Abstract
We consider spectral properties and the edge universality of sparse random matrices, the class of random matrices that includes the adjacency matrices of the Erdős–Rényi graph model G(N, p). We prove a local law for the eigenvalue density up to the spectral edges. Under a suitable condition on the sparsity, we also prove that the rescaled extremal eigenvalues exhibit GOE Tracy–Widom fluctuations if a deterministic shift of the spectral edge due to the sparsity is included. For the adjacency matrix of the Erdős–Rényi graph this establishes the Tracy–Widom fluctuations of the second largest eigenvalue when p is much larger than \(N^{-2/3}\) with a deterministic shift of order \((Np)^{-1}\).
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1 Introduction
We consider spectral properties of sparse random matrices. One of the most prominent examples in the class of sparse random matrices is the (centered) adjacency matrix of the Erdős–Rényi graph on N vertices, where an edge is independently included in the graph with a fixed probability \(p\equiv p(N)\). Introduced in [11, 12, 24], the Erdős–Rényi graph model G(N, p) serves as a null model in the theory of random graphs and has numerous applications in many fields including network theory. Information about a random graph can be obtained by investigating its adjacency matrix, especially the properties of its eigenvalues and eigenvectors.
The sparsity of a real symmetric N by N random matrix may be measured by a sparsity parameter \(q\equiv q(N)\), with \(0\le q\le N^{1/2}\), such that the expected number of non-vanishing entries is asymptotically equal to \(Nq^2\). For example, for the adjacency matrices of the Erdős–Rényi graph we have \(q^2 =Np\) as the expected number of non-vanishing entries is \(N(N-1)p=(N-1)q^2\). For the Gaussian orthogonal ensemble (GOE) we have \(q=N^{1/2}\) as the expected number of non-vanishing entries is \(N^2\). We call a random matrix sparse if q is much smaller than \(N^{1/2}\).
For Wigner matrices, one of the fundamental inputs in the proof of universality results is the local semicircle law [18,19,20, 23], which provides an estimate of the local eigenvalue density down to the optimal scale. The framework built on the local law can also help understanding the spectral properties of sparse random matrices [15]. However, in contrast to Wigner matrices, the local eigenvalue density for a sparse random matrix depends on its sparsity. For this reason, the universality of the local eigenvalue statistics for sparse random matrices was proved at first only for \(q\ge N^{1/3}\) in [15, 16]. Recently, bulk universality was proved in [29] under the much weaker condition \(q \ge N^{\phi }\), for any \(\phi > 0\). The main obstacle in the proof of the edge universality is that the local law obtained in [15] deteriorates at the edge of the spectrum.
Our first main result is a local law for sparse random matrices up to the edges. More precisely, we prove a local law for the eigenvalue density in the regime \(q\ge N^\phi \), for arbitrarily small \(\phi >0\). The main observation is that, although the empirical spectral measure of sparse random matrices converges in the large N limit to the semicircle measure, there exists a deterministic correction term that is not negligible for large but finite N. As a result, we establish a local law that compares the empirical spectral measure not with the semicircle law but with its refinement. We also explicitly compute the leading term of the deterministic shift of the spectral edge, which equals 1 / (Np) in case of the (centered) adjacency matrix of the Erdős–Rényi graph (N, p). (See Theorem 2.4 and Corollary 2.5 for more detail.) This information on the shift of the spectral edge is crucial in understanding the fluctuations of the largest eigenvalues.
The largest eigenvalue \(\mu _1\) of a real symmetric N by N Wigner matrix (whose entries are centered and have variance 1 / N) converges almost surely to two under the finite fourth-moment condition, and \(N^{2/3}(\mu _1 -2)\) converges in distribution to the GOE Tracy–Widom law. However, for sparse random matrices with sparsity \(q\le N^{1/3-\phi }\), \(\phi >0\), the deterministic shift of the edge is far greater than \(N^{-2/3}\), the typical size of the Tracy–Widom fluctuations.
Our second main result is the edge universality that states the limiting law for the fluctuations of the rescaled largest eigenvalues of a (centered) sparse random matrices is given by the Tracy–Widom law if the shift is taken into consideration and if \(q\ge N^{1/6+\phi }\), where \(\phi >0\) is arbitrarily small. We expect that the exponent one-sixth is critical, i.e. we expect a different limiting law for the fluctuations of the rescaled largest eigenvalues when \(q\le N^{1/6-\phi }\). (See Theorem 2.10 and the discussion below it for more detail.) For the adjacency matrices of the Erdős–Rényi graphs, the sparsity conditions corresponds to \(p\ge N^{-2/3 + \phi }\), for any \(\phi > 0\), and our result then assures that the rescaled second largest eigenvalue has GOE Tracy–Widom fluctuations; see Corollary 2.13.
In the proof of the local law, we introduce a new method based on a recursive moment estimate for the normalized trace m of the Green function, i.e. we recursively control high moments of |P(m)|, for some specifically chosen polynomial P, by using lower moments of |P(m)|. This recursive moment estimate allows us to track the effects of the sparsity efficiently and it directly establishes the fluctuation averaging mechanism without fully expanding all powers of m; see Sect. 3 for details. To establish the recursive moment estimate we use a cumulant expansion of P(m) replacing customary resolvent expansions of m using Schur’s complement or finite-rank perturbation formulas.
Our proof of the Tracy–Widom limit of the extremal eigenvalues relies on the Green function comparison method [21, 23]. However, instead of applying the conventional Lindeberg replacement approach, we use a continuous flow that interpolates between the sparse random matrix and the Gaussian orthogonal ensemble (GOE). The main advantage of using a continuous interpolation is that we may estimate the rate of change of m along the flow even if the moments of the entries in the sparse matrix are significantly different from those of the entries in the GOE matrix. The change of m over time is offset by the deterministic shift of the edge. A similar idea was used in the proof of edge universality of other random matrix models in [37, 38].
This paper is organized as follows: In Sect. 2, we define the model, present the main results and outline applications to adjacency matrices of the Erdős–Rényi graph ensemble. In Sect. 3, we explain the main strategy of our proofs. In Sect. 4, we prove several properties of the deterministic refinement of Wigner’s semicircle law. In Sect. 5, we prove the local law using our technical result on the recursive moment estimate, Lemma 5.1. In Sect. 6, we prove Lemma 5.1 with technical detail. In Sect. 7, we prove our second main result on the edge universality.
Notational conventions We use the symbols \(O(\,\cdot \,)\) and \(o(\,\cdot \,)\) for the standard big-O and little-o notation. The notations O, o, \(\lll \), \(\ggg \), refer to the limit \(N\rightarrow \infty \) unless otherwise stated. Here \(a\lll b\) means \(a=o(b)\). We use c and C to denote positive constants that do not depend on N, usually with the convention \(c\le C\). Their value may change from line to line. We write \(a\asymp b\), if there is \(C\ge 1\) such that \(C^{-1}|b|\le |a|\le C |b|\). Throughout the paper we denote for \(z\in \mathbb {C}^+\) the real part by \(E={{\mathrm{\mathrm {Re}}}}z\) and the imaginary part by \(\eta ={{\mathrm{\mathrm {Im}}}}z\). For \(a\in \mathbb {R}\), we let \((a)_+=\max (0,a)\), and \((a)_-= -\min (a,0)\). Finally, we use double brackets to denote index sets, i.e. for \(n_1,n_2\in \mathbb {R}\), \(\llbracket n_1,n_2\rrbracket :=[n_1,n_2]\cap \mathbb {Z}\).
2 Definitions and main results
2.1 Motivating examples
2.1.1 Adjacency matrix of Erdős–Rényi graph
One motivation for this work is the study of adjacency matrices of the Erdős–Rényi random graph model G(N, p). The off-diagonal entries of the adjacency matrix associated with an Erdős–Rényi graph are independent, up to the symmetry constraint, Bernoulli random variables with parameter p, i.e. the entries are equal to 1 with probability p and 0 with probability \(1-p\). The diagonal entries are set to zero, corresponding to the choice that the graph has no self-loops. Rescaling this matrix ensemble so that the bulk eigenvalues typically lie in an order one interval we are led to the following random matrix ensemble. Let A be a real symmetric \(N\times N\) matrix whose entries, \(A_{ij}\), are independent random variables (up to the symmetry constraint \(A_{ij}=A_{ji}\)) with distributions
Note that the matrix A typically has \(N(N-1)p\) non-vanishing entries. For our analysis it is convenient to extract the mean of the entries of A by considering the matrix \(\widetilde{A}\) whose entries, \(\widetilde{A}_{ij}\), have distribution
with \(i\not =j\). A simple computation then reveals that
and
with \(i\not =j\), where \(d\mathrel {\mathop :}=pN\) denotes the expected degree of a vertex, which we allow to depend on N. As already suggested by (2.3), we will assume that \(p\lll 1\).
2.1.2 Diluted Wigner matrices
Another motivation for this work are diluted Wigner matrices. Consider the matrix ensemble of real symmetric \(N\times N\) matrices of the form
where \((B_{ij}:i\le j)\) and \((V_{ij}:i\le j)\) are two independent families of independent and identically distributed random variables. The random variables \((V_{ij})\) satisfies \(\mathbb {E}V_{ij}^2=1\) and \(\mathbb {E}V_{ij}^{2k}\le (Ck)^{ck}\), \(k\ge 4\), for some constants c and C, and their distribution is, for simplicity, often assumed to be symmetric. The random variables \((B_{ij})\) are chosen to have a Bernoulli type distribution given by
with \(i\not =j\). We introduce the sparsity parameter q through
with \(0<q\le N^{1/2}\). We allow q to depend on N. We refer to the random matrix \(D=(D_{ij})\) as a diluted Wigner matrix whenever \(q\lll N^{1/2}\). For \(q=N^{1/2}\), we recover the usual Wigner ensemble.
