I = [a , b], I_1 = I^c \cap [ a - | I | , b + | I | ] \cap K\) and \(I_2 = I^c \cap [ a - | I | , b + | I | ]^c \cap K \),
However, by assumption if \(J = [ E - \eta /2, E + \eta /2] \) is an interval of size \(\eta \) with \(E \in K\),
We deduce that \( \mu ( J) \le \frac{ 4L}{3} \eta .\) Now, consider a partition \({\mathcal{P }}\) of \(\mathbb{R }\) into intervals of size \( \eta \). We get from this last upper bound
The other terms are bounded similarly. \(\square \)
Proof of Theorem 3.5
In view of Proposition 3.6 and Lemma 3.7 applied to \(\mathbb{E }\mu _A\) and \(\mu _\alpha \), it remains to prove statement \((i)\) of Theorem 3.5. We thus assume in the sequel that \(1 < \alpha < 2\). The proof is divided into five steps. Throughout the proof, we assume that \(n \ge 3\) (without loss of generality) and we denote by \(\mathbb{E }_1 [ \cdot ] \) and \(\mathbb{P }_1 ( \cdot ) \) the conditional expectation and probability given \({\mathcal{F }}_1\), the \(\sigma \)-algebra generated by the random variables \((X_{ij})_{ i \ge j \ge 2}\).
Step one: Lower bound on the Stieltjes transform Let \(K = [ a , b]\) be an interval which does not intersect the finite set \({\mathcal{E }}_\alpha \), defined in Proposition 3.6. The limit spectral measure \(\mu _\alpha \) has a positive density on \(\mathbb{R }\). In particular, there exists a constant \( c_0 = c_0(K_0,\alpha ) > 0\) such that for all \(0 \le \eta \le 1\) and \(x \in K\),
Consequently, if there exists \(0 \le \eta \le 1\) such that for all \(x \in K\),
then
Note that Proposition 3.6 already proves that (45) holds if \(n \ge n_0\) is large enough and
for some \(\varepsilon > 0\). By an inductive argument, we aim at proving that (45) holds for the same constants \(n_0\) but for some \(\eta \ll \eta _0\).
For some constant \(\delta > 0 \) to be defined later on, we set for \(1 < \alpha < 2\),
Note that \(\eta _\infty \ge n ^{- \frac{ \alpha +2 }{ 4}}\) for all \(n\) large enough (say again \(n_0\)).
Step two: Start of the induction We assume that (45) holds for some \(\eta _1 \in [ n ^{- \frac{ \alpha +2 }{ 4}} , \eta _0]\) and that
where \(\tau \) was defined in (42). Let \(0 < \tau ^{\prime } < \tau \) to be chosen later on. We are going to prove that (45)–(47) hold also for
provided that \(\eta _1 \ge t \eta _\infty , n \ge n_0\) and \(t\) large enough. As in the proof of Proposition 3.6, cf. (44), if \(\tau ^{\prime } \) is small enough, we note that (47) implies that
First, by Weyl’s interlacing property (45) holds for \(A^{(1)}\) with \(c_0/2\) replaced by \(c_0 /4\) (\(\eta _1\gg n^{-1}\)). Also it follows by Jensen’s inequality that for \( z = x + i \eta \), with \(x \in K\),
where we have used the fact \( [\eta / ( (\lambda ^{(1)}_i - x)^2 + \eta ^2 )]^{\alpha /2 -1}\ge \eta ^{1-\alpha /2}\) for \(\alpha \in [0,2]\). Note also that from (44) and (48)
Hence, if \(\tau ^{\prime }\) is chosen small enough so that the above is less than \(c_0 / 8\), we deduce from (46) and (50) that with \(c_1 =c_0 / 16\),
Now, for bounding from below
we observe by Lemma 8.1, since \( x^{\alpha /2}\) has total variation on \([0,\eta ^{-1}]\) equal to \(\eta ^{-\alpha /2}\), that for \(r \ge 0\),
Applying the above with \(r=c_1 \eta ^{ 1 - \frac{\alpha }{2}}\) shows with (51) that for some \(\delta >0\),
where
Note that this probabilistic bound is non trivial only if \(\eta _1 \ge n^{-\frac{1}{2}} \ge n ^{- \frac{ \alpha +2 }{ 4}}\) (recall that \(1 < \alpha < 2\)).
Step three: Gaussian concentration for quadratic forms For any \(z = x + i \eta \in \mathbb{C }_+\), we may bound from below the imaginary part of
on the event \(\Lambda (z) \in {\mathcal{F }}_1\). Indeed, as the \(\mathrm{Im}R^{(1)}_{kk},1\le k\le n-1,\) are non negative, we can use Lemma 7.1 to see that conditionally on \({\mathcal{F }}_1\),
where the equality holds in law and \(S\) is a positive \(\alpha /2\)-stable law whereas the \(G_k\) are independent standard Gaussian variables, independent from \(S\). Moreover, if \(\Lambda (z)\) holds then from (52)
Hence, if \(\Lambda (z)\) holds and if \(\eta \ge c n^{-\frac{\alpha }{2}}\), we may apply Corollary 6.2 to \(p = \alpha \) and \(A\) the diagonal matrix with diagonal entries \(\sqrt{\mathrm{Im}R_{kk} ^{(1)} (z)}, 2 \le k \le n\). We deduce that for some universal constants \(c > 0, 0 < \delta < 1\), if \(\Lambda (z)\in {\mathcal{F }}_1\) holds, then
Taking the square, this yields to
Finally, we observe similarly that by Lemma 7.1,
where \(T(z)\) was defined by (28) and, conditionally on \({\mathcal{F }}_1, \tilde{S}\) is a positive \(\alpha /2\)-stable law, independent of \(\tilde{L}(z) \). Also, \(\tilde{L}(z) = \Vert A G \Vert _\alpha ^2\), where \(A = \left( \mathrm{Im}R^{(1)}\right)^{1/2}\) and \(G \in \mathbb{R }^{n-1}\) is a standard Gaussian vector. If \(\Lambda (z)\) holds, we may again apply Corollary 6.2 to \(p = \alpha \) and \(A\), we find that if \(\eta \ge c n^{-\frac{\alpha }{2}}\), the random variable \(\tilde{L}(z)\) satisfies if \( \Lambda (z)\) holds, the probabilistic bound
We may thus summarize the last two steps by stating that if \(n ^{-1/2} \le \eta _1 \le 1\) holds then
where \(z = x + i \eta , x \in K\) and
(recall that \(1 < \alpha < 2\)).
Step four: Improved convergence estimates We next improve the results of Proposition 3.1 for our choice of \(z = x+ i \eta , x \in K\). We write instead of (29)
Then from Lemma 3.3 and (27), there exists \(p > 1\) (depending on \(\alpha \)) such that
From Hölder’s inequality and Lemma 7.3, we deduce that for some new constant \(c>0\),
From (56)–(56), it follows that for \(n ^{-1/2} \le \eta _1 \le 1\) and some new constant \(c>0\),
Note that \( \eta ^{- 2(\frac{2}{\alpha } -1) }n^{-\frac{1}{\alpha }} \le 1\) if \(\eta \ge n^{- \frac{1}{2 ( 2 - \alpha )}}\ge n ^{-1/2} \) while \(\eta ^{-1} \exp ( - \delta n \eta ^2 ) \le 1\) if \(( 2 \delta n / \log n )^{-1/2} \le \eta \) . Note also that this last expression improves upon Lemma 3.2 and then Proposition 3.1 can be improved into
Then, by (49) we may use the bound (43). From (40), we thus obtain, for \(( 2 \delta n / \log n )^{-1/2} \le \eta _1 \le 1\),
We deduce that, for some constant \(c > 0\), if \( \eta _\infty \le \eta _1 \le 1\),
So finally, from (43)–(57), we find that for \(\eta _\infty \le \eta _1 \le 1\),
Step five: End of the induction From (42)–(58), we deduce that if \( t \eta _\infty \le \eta _1 \le 1\) and \(t\) large enough, then
We have thus proved that (45)–(47) holds also for our choice of \(\eta \).
The argument is completed as follow. Let \(K = [a , b]\) be a compact interval that does not intersect \({\mathcal{E }}_\alpha \). Starting for \(\eta _0 = n^{-\varepsilon }\), by applying \(m\) times the induction, we deduce that (45) holds for \(\eta _m = \eta _{m-1} - \tau ^{\prime } \eta _{m-1}^2\). Since this sequence vanishes as \(m\) goes to infinity, and, for some \(m\), we have \(t \eta _\infty \le \eta _m < 2 t \eta _\infty \). We deduce that for all \(n\) large enough (say \(n_0\)), (58) holds for \(z = x + i \eta \), with \(t \eta _\infty \le \eta \le 1\) and \(K = [a , b]\). The statement follows. \(\square \)
4 Weak delocalization of eigenvectors
Following Erdős–Schlein–Yau [20], from local convergence of the empirical spectral distribution (Theorem 3.5), it is possible to deduce the delocalization of eigenvectors. Using the union bound, Theorem 1.2 follows from the next proposition.
Proposition 4.1
(Delocalization of the eigenvectors) For any \(1 < \alpha < 2\), there exist \(\delta , c >0\) and a finite set \({\mathcal{E }}_\alpha \subset \mathbb{R }\) such that if \(I\) is a compact interval with \(I \cap {\mathcal{E }}_\alpha = \emptyset \), then for any unit eigenvector \(v\) with eigenvalue \(\lambda \in I\) and any \(1 \le i \le n\),
where \(\rho \) is as in Theorem 1.1 and \(Z\) is a non-negative random variable whose law depends on \((\alpha , I)\) and which satisfies
Proof
Let \({\mathcal{E }}_\alpha \) be as in Theorem 1.1. The density of \(\mu _\alpha \) is uniformly lower bounded on \(I\) by say \(4\varepsilon >0\). We set
where the constant \(c_1\) is large enough to guarantee that for any interval \(J\) of length at least \(\eta \) in \(I\) we have \(|\mathbb{E }\mu _A (J) - \mu _\alpha (J) | \le 2 \varepsilon | J|\). Then, we partition the interval \(I = \cup _\ell I_\ell \) into \(c_2 \eta ^{-1} \) intervals of length \( \eta \). From what precedes we have for any \( 1 \le \ell \le c_2 \eta ^{-1}, \mathbb{E }\mu _A (I_\ell ) > 2 \varepsilon | I_\ell |\).