2.2 Notation
In this subsection we introduce some of the notation and conventions used.
2.2.1 Probability estimates
We first introduce a suitable notion for high-probability estimates.
Definition 2.1
(High probability event) We say that an N-dependent event \(\Xi \equiv \Xi ^{(N)}\) holds with high probability if, for any (large) \(D>0\),
for sufficiently large \(N\ge N_0(D)\).
Definition 2.2
(Stochastic domination) Let \(X\equiv X^{(N)}\), \(Y\equiv Y^{(N)}\) be N-dependent non-negative random variables. We say that Y stochastically dominates X if, for all (small) \(\epsilon >0\) and (large) \(D>0\),
for sufficiently large \(N\ge N_0(\epsilon ,D)\), and we write \(X \prec Y\). When \(X^{(N)}\) and \(Y^{(N)}\) depend on a parameter \(u\in U\) (typically an index label or a spectral parameter), then \(X(u) \prec Y (u)\), uniformly in \(u\in U\), means that the threshold \(N_0(\epsilon ,D)\) can be chosen independently of u. A slightly modified version of stochastic domination appeared first in [14].
In Definition 2.2 and hereinafter we implicitly choose \(\epsilon >0\) strictly smaller than \(\phi /10>0\), where \(\phi >0\) is the fixed parameter appearing in (2.13) below.
The relation \(\prec \) is transitive and it satisfies the following arithmetic rules: If \(X_1\prec Y_1\) and \(X_2\prec Y_2\) then \(X_1+X_2\prec Y_1+Y_2\) and \(X_1 X_2\prec Y_1 Y_2\). Furthermore, the following property will be used on many occasions: If \(\Phi (u)\ge N^{-C}\) is deterministic, if Y(u) is a nonnegative random variable satisfying \(\mathbb {E}[Y(u)^2]\le N^{C'}\) for all u, and if \(Y(u) \prec \Phi (u)\) uniformly in u, then, for any \(\epsilon >0\), we have \(\mathbb {E}[Y(u)] \le N^\epsilon \Phi (u)\) for \(N\ge N_0(\epsilon )\), with a threshold independent of u. This can easily be checked since
for any (large) \(D > 0\), and \(\mathbb {E}[Y(u) \mathbbm {1}(Y(u) \le N^{\epsilon /2} \Phi (u))] \le N^{\epsilon /2} \Phi (u)\), hence \(\mathbb {E}[Y(u)] \le N^{\epsilon } \Phi (u)\).
2.2.2 Stieltjes transform
Given a probability measure \(\nu \) on \(\mathbb {R}\), we define its Stieltjes transform as the analytic function \(m_\nu \,:\,\mathbb {C}^+\rightarrow \mathbb {C}^+\), with \(\mathbb {C}^+\mathrel {\mathop :}=\{ z=E+\mathrm {i}\eta \,:\, E\in \mathbb {R}, \eta >0\}\), defined by
Note that \(\lim _{\eta \nearrow 0}\mathrm {i}\eta \, m_{\nu }(\mathrm {i}\eta )=-1\) since \(\nu \) is a probability measure. Conversely, if an analytic function \(m\,:\,\mathbb {C}^+\rightarrow \mathbb {C}^+\) satisfies \(\lim _{\eta \nearrow 0}\mathrm {i}\eta \, m(\mathrm {i}\eta )=-1\), then it is the Stieltjes transform of a probability measure.
Choosing \(\nu \) to be the standard semicircle law with density \(\frac{1}{2\pi }\sqrt{4-x^2}\) on \([-2,2]\), on easily shows that \(m_{\nu }\), for simplicity hereinafter denoted by \(m_\mathrm {sc}\), is explicitly given by
where we choose the branch of the square root so that \(m_\mathrm {sc}(z) \in \mathbb {C}^+\), \(z\in \mathbb {C}^+\). It directly follows that
2.3 Main results
In this section we present our main results. We first generalize the matrix ensembles derived from the Erdős–Rényi graph model and the diluted Wigner matrices in Section 2.1.
Assumption 2.3
Fix any small \(\phi >0\). We assume that \(H = (H_{ij})\) is a real symmetric \(N \times N\) matrix whose diagonal entries are almost surely zero and whose off-diagonal entries are independent, up to the symmetry constraint \(H_{ij} = H_{ji}\), identically distributed random variables. We further assume that \((H_{ij})\) satisfy the moment conditions
with sparsity parameter q satisfying
We assume that the diagonal entries satisfy \(H_{ii}=0\) a.s., yet this condition can easily be dropped. For the choice \(\phi =1/2\) we recover the real symmetric Wigner ensemble (with vanishing diagonal). For the rescaled adjacency matrix of the Erdős–Rényi graph, the sparsity parameter q, the edge probability p and the expected degree of a vertex d are linked by \(q^2=pN=d\).
We denote by \(\kappa ^{(k)}\) the k-th cumulant of the i.i.d. random variables \((H_{ij}:i<j)\). Under Assumption 2.3 we have \(\kappa ^{(1)}=0\), \(\kappa ^{(2)}=1/N\), and
We further introduce the normalized cumulants, \(s^{(k)}\), by setting
In case H is given by the centered adjacency matrix \(\widetilde{A}\) introduced in Sect. 2.1.1, we have that \(s^{(k)}=1+O(d/N)\), \(k\ge 3\), as follows from (2.3).
We start with the local law for the Green function of this matrix ensemble.
2.3.1 Local law up to the edges for sparse random matrices
Given a real symmetric matrix H we define its Green function, \(G^H\), and the normalized trace of its Green function, \(m^H\), by setting
The matrix entries of \(G^H(z)\) are denoted by \(G^{H}_{ij}(z)\). In the following we often drop the explicit z-dependence from the notation for \(G^H(z)\) and \(m^H(z)\).
Denoting by \(\lambda _1 \ge \lambda _2 \ge \dots \ge \lambda _N\) the ordered eigenvalues of H, we note that \(m^H\) is the Stieltjes transform of the empirical eigenvalue distributions, \(\mu ^H\), of H given by
We further introduce the following domain of the upper-half plane
Our first main result is the local law for \(m^H\) up to the spectral edges.
Theorem 2.4
Let H satisfy Assumption 2.3 with \(\phi >0\). Then, there exists an algebraic function \(\widetilde{m} : \mathbb {C}^+ \rightarrow \mathbb {C}^+\) and \(2<L<3\) such that the following hold:
-
(1)
The function \(\widetilde{m}\) is the Stieltjes transform of a deterministic symmetric probability measure \(\widetilde{\rho }\), i.e. \(\widetilde{m}(z)=m_{\widetilde{\rho }}(z)\). Moreover, \({{\mathrm{supp}}}\widetilde{\rho }=[-L,L]\) and \(\widetilde{\rho }\) is absolutely continuous with respect to the Lebesgue measure with a strictly positive density on \((-L,L)\).
-
(2)
The function \(\widetilde{m}\equiv \widetilde{m}(z)\) is a solution to the polynomial equation
$$\begin{aligned} \begin{aligned} P_{ z}(\widetilde{m})&\mathrel {\mathop :}=1 + z \widetilde{m} + \widetilde{m}^2 + \frac{{ s^{(4)}}}{q^2}\widetilde{m}^4=0\,,\qquad \qquad (z\in \mathbb {C}^+)\,. \end{aligned} \end{aligned}$$(2.19) -
(3)
The normalized trace \(m^H\) of the Green function of H satisfies
$$\begin{aligned} |m^H (z) - \widetilde{m} (z)| \prec \frac{1}{q^2}+\frac{1}{N\eta }\,, \end{aligned}$$(2.20)uniformly on the domain \({\mathcal E}\), \(z=E+\mathrm {i}\eta \).
Some properties of \(\widetilde{\rho }\) and its Stieltjes transform \(\widetilde{m}\) are collected in Lemma 4.1 below.
The local law (2.20) implies estimates on the local density of states of H. For \(E_1< E_2\) define
Corollary 2.5
Suppose that H satisfies Assumption 2.3 with \(\phi >0\). Let \(E_1,E_2\in \mathbb {R}\), \(E_1<E_2\). Then,
The proof of Corollary 2.5 from Theorem 2.4 is a standard application of the Helffer-Sjöstrand calculus; see e.g. Section 7.1 of [17] for a similar argument.
An interesting effect of the sparsity of the entries of H is that its eigenvalues follow, for large N, the deterministic law \(\widetilde{\rho }\) that depends on the sparsity parameter q. While this law approaches the standard semicircle law \(\rho _{sc}\) in the limit \(N\rightarrow \infty \), its deterministic refinement to the standard semicircular law for finite N accounts for the non-optimality at the edge of results obtained in [15], i.e. when (2.20) is compared with (2.23) below.
Proposition 2.6
(Local semicircle law, Theorem 2.8 of [15]) Suppose that H satisfies Assumption 2.3 with \(\phi >0\). Then, the following estimates hold uniformly for \(z \in {\mathcal E}\):
where \(m_\mathrm {sc}\) denote the Stieltjes transform of the standard semicircle law, and
where \(\varkappa \equiv \varkappa (z)\mathrel {\mathop :}=|E-2|\), \(z=E+\mathrm {i}\eta \).
We remark that the estimate (2.23) is essentially optimal as long as the spectral parameter z stays away from the spectral edges, e.g. for energies in the bulk \(E\in [-2+\delta ,2-\delta ]\), \(\delta >0\). For the individual Green function entries, \(G_{ij}\), we believe that the estimate (2.24) is already essentially optimal (\(m_\mathrm {sc}\) therein may be replaced by \(\widetilde{m}\) without changing the error bound). A consequence of Proposition 2.6 is that all eigenvectors of H are completely delocalized.
Proposition 2.7
(Theorem 2.16 and Remark 2.18 in [15]) Suppose that H satisfies Assumption 2.3 with \(\phi >0\). Denote by \(({\varvec{u}}_i^H)\) the \(\ell ^2\)-normalized eigenvectors of H. Then,
Using (2.23) as a priori input it was proved in [29] that the local eigenvalue statistics in the bulk agree with the local statistics of the GOE, for \(\phi >0\); see also [16] for \(\phi >1/3\). When combined with a high moment estimates of H (see Lemma 4.3 in [15]), the estimate in (2.23) implies the following bound on the operator norm of H.
Proposition 2.8
(Lemma 4.4 of [15]) Suppose that H satisfies Assumption 2.3 with \(\phi >0\). Then,
The following estimates of the operator norm of H sharpens the estimates of Proposition 2.8 by including the deterministic refinement to the semicircle law as expressed by Theorem 2.4.
Theorem 2.9
Suppose that H satisfies Assumption 2.3 with \(\phi >0\). Then,
where \(\pm L\) are the endpoints of the support of the measure \(\widetilde{\rho }\) given by
Here and above, we restricted the choice of the sparsity parameter q to the range \(N^\phi \le q\le N^{1/2}\) for arbitrary small \(\phi >0\). Yet, pushing our estimates and formalism we expect also to cover the range \((\log N)^{A_0\log \log N}\le q\le N^{1/2}\), \(A_0\ge 30\), considered in [15]. In fact, Khorunzhiy showed for diluted Wigner matrices (cf. Sect. 2.1.2) that \(\Vert H\Vert \) converges almost surely to 2 for \(q\ggg (\log N)^{1/2}\), while \(\Vert H\Vert \) diverges for \(q\lll (\log N)^{1/2}\); see Theorem 2.1 and Theorem 2.2 of [32] for precise statements.
As noted in Theorem 2.9, the local law allows strong statements on the locations of the extremal eigenvalues of H. We next discuss implications for the fluctuations of the rescaled extremal eigenvalues.
2.3.2 Tracy–Widom limit of the extremal eigenvalues
Let W be a real symmetric Wigner matrix and denote by \(\lambda _1^{W}\) its largest eigenvalue. The edge universality for Wigner matrices asserts that
where \(F_1\) is the Tracy–Widom distribution function [49, 50] for the GOE. Statement (2.29) holds true for the smallest eigenvalue \(\lambda _N^{W}\) as well. We henceforth focus on the largest eigenvalues, the smallest eigenvalues can be dealt with in exactly the same way.
The universality of the Tracy–Widom laws for Wigner matrices was first proved in [45, 46] for real symmetric and complex Hermitian ensembles with symmetric distributions. The symmetry assumption on the entries’ distribution was partially removed in [42, 43]. Edge universality without any symmetry assumption was proved in [48] under the condition that the distribution of the matrix elements has subexponential decay and its first three moments match those of the Gaussian distribution, i.e. the third moment of the entries vanish. The vanishing third moment condition was removed in [23]. A necessary and sufficient condition on the entries’ distribution for the edge universality of Wigner matrices was given in [35].
Our second main result shows that the fluctuations of the rescaled largest eigenvalue of the sparse matrix ensemble are governed by the Tracy–Widom law, if the sparsity parameter q satisfies \(q\ggg N^{1/6}\).
Theorem 2.10
Suppose that H satisfies Assumption 2.3 with \(\phi >1/6\). Denote by \(\lambda ^H_1\) the largest eigenvalue of H. Then,
where L denotes the upper-edge of the deterministic measure \(\widetilde{\rho }\) given in (2.28).
The convergence result (2.30) was obtained in Theorem 2.7 of [16] under the assumption that the sparsity parameter q satisfies \(q\ggg N^{1/3}\), i.e. \(\phi > 1/3\) (and with 2 replacing L).
In the regime \(N^{1/6}\lll q\le N^{1/3}\), the deterministic shift of the upper edge by \(L-2=O(q^{-2})\) is essential for (2.30) to hold since then \(q^{-2} \ge N^{-2/3}\), the latter being the scale of the Tracy–Widom fluctuations. In other words, to observe the Tracy–Widom fluctuations in the regime \(N^{1/6}\lll q\le N^{1/3}\) corrections from the fourth moment of the matrix entries’ distribution have to be accounted for. This is in accordance with high order moment computations for diluted Wigner matrices in [33].
It is expected that the order of the fluctuations of the largest eigenvalue exceeds \(N^{-2/3}\) if \(q \lll N^{1/6}\). The heuristic reasoning is that, in this regime, the fluctuations of the eigenvalues in the bulk of the spectrum are much larger than \(N^{-2/3}\) and hence affect the fluctuations of the eigenvalues at the edges. Indeed, the linear eigenvalue statistics of sparse random matrices were studied in [6, 44]. For any sufficiently smooth function \(\varphi \), it was shown there that
converges to a centered Gaussian random variable with variance of order one. This suggests that the fluctuations of an individual eigenvalue in the bulk are of order \(N^{-1/2} q^{-1}\), which is far greater than the Tracy–Widom scale \(N^{-2/3}\) if \(q \lll N^{1/6}\). Moreover, comparing the size of the fluctuations with the estimate in (2.23), this also suggests that, for large \(\eta \), \(\mathbb {E}m^H(z)\) is not well approximated by \(m_{sc}(z)\).
Remark 2.11
Theorem 2.10 can be extended to correlation functions of extreme eigenvalues as follows: For any fixed k, the joint distribution function of the first k rescaled eigenvalues converges to that of the GOE, i.e. if we denote by \(\lambda _1^{\mathrm {GOE}} \ge \lambda _2^{\mathrm {GOE}} \ge \cdots \ge \lambda _N^{\mathrm {GOE}}\) the eigenvalues of a GOE matrix independent of H, then
We further mention that all our results also hold for complex Hermitian sparse random matrices with the GUE Tracy–Widom law describing the limiting edge fluctuations.
2.3.3 Applications to the adjacency matrix of the Erdős–Rényi graph
We briefly return to the adjacency matrix A of the Erdős–Rényi graph ensemble introduced in Sect. 2.1.1. Since the entries of A are not centered, the largest eigenvalue \(\lambda _1^A\) is an outlier well-separated from the other eigenvalues. Recalling the definition of the matrix \(\widetilde{A}\) whose entries are centered, we notice that
with \(f\mathrel {\mathop :}=q(1-q^2/N)^{-1/2}\), \(a\mathrel {\mathop :}=f/N\) and \(\varvec{e}\mathrel {\mathop :}=N^{-1/2} (1,1,\ldots ,1)^\mathrm {T}\in \mathbb {R}^N\). (Here, \(| \varvec{e} \rangle \langle \varvec{e} |\) denotes the orthogonal projection onto \(\varvec{e}\).) The expected degree d and the sparsity parameter q are linked by
Applying a simple rank-one perturbation formula and shifting the spectrum by a we get from Theorem 2.4 the following corollary whose proof we leave aside.
Corollary 2.12
Fix \(\phi >0\). Let A satisfies (2.2) and (2.3) with expected degree \(N^{2\phi }\le d\le N^{1-2\phi }\). Then the normalized trace \(m^A\) of the Green function of A satisfies
uniformly on the domain \({\mathcal E}\), where \(z=E+\mathrm {i}\eta \) and \(a=\frac{q}{N}{(1-q^2/N)^{-1/2}}\).
Let \(\lambda ^A_1\ge \lambda ^A_2\ge \cdots \ge \lambda ^A_N\) denote the eigenvalues of A. The behavior of the largest eigenvalue \(\lambda _1^A\) was fully determined in [15], where it was shown that it has Gaussian fluctuations, i.e.
in distribution as \(N\rightarrow \infty \), with \(\mathbb {E}\lambda _1^A=f-a+\frac{1}{f}+O({q^{-3}})\); see Theorem 6.2 in [15].
Combining Theorem 2.10 with the reasoning of Section 6 of [16], we have the following corollary on the behavior of the second largest eigenvalue \(\lambda _2^A\) of the adjacency matrix A.
Corollary 2.13
Fix \(\phi >1/6\) and \(\phi '>0\). Let A satisfies (2.2) and (2.3) with expected degree \(N^{2\phi }\le d\le N^{1-2\phi '}\). Then,
where \(L=2+d^{-1}+O(d^{-2})\) is the upper edge of the measure \(\widetilde{\rho }\); see (2.28).
We skip the proof of Corollary 2.13 from Theorem 2.10, since it is essentially the same as the proof of Theorem 2.7 in [16], where the result was obtained for \(\phi >1/3\), with L replaced by 2. In analogy with Remark 2.11, the convergence result in (2.35) extends in an obvious way to the eigenvalues \(\lambda _{k-1}^A,\ldots ,\lambda _2^A\), for any fixed k. The analogous results apply to the k-smallest eigenvalues of A. We leave the details to the interested reader.
Remark 2.14
The largest eigenvalues of sparse random matrices, especially the (shifted and normalized) adjacency matrices of the Erdős–Rényi graphs, can be used to determine the number of clusters in automated community detection algorithms [7, 40] in stochastic block models. Corollary 2.13 suggests that the test statistics for such algorithms should reflect the shift of the largest eigenvalues if \(N^{-2/3} \lll p \lll N^{-1/3}\), or equivalently, \(N^{1/3} \lll d \lll N^{2/3}\). If \(p \lll N^{-2/3}\), the test based on the edge universality of random matrices may fail as we have discussed after Theorem 2.10.
In applications, the sparsity should be taken into consideration due to small N even if p is reasonably large. For example, in the Erdős–Rényi graph with \(N \simeq 10^3\), the deterministic shift is noticeable if \(p \simeq 0.1\), which is in the vicinity of the parameters used in numerical experiments in [7, 40].
3 Strategy and outline of proofs
In this section, we outline the strategy of our proofs. We begin with the local law of Theorem 2.4.