Now, by Lemma 8.1, the event \(F_n\) that for all \( 1 \le \ell \le c_2 \eta ^{-1}\),
has probability at least \( 1 - c_2 \eta ^{-1} \exp ( - n \varepsilon ^2 c^2_3 \eta ^2 / 2 ) \ge 1 - c \exp ( - c n^{\delta })\) for some constants \(c , \delta >0\).
Let \(v\) be a unit eigenvector and \(\lambda \in I\) such that \(A v = \lambda v\). Set \(v_i = \langle v , e_i \rangle \). We recall the formula
with \(X_1 = ( X_{1 2}, \ldots , X_{1 n}) \in \mathbb{R }^{n-1}\) and \(A^{(1)}\) is the principal minor matrix of \(A\) where the first row and column have been removed. We may now argue as in the proof of Proposition 2.1 : for some \(1 \le \ell \le c_2 \eta ^{-1}, \lambda \in I_\ell \), and it follows
where \((\lambda _i ^{(1)},u_i ^{(1)}), 1 \le i \le n-1\) denotes the eigenvalues and an eigenvectors basis of \(A^{(1)}\). We rewrite the above expression as
where \(W^{(1)} = \mathrm vect \left\{ u_{i}^{(1)} : 1 \le i \le n-1 , \lambda _i^{(1)} \notin I_\ell \right\} ,\) and \(P_1\) is the orthogonal projection onto the orthogonal of \(W^{(1)}\). The rank of \(P_1\) is equal to
From Weyl interlacement theorem, we get
From Lemma 7.1, there exists a positive \(\alpha /2\)-stable random variable \(S\) and a standard Gaussian vector \(G\) such that
By Corollary 6.2 and (12), if \(n\) is large enough, on the event \(F_n\), see (59), with probability at least
the lower bound
holds. Let us denote this enlarged event by \(\bar{F}_n\). Hence, for some \(c >0\), on \(\bar{F}_n\), we have from (60)
In summary, we have shown that
where \(\bar{F}_n^c\) has probability at most \(c \exp ( - c n^{\delta })\), for some \(c >1\). For \(0 < \delta ^{\prime } < 1\), it yields,
where we have used that \(\eta \ge 1/n\) and \(\alpha < 2\). Using, Lemma 7.3, if \(\delta ^{\prime }\) is small enough, the above is uniformly bounded in \(n\). This gives our statement for any \(\delta ^{\prime \prime } < \delta ^{\prime }\). \(\square \)
5 Analysis of the limit recursive equation
We next turn to the analysis of the limiting equation describing the resolvent, in case \(\alpha <1\). Let \({\mathcal{H }}\) be the set of analytic functions \(h : \mathbb{C }_+ \rightarrow \mathbb{C }_+\) such that for all \(z \in \mathbb{C }_+, |h(z)| \le \mathrm{Im}(z)^{-1}\). We also consider the subset \({\mathcal{H }}_0\) of functions of \({\mathcal{H }}\) such that for all \(z \in \mathbb{C }_+, h( - \bar{z}) = - \bar{h} ( z)\). For every \(n\) and \(1 \le k \le n\), the function \(z \mapsto R (z)_{kk}\) is in \({\mathcal{H }}\). It is proved in [10] that \(R_{kk}\) converges weakly for the finite dimensional convergence to the random variable \(R_0\) in \({\mathcal{H }}_0\) which is the unique solution of the recursive distributional equation for all \(z \in \mathbb{C }_+\),
where \(\{\xi _k\}_{k \ge 1}\) is a Poisson process on \(\mathbb{R }_+\) of intensity measure \(\frac{\alpha }{2} x^{ \frac{\alpha }{2} - 1} dx\), independent of \((R_k)_{k\ge 1}\), a sequence of independent copies of \(R_0\). In [10], \(R_0(z)\) is shown to be the resolvent at a vector of a random self-adjoint operator defined associated to Aldous’ Poisson Weighted Infinite Tree. We define \(\bar{\mathbb{C }}_+ = \{z \in \mathbb{C }: \mathrm{Im}(z) \ge 0 \} = \mathbb{C }_+ \cup \mathbb{R }\). In the following statement, we establish a new property of this resolvent.
Theorem 5.1
(Unicity for the resolvent recursive equation) Let \(0 < \alpha < 2/3\). There exists \(E_\alpha > 0 \) such that for any \(z \in \bar{\mathbb{C }}_+\) with \(|z| \ge E_\alpha \), there is a unique random variable \(R_0 (z)\) on \(\bar{\mathbb{C }}_+\) which satisfies the distributional equation (62) and \(\mathbb{E }| R_0(z) |^ {\frac{\alpha }{2} } < + \infty \). Moreover, for any \(0 < \kappa < \alpha / 2\), there exist \(E_{\alpha ,\kappa } \ge E_\alpha \) and \(c>0\), such that for any \(z \in \mathbb{C }_+\) with \(|z| \ge E_{\alpha ,\kappa }\),
In particular, if \(\mathrm{Im}( z) = 0, R_0(z)\) is a real random variable.
The main part of this section is devoted to the proof of Theorem 5.1. We will then analyze its consequence on our random matrix and prove Theorem 1.3 in Sect. 5.4. As usual, it is based on a fixed point argument. However, as \(R_0\) is complex-valued, it is not enough to get a fixed point argument for the moments of \(R_0\) as was done previously in [6, 7]. Instead, we prove that moments of linear combinations of \(R_0\) and its conjugate satisfy a fixed point equation. We then show that this new fixed point equation is well defined and for sufficiently large \(z\), has a unique solution.
5.1 Proof of Theorem 5.1
We shall give the proof of Theorem 5.1 in this section, but postpone the proofs of technical lemmas to the next subsection. By construction \( H (z) = - i R_0 ( z) \in {\mathcal{K }}_{1}\) as well as \(\bar{H} (z) = i \bar{R}_0 ( z) \in {\mathcal{K }}_{1}\) (recall that for \(\beta \in [0,2]\) , \({\mathcal{K }}_\beta = \{z \in \mathbb{C }: | \arg (z) | \le \frac{\pi \beta }{2} \}\)). For ease of notation, we define the bilinear form \(h.u\) for \( h \in \mathbb{C }\) and \(u \in {\mathcal{K }}_1^+ = {\mathcal{K }}_1 \cap \bar{\mathbb{C }}_+\) given by
For example, \(| i.u | = | \mathrm{Re}(u) - \mathrm{Im}(u)|\). Note also that if \(h \in {\mathcal{K }}_1\) and \(u\in {\mathcal{K }}_1^+\), then \(h.u \in {\mathcal{K }}_1\). We set
We let \({\mathcal{C }}_\alpha \) (resp. \({\mathcal{C }}_\alpha ^{\prime }\)) denote the set of continuous functions \(g\) from \( {\mathcal{K }}_1^+\) to \({\mathcal{K }}_{\alpha /2}\) (resp. \(\mathbb{C }\)) such that \(g( \lambda u ) = \lambda ^{\alpha /2 } g(u)\), for all \(\lambda > 0\). Then, for \(\alpha /2 \le \beta \le 1\), we introduce the norm
where \(S_+^1=\{u\in {\mathcal{K }}_1^+, |u|=1\}\). We then define \( {\mathcal{H }}_{\beta }\) (resp. \( {\mathcal{H }}^{\prime }_{\beta }\)) as the set of functions \(g\) in \({\mathcal{C }}_\alpha \) (resp. \({\mathcal{C }}_\alpha ^{\prime }\)) such that \(\Vert g \Vert _\beta \) is finite. Note that \(\Vert g \Vert _{\beta } \) contains two parts : the infinite norm and a weighted \(\beta \)-Hölder norm which get worse as the argument of \(u\) or \(v\) gets close to \(\pi /4\). Notice also that \( {\mathcal{H }}^{\prime }_{\beta }\) is a real vector space and \( {\mathcal{H }}_{\beta }\) is a cone.
The starting point of our analysis is that \(\gamma _z\) belongs to \({\mathcal{H }}_{\beta }\).
Lemma 5.2
(Regularity of fractional moments) Let \(0 < \alpha < 2\) and \(z \in \bar{\mathbb{C }}_+\),
-
Let \(R_0(z)\) be a solution of (62) such that \(\mathbb{E }| R_0(z) |^ {\frac{\alpha }{2} } < + \infty \). Then for all \(0 < \beta <1\), all \(z\in \bar{\mathbb{C }}_+ \backslash \{ 0\} \), \(\mathbb{E }| R_0(z)| ^\beta \le c | \mathrm{Re}( z) | ^{-\beta } \) for some constant \(c = c(\alpha ,\beta )\).
-
Let \(H\) be a random variable in \({\mathcal{K }}_{1}\) such that \(\mathbb{E }| H|^ { \frac{\alpha }{2}} \) is finite. If we define for \(u \in {\mathcal{K }}_1 ^ +, \gamma ( u ) = \mathbb{E }( H.u )^{\frac{\alpha }{2}} \), then \(\gamma \in {\mathcal{H }}_{\beta }\) for all \(\alpha /2 \le \beta \le 1\) and \(\Vert \gamma \Vert _\beta \le c \mathbb{E }| H|^ { \frac{\alpha }{2}}\) for some universal constant \(c > 0\).
Let \( h \in {\mathcal{K }}_1\) and \(g \in {\mathcal{H }}_{\beta }\). We define formally the function given for all \( u \in S^1_+\) by
We next see that \(F_{-iz}\) is closely related to a fixed point equation satisfied by \(\gamma _z\).
Lemma 5.3
(Fixed point equation for fractional moments) Let \(z \in \bar{\mathbb{C }}_+, 0 < \alpha < 2\) and \(R_0(z)\) solution of (62) such that \(\mathbb{E }| R_0(z) |^ {\frac{\alpha }{2} } < + \infty \). Then for all \(u \in {\mathcal{K }}_1^+\),
where
To prove this lemma, we will properly define and study the function \(F_h\) on \( {\mathcal{H }}_{\beta }\), at least for some values of \((h,\alpha ,\beta )\). We shall prove that
Lemma 5.4
(Domain of definition of \(F_h\)) Let \(h \in {\mathcal{K }}_1\) with \(|h| \ge 1, 0 < \alpha < 1\) and \(\beta \) such that
Then \(F_h\) defines a map from \({\mathcal{H }}_{\beta }\) to \({\mathcal{H }}^{\prime }_{\beta }\), and there exists a constant \(c = c(\alpha )\) such that
We could not prove unfortunately that \(F_h\) is a contraction for \(\Vert .\Vert _\beta \) but for a weaker and less appealing norm on \({\mathcal{H }}^{\prime }_{\beta }\) which is given for \(\varepsilon > 0\) by:
It turns out that the map \(F_h\) is Lipschitz for this new norm if \(\alpha \) is small enough.