3.1 Local laws for the Green function
We start by recalling the approach to the local law for Wigner matrices initiated in [18,19,20]. Using Schur’s complement (or the Feshbach formula) and large deviation estimates for quadratic forms by Hanson and Wright [27], one shows that the normalized trace \(m^W(z)\) approximately satisfies the equation \(1+zm^W(z)+m^W(z)^2\simeq 0\), with high probability, for any z in some appropriate subdomain of \(\mathcal {E}\). Using that \(m_\mathrm {sc}\) satisfies (2.11), a local stability analysis then yields \(|m^W(z)-m_{sc}(z)|\prec (N\eta )^{-1/2}\), \(z\in \mathcal {E}\). In fact, the same quadratic equation is approximately satisfied by each diagonal element of the resolvent, \(G_{ii}^W\), and not only by their average \(m^W\). This observation and an extension of the stability analysis to vectors instead of scalars then yields the entry-wise local law [21,22,23], \(|G_{ij}^W(z)-\delta _{ij} m_\mathrm {sc}(z)|\prec (N\eta )^{-1/2}\), \(z\in \mathcal {E}\). (See Sect. 3.2 for some details of this argument.) Taking the normalized trace of the Green function, one expects further cancellations of fluctuations to improve the bound. Exploring the fluctuation averaging mechanism for \(m^W\) and refining the local stability analysis, one obtains the strong local law up to the spectral edges [23], \(|m^W(z)-m_\mathrm {sc}(z)|\prec (N\eta )^{-1}\), \(z\in \mathcal {E}\). The fluctuation averaging mechanism was first introduced in [22] and substantially extended in [14, 17] to generalized Wigner matrices. We refer to [13, 17] for reviews of this general approach. Parallel results were obtained in [47, 48]. For further improvements of convergence rates we refer to [9] and [25, 26]. For other recent developments on local laws we refer e.g. to [2, 3, 10] for Wigner matrices with general variance profile and Wigner matrices with correlated entries, and to [8] for heavy-trailed Wigner matrices. Local laws for additive random matrix models were studied in [4, 30, 34, 36, 39]. Entry-wise local laws for random graphs models other than the Erdős-Rényi graph were obtained in [5] for d-regular graphs and in [1] for random graphs with given expected degree sequences.
The strategy outlined in the preceding paragraph was applied to sparse random matrices in [15]. The sparsity of the entries manifests itself in the large deviation estimate for quadratic forms, e.g. letting \((H_{ij})\) satisfy (2.12) and choosing \((B_{ij})\) to be any deterministic \(N\times N\) matrix, Lemma 3.8 of [15] assures that
for all \(i\in \llbracket 1,N\rrbracket \). Using the above ideas the entry-wise local law in (2.24) was obtained in [15]. Exploiting the fluctuation averaging mechanism for the normalized trace of the Green function, an additional power of \(q^{-1}\) can be gained, leading to (2.23) with the deteriorating factor \((\varkappa +\eta )^{-1/2}\).
To establish the local law for the normalized trace of the Green function which does not deteriorate at the edges, we propose in this paper a novel recursive moment estimate for the Green function. When applied to the proof of the strong local law for a Wigner matrix, it is estimating \(\mathbb {E}|1+zm^W(z)+m^W(z)^2|^D\) by the lower moments \(\mathbb {E}|1+zm^W(z)+m^W(z)^2|^{D-l}\), \(l\ge 1\). The use of recursive moment estimate has three main advantages over the previous fluctuation averaging arguments: (1) it is more convenient in conjunction with the cumulant expansion in Lemma 3.2, (2) it is easier to track the higher order terms involving the fourth and higher moments if needed, and (3) it does not require to fully expand the higher power terms and thus simplifies bookkeeping and combinatorics. The same strategy can also be applied to individual entries of the Green function by establishing a recursive moment estimate for \(\mathbb {E}|1+zG^W_{ii}(z)+m_\mathrm {sc}(z) G^W_{ii}(z)|^D\) and \(\mathbb {E}|G_{ij}|^D\) leading to the entry-wise local law. We remark that estimating \(\mathbb {E}|1+zm^W(z)+m^W(z)^2|^D\) is also a key step for the proof of the local law in [25, 26], where the estimate is obtained by establishing a bound for the diagonal resolvent entries instead of directly relating it with the lower moments \(\mathbb {E}|1+zm^W(z)+m^W(z)^2|^{D-l}\).
We illustrate this approach for the simple case of the GOE next.
3.2 Local law for the GOE
Choose W to be a GOE matrix. Since \(m_\mathrm {sc}\equiv m_\mathrm {sc}(z)\), the Stieltjes transform of the semicircle law, satisfies \(1+zm_\mathrm {sc}+m_\mathrm {sc}^2=0\), we expect that the moments of the polynomial \(1+zm(z)+m(z)^2\), with \(m\equiv m^W(z)\) the normalized trace of the Green function \(G\equiv G^W(z)\), are small. We introduce the subdomain \({\mathcal D}\) of \({\mathcal E}\) by setting
We are going to derive the following recursive moment estimate for m. For any \(D\ge 2\),
for \(z\in \mathbb {C}^+\). Fix now \(z\in {\mathcal D}\). Using Young’s inequality, the second order Taylor expansion of m(z) around \(m_\mathrm {sc}(z)\) and the a priori estimate \(|m(z)-m_\mathrm {sc}(z)|\prec 1\), we conclude from (3.3) with Markov’s inequality that
where \(\alpha _{\mathrm {sc}}(z)\mathrel {\mathop :}=z+2m_\mathrm {sc}(z)\); see Sect. 5.1 for a similar computation. An elementary computation reveals that \(|\alpha _{\mathrm {sc}}(z)| \asymp {{\mathrm{\mathrm {Im}}}}m_\mathrm {sc}(z)\). Equation (3.4) is a self-consistent equation for the quantity \(m(z)-m_\mathrm {sc}(z)\). Its local stability properties up to the edges were examined [21, 22]. From these stability properties and (3.4) it follows that, for fixed \(z\in {\mathcal D}\), \(|m(z)-m_\mathrm {sc}(z)|\prec 1\) implies \(|m(z)-m_\mathrm {sc}(z)|\prec \frac{1}{N\eta }\). To obtain the local law on all of \({\mathcal D}\) one applies a continuity or bootstrapping argument [18, 21, 22] by decreasing the imaginary part of the spectral parameter from \(\eta \asymp 1\) to \(\eta \ge N^{-1}\). Using the monotonicity of the Stieltjes transform, this conclusion is extended to all of \({\mathcal E}\). This establishes that
uniformly on the domain \(\mathcal {E}\), which is the local law for the GOE.
Hence, to obtain the strong local law for the GOE, it suffices to establish (3.3) for fixed \(z\in {\mathcal D}\). By the definition of the normalized trace, \(m\equiv m(z)\), of the Green function we have
We expand the diagonal Green function entry \(G_{ii}\equiv G^W_{ii}\) using the following identity:
which follows directly from the defining relation \((W-z\mathrm {I})G = \mathrm {I}\). To some extent (3.7) replaces the conventional Schur complement formula. We then obtain from (3.6) and (3.7) that
Using that the matrix entries \(W_{ik}\) are Gaussian random variables, integration by parts shows that
for differentiable functions \(F\,:\,\mathbb {R}\rightarrow \mathbb {C}\), where \(\partial _{ik} \equiv \partial /(\partial W_{ik})\). Here we used that \(\mathbb {E}W_{ij}=0\) and \(\mathbb {E}W^2_{ij}=(1+\delta _{ij})/N\) for the GOE. Identity (3.9) is often called Stein’s lemma in the statistics literature. Combining (3.8) and (3.9) we obtain
We next expand and estimate the first term on the right side of (3.10). It is easy to see that
After averaging over the indices i and k, the first term on the right side of (3.11) becomes
which exactly cancels with the second term on the right side of (3.10). The second term on the right side of (3.11) can be estimated as
where we used the identity
which we refer to as the Ward identity below. It follows from the spectral decomposition of W.
For the third term on the right side of (3.11) we have that
where we used that
The fourth term on the right side of (3.11) can be estimated in a similar manner since
Returning to (3.10), we hence find, for \(z\in \mathbb {C}^+\), that
which is the recursive moment estimate for the GOE stated in (3.3).
Remark 3.1
The above presented method can also be used to obtain the entry-wise local law for the Green function of the GOE. Assuming the local law for \(m^W\) has been obtained, one may establish a recursive moment estimate for \(1+zG^W_{ii}(z)+m_\mathrm {sc}(z) G^W_{ii}(z)\) to derive that
uniformly in \(i\in \llbracket 1,N\rrbracket \) and \(z\in \mathcal {E}\). (One may also consider high moments of \(1+z G^W_{ii}(z)+m^W(z)W_{ii}(z)\) to arrive at the same conclusion.) We leave the details to the reader. Yet, for later illustrative purposes in Sects. 6 and 7, we sketch the derivation of the recursive moment estimate for the off-diagonal Green function entries \(G_{ij}\equiv G^W_{ij}(z)\), \(i\not =j\). Let \(z\in {\mathcal D}\). Using the relation \((W-z\mathrm {I})G = \mathrm {I}\), we get
where we used Stein’s lemma in (3.9) in the last step. Upon computing the derivative we get, for \(i\not =j\),
The first term in the right side of (3.17) equals \(-\mathbb {E}\big [m|G_{ij}|^{2D} \big ]\). Using the Ward identity (3.12) we have
Thus using (3.16), we get the bound
uniformly on \(\mathcal {E}\). We can now easily bound the right side of (3.17), e.g. for any given \(\epsilon > 0\),
for N sufficiently large. Thus we get from (3.17) that, for \(i\not =j\),
uniformly on \(\mathcal {E}\), for N sufficiently large. Since \(|z+m_\mathrm {sc}| > c\) on \(\mathcal {E}\), for some N-independent constant \(c>0\), we find from (3.18) and (3.5) by Young’s and Markov’s inequality that
for fixed \(z\in \mathcal {D}\). Using continuity and monotonicity of \(G_{ij}(z)\) the bound can be made uniform on the domain \(\mathcal {E}\). Together with (3.16), this shows the entry-wise local law for the GOE.