Lemma 5.5
(Contraction property of \(F_h\)) Let \(h \in {\mathcal{K }}_1\) with \(|h| \ge 1, 0 < \alpha < 2/3, \alpha / 2 < \beta < 1 - \alpha /2\) and \(0 < \varepsilon < (1 - 3 \alpha /2)\wedge (\beta -\alpha /2) \). Then there exists a finite constant \(c = c(\alpha ,\beta ,\varepsilon )\) such that, for all \(f, g \in {\mathcal{H }}_{\beta }\),
We can now turn to the proof of Theorem 5.1. To this end define the map \(G_z\) which maps \(g \in {\mathcal{H }}_{\beta }\) to the function
where \(c_\alpha \) and \(\check{u}\) are given by Lemma 5.3. Then by Lemma 5.4, if \(|z|\) is large enough, any fixed point \(g\) of \(G_z\) satisfies \(\Vert g \Vert _\beta \le c_0 /2\) for some constant \(c_0= c_0(\alpha , \beta )\). By Lemma 5.5, for any \(0 < \varepsilon < 1 - 3 \alpha /2 \), if \(|z| \ge E_{\alpha , \varepsilon }\) is large enough, \(G_z\) satisfies
Thus, by Lemma 5.3, \(\gamma _z\) is the unique solution in \({\mathcal{H }}_\beta \) of the fixed point equation \( \gamma _z = G_z (\gamma _z). \) However, by Lemma 5.7 below, the law of \(R_0(z)\) which satisfies (62) is uniquely characterized by its fractional moments \(\gamma _z\). Therefore, there is a unique solution to this recursive distributional equation.
To prove the estimate on \(\mathbb{E }[\mathrm{Im}R_0(z)^{\alpha /2}]\), we start by proving that \(\mathrm{Im}R_0(E)\) vanishes almost surely. Indeed, we first note that when \(z = E \ne 0\) is real, there is a real solution of the fixed point equation \(\gamma _z = G_z (\gamma _z)\). Let us seek for a probability distribution \(P_E\) in \(\mathbb{R }\) such that (62) holds. We recall that if \(y_k\) are non-negative i.i.d. random variables, independent of \(\{\xi _k\}_{k\ge 1}\), then \(\sum _{k} y_k \xi _k \) is equal in law to \( ( \mathbb{E }y_1^{\frac{\alpha }{2}} ) ^{\frac{2}{\alpha } } \sum _{k} \xi _k\) and \(S = \sum _{k} \xi _k\) is a non-negative \(\alpha /2\)-stable law. Thus, using the Poisson thinning property, by definition \(P_E\) has to be the law of
if \(S\) and \(S^{\prime }\) are independent \(\alpha /2\)-stable positive laws and \(a = \int \max ( x, 0)^{\alpha /2}dP_E(x), b = \int \max ( -x, 0)^{\alpha /2}dP_E(x)\). We find the system of equations
(where we have used the notation \((x)_+ = \max ( x, 0)\), \((x)_- = \max ( -x, 0)\)). Notice that \(a^{2/\alpha } S - b ^{2/\alpha } S^{\prime }\) is an \(\alpha /2\)-stable variable, it has a bounded density. Hence, for any \(0 < \alpha < 2\), \( | E + a^{ \alpha /2} S - b^{ \alpha /2} S^{\prime } |^{-\alpha /2}\) is perfectly integrable. Thus, by construction \(\tilde{\gamma }_E(u)=\Gamma (1-\frac{\alpha }{2}) \int (-i u. x)^{\frac{\alpha }{2}}dP_E(x)\) belongs to \({\mathcal{H }}_\beta \) and it is a fixed point of \(G_E\). This insures the existence of \(a,b\ge 0\) and also the fact that \(P_E\) is the law of \(R_0(E)\) as soon as \(E\) is large enough so that \(G_E\) is a contraction.
To consider \(\gamma _z\) with small imaginary part, we need the additional following lemma.
Lemma 5.6
(Continuity of the maps \(F_h\)) Let \( 0 < \alpha < 2/3, \alpha / 2 < \beta < 1 - \alpha /2, 0<\varepsilon \le \beta -\frac{\alpha }{2}\) and \(0 < \kappa < \alpha /2\). There exists a constant \(c = c( \alpha , \beta , \kappa ) >0\), such that for any \(h,k \in {\mathcal{K }}_1, |h|, |k| \ge 1\), and \(g \in {\mathcal{H }}_{\beta }\)
Set \(z = E + i \eta \) with \( |E| \ge E_{\alpha , \varepsilon }\) and let \(0 < \kappa < \alpha /2\). Then, by Lemma 5.6, we have
Hence, for some \(c^{\prime } = c^{\prime } (\alpha ,\beta , \kappa )\), we deduce from Lemma 5.3 that if \(z=E+i\eta \)
Then, since \(\gamma _E ( e^{i \frac{\pi }{4}} )=0\) as \(\gamma _E\) is real, for any \(u \in S^1_+\),
Choosing \(u\) such that \( | u - e^{i \frac{\pi }{4}} |\) is of order \(\eta ^{\frac{2 \kappa }{\alpha + 2 \varepsilon } } \), we deduce that for all \(z = E + i \eta \) with \( |E| \ge E_{\alpha , \varepsilon }, \mathbb{E }\mathrm{Im}R_0(z) ^{\frac{\alpha }{2}} \) is bounded by \(\eta ^{\frac{ \kappa \alpha }{\alpha + 2 \varepsilon } } \) up to a multiplicative constant. Since \(\varepsilon > 0\) can be arbitrarily small, this concludes the proof of Theorem 5.1.
5.2 Proofs of technical lemmas
We collect in this part the proofs of a few technical results used in the proof of Theorem 5.1.
5.2.1 Proof of Lemma 5.2 (Regularity of fractional moments)
If \(\mathrm{Re}(z)=E\),
where \(\sum \xi _k \mathrm{Re}(R_{k}(z))\) is equal in law to \(aS-bS^{\prime }\) for two non-negative constants \(a,b\) and two independent stable laws \(S,S^{\prime }\). Assume for example that \(E > 0\). By conditioning on \(S^{\prime }\) and integrating over \(S\), we deduce from Lemma 7.4 that there exists a finite constant \(C = c(\alpha ,\beta )\) so that
In particular, as \(\eta \) goes to \(0\), any limit point \(R_0(E)\) of \(R_0(E + i \eta )\), solution of (62), satisfies the above inequality. The conclusion of the first point follows.
To prove the second point, we notice that it is straightforward that \(\gamma \) belongs to \({\mathcal{C }}_\alpha \). Moreover, for any \(\beta \in [\frac{\alpha }{2},1]\), there exists a constant \(c = c(\alpha , \beta )\) such that for any \(x, y\) in \({\mathcal{K }}_1\),
Also, we have, for \(u \in {\mathcal{K }}_1 ^ +\) and \(h \in {\mathcal{K }}_1\),
Indeed, if \( u = s + i t, s , t \ge 0\), then \(|h.u|^ 2 = \mathrm{Re}(h)^ 2 ( s + t)^2 + \mathrm{Im}(h)^ 2 ( s - t)^2\). This last expression is bounded from below by \( |h|^2 ( (s+ t)^ 2 \wedge (s- t)^ 2 ) = |h|^2 (s- t)^ 2 = |i.u |^2 |h|^2\). While it is bounded from above by \( |h|^ 2 ( ( s + t)^2 + ( s - t)^2 )= 2 |h|^2| u |^2\).
Now, using Jensen inequality and (65), we find
whereas (64) and (65) imply for \(\beta \in [\frac{\alpha }{2}, 1]\),
This completes the proof with \(\Vert \gamma \Vert _\beta \le \left( \sqrt{2}^{\frac{\alpha }{2}} +c2^{\frac{\beta }{2}} \right) \mathbb{E }| H | ^{\frac{\alpha }{2}}\).
5.2.2 Proof of Lemma 5.3 (Fixed point equation for fractional moments)
Write \(u = u_1 + i u_2\) and \(-iz = h \in {\mathcal{K }}_1\). By definition, we have
We use the formulas (33) : for all \(w \in {\mathcal{K }}_1, \delta > 0\),
and for \(0 < \delta < 1\),
Formally, we find that \(\gamma _z ( u) \) is equal to
If we perform the change of variable \(x = r y\) and apply Levy-Kintchine formula, we obtain the stated formula. The exchange of expectation and integrals is then justified by invoking Lemmas 5.2 and 5.4. We next prove Lemma 5.4.
5.3 Properties of the map \(F\)
Proof of Lemma 5.4
We start by proving that for all \(u \in S^1_+\), for \(h\in {\mathcal{K }}_1, |h|\ge 1\),
By Lemma 9.1, for \(h \in {\mathcal{K }}_1\), the map on \({\mathcal{K }}_{\frac{\alpha }{2}}\) given by
is bounded by \(c |h|^{-\alpha /2}\) and Lipschitz with constant \(c |h|^{-\alpha }\). Let \(T > 0 \) to be chosen later on. From (65) and (100), for \(\theta \in [0,\frac{\pi }{2}]\) and \(h\in {\mathcal{K }}_1, u\in S_1^+, g:{\mathcal{K }}_1\rightarrow {\mathcal{K }}_{\frac{\alpha }{2}}\), we have
where we have used the fact that
and the control, for any real \(t , \delta , T>0\), any \(\gamma _1<\gamma _2<1, \gamma _1\ne 0\), (here \(\gamma _2=-\gamma _1=\alpha /2)\),
where the sign depends on whether or not \(t, \delta \) have a different sign.
For the integration over \(y\) in the interval \([0,T]\), we find similarly by (101) that
Recalling that \(g(z )=|z|^{\frac{\alpha }{2}} g(\frac{z}{|z|})\) (by definition of \({\mathcal{C }}_\alpha \) and thus \({\mathcal{H }}_\beta \)) and using (64)–(65) we find that there exist finite constants \(C,C^{\prime }\) so that for all \(z,z^{\prime }\in \mathcal{K }_1^+\),
Using the fact that \(|e^{i \theta } + y u|\ge 1\) as \(u, e^{i\theta }\in S_1^+, y\ge 0\), we find with (65) that
where we have used that \(\beta \in ( \alpha /2,1]\) to obtain a convergent integral.
In the integration over \(y\) on the interval \([0,T]\), we have left aside the term
We shall use this time the third statement (102) of Lemma 9.1 with \(\kappa = 1\). We choose \(T = |i.e^{i \theta } | /2\) so that for all \(y \in [0,T]\) from (65)
For this choice of \(T\), we get
Finally, using our choice of \(T\), we deduce from (67), (70), (71) that
is bounded by \(c | h | ^{ - \frac{\alpha }{2}} | \theta - \frac{\pi }{4} | ^{ -\alpha } ( 1+ \Vert g \Vert _\beta )\) for \(|h|\ge 1\). We obtain (66) since
is integrable over \([0, \pi /2]\).
The proof of the lemma will be complete if we prove that for all \(u, v \in S_1 ^ + \),
To do so, we fix \(\theta \in [0, \pi /2]\) and assume for example that \(| i. ( e^{i \theta } + y u ) | \le | i. ( e^{i \theta } + y v ) |\). We first use the Lipschitz bound (101), together with (69), and write
where we have used (68) with \(\gamma _1=\beta -\alpha /2>0\) and \(\gamma _2=\kappa >\gamma _1\).
Now, in our control of
we have so far left aside
where \(| i. ( e^{i \theta } + y u ) | \le | i. ( e^{i \theta } + y v ) |\). By (102) applied to \(\kappa = \beta \),
is bounded up to multiplicative constant by
Using again (68) with \(0<\gamma _1=\beta -\alpha /2<\gamma _2=\beta +\alpha /2\) yields
We may conclude the proof of (72) by noticing that the bounds given by (73)–(74) and multiplied by \(( \sin 2 \theta ) ^{\frac{\alpha }{2} - 1}\) are uniformly integrable on \([0, \pi /2]\). \(\square \)
We can now build upon the proof of Lemma 5.4 to get proofs for Lemmas 5.5 and 5.6.
Proof of Lemma 5.5
We shall now use the norm \(\Vert .\Vert _{\beta ,\varepsilon }\) for which we have the following analogue of (69): if \(0 \le \varepsilon \le \beta -\frac{\alpha }{2}\), for all \(z,z^{\prime }\in \mathcal{K }_1^+\),
We start by showing that for any \(u \in S^1_+\),
The proof is similar to the argument in Lemma 5.4. We notice that \(F_h ( g) (u ) - F_h (f) (u)\) is equal to
where, with \(g_u = g ( e^{i \theta } + y u ), f_u = f ( e^{i \theta } + y u ), h_u = h . (e^{i\theta } + y u ) \),
We set \(\delta = - i . e^{i\theta } \) and \( t = i . u\). On the integration interval \([T, \infty )\) of \(y\) we use the first form of \(Z\) and treat the two terms separately. As in Lemma 5.4, we use (101) and (75) to find
The above computation requires the hypothesis \(\varepsilon >0\) to insure that the control of the integrals hold following (68) with \(\gamma _1\ne 0\).
For the integration interval \([0, T)\) of \(y\) we use the second form of \(Z\) and choose \( T = | i . e^{i \theta } | / 2 = |\delta | /2\). We use (104), (75) and \(\kappa =\alpha /2 + \varepsilon \), we find
Similarly, by (103) and (75), (76) and (69), our choice of \(T\) gives
Since \(3 \alpha / 2 + \varepsilon < 1\), we may integrate our bounds over \(\theta \) and obtain (77).
The proof of the lemma will be complete if we show that for any \(u \ne v \in S_1^+\), with \(|i.u| \le |i.v|\),
The proof is simpler than in the previous case as we do not need to consider separately the cases where \(y\) is small or large. We set \(\delta = - i . e^{i\theta } , t = i . u, t^{\prime } = i.v, |t| \le |t^{\prime }|\). Using (104) with \(\kappa =\beta \) and (75), we find
Moreover, using again (103), (75), (76) and (69) we find
Now, by assumption, \( 3 \alpha /2 + \varepsilon < 1\) and we may integrate our bounds over \(\theta \) and obtain (78). \(\square \)
Proof of Lemma 5.6
The proof is very close to the previous one, and we simply outline it. We assume for example \(| h | \le |k|\). By Lemma 5.4, we can also assume that \( | h - k | \le |h|\) and in particular \(|k| \le 2 |h|\). We first prove that for any \(u \in S^1_+\),
The expression \(F_h ( g) (u ) - F_k (g) (u)\) is equal to
where, with \(g_u = g ( e^{i \theta } + y u ), h_u = h . (e^{i\theta } + y u ) , k_u = k . (e^{i\theta } + y u ) \),
Let \(T > 0\). We set \(\delta = - i . e^{i\theta } \) and \( t = i . u\). On the integration interval \([T,\infty )\) for \(y\), we use the first form of \(Z\). Then, from (102) in Lemma 9.1, we find
where we have used that \(\kappa < \alpha /2\). On the integration interval \([0,T)\) for \(y\), we use the second form of \(Z\). We choose \(T = |i.e^{i\theta }|/2 = | \delta | /2\). For the first term, by (104) in Lemma 9.1,
The second term is easily bounded by (105) with \(\kappa _1=\kappa \) and \(\kappa _2=0\). Integrating our bounds over \(\theta \), we obtain (79). The proof that for all \(u \ne v \in S^+_1\), with \(| i. u | \le | i. v | \).
is easier as it does not require to consider separately small and large \(y\); we leave it to the reader. \(\square \)
5.3.1 Computation of characteristic function
With the notation of proof of Theorem 5.1, we define for \(z \in \bar{\mathbb{C }}_+, u \in {\mathcal{K }}_1^+\),
We note that the distribution of \(R_0(z)\) is characterized by the value of \(\chi _z\) on any open neighborhood in \({\mathcal{K }}_1^+\). The next lemma asserts that the distribution of \(R_0(z)\) is also characterized by the value of \(\gamma _z\) on \({\mathcal{K }}_1^+\) (that is on \(S_+^1\) by homogeneity).
Lemma 5.7
(From fractional moment to characteristic function) Let \(z \in \bar{\mathbb{C }}_+, 0 < \alpha < 2\) and \(R_0(z)\) solution of (62) such that \(\mathbb{E }| R_0(z) |^ {\frac{\alpha }{2} } < + \infty \). For all \(u = u_1+ i u_2 \in {\mathcal{K }}_1^+ \),
where \(J_1 (x) = \frac{x}{2} \sum _{k \ge 0} \frac{ (-x^2 / 4) ^k }{k ! (k+1)!}\) is a Bessel function of the first kind.
Proof
We use the formulas for \(w \in {\mathcal{K }}_1\),
(see [1]) and for \(z,z^{\prime } \in \mathbb{C }\),
Then, it follows from (62) that
Since \(J_1\) is bounded on \(\mathbb{R }_+\), we may safely take expectation. The conclusion follows from Levy-Khintchine formula. \(\square \)
5.4 Proof of Theorem 1.3
We start with a simple lemma which relates \(W_I (i)\) to the diagonal of resolvent.
Lemma 5.8
(From eigenvectors to diagonal of resolvent) Let \( \alpha > 0 \) and \(I = [E - \eta , E+\eta ]\) be an interval. Setting \(z = E + i \eta \in \mathbb{C }_+\), we have
Proof
From the spectral theorem, we have
It remains to sum the above inequality. \(\square \)
At this stage, it should be clear that the proof of Theorem 1.3 will rely on Theorem 5.1 and on an extension of the previous fixed point argument to finite \(n\) system. The bottleneck in the proof will be on the lower bound of \(| \Lambda _I |/n\eta \) which in particular requires according to Lemma 3.7 that \(\mu _A(I)\le L |I|\). This last control is difficult when \(\alpha <1\) as in this case \(n^{-1}\sum _{k=1}^n( \mathrm{Im}R(z)_{kk})^{\alpha /2}\) goes to zero like \(\eta ^{\alpha /2}\) so that arguments such as those used in the proof of Theorem 3.5 do not hold. It will be responsible for the restrictive condition \(\eta \ge n ^{-\rho + o(1)}\) in the statement of Theorem 1.3. For completeness we will also prove in this subsection a vanishing upper bound on
for \(\eta \) of order \(n^{ - 1/6}\) for all \( \alpha > 0\). More precisely, we have
Theorem 5.9
(Vanishing fractional moment for the resolvent) Let \( 0 < \alpha < 2/3 , 0 < \varepsilon < \frac{\alpha ^2}{2 (4 - \alpha ) }, \rho ^{\prime } = \frac{2 + \alpha }{4 ( 3 +\alpha ) }\) and \(c_0 = \frac{ (2 + \alpha )^2}{ 16 ( 3 + \alpha ) }\). There exist \(c_1 = c_1 ( \alpha ), c = c ( \alpha , \varepsilon ) > 0\) such that if \(n \ge 1, z = E + i \eta \in \mathbb{C }_+, |z| \ge c, n^{-\rho ^{\prime }} ( \log n )^ {c_0} \le \eta \le 1 \),
Moreover, if \(n^{-\rho } ( \log n )^{\frac{4}{2 + 3 \alpha } } \le \eta \le 1 \),
Theorem 1.3 is a consequence of the second statement of Theorem 5.9 together with Theorem 3.5, Lemma 5.8 and Lemma 8.4 which asserts that
We consider for \(u\in {\mathcal{K }}_1^+\),
Lemma 5.10
(Bound on fractional moments of the resolvent) Let \( 0 < \alpha / 2 \le \beta < 2 \alpha / ( 4 - \alpha )\) and \(\rho ^{\prime }, c_0\) as in Theorem 5.9. There exists \(c> 0\) such that if \(n \ge 1, z = E + i \eta \in \mathbb{C }_+, |z| \ge 1, \eta \ge n^{-\rho ^{\prime }} ( \log n )^ {c_0} \), then
The proof of Theorem 5.9 provides also the local convergence of the fractional moments \(\gamma _z^n\) for the norm \(\Vert .\Vert _{\frac{\alpha }{2},\varepsilon }\). Indeed it is based again on an approximate fixed point argument for these quantities.