3.3 Local law for sparse matrices
When applying the strategy of Sect. 3.2 to sparse matrices we face two difficulties. First, since the matrix entries are not Gaussian random variables, the simple integration by parts formula (3.9) needs to replaced by a full-fletched cumulant expansion. Second, since the higher order cumulants are not small (in the sense that the \((\ell +2)\)-nd cumulant is only \(O(N^{-1} q^{-\ell })\)) we need to retain higher orders in the cumulant expansion. The following result generalizes (3.9).
Lemma 3.2
(Cumulant expansion, generalized Stein lemma) Fix \(\ell \in \mathbb {N}\) and let \(F\in C^{\ell +1}(\mathbb {R};\mathbb {C}^+)\). Let Y be a centered random variable with finite moments to order \(\ell +2\). Then,
where \(\mathbb {E}\) denotes the expectation with respect to Y, \(\kappa ^{(r+1)}(Y)\) denotes the \((r+1)\)-th cumulant of Y and \(F^{(r)}\) denotes the r-th derivative of the function F. The error term \(\Omega _\ell (YF(Y))\) in (3.20) satisfies
where \(Q\ge 0\) is an arbitrary fixed cutoff and \(C_\ell \) satisfies \(C_\ell \le \frac{(C\ell )^\ell }{\ell !}\) for some numerical constant C.
In case Y is a standard Gaussian we recover (3.20) and we thus sometimes refer to Lemma 3.2 as generalized Stein lemma. The cumulant expansion in combination with the Green function was successfully used to study linear eigenvalue statistics of Wigner matrices in [31, 41] and more recently in [28]. For a proof of Lemma 3.2, we refer to Proposition 3.1 in [41].
Let H be a sparse matrix satisfying Assumption 2.3 with \(\phi >0\). Recall the polynomial \(P\equiv P_z\) and the function \(\widetilde{m}\equiv \widetilde{m}(z)\) of Theorem 2.4 which satisfy \(P(\widetilde{m})=0\). Let \(m\equiv m^{H}(z)\) be given by (2.16). Following the ideas of Sect. 3.2, we derive in Sect. 6 a recursive estimate for \(\mathbb {E}|P(m)|^{2D}\), for large D with \(z\in \mathcal {D}\); see Lemma 5.1 for the precise statement and Sect. 6 for its proof. We start with
for \(D\ge 2 \), and expand zm using the identity
which follows from the definition of the Green function. We then obtain the identity
Using the generalized Stein lemma, Lemma 3.2, we get
where \(\partial _{ik} = \partial /(\partial H_{ik})\) and \(\kappa ^{(k)}\) are the cumulants of \(H_{ij}\), \(i\not =j\). The detailed form of the error \(\mathbb {E}\Omega _\ell (\cdot )\) is discussed in Sect. 6.1. Anticipating the outcome, we mention that we can truncate the expansion at order \(\ell \ge 8D\) so that the error term becomes sufficiently small for our purposes.
In Sect. 6, we prove that the relevant terms in (3.25) cancel with the third and fourth term on the right side of (3.22). This yields the recursive moment estimate for P(m), respectively m. The proof is based on the following observations:
-
The leading term on the right side of (3.25) comes from the \(r=1\) term and is identified to be \(\mathbb {E}[-m^2 P^{D-1}\overline{P^D}]\), which cancels with the third term on the right side of (3.22). See the first part of Lemma 6.4 and Sect. 6.3, where the estimate is obtained by following the discussion in Sect. 3.2.
-
The second largest term on the right side of (3.25) comes from the \(r=3\) term and is identified to be \(s^{(4)}q^{-2}\mathbb {E}[-N^{-2} (\sum _i (G_{ii})^2)^2 P^{D-1}\overline{P^D}]\). In Lemma 6.12 we prove that it cancels with the fourth term on the right side of (3.22) after removing some negligible contributions (see the beginning of Sect. 6 for quantitative statement of negligible contributions). See the second part of Lemma 6.4 and Sect. 6.7.
-
The \(r=2\) term is negligible. The main ideas in this estimate are the identification of an unmatched index and the cumulant expansion of the term in the unmatched index. See Sect. 6.4.
-
The negligible contributions, including the terms with \(r \ge 4\), can be identified by power counting or further expansions using cumulant series. For this analysis, we rely on the entry-wise local law (2.24) and ideas inspired by the GOE computation in Remark 3.1 above.
As we will see in Sect. 4, the inclusion of the fourth moment \(s^{(4)}/q^2\) in P(m) enables us to compute the deterministic shift of edge which is of order \(q^{-2}\). While it is possible to include a higher order correction term involving the sixth moment, \(s^{(6)}/q^4\), this would not improve the local law in our proof since the largest among the negligible contributions originates from the \(r=3\) term in (3.25). (More precisely, it is \(I_{3, 2}\) of (6.6).)
In Sect. 5, we then prove Theorem 2.4 and Theorem 2.9 using the recursive moment estimate for m and a local stability analysis. The local stability analysis relies on some properties of the Stieltjes transform \(\widetilde{m}\) of the deterministic distribution \(\widetilde{\rho }\) obtained in Sect. 4.
3.4 Tracy–Widom limit and Green function comparison
To establish the edge universality (for \(\phi >1/6\)), we first show in Sect. 7.1 that the distribution of the largest eigenvalue of H may be obtained as the expectation (of smooth functions) of the imaginary part of m(z), for appropriately chosen spectral parameters z. Such a relation was the basic structure for proving the edge universality in [16, 23], and the main ingredients in the argument are the local law, the square-root decay at the edge of the limiting density, and an upper bound on the largest eigenvalue, which are Theorems 2.4, 2.9 and Lemma 4.1 for the case at hand. For the sake of completeness, we redo some parts of these estimates in Sect. 7.1.
In Sect. 7.2, we then use the Green function comparison method [21, 23] to compare the edge statistics of H with the edge statistics of a GOE matrix. Together with the argument of Sect. 7.1, this will yield the Tracy–Widom limit of the largest eigenvalue. However, the conventional discrete Lindeberg type replacement approach to the Green function comparison does not work due to the slow decaying moments of the sparse matrix. We therefore use a continuous flow that interpolates between the sparse matrix ensemble and the GOE. Such an approach has shown to be effective in proving edge universality for deformed Wigner matrices [37] and for sample covariance matrices [38].
More concretely, we consider the Dyson matrix flow with initial condition \(H_0\) defined by
where \(W^{\mathrm {GOE}}\) is a GOE matrix independent of \(H_0\). In fact, since we will choose \(H_0\) to be a sparse matrix H with vanishing diagonal entries, we assume with some abuse of terminology that \(W^{\mathrm {GOE}}=(W^{\mathrm {GOE}})^*\) has vanishing diagonal, i.e. we assume that \(W^{\mathrm {GOE}}_{ii}=0\) and that \((W^{\mathrm {GOE}}_{ij},i<j)\) are independent centered Gaussian random variables of variance 1 / N. It was shown in Lemma 3.5 of [35] that the local edge statistics of \(W^{\mathrm {GOE}}\) is described by the GOE Tracy–Widom statistics.
Let \(\kappa _t^{(k)}\) be the k-th cumulant of \((H_t)_{ij}\), \(i\not =j\). Then, by the linearity of the cumulants under the addition of independent random variables, we have \(\kappa _t^{(1)} =0\), \(\kappa _t^{(2)}=1/N\) and \(\kappa _t^{(k)}=\mathrm {e}^{-kt/2}\kappa ^{(k)}\), \(k\ge 3\). In particular, we have the bound
where we introduced the time-dependent sparsity parameter
Choosing \(t=6\log N\), a straightforward perturbation argument shows that the local statistics, at the edges and in the bulk, of \(H_t\) and \(W^{\mathrm {GOE}}\) agree up to negligible error. It thus suffices to consider \(t\in [0,6\log N]\).
We first establish the local law for the normalized trace of the Green function of \(H_t\). Let
Proposition 3.3
Let \(H_0\) satisfy Assumption 2.3 with \(\phi >0\). Then, for any \(t\ge 0\), there exists an algebraic function \(\widetilde{m}_t : \mathbb {C}^+ \rightarrow \mathbb {C}^+\) and \(2\le L_t<3\) such that the following holds:
-
(1)
\(\widetilde{m}_t\) is the Stieltjes transform of a deterministic symmetric probability measure \(\widetilde{\rho }_t\), i.e. \(\widetilde{m}_t(z)=m_{\widetilde{\rho }_t}(z)\). Moreover, \({{\mathrm{supp}}}\widetilde{\rho }_t=[-L_t,L_t]\) and \(\widetilde{\rho }_t\) is absolutely continuous with respect to the Lebesgue measure with a strictly positive density on \((-L_t,L_t)\).
-
(2)
\(\widetilde{m}_t \equiv \widetilde{m}_t (z)\) is a solution to the polynomial equation
$$\begin{aligned} \begin{aligned} P_{t, z}(\widetilde{m}_t)&\mathrel {\mathop :}=1 + z \widetilde{m}_t + \widetilde{m}_t^2 + \mathrm {e}^{-t} q_t^{-2} s^{(4)}\widetilde{m}_t^4 \\&=1 + z \widetilde{m}_t + \widetilde{m}_t^2 + \mathrm {e}^{-2t} q^{-2}s^{(4)}\widetilde{m}_t^4 = 0\,. \end{aligned}\end{aligned}$$(3.30) -
(3)
The normalized trace of the Green function satisfies
$$\begin{aligned} |m_t (z) - \widetilde{m}_t (z)| \prec \frac{1}{q_t^2}+\frac{1}{N\eta }\,, \end{aligned}$$(3.31)uniformly on the domain \({\mathcal E}\) and uniformly in \(t\in [0,6\log N]\).
Note that Theorem 2.4 is a special case of Proposition 3.3. Given Proposition 3.3, Corollary 2.5 extends in the obvious way from H to \(H_t\). Proposition 3.3 is proved in Sect. 5.1.
The endpoints \(\pm L_t\) of the support of \(\widetilde{\rho }_t\) are given by \(L_t = 2 + \mathrm {e}^{-t} s^{(4)}q_t^{-2} + O(\mathrm {e}^{-2t}q_t^{-4})\) and satisfy
where \(\dot{L}_t\) denotes the derivative with respect to t of \(L_t\); cf. Remark 4.2 below.
Choose now \(q\ge N^{\phi }\) with \(\phi >1/6\). In our proof of the Green function comparison theorem, Proposition 7.2, we estimate the rate of change of \(m_t\) along the Dyson matrix flow over the time interval \([0,6\log N]\), where it undergoes a change of o(1). The continuous changes in \(m_t\) can be compensated by letting evolve the spectral parameter \(z\equiv z(t)\) according to (3.32). This type of cancellation argument appeared first in [37] in the context of deformed Wigner matrices. However, one cannot prove the Green function comparison theorem for sparse random matrices by directly applying the cancellation argument since the error bound for the entry-wise local law in Proposition 2.6 is not sufficiently small. Thus the proof of the Green function comparison theorem requires some non-trivial estimates on functions of Green functions as is explained in Sect. 7.2.
4 The measure \(\widetilde{\rho }\) and its Stieltjes transform
In this section, we prove important properties of \(\widetilde{m}_t\equiv \widetilde{m}_t(z) \) introduced in Proposition 3.3. Recall that \(\widetilde{m}_t\) is a solution to the polynomial equation \(P_{t,z}(\widetilde{m}_t) = 0\) in (3.30) and that we set \(q_t\mathrel {\mathop :}=\mathrm {e}^{t/2}q\). Recall moreover the domain \(\mathcal {E}\) defined in (2.18).
Lemma 4.1
For any fixed \(z = E + \mathrm {i}\eta \in {\mathcal E}\) and any \(t\ge 0\), the polynomial equation \(P_{t,z}(w_t) = 0\) has a unique solution \(w_t \equiv w_t(z)\) satisfying \({{\mathrm{\mathrm {Im}}}}w_t > 0\) and \(|w_t| \le 5\). Moreover, \(w_t\) has the following properties:
-
(1)
There exists a probability measure \(\widetilde{\rho }_t\) such that the analytic continuation of \(w_t\) coincides with the Stieltjes transform of \(\widetilde{\rho }_t\).
-
(2)
The probability measure \(\widetilde{\rho }_t\) is supported on \([-L_t, L_t]\), for some \(L_t \ge 2\), it has a strictly positive density inside its support and it vanishes as a square-root at the edges, i.e. letting
$$\begin{aligned} \varkappa _t\equiv \varkappa _t(E) \mathrel {\mathop :}=\min \{ |E+L_t|, |E-L_t| \}\,, \end{aligned}$$(4.1)we have
$$\begin{aligned} \widetilde{\rho }_t(E) \asymp \varkappa _t^{1/2}(E)\,,\qquad \qquad \quad (E\in (-L_t,L_t))\,. \end{aligned}$$(4.2)Moreover, \(L_t = 2 + \mathrm {e}^{-t} q_t^{-2}s^{(4)}+ O(\mathrm {e}^{-2t}q_t^{-4})\).
-
(3)
The solution \(w_t\) satisfies
$$\begin{aligned} \begin{aligned} {{\mathrm{\mathrm {Im}}}}w_t(E+\mathrm {i}\eta )&\asymp \sqrt{\varkappa _t + \eta } \qquad \text { if }\quad E \in [-L_t, L_t]\,, \\ {{\mathrm{\mathrm {Im}}}}w_t(E+\mathrm {i}\eta )&\asymp \frac{\eta }{\sqrt{\varkappa _t + \eta }} \qquad \! \text { if }\quad E \notin [-L_t, L_t]\,. \end{aligned} \end{aligned}$$(4.3)
Proof
For simplicity, we abbreviate \(P \equiv P_{t, z}\). Let
By definition, \(P(w) = 0\) if and only if \(z = Q(w)\). It is easily checked that the derivative
is monotone increasing on \((-\infty , 0)\). Furthermore, we have
Hence, \(Q'(w) = 0\) has a unique solution on \((-1, -1+ 2 q_t^{-2}s^{(4)})\), which we will denote by \(\tau _t\), and \(Q(w) \equiv Q_t(w_t)\) attains its minimum on \((-\infty , 0)\) at \(w_t = \tau _t\). We let \(L_t\mathrel {\mathop :}=Q_t(\tau _t)\), or equivalently, \(w_t=\tau _t\) if \(z=L_t\). A direct calculation shows that
For simplicity we let \(L \equiv L_t\) and \(\tau \equiv \tau _t\). Choosing now \(w = Q^{-1}(z)\) we have the expansion
in a \(q_t^{-1/2}\)-neighborhood of \(\tau _t\). We hence find that
in that neighborhood. In particular, choosing the branch of the square root so that \(\sqrt{z-L} \in \mathbb {C}^+\), we find that \({{\mathrm{\mathrm {Im}}}}w > 0\) since \(Q''(\tau )>0\). We note that there exists another solution with negative imaginary part, which corresponds to the different branch of the square root.
We can apply the same argument with a solution of the equation \(Q'(w) = 0\) on \((1-2q_t^{-2}s^{(4)}, 1)\) leading us to the relation
in a \(q_t^{1/2}\)-neighborhood of \(-\tau \).
For uniqueness, we consider the disk \(B_5 = \{ w \in \mathbb {C}: |w| < 5 \}\). On its boundary \(\partial B_5\)
for \(z \in {\mathcal E}\). Hence, by Rouché’s theorem, the equation \(P(w) = 0\) has the same number of roots as the quadratic equation \(w^2 + zw + 1 = 0\) in \(B_5\). Since \(w^2 + zw + 1 = 0\) has two solutions on \(B_5\), we find that \(P(w) = 0\) has two solutions on it. For \(z = L + \mathrm {i}q_t^{-1}\), we can easily check that one solution of \(P(w) = 0\) has positive imaginary part (from choosing the branch of the square root as in (4.8)) and the other solution has negative imaginary part. If both solutions of \(P(w) = 0\) are in \(\mathbb {C}^+ \cup \mathbb {R}\) (or in \(\mathbb {C}^- \cup \mathbb {R}\)) for some \(z = \widetilde{z} \in \mathbb {C}^+\), then by continuity, there exists \(z'\) on the line segment joining \(L + \mathrm {i}q_t^{-1}\) and \(\widetilde{z}\) such that \(P_{t, z'}(w') = 0\) for some \(w' \in \mathbb {R}\). By the definition of P this cannot happen, hence one solution of \(P(w)=0\) is in \(\mathbb {C}^+\) and the other in \(\mathbb {C}^-\), for any \(z \in {\mathcal E}\). This shows the uniqueness statement of the lemma.
Next, we extend \(w \equiv w(z)\) to cover \(z \notin {\mathcal E}\). (With slight abuse of notation, the extension of w will also be denoted by \(w \equiv w(z)\).) Repeating the argument of the previous paragraph, we find that \(P(w) = 0\) has two solutions for \(z \in (-L, L)\). Furthermore, we can also check that exactly one of them is in \(\mathbb {C}^+\) by considering \(z = \pm L \mp q_t^{-1}\) and using continuity. Thus, w(z) forms a curve on \(\mathbb {C}^+\), joining \(-\tau \) and \(\tau \), which we will denote by \(\Gamma \). We remark that, by the inverse function theorem, w(z) is analytic for \(z \in (-L, L)\) since \(Q'(w) \ne 0\) for such z.
By symmetry, \(\Gamma \) intersects the imaginary axis at w(0). On the imaginary axis, we find that
Thus, we get from \(Q(w)=z\) that
which shows in particular that \(w(\mathrm {i})\) is purely imaginary and \({{\mathrm{\mathrm {Im}}}}w(\mathrm {i}) < {{\mathrm{\mathrm {Im}}}}w(0)\). By continuity, this shows that the analytic continuation of w(z) for \(z \in \mathbb {C}^+\) is contained in the domain \(D_{\Gamma }\) enclosed by \(\Gamma \) and the interval \([-L, L]\). We also find that \(|w(z)| < 5\), for all \(z \in \mathbb {C}^+\).
To prove that w(z) is analytic in \(\mathbb {C}^+\), it suffices to show that \(Q'(w) \ne 0\) for \(w \in D_{\Gamma }\). If \(Q'(w) = 0\) for \(w \in D_{\Gamma }\), we have
On the circle \(\{ w \in \mathbb {C}: |w| = 5 \}\),
Hence, again by Rouché’s theorem, \(w^2 Q'(w) = 0\) has two solutions in the disk \(\{ w \in \mathbb {C}: |w| < 5 \}\). We already know that those two solutions are \(\pm \tau \). Thus, \(Q'(w) \ne 0\) for \(w \in D_{\Gamma }\) and w(z) is analytic.