Lemma 5.11
(Approximate fixed point for fractional moments of the resolvent) Let \( 0 < \alpha < 2/3 \) and \(\rho ^{\prime }, c_0\) as in Theorem 5.9 and \(G_z\) as in (63). For all \(0 < \varepsilon < \frac{\alpha ^2}{2 (4 - \alpha ) }\), there exists \(c = c ( \alpha , \varepsilon ) > 0\) such that if \(n \ge 1\), \(z = E + i \eta \in \mathbb{C }_+, |z| \ge c, n^{-\rho ^{\prime }} ( \log n )^ {c_0} \le \eta \le 1\),
Moreover, if \(n^{-\rho } ( \log n )^{\frac{4}{2 + 3 \alpha } } \le \eta \le 1 \),
We now check that the above two lemmas imply Theorem 5.9. Note in the proof below that they also imply the convergence of \(\gamma _z^n\) to \(\gamma _z\) for \(\eta \ge n^{-\rho ^{\prime }}(\log n)^{c_0}\).
Proof of Theorem 5.9
We prove the first statement. Let \(0 < \varepsilon < \frac{\alpha ^2}{2 (4 - \alpha ) }\) and \(\delta = \eta ^ { -\frac{\alpha ( 3 + \alpha ) }{2 + \alpha } } n^{-\frac{\alpha }{4} }\). Now since \(\Vert \gamma _{z}^n\Vert _{\frac{\alpha }{2} + \varepsilon } \) and \(\Vert \gamma _{z} \Vert _{\frac{\alpha }{2} + \varepsilon }\) are uniformly bounded, we have by Lemma 5.5 and Lemma 5.11,
as long as \(|z| \ge c\) with imaginary part \( n^{-\rho ^{\prime }} ( \log n )^ {c_0} \le \eta \le 1 \). Hence, if \(|z|\) is large enough, \(c|z|^{-\alpha }\) is less than \(2\) and it follows that
Now, we may argue as in the proof of Theorem 5.1. By Theorem 5.1, for \(|z|\) large enough, \(| \gamma _z ( e^{i \frac{\pi }{4}} ) | \le c^{\prime } \eta ^{\frac{\alpha }{2} - \varepsilon } \), for some constant \(c^{\prime } > 0\). Then, for any \(u \in S^1_+\), using Lemma 5.10, Lemma 5.2 and (83),
Choosing \(u\) such that \( | u - e^{i \frac{\pi }{4}} |\) is of order \(\delta ^{\frac{2 }{\alpha + 2 \varepsilon } } \), we deduce that for all \(z = E + i \eta \) with \( |E| \ge E_{\alpha , \varepsilon }, \mathbb{E }\mathrm{Im}R(z) ^{\frac{\alpha }{2}} \) is bounded up to a multiplicative constant, by
Since \(\varepsilon > 0\) can be arbitrarily small, this concludes the proof for the case \( n^{-\rho ^{\prime }} ( \log n )^ {c_0} \le \eta \le 1 \). The proof for \(n^{-\rho } ( \log n )^{\frac{4}{2 + 3 \alpha } } \le \eta \le 1 \) is identical: we find, by using the second part of Lemma 5.11 in (82)
It remains to notice that for \(\varepsilon \) small enough, in our range of \(\eta \), the second term dominates the first term. \(\square \)
Proof of Lemma 5.10
As in the proof of Lemma 5.2, it is sufficient to check that for some constant \(c = c( \alpha , \beta )\),
As usual, from (10), we have
We first get rid of the non-diagonal term in the scalar product \(\langle X_1 , R^{(1)}X_1 \rangle \). We perform this as in the proof of Lemma 3.2. Using the definition (28) and (64) with \(\alpha / 2 = \beta \), we find
In particular, since \( | z |^{-\beta } \le | \mathrm{Re}(z) |^{-\beta }\), we find
Now, we decompose the sum into a positive and a negative part
Note that, conditioned on \(R^{(1)}\), the two sums are independent. We invoke Lemma 7.1
where, conditioned on \(R^{(1)}\), \(a,b,S,S^{\prime }\) are independent non-negative random variables, \(S,S^{\prime }\) being \(\alpha /2\)-stable random variables. Hence from what precedes, we have
Assume for example that \(E > 0\). Let \({\mathcal{F }}\) be the filtration generated by \((R^{(1)}, a,b,S)\) and \(\mathbb{E }^{\prime } = \mathbb{E }[ \cdot | {\mathcal{F }}] \). Using Lemma 7.4 conditionally to \({\mathcal{F }}\) yields that for some constants \(c=C^2> 0\),
If \( E < 0\), we repeat the same argument with the filtration generated by \((R^{(1)}, a,b ,S^{\prime })\).
Now, if \(0 < \beta < 2 \alpha / ( 4 - \alpha )\), using the tail bound (27), we find
We now use the bound given by (19) on \(M_n\) which is valid for all \(\eta \ge n^{-\frac{\alpha +2 }{4}}\),
This concludes the proof of the lemma, since for \(\eta \ge n^{-\rho ^{\prime }} ( \log n )^ { \frac{ (2 + \alpha )^2}{ 16 ( 3 + \alpha ) } } \), the above expression is uniformly bounded. \(\square \)
Note that in the proof of Lemma 5.10 we have used the bound (19) instead of the bound \(M_n \le c\eta ^{-1}\) given by the proof of Proposition 3.6 because it is valid for a wider range of \(\eta \).
Proof of Lemma 5.11
Set \(h = -i z \in {\mathcal{K }}_1, H_{k}(h)=-i R^{(1)}(ih)_{kk}\) and define
where we recall that \(\check{u} = \mathrm{Im}(u) + i \mathrm{Re}(u)\).
Step one: Diagonal approximation. In this first step, we generalize Lemma 3.2. We will upper bound the expression \( \Vert \gamma _{ih}^n-I^n_{h}\Vert _{\beta ,\varepsilon }. \) Using the definition (28), we find that for any \(u\in S_1^+\), with \(\eta =\mathrm{Re}(h)>0\),
where we have used (64) with \(\beta =\frac{\alpha }{2}\). Using (85) for \(\beta = \alpha / 2\), we deduce that
Whereas using (19) to bound \(M_n\), we find for \(n^{ - \frac{\alpha +2 }{4}} \le \eta \le 1\) that
To bound
we first observe that for \(x_1,x_2,y_1,y_2\in {\mathcal{K }}_1\), by using the standard interpolation trick, for \(\kappa _1,\kappa _2,\beta \in [0,1]\), we have if \(N=|x_1|\wedge |x_2|\wedge |y_1|\wedge |y_2|\), and \(\kappa _1+\kappa _2\ge \alpha /2\)
We use this inequality with
and in \(y_j, v\) replaces \(u\). For \(j \in \{ 1, 2\} \), one can check that, with \(D_j=( h +a_n^{-2} \sum _{k=2}^n X_{1k}^2 H_k -i(j-1) T(z))^{-1}\),
and
Moreover, using (65), we find \(N\ge ( |i.u|\wedge |i.v| ) ( |D_1| \wedge |D_2| ) \). Recall finally that \(|D_1|\) and \(|D_2|\) are bounded by \(\eta ^{-1}\). Hence, choosing \(\kappa _1=\beta ,\kappa _2=\frac{\alpha }{2} +\varepsilon \) (with \(\varepsilon \) small enough so that \(\frac{\alpha }{2}+\varepsilon <\frac{2\alpha }{4-\alpha }\)) we deduce that,
We naturally choose \(\beta =\frac{\alpha }{2}+\varepsilon < \frac{2\alpha }{4-\alpha }\). From (27), \(T(z) \in L^\beta \) and
where we have finally assumed that \(n^{-\frac{2 + \alpha }{4} } \le \eta \le 1\) and used (19). This gives for \(n^{-\frac{2 + \alpha }{4} } \le \eta \le 1\)
Now it easy to check that for \(\eta \ge n^{-\frac{2 + \alpha }{2 (4+\alpha )} } \), we have \(\eta ^{ - \beta - \frac{\alpha }{2} - \frac{2 \beta }{2 + \alpha } } n^{-\frac{\beta }{2}} < \eta ^ { -\frac{\alpha ( 3 + \alpha ) }{2 + \alpha } } n^{-\frac{\alpha }{4} }\). It follows for \(n^{-\rho ^{\prime } } \le \eta \le 1\) and a new constant \(c > 0\), depending on \(\varepsilon \), that
If instead we assume that \(n^{-\rho } ( \log n )^{\frac{4}{2 + 3 \alpha } } \le \eta \le 1 \), then, from the proof of Proposition 3.6, we may use the stronger bound \(M_n \le c \eta ^{-1}\) if \(|z|\) large enough. We then find
(where, for the last inequality, we have used the fact that \(\eta \ge n ^{-1/3}\) for \(n^{-\rho }\) \(( \log n )^{\frac{4}{2 + 3 \alpha } } \le \eta \le 1 \) and \(n\) large enough).