Let \(\widetilde{\rho }\) be the measure obtained by the Stieltjes inversion of \(w \equiv w(z)\). To show that \(\widetilde{\rho }\) is a probability measure, it suffices to show that \(\lim _{y \rightarrow \infty } \mathrm {i}y \, w(\mathrm {i}y) = -1\). Since w is bounded, one can easily check from the definition of w that \(|w| \rightarrow 0\) as \(|z| \rightarrow \infty \). Thus, by considering \(z = \mathrm {i}y\) and letting \(y \rightarrow \infty \),
which implies that \(\lim _{y \rightarrow \infty } \mathrm {i}y \, w(\mathrm {i}y) = -1\). This proves the first property of \(\widetilde{\rho }\). Other properties can be easily proved from the first property and Eqs. (4.8) and (4.9). \(\square \)
Remark 4.2
Recall that \(q_t=\mathrm {e}^{t/2}q\). As we have seen in (4.6),
Moreover, the time derivative of \(L_t\) satisfies
hence, referring once more to (4.6),
Remark 4.3
It can be easily seen from the definition of \(P_{t, z}\) that \(w_t \rightarrow m_\mathrm {sc}\) as \(N \rightarrow \infty \) or \(t \rightarrow \infty \). For \(z \in {\mathcal E}\), we can also check the stability condition \(|z+w_t| > 1/6\) since
and \(|w_t|<5\), as we have seen in the proof of Lemma 4.1.
5 Proof of Proposition 3.3 and Theorem 2.9
5.1 Proof of Proposition 3.3
In this section, we prove Proposition 3.3. The main ingredient of the proof is the recursive moment estimate for \(P(m_t)\). Recall the subdomain \({\mathcal D}\) of \({\mathcal E}\) defined in (3.2) and the matrix \(H_t\), \(t\ge 0\), defined in (3.26). We have the following result.
Lemma 5.1
(Recursive moment estimate) Fix \(\phi >0\) and suppose that \(H_0\) satisfies Assumption 2.3. Fix any \(t\ge 0\). Recall the definition of the polynomial \(P\equiv P_{t,z}\) in (3.30). Then, for any \(D > 10\) and (small) \(\epsilon > 0\), the normalized trace of the Green function, \(m_t \equiv m_t (z)\), of the matrix \(H_t\) satisfies
uniformly on the domain \({\mathcal D}\), for N sufficiently large.
The proof of Lemma 5.1 is postponed to Sect. 6. We are now ready to prove Proposition 3.3.
Proof of Proposition 3.3 and Theorem 2.4
Let \(\widetilde{m}_t\) be the solution \(w_t\) in Lemma 4.1. Parts (1) and (2) of Proposition 3.3 and Theorem 2.4 were already proved in Lemma 4.1, so it suffices to prove the statements (3). For simplicity we omit the z-dependence from the notation. Fix \(t\in [0,6\log N]\). Let
We remark that from the local law in Lemma 2.6, we have \(\Lambda _t\prec 1\), \(z\in {\mathcal D}\). We also define the following z-dependent deterministic parameters
with \(z=E+\mathrm {i}\eta \). We note that
since \(|\widetilde{m}_t| \le 5\) as proved in Lemma 4.1. Recall that \(\widetilde{m}_t(L) = \tau \) in the proof of Lemma 4.1. Recalling the definition of \(\varkappa _t\equiv \varkappa _t(E)\) in (4.1) and using (4.8) we have
By the definitions of \(\tau \) and L in the proof of Lemma 4.1, we also have that
hence
and we find from (4.8) that
We remark that the parameter \(\alpha _1\) is needed only for the proof of Theorem 2.9; the proof of Proposition 3.3 can be done simply by substituting every \(\alpha _1\) below with \(|\alpha _2|\).
Recall that, for any \(a, b \ge 0\) and \({\textsf {p}},{\textsf {q}} > 1\) with \({\textsf {p}}^{-1} + {\textsf {q}}^{-1} = 1\), Young’s inequality states
Let \(D\ge 10\) and choose any (small) \(\epsilon >0\). All estimates below hold for N sufficiently large (depending on D and \(\epsilon \)). For brevity, N is henceforth implicitly assumed to be sufficiently large. Using first that \({{\mathrm{\mathrm {Im}}}}m_t \le {{\mathrm{\mathrm {Im}}}}\widetilde{m}_t + |m_t -\widetilde{m}_t| = \alpha _1 + \Lambda _t \) and then applying (5.5) with \({\textsf {p}}=2D\) and \({\textsf {q}} =2D/(2D-1)\), we get for the first term on the right side of (5.1) that
Similarly, for the second term on the right side of (5.1), we have
From the Taylor expansion of \(P'(m_t)\) around \(\widetilde{m}_t\), we have
and \(|P'(m_t)| \le |\alpha _2| + 3\Lambda _t\), for all \(z\in \mathcal {D}\), with high probability since \(P''(\widetilde{m}_t) = 2 + O(q_t^{-2})\) and \(\Lambda _t\prec 1\) by assumption. We note that, for any fixed \(s\ge 2\),
with high probability, uniformly on \({\mathcal D}\), since \(\alpha _1 \le |\alpha _2| \le C\) and \(\Lambda _t \prec 1\). In the third term on the right side of (5.1), note that \(2s-s'-2 \ge s\) since \(s' \le s-2\). Hence, for \(2 \le s \le 2D\),
uniformly on \(\mathcal {D}\) with high probability. For the last term on the right side of (5.1), we note that
uniformly on \({\mathcal D}\). Thus, similar to (5.9) we find that, for \(2 \le s \le 2D\),
for all \(z\in {\mathcal D}\), with high probability. We hence have from (5.1), (5.6), (5.7), (5.9) and (5.11) that
for all \(z\in \mathcal {D}\). Note that the last term on the right side can be absorbed into the left side. Thence
uniformly on \(\mathcal {D}\), where we used that \(D> 10\) and the inequality
for any \(a, b \ge 0\) and \({\textsf {p}} \ge 1\), to get the second line.
Next, from the third order Taylor expansion of \(P(m_t)\) around \(\widetilde{m}_t\), we have
since \(P(\widetilde{m}_t) = 0\) and \(P'''(\widetilde{m}_t)=4!\mathrm {e}^{-t} q_t^{-2}s^{(4)}\widetilde{m}_t\). Thus, using \(\Lambda _t\prec 1\) and \(P''(\widetilde{m}_t)=2+O(q_t^{-2}) \) we get
Taking the 2D-power of the inequality and using once more (5.14), we get after taking the expectation
Replacing from (5.13) for \(\mathbb {E}[ |P(m_t)|^{2D}]\) we obtain, using that \(4^{2D}\le N^{\epsilon /2}\), for N sufficiently large,
uniformly on \(\mathcal {D}\). Applying the Schwarz inequality to the first term and the third term on the right, absorbing the terms \(o(1)\mathbb {E}[\Lambda _t^{4D}]\) into the left side and using \(q_t^{-2}\le \beta \) in the fourth term, we get
uniformly on \(\mathcal {D}\). Feeding (5.19) back into (5.13) we get, for any \(D\ge 10\) and (small) \(\epsilon >0\),
uniformly on \(\mathcal {D}\), for N sufficiently large, where we used the Schwarz inequality and once more (5.19) to get the second line.
By Markov’s inequality, we therefore obtain from (5.20) that for fixed \(z\in \mathcal {D}\), \(|P(m_t)|\prec |\alpha _2|\beta +\beta ^2 + q_t^{-1} |\alpha _2|^2\). It then follows from the Taylor expansion of \(P(m_t)\) around \(\widetilde{m}_t\) in (5.15) that
for each fixed \(z\in \mathcal {D}\), where we used that \(q_t^{-2}\le \beta \). To get a uniform bound on \(\mathcal {D}\), we choose \(18 N^{8}\) lattice points \(z_1, z_2,\ldots , z_{18 N^{8}}\) in \({\mathcal D}\) such that, for any \(\widetilde{z} \in {\mathcal D}\), there exists \(z_n\) satisfying \(|\widetilde{z}-z_n| \le N^{-4}\). Since
and since a similar estimate holds for \(|\widetilde{m}_t(\widetilde{z}) - \widetilde{m}_t(z)|\), a union bound yields that (5.21) holds uniformly on \(\mathcal {D}\) with high probability. In particular, for any (small) \(\epsilon >0\) and (large) D there is an event \(\widetilde{\Xi }\) with \(\mathbb {P}(\widetilde{\Xi })\ge 1-N^D\) such that, for all \(z\in \mathcal {D}\),
on \(\widetilde{\Xi }\), for N sufficiently large.
Recall next that there is a constant \(C_0>1\) such that \(C_0^{-1}\sqrt{\varkappa _t(E)+\eta }\le |\alpha _2|\le C_0 \sqrt{\varkappa _t(E)+\eta }\), where we can choose \(C_0\) uniform in \(z\in \mathcal {E}\). Note further that \(\beta =\beta (E+\mathrm {i}\eta )\) is for fixed E a decreasing function of \(\eta \) while \(\sqrt{\kappa _t(E)+\eta }\) is increasing. Thus, there exists \(\widetilde{\eta }_0 \equiv \widetilde{\eta }_0(E)\) such that \(\sqrt{\varkappa (E)+ \widetilde{\eta }_0} = C_0 q_t \beta (E+\mathrm {i}\widetilde{\eta }_0)\). We then consider the subdomain \(\widetilde{\mathcal D}\subset {\mathcal D}\) defined by
On \(\widetilde{\mathcal D}\), \(\beta \le q_t^{-1} |\alpha _2|\), hence we obtain from the estimate (5.22) that
and thus
uniformly on \(\widetilde{\mathcal D}\) on \(\widetilde{\Xi }\). Hence, we get on \(\widetilde{\Xi }\) that either
Note that any \(z \in {\mathcal E}\) with \(\eta ={{\mathrm{\mathrm {Im}}}}z = 3\) is in \(\widetilde{\mathcal D}\). When \(\eta =3\), we easily see that
for sufficiently large N. In particular we have that either \(3/4\le \Lambda _t\) or \(\Lambda _t\le 6N^{\epsilon } q_t^{-1}|\alpha _2|\) on \(\widetilde{\Xi }\) for \(\eta =3\). Moreover, since \(m_t\) and \(\widetilde{m}_t\) are Stieltjes transforms, we have
We conclude that, for \(\eta =3\), the second possibility, \(\Lambda _t\le 6N^{\epsilon } q_t^{-1}|\alpha _2|\) holds on \(\widetilde{\Xi }\). Since \(6N^{\epsilon } q_t^{-1} \lll 1\) on \(\widetilde{\mathcal D}\), in particular \(6N^{\epsilon } q_t^{-1}|\alpha _2| < |\alpha _2|/8\), we find from (5.24) by continuity that
holds on the event \(\widetilde{\Xi }\). Putting the estimate (5.25) back into (5.13), we find that
for any (small) \(\epsilon >0\) and (large) D, uniformly on \(\widetilde{\mathcal D}\). Note that, for \(z \in {\mathcal D}\backslash \widetilde{\mathcal D}\), the estimate \(\mathbb {E}[|P(m_t)|^{2D}] \le N^{6D\epsilon } \beta ^{2D} |\alpha _2|^{2D} + N^{6D \epsilon } \beta ^{4D}\) can be directly checked from (5.20). Considering lattice points \(\{ z_i \} \subset {\mathcal D}\) again, a union bound yields for any (small) \(\epsilon >0\) and (large) D there is an event \(\Xi \) with \(\mathbb {P}(\Xi )\ge 1-N^D\) such that
on \(\Xi \), uniformly on \({\mathcal D}\) for N sufficiently large.