Step two: approximate fixed point equation Next, we extend the proof of Proposition 3.1. We denote by \(\mathbb{E }_1 [ \cdot ] \) and \(\mathbb{P }_1 ( \cdot ) \) the conditional expectation and probability given \({\mathcal{F }}_1\), the \(\sigma \)-algebra generated by the random variables \((X_{ij})_{ i \ge j \ge 2}\). We assume that \(\alpha / 2 < \beta < 1 - \alpha / 2 \) and \(0 < \varepsilon < 1 - 3 \alpha / 2\). We first remark that by arguments similar to the proof of Lemma 5.3 and by Corollary 7.2, we have
where, conditioned on \({\mathcal{F }}_1, Z_n(u)=\frac{\kappa }{n}\sum _{k=2}^n ( H_k . u ) ^{\frac{\alpha }{2}}|g_k|^\alpha , g_k\) are i.i.d standard normal variables and \(\kappa = \Gamma ( 1 - \alpha / 2 ) / \mathbb{E }|g_1 | ^\alpha \). Note that, from Lemma 5.2, \(\Vert Z_n\Vert _\beta \le \frac{c}{n }\sum _{k=2}^n | H_k | ^{\frac{\alpha }{2}} |g_k|^\alpha \) which belongs to \(L^p\) for any \(p >0\). Therefore, we can use Lemma 5.5 and the Hölder inequality to insure that,
where \(1 / p + 1 /q = 1\) and
From the triangle inequality,
But, using Lemma 8.5 from the Appendix,
Similarly, by Lemma 8.5,
Whereas using (97) as we did in the proof of Proposition 3.1, we have, with \(c_0 = 2 \Gamma ( 1 - \frac{\alpha }{2} )\),
Hence, there exists a new constant \(c(q)\) such that
Similarly, using the triangle inequality at the first line, (97) at the second line and the Jensen inequality at the third,
We choose \(p > 1\) such that \(p \alpha /2 < 2 \alpha / (4-\alpha )\) and we finally use (84) and Lemma 5.10. Then, for our range of \(\eta \), the right hand side of the above inequality is of order \(1\). Putting these estimates in (89), we find finally that for any \(\alpha / 2 < \beta < 2 \alpha / ( 4 - \alpha )\), any \(0 < \varepsilon < 1 - 3 \alpha / 2\) and \(n^{-\rho ^{\prime } } (\log n ) ^{c_0} \le \eta \le 1\), there exists a constant \(c(\alpha , \beta , \varepsilon )\) such that
Putting this together with (87), this conclude our proof (the above term is negligible compared to the right hand side of (87) or (88)). \(\square \)
References
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, vol. 55. For sale by the Superintendent of Documents. U.S. Government Printing Office, Washington, DC (1964)
Aizenman, M., Schenker, J.H., Friedrich, R.M., Hundertmark, D.: Finite-volume fractional-moment criteria for Anderson localization. Dedicated to Joel L. Lebowitz. Commun. Math. Phys. 224(1), 219–253 (2001)
Anderson, G.W., Guionnet, A., Zeitouni, O.: An introduction to random matrices. Cambridge Studies in Advanced Mathematics, vol. 118. Cambridge University Press, Cambridge (2010)
Bai, Z.: Methodologies in spectral analysis of large-dimensional random matrices: a review. Stat. Sin. 9(3), 611–677. With comments by G. J. Rodgers and Jack W. Silverstein; and a rejoinder by the author (1999)
Bai, Z., Silverstein, J.W.: Spectral Analysis of Large Dimensional Random Matrices. Springer Series in Statistics, 2nd edn. Springer, New York (2010)
Belinschi, S., Dembo, A., Guionnet, A.: Spectral measure of heavy tailed band and covariance random matrices. Commun. Math. Phys. 289(3), 1023–1055 (2009)
Ben Arous, G., Guionnet, A.: The spectrum of heavy tailed random matrices. Commun. Math. Phys. 278(3), 715–751 (2008)
Ben Arous, G., Guionnet, A.: Wigner matrices. In: Akemann G., Baik J., Di Francesco, P. (eds.) Handbook in Random Matrix Theory, Chap. 21. Oxford University Press (2010)
Benaych Georges, F., Guionnet, A., Male, C.: Central limit theorem for heavy tailed random matrices. Preprint (2012)
Bordenave, C., Caputo, P., Chafai, D.: Spectrum of large random reversible Markov chains: heavy-tailed weights on the complete graph. Ann. Probab. 39(4), 1544–1590 (2011)
Bordenave, C., Caputo, P., Chafai, D.: Spectrum of non-Hermitian heavy tailed random matrices. Commun. Math. Phys. 307(2), 513–560 (2011)
Bouchaud, J.-P., Cizeau, P.: Theory of Lévy matrices. Phys. Rev. E 3, 1810–1822 (1994)
de la Peña, V.H., Giné, E.: Decoupling. Probability and its Applications. Springer, New York (1999)
Dumitriu, I., Pal, S.: Sparse regular random graphs: spectral density and eigenvectors. Ann. Probab. (2012, to appear)
Erdős, L.: Universality of Wigner random matrices: a survey of recent results. Uspekhi Mat. Nauk. 66(3), 67–198 (2011)
Erdős, L., Knowles, A., Yau, H.-T.: Spectral statistics of Erdős-Rényi graphs I: local semicircle law. arXiv:1103.1919 (2011)
Erdős, L., Knowles, A., Yau, H.-T., Yin, J.: Spectral statistics of Erdős-Rényi graphs II: eigenvalue spacing and the extreme eigenvalues. Commun. Math. Phys. 314(3), 587–640 (2012)
Erdős, L., Schlein, B., Yau, H.-T.: Local semicircle law and complete delocalization for Wigner random matrices. Commun. Math. Phys. 287(2), 641–655 (2009)
Erdős, L., Schlein, B., Yau, H.-T.: Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices. Ann. Probab. 37(3), 815–852 (2009)
Erdős, L., Schlein, B., Yau, H.-T.: Wegner estimate and level repulsion for Wigner random matrices. Int. Math. Res. Not. IMRN 3, 436–479 (2010)
Erdős, L., Schlein, B., Yau, H.-T.: Universality of random matrices and local relaxation flow. Invent. Math. 185(1), 75–119 (2011)
Guionnet, A.: Grandes matrices aléatoires et théorèmes d’universalité. Astérisque, Exp. No. 1018, Séminaire Bourbaki, vol. 2009/2011 (2010)
Johansson, K.: Universality for certain Hermitian Wigner matrices under weak moment conditions. Ann. Inst. Henri Poincaré Probab. Stat. 48(1), 47–79 (2012)
Ledoux, M.: The concentration of measure phenomenon. Mathematical Surveys and Monographs, vol. 89. American Mathematical Society, Providence (2001)
McDiarmid, C.: On the method of bounded differences. Surveys in Combinatorics, 1989 (Norwich, 1989). London Math. Soc. Lecture Note Ser., vol. 141, pp. 148–188. Cambridge University Press, Cambridge (1989)
Shcherbina, M., Tirozzi, B.: Central limit theorem for fluctuations of linear eigenvalue statistics of large random graphs. J. Math. Phys. 51(2), 023523 (2010)
Tao, T., Vu, V.: Random matrices: universality of local eigenvalue statistics up to the edge. Commun. Math. Phys. 298(2), 549–572 (2010)
Tao, T., Vu, V.: Random matrices: the four moment theorem for Wigner ensembles. Preprint, arXiv:1112.1976 (2011)
Tao, T., Vu, V.: Random matrices: universality of local eigenvalue statistics. Acta Math. 206(1), 127–204 (2011)
Tran, L., Vu, V., Wang, K.: Sparse random graphs: eigenvalues and eigenvectors. Preprint, arXiv:1011.6646 (2010)
Acknowledgments
The authors thank Gérard Ben Arous, Amir Dembo and Terence Tao for a stimulating discussion on the topic held at the Institute of Pacific for Pure and Applied Mathematics, Los Angeles. We are very grateful to Michael Aizenman, Djalil Chafaï and Benjamin Schlein, as this work benefited a lot from their comments. This work was partially supported by ANR project GranMa ANR-08-BLAN-0311-01.
Author information
Authors and Affiliations
Corresponding author
Appendices
Appendix A: Concentration of Gaussian measure
In this paragraph, we recall a well-known concentration phenomenon of the Gaussian measure. The following classical result is contained in Ledoux [24]. It is a consequence of the Logarithmic Sobolev inequality for the Gaussian measure and the Herbst argument.
Theorem 6.1
(Concentration of Gaussian measure) Let \(F\) be a \(1\)-Lipschitz function on the Euclidean space \(\mathbb{R }^n\) and \(G\) be a standard Gaussian vector in \(\mathbb{R }^n\), then for every \( r \ge 0\),
For \(p , q > 0\), we define for \(x \in \mathbb{R }^n\)
and for a matrix \(A\)
(this is a norm for \(p,q \ge 1\)) The usual operator norm is denoted by
where the \(s_i\)’s are the singular values of \(A\). Recall that if \(0 < p \le 2\),
Corollary 6.2
Let \(A\) be a \(n\times n\) non-negative matrix, \(0\le p \le 2\) and \(G\) be a standard Gaussian vector in \(\mathbb{R }^n N(0,I)\). There exist positive constants \(c, \delta > 0\) depending only on \(p\), such that if \( \left( {\mathrm{tr }}A^{p} \right) ^\frac{1}{p} \ge c \Vert A \Vert n^{ \frac{1}{p} - \frac{1}{2}} \) then
with probability at least
Proof
We first consider the case \(1 \le p\le 2\). We define \(F (x) = \Vert A x \Vert _p\). From the triangle inequality (valid for all \(p\ge 1\))
Since \(\Vert A \Vert _{2 \rightarrow p } \le \Vert A \Vert _{2 \rightarrow 2} \Vert I_n \Vert _{ 2 \rightarrow p}\), we deduce that \(F\) is Lipschitz with constant
It follows by Theorem 6.1 that for every \( r \ge 0\),
The corollary will follow by applying the above inequality to \(r =\mathbb{E }\Vert A G \Vert _ p /2\) and by showing that, for some constant \(c_0>0\),
From (90), for some \(c_1 >0\),
Hence
Now, let \((\lambda _k,u_k)_{1 \le k \le n}\) be the eigenvalues and normalized eigenvectors of \(A\). We note that
In particular, \((A G)_i\) has distribution \(N ( 0, \sum _k \lambda _k^2 \langle u_k , e_i \rangle ^2)\) and for some \(c_2 >0\),
For \(0 < p \le 2\) and \(\sum _{k=1}^n \langle u_k , e_i \rangle ^2=1\) for all \(i\in \{1,\ldots ,n\}\), we may use the Jensen inequality to deduce from the above inequality that
Then, from (92) and the value of \(\sigma \), we deduce that (91) holds with \(c_0 = c_2 ^{ 1/ p} /2 \) if \(c\) is chosen large enough so that \( c^{1/p}_2 c \ge 2 c_1\).
We next consider the case \(0 \le p < 1\). We denote, for \(R=(\frac{\kappa }{n}{\mathrm{tr }}A^p)^{1/p}\) with some positive constant \(\kappa \) to be chosen later, \(\phi ( x) = |x|^p \wedge (R^{p-1} |x| ) \). Then \(\phi \) has Lipschitz constant equal to \(R^{p-1}\). In particular, the \(\mathbb{R }^n \rightarrow \mathbb{R }_+\) function \(x \mapsto \sum _i \phi (x_i)\) is Lipschitz with constant bounded by \( \sqrt{n} R^{p-1}\). It follows that the \(\mathbb{R }^n \rightarrow \mathbb{R }_+\) function \(F (x) = \sum _i \phi ((A x)_i)\) is Lipschitz with constant bounded by \( \Vert A \Vert \sqrt{n} R^{p-1}\). Hence, by Theorem 6.1, for any \(r>0\),
Now, we observe that as \(\phi \) is upper bounded by \(|x|^p\),
whereas, since \(\phi (x ) \ge |x|^p - R^p\), by the choice of \(R\),
Therefore from (93), if we choose \(\kappa <c_2/4\),
Finally, we set \(r= c_2{\mathrm{tr }}(A^p)/4\) and conclude that
\(\square \)
Remark 6.3
Consider the special case where \(A\) is the projector on a vector space \(W\) of dimension \(d\). Then \(A^p = A\) for all \(p >0\). Corollary 6.2 gives a lower bound for \(\Vert A G \Vert _p\) of order \(d^{\frac{1}{p}}\) when \(d\ge c^p n^{1-p/2}\). However, if \((u_1, \ldots , u_d)\) is an orthonormal basis of \(W\) such that \(\langle u_k , e_i \rangle ^2 \ge \varepsilon ^2 / n\) then for \(p\le 1\) we have a lower bound for \(\Vert A G \Vert _p\) of order \(\varepsilon n^{ \frac{1}{p} - \frac{1}{2}} d^{\frac{1}{2}}\) which can be significantly larger. Hence, we expect that Corollary 6.2 is sharp if \(W\) has a localized basis and not sharp if \(W\) has a delocalized basis.
Appendix B: Stable distributions
In this paragraph, we give some properties of stable distributions.
Let \(\sigma >0, 0 < \alpha < 2\) and \(\beta \in [-1,1]\). A real random variable \(X\) has \(\alpha \)-stable distribution \({\mathrm{Stab }}_\alpha (\beta ,\sigma )\) if its Fourier transform is given for all \(t \in \mathbb{R }\), by
where \(\mathrm sgn (t)\) is the sign of \(t\) and \(u_\alpha =\tan (\pi \alpha /2)\, \) for all \(\alpha \) except \(\alpha = 1\) in which case \(u_1=-(2/\pi )\log |t|.\,\)
If \(0 < \alpha < 1\) and \(\beta =1\), the distribution \({\mathrm{Stab }}_\alpha (1,\sigma )\) has support \(\mathbb{R }_+\) and its Laplace transform is conveniently given for all \(t \in \mathbb{R }_+\), by
with \(v_\alpha = \frac{2}{\pi }\sin \left( \frac{\pi \alpha }{2} \right) \Gamma ( 1 - \alpha ) \Gamma ( \alpha )\).
Lemma 7.1
(Decomposition of quadratic form) Let \(X = (X_i)_{1 \le i \le n}\) be iid symmetric \(\alpha \)-stable random variables with distribution \({\mathrm{Stab }}_\alpha (0,\sigma )\). Let \(A\) be a \(n\times n\) positive definite matrix, then
where \(G\) is a standard gaussian vector \(N( 0 , I ) \) independent of \(S\), a positive \(\alpha / 2\)-stable \({\mathrm{Stab }}_{\frac{\alpha }{2}}(1, 2 \sigma ^2 v_{\frac{\alpha }{2}}^{-\frac{2}{\alpha }})\).
Proof
We use the identity, for \(y \in \mathbb{R }^n\),
Applied to \(y = A^{1/2} X\), we get, for \(t \ge 0\),
Then, since \(X\) is stable vector, \(\langle X , A^{1/2} G \rangle \) has distribution \({\mathrm{Stab }}_\alpha (0 , \sigma \Vert A^{1/2} G \Vert _\alpha )\). From (94), it follows
Then, we conclude by applying (95). \(\square \)
Corollary 7.2
(Sum of weighted squares) Let \(X = (X_k)_{1 \le k \le n}\) be iid symmetric \(\alpha \)-stable random variables with distribution \({\mathrm{Stab }}_\alpha (0,\sigma )\) and let \((w_k)_{1 \le k \le n} \in \mathbb{C }^n_+\). Then
where \(G = (g_1, \ldots , g_n)\) is a standard gaussian vector \(N( 0 , I ) \).
Proof
We set \(\rho _k = - i w_k\), we shall prove that
We write \(\rho _k = i ( a_k - b_k ) + c_k\), where \(a_k, b_k\) are the negative and positive parts of \(\mathrm{Re}(w_k)\) and \(c_k = \mathrm{Im}(w_k) >0\). We set \(\rho _k (t,s) = t a_k + s b_k + c_k, D = \{z \in \mathbb{C }: \mathrm{Re}(z ) > 0 \} = - i \mathbb{C }_+\) and \(D_\varepsilon = \{z \in \mathbb{C }: \mathrm{Re}(z ) > - \varepsilon , | \mathrm{Im}(z) | < 2 \}\), where \(2 \varepsilon = \min ( c_k / ( a_k + b_k) )\). Then, the \(D^2_\varepsilon \rightarrow D\) function \((t,s) \mapsto \sum _{k=1}^n \rho _k (t,s) X_k^2\) is analytic in each of its coordinates. Since the function \(z \mapsto \exp ( - z )\) is analytic and bounded on \(D\), from Montel’s Theorem, we deduce that the \(D_\varepsilon ^2 \rightarrow \mathbb{C }\) function
is analytic in each of its coordinates in \(D_\varepsilon \). However, for \(s,t \in \mathbb{R }_+\), we notice that \(\rho _k(s,t) \in \mathbb{R }_+\). Hence by Lemma 7.1 applied to a diagonal matrix, we have
The \(D \rightarrow D\) function \(z \mapsto z^{\alpha / 2}\) is analytic. We may thus again apply Montel theorem and deduce that the right hand side of the above identity is analytic in \((s,t)\) on \(D_\varepsilon ^2\). So finally, the above equality holds true for all \((s,t) \in D^2_\varepsilon \). Applied to \((s,t) = (i,-i)\), we obtain precisely (96). \(\square \)
The next lemma looks at the behavior of a positive stable random variable near 0.
Lemma 7.3
(Tail of inverse positive stable variable) Let \(\sigma >0, 0 < \alpha <1\) and \(S\) be a positive \(\alpha \)-stable \({\mathrm{Stab }}_{\alpha }(1, \sigma )\) random variable. There exists a positive constant \(c_0 (\alpha )\) such that for all \(0 < c < \sigma ^{\frac{\alpha }{1 - \alpha }} c_0 ( \alpha )\),
while the above is infinite for \(c > \sigma ^{\frac{\alpha }{1 - \alpha }} c_0 ( \alpha )\).
Proof
From the identity, for \(m >0, x > 0\),
we deduce that, for \(p \ge 0\),
In particular, from (95), with \(\hat{\sigma }= \sigma v_{\alpha }^{1/\alpha }\), and Fubini’s theorem,
The conclusion follows easily from Stirling’s formula, \(\Gamma (x) \sim _{x \rightarrow \infty } \sqrt{\frac{2 \pi }{ x}}~{\left( \frac{x}{e} \right)}^x\). \(\square \)
lemma 7.4
(Negative fractional moments of smooth random variable) Let \( \alpha > 0\) and \(S\) be a real-valued random variable with law which has a uniformly bounded density on \([-1,1]\) and is bounded by \(c |x|^{-\alpha -1}\) on \([-1,1]^c\) for some finite positive constant \(c\). Then, for any \(0 < \beta <1\), there exists a finite constant \(C \) so that for any \(x\in \mathbb R \), any \(\sigma \ge 0\), we have
Proof
Let us first assume that \(\sigma \ge 2|x|\). If \(C\) is a bound on the density of the law of \(S\) on \([-1,1]\), for \(T\ge (2/\sigma )^\beta \),
Choosing \(T=(2/\sigma )^\beta \le |x|^{-\beta }\) provides the desired estimate. In the case \(\sigma \le 2|x|\) and \(t^{-1/\beta }\le |x|/2\), we have
Therefore if \(\sigma \le 2|x|\) and \(T=(2/|x|)^\beta \),
which completes the proof of the lemma. \(\square \)
Appendix C: Concentration of random matrices with independent rows
The total variation norm of \(f:\mathbb{R }\rightarrow \mathbb{R }\) is
where the supremum runs over all sequences \((x_k)_{k \in \mathbb{Z }}\) such that \(x_{k+1} \ge x_k\) for any \(k \in \mathbb{Z }\). If \(f = {\mathbf{1 }}_{(-\infty ,s]}\) for some real \(s\) then \(\Vert f \Vert _\textsc {TV}=1\), while if \(f\) has a derivative in \(\mathrm L ^1(\mathbb{R })\), we get
Lemma 8.1
(Concentration for spectral measures [11]) Let \(A\) be an \(n\times n\) random Hermitian matrix. Let us assume that the vectors \((A_i)_{1 \le i \le n}\), where \(A_i := (A_{ij})_{1 \le j \le i} \in \mathbb{C }^i\), are independent. Then for any \(f:\mathbb{R }\rightarrow \mathbb{C }\) such that \(\Vert f \Vert _\textsc {TV}\le 1\) and \(\mathbb{E }|\int \!f\,d\mu _A |<\infty \), and every \(t\ge 0\),
The next lemma is an easy consequence of Cauchy–Weyl interlacing Theorem. It is an ingredient of the proof of Lemma 8.1.
Lemma 8.2
(Interlacing of eigenvalues) Let \(A\) be an \(n \times n\) hermitian matrix and \(B\) a principal minor of \(A\). Then for any \(f:\mathbb{R }\rightarrow \mathbb{C }\) such that \(\Vert f \Vert _{TV} \le 1\) and \(\lim _{|x| \rightarrow \infty } f(x) = 0\),
The Lipschitz norm of \(f:\mathbb{C }\rightarrow \mathbb{C }\) is
Lemma 8.3
(Concentration for the diagonal of the resolvent) Let \(A\) be an \(n\times n\) random Hermitian matrix and consider its resolvent matrix \(R (z) = (A - z)^{-1} , z \in \mathbb{C }_+\). Let us assume that the vectors \((A_i)_{1 \le i \le n}\), where \(A_i := (A_{ij})_{1 \le j \le i} \in \mathbb{C }^i\), are independent. Then for any \(f:\mathbb{C }\rightarrow \mathbb{R }\) such that \(\Vert f \Vert _{L} \le 1\), and every \(t\ge 0\),
Proof
The proof is close to the proof of Lemma 8.1 as done in [11] and relies on the method of bounded martingale difference. We start by showing that for every \(n\times n\) deterministic Hermitian matrices \(B\) and \(C\) and any measurable function \(f\) with \(\Vert f \Vert _{L} \le 1\),
where \(R_B = (B - z)^{-1} \) and \(R_C = (C - z)^{-1} \) are their resolvent matrices. Indeed, by assumption
The resolvent identity asserts that
It follows that \(r = \mathrm rank (M)\le \mathrm rank (B - C)\). We notice also that \( \Vert M \Vert \le 2 \mathrm{Im}(z)^{-1}\). Hence, in the singular value decomposition of \(M = U D V\), at most \(r\) entries of \(D = \mathrm diag ( s_1, \ldots , s_n)\) are non zero and they are bounded by \(\Vert M \Vert \). We denote by \(u_1, \ldots , u_r\) and \(v_1, \ldots , v_r\) the associated orthonormal vectors so that
and
We obtain from Cauchy–Schwarz,
Equation (97) is thus proved.
Next, for any \(x = (x_1, \ldots , x_n) \in {\mathcal{X }}:= \{(x_i)_{1 \le i \le n} : x_i \in \mathbb{C }^{i-1} \times \mathbb{R }\}\), let \(B(x)\) be the \(n \times n\) Hermitian matrix given by \(B(x)_{ij} = x_{i,j}\) for \(1 \le j \le i \le n\) and \(R_x (z) = (B(x) - z) ^{-1}\). We thus have \(R (z) = R _{(A_1, \ldots , A_n)} (z) \). For all \(x \in {\mathcal{X }}\) and \(x^{\prime }_i \in \mathbb{C }^{i-1} \times \mathbb{R }\), the matrix
has only the \(i\)th row and column possibly different from \(0\), and thus
Therefore from (97), we obtain, for every \(f:\mathbb{R }\rightarrow \mathbb{R }\) with \(\Vert f \Vert _{L} \le 1\),
The desired result follows now from the Azuma–Hoeffding inequality, see e.g. [25, Lemma 1.2]. \(\square \)
Lemma 8.4
(Concentration for the diagonal of the resolvent) Let \(A\) be an \(n\times n\) random Hermitian matrix and consider its resolvent matrix \(R (z) = (A - z)^{-1} , z \in \mathbb{C }_+\). Let us assume that the vectors \((A_i)_{1 \le i \le n}\), where \(A_i := (A_{ij})_{1 \le j \le i} \in \mathbb{C }^i\), are independent. Then for any \(\gamma \in [0,1]\), there exists a positive constant \(c\) so that for every \(t\ge 0\),
Proof
Let \(\varepsilon \) be a positive real number and \(\phi _\varepsilon :\mathbb{C }\rightarrow \mathbb{C }\) be equal to one on \(|z|\ge 2 \varepsilon \), vanishing on \(|z|\le \varepsilon \) and growing linearly with the modulus in between. Thus, \(\phi _\varepsilon \) is Lipschitz with constant bounded by \(1/\varepsilon \). We decompose \(x^\gamma \) as
By definition, \(x^\gamma (1-\phi _\varepsilon (x))\) has modulus uniformly bounded above by \((2\varepsilon )^\gamma \) so that if we choose \(\varepsilon >0\) so that \((2\varepsilon )^\gamma =t/4\) then with \(f(x)=x^\gamma \phi _\varepsilon (x)\) we have
On the other hand, \(f\) is Lipschitz with constant bounded by \(2 \varepsilon ^{\gamma -1}=2^{\frac{2}{\gamma }-\gamma } t^{1-\frac{1}{\gamma }}\). Hence, Lemma 8.3 yields
This concludes the proof. \(\square \)
We conclude this Appendix by deviations inequalities for the norm \(\Vert \cdot \Vert _{\beta , \varepsilon }\) introduced in Sect. 5.
Lemma 8.5
Let \(0 < \alpha \le 1, \varepsilon \ge 0, \alpha / 2 < \beta < 1\) and assume that \(\alpha / 2 + \beta + \varepsilon \le 1\). Let \((g_k), 1 \le k \le n,\) be iid standard Gaussian variables and \((h_k), 1 \le k \le n \in {\mathcal{K }}_1\) with \(|h_k| \le \eta ^{-1}\). Define, \(\gamma (u) = \frac{1}{n} \sum _{k= 1} ^ n (h_k. u )^{\frac{\alpha }{2}} ( |g_k|^\alpha - \mathbb{E }| g_k |^ \alpha ) \). Then there exists constant \(c_0,c_1\), such that for all \(t \ge 1\), all \(n \ge 2\),
Similarly, let \(A\) be an \(n\times n\) random Hermitian matrix and consider its resolvent matrix \(R (z) = (A - z)^{-1} , z = E +i \eta \in \mathbb{C }_+\). Let \(H_k = - i R _{kk} (z)\) be as above and \(\gamma ^{\prime } (u) = \frac{1}{n} \sum _{k= 1} ^ n (H_k. u )^{\frac{\alpha }{2}} - \mathbb{E }\frac{1}{n} \sum _{k= 1} ^ n (H_k. u )^{\frac{\alpha }{2}} \), for all \(t \ge 1\), all \(n \ge 2\),
Proof
Set \(L = \frac{1}{n} \sum _{k=1} ^ n | |g_k|^\alpha - \mathbb{E }| g_k |^ \alpha |\). We first use a net argument. For any \(u, v \in S_+^ 1\), from (64), for some constant \(c = c(\alpha )\).
In particular, choosing \(\beta =\alpha /2\) and setting, for integer \(m\) and \(1 \le k \le m, u_k = e^{ i 2 \pi k / m}\), we find
Notice also that if \(| u - v | \le 4 / m \), then, with \(\beta ^{\prime } = \alpha / 2 + \beta + \varepsilon \le 1\),
While if \(| u - v | \ge 4 / m \), we denote by \(u_*\) and \(v_*\), the element of \(\{ u_k : 1 \le k \le m\} \) at distance at most \(1 / m\) of \(u\) and \(v\) and with \(| i . u_* | \ge |i . u|, | i . v_* | \ge |i . v|\). We get \(| u - v | \ge 2 | u_* - v_* |\) and
We deduce that, for some constant \(c_0 \ge 1\),
On the other hand, since \(0 < \alpha \le 1\), the random variable \(|g_k|^\alpha \) is sub-gaussian. It follows from Hoeffding’s inequality, that for any \(s \ge 0\),
and for any \(u, v \in S_+^ 1\),
From the union bound, we get from (99),
We take \(m = [n^{1/\alpha } ( ( \mathbb{E }L + s ) ) ^ { 2 / \alpha } t ^ { - 2 / \alpha }]\) and \( s = t \), we find for all \(t \ge 2 / c_0\),
This prove the first statement. For the second statement, the proof is similar. First, the above net argument gives that (99) holds for \(\gamma ^{\prime }\) with \(L= 1\). Also the proof of Lemma 8.4 implies that for any \(u, v \in S_+^ 1\),
From the union bound, we deduce that for all \(s \ge 0\),
Taking \( s = t / ( \eta ^ 2 n ) ^{\frac{\alpha }{4} } \) and \( m = \sqrt{n} t ^{- 2 / \alpha } (2 c_0 ) ^{ 2 / c_0} \), this concludes the proof. \(\square \)
Appendix D: Key bounds on relevant integrals
The next lemma is used repeatedly in the proofs of Sect. 5.
Lemma 9.1
(Consequences of [6, Lemma 3.6]) Let \(0 < \alpha < 2, \gamma > 0\) and \(0 < \kappa \le 1\), there exists a constant \(c = c ( \alpha , \gamma ) >0\) such that for all \(h \in {\mathcal{K }}_1\) and \(x \in {\mathcal{K }}_{\frac{\alpha }{2}}\),
For all \(h , k \in {\mathcal{K }}_1\) and \(x , y \in {\mathcal{K }}_{\frac{\alpha }{2}}\),
and
For all \(x_1, x_2, y_1, y_2 \in {\mathcal{K }}_{\frac{\alpha }{2}}\),
Moreover, for all \(h , k \in {\mathcal{K }}_1, x, y \in {\mathcal{K }}_{\frac{\alpha }{2}}\), for \(0 < \kappa \le 1\), we have
and finally, for all \(0 \le \kappa _1, \kappa _2 \le 1, h_1 , k_1 , h_2, k_2 \in {\mathcal{K }}_1\), with \(N = | h_1 | \wedge |k_1| \wedge |h_2| \wedge |k_2|\),
Proof
The bound (100) is [6, Lemma 3.6]. The bound (101) follows from (100) by taking derivative and using the convexity of \({\mathcal{K }}_{\frac{\alpha }{2}}\). For (102), assume for example that \(|h| \le |k|\). We may assume that \( |h - k | \le |h| / 2\) since otherwise we get the estimate by bounding independently both terms of the difference and using (100). Then taking derivative, we find
where \(N = \min \{ | t h + (1 - t) k | : 0 \le t \le 1\}\). Since \( |h - k | \le |h| / 2\) then \( N \ge |h| / 2\) and
To prove (103), (104) and (105), we need to take derivatives and use the first inequalities. Namely, for (103), we define the function on \([0,1]\),
with \(x(t) = t x_1 + (1 - t) x_2\) and \(y(t) = t y_1 + (1 - t) y_2\). We are looking for an upper bound on \(|\varphi (1) - \varphi (0)|\). A straightforward computation yields
By convexity, we note that \(x(t), y(t)\) are in \({\mathcal{K }}_{\frac{\alpha }{2}}\). Hence using (100) and (101), we may upper bound \(|\varphi ^{\prime } (t)| \), up to a multiplicative constant, by
This completes the proof of (103). For (104), we first first notice that splitting \(h\) and \(k\) and arguing as for the proof of (102), it is sufficient to prove the statement for \(|h-k| \le (|h| \wedge |k| )/ 2\) and \(\kappa = 1\). Then, with \(h(t) = t h + (1 - t)k\), we consider this time the function
and take the derivative. We find that it is proportional to \((h-k)(\varphi (1)-\varphi (0))\) with the previous function \(\varphi \) with \(\gamma \) replaced by \(\gamma +1\) and \(x_1=x_2=x,y=y_1=y_2\). The bound is therefore clear. The proof is identical for the third statement for \(\kappa _1 = \kappa _2 =1\). As above, we then generalize it to any \(0 \le \kappa _1, \kappa _2 \le 1\) by using the rough bound given by (102). \(\square \)
Rights and permissions
About this article
Cite this article
Bordenave, C., Guionnet, A. Localization and delocalization of eigenvectors for heavy-tailed random matrices. Probab. Theory Relat. Fields 157, 885–953 (2013). https://doi.org/10.1007/s00440-012-0473-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-012-0473-9