Next, recall that \(\beta =\beta (E+\mathrm {i}\eta )\) is for fixed E a decreasing function of \(\eta \) while \(\sqrt{\varkappa _t(E)+\eta }\) is increasing. Thus there is \(\eta _0\equiv \eta _0(E)\) such that \(\sqrt{\varkappa (E)+\eta _0}=10C_0N^\epsilon \beta (E+\mathrm {i}\eta _0)\). Further notice that \(\eta _0(E)\) is a continuous function. We consider the three subdomains of \({\mathcal E}\) defined by
Note that \({\mathcal E}_1\cup {\mathcal E}_2\subset \mathcal {D}\). We split the stability analysis of (5.27) according to whether \(z\in {\mathcal E}_1\), \({\mathcal E}_2\) or \({\mathcal E}_3\). Case 1 If \(z\in {\mathcal E}_1\), we note that \(|\alpha _2|\le C_0 \sqrt{\varkappa (E)+\eta }\le 10C_0^2 N^\epsilon \beta (E+\mathrm {i}\eta )\). We then obtain from (5.27) that
on \(\Xi \). Thus,
on \(\Xi \), for some finite constant C.
Case 2 If \(z\in {\mathcal E}_2\), we obtain from (5.27) that
on \(\Xi \). We then note that \(C_0|\alpha _2|\ge \sqrt{\varkappa _t(E)+\eta }\ge 10C_0 N^\epsilon \beta \), i.e. \(N^\epsilon \beta \le |\alpha _2|/10\), so that
on \(\Xi \), where we used that \(N^\epsilon \beta \le 1\). Hence, we get on \(\Xi \) that either
We follow the dichotomy argument and the continuity argument that were used to obtain (5.25). Since \(3N^{\epsilon } \beta \le |\alpha _2|/8\) on \({\mathcal E}_2\), we find by continuity that
holds on the event \(\Xi \). Case 3 For \(z\in {\mathcal E}_3={\mathcal E}\backslash ({\mathcal E}_1\cup {\mathcal E}_2)\) we use that \(|m'_t(z)|\le {{\mathrm{\mathrm {Im}}}}m_t(z)/{{\mathrm{\mathrm {Im}}}}z\), \(z\in \mathbb {C}^+\), since \(m_t\) is a Stieltjes transform. Set now \( \widetilde{\eta }\mathrel {\mathop :}=10N^{-1+\epsilon }\). By the fundamental theorem of calculus we can estimate
Using that \(s\rightarrow s {{\mathrm{\mathrm {Im}}}}m_t(E+\mathrm {i}s)\) is a monotone increasing function as is easily checked from the definition of the Stieltjes transform, we find that
for some C where we used \(\widetilde{\eta }\mathrel {\mathop :}=10N^{-1+\epsilon }\) to get the second line. Noticing that \(z=E+\mathrm {i}\widetilde{\eta }\in {\mathcal E}_1\cup {\mathcal E}_2\), we have on the event \(\Xi \) that \(\Lambda _t(E+\mathrm {i}\widetilde{\eta })\le CN^\epsilon \beta (E+\mathrm {i}\widetilde{\eta }) \le C\) as follows from (5.28) and (5.32). Using moreover that \(\widetilde{m}_t\) is uniformly bounded by a constant on \({\mathcal E}\), we then get that, on the event \(\Xi \),
Combining (5.28), (5.32) and (5.34), and recalling the definition of the event \(\Xi \), we get \(\Lambda _t\prec \beta \), uniformly on \({\mathcal E}\) for fixed \(t\in [0,6\log N]\). Choosing \(t=0\), we have completed the proof of Theorem 2.4. To extend this bound to all \(t\in [0,6\log N]\), we use the continuity of the Dyson matrix flow. Choosing a lattice \(\mathcal {L}\subset [0,6\log N]\) with spacings of order \(N^{-3}\), we get \(\Lambda _t\prec \beta \), uniformly on \(\mathcal {E}\) and on \(\mathcal {L}\), by a union bound. By continuity we extend the conclusion to all of \([0,6\log N]\). This proves Proposition 3.3.
5.2 Proof of Theorem 2.9
We start with an upper bound on the largest eigenvalue \(\lambda _1^{H_t}\) of \(H_t\).
Lemma 5.2
Let \(H_0\) satisfy Assumption 2.3 with \(\phi >0\). Let \(L_t\) be deterministic number defined in Lemma 4.1. Then,
uniformly in \(t\in [0,6\log N]\).
Proof
To prove Lemma 5.2 we follow the strategy of the proof of Lemma 4.4 in [15]. Fix \(t\in [0,6\log N]\). Recall first from (5.3) the deterministic z-dependent parameters
We mostly drop the z-dependence for brevity. We further introduce the z-independent quantity
Fix a small \(\epsilon > 0\) and define the domain \({\mathcal D}_{\epsilon }\) by
where \(\varkappa _t\equiv \varkappa _t(E)=E-L_t\). Note that on \({\mathcal D}_\epsilon \),
In particular we have \(N^\epsilon \widetilde{\beta }\le (N\eta )^{-1}\), hence \(N^\epsilon q_t^{-2}\le C (N\eta )^{-1}\) so that \(q_t^{-2}\) is negligible when compared to \((N\eta )^{-1}\) and \(\beta \) on \({\mathcal D}_\epsilon \). Note moreover that
In particular we have \(\alpha _1\lll |\alpha _2|\) on \({\mathcal D}_\epsilon \).
We next claim that
with high probability on the domain \({\mathcal D}_{\epsilon }\).
Since \({\mathcal D}_{\epsilon } \subset {\mathcal E}\), we find from Proposition 3.3 that \(\Lambda _t \le N^{\epsilon '}\beta \) for any \(\epsilon ' > 0\) with high probability. Fix \(0< \epsilon ' < \epsilon /7 \). From (5.13), we get
for N sufficiently large, where we used that \(\Lambda _t\le N^{\epsilon '}\beta \lll N^\epsilon \beta \) with high probability and, by (5.39), \(\alpha _1\lll |\alpha _2|\), \(|\alpha _2|\le CN^\epsilon \beta \) on \({\mathcal D}_\epsilon \). Applying (2D)-th order Markov inequality and a simple lattice argument combined with a union bound, we get \(|P(m_t)| \le CN^{3\epsilon '} \beta ^{2}\) uniformly on \({\mathcal D}_\epsilon \) with high probability. From the Taylor expansion of \(P(m_t)\) around \(\widetilde{m}_t\) in (5.15), we then get that
uniformly on \({\mathcal D}_\epsilon \) with high probability, where we also used that \(\Lambda _t\lll 1\) on \({\mathcal D}_\epsilon \) with high probability.
Since \(\Lambda _t\le N^{\epsilon '} \beta \le C N^{\epsilon '-\epsilon }|\alpha _2|\) with high probability on \({\mathcal D}_\epsilon \), we have \(|\alpha _2|\Lambda _t\ge CN^{\epsilon -\epsilon '}\Lambda _t^2\ggg 2\Lambda _t^2\). Thus the first term on the right side of (5.40) can be absorbed into the left side and we conclude that
must hold with high probability on \({\mathcal D}_\epsilon \). Hence, using that \(0<\epsilon '<\epsilon /7\), we obtain that
with high probability on \({\mathcal D}_\epsilon \). This proves the claim that \(\Lambda _t \lll (N\eta )^{-1}\) on \({\mathcal D}_{\epsilon }\) with high probability. Moreover, this also shows that
on \({\mathcal D}_{\epsilon }\) with high probability, where we used (5.39).
Now we prove the estimate (5.35). If \(\lambda _1^{H_t} \in [E-\eta , E+\eta ]\) for some \(E \in [L_t +N^\epsilon (q_t^{-4}+ N^{-2/3}), L_t +{q}^{-1/3}]\) with \(z = E + \mathrm {i}\eta \in {\mathcal D}_{\epsilon }\),
which contradicts the high probability bound \({{\mathrm{\mathrm {Im}}}}m_t \lll (N\eta )^{-1}\) in (5.41). The size of each interval \([E-\eta , E+\eta ]\) is at least \(N^{{ -1+\epsilon }} q_t^{1/6}\). Thus, considering O(N) such intervals, we can conclude that \(\lambda _1 \notin [L + N^\epsilon (q_t^{-4}+N^{-2/3}), L +q_t^{-1/3}]\) with high probability. From Proposition 2.8, we find that \(\lambda _1^{H_t}-L_t\prec q_t^{-1/3}\) with high probability, hence we conclude that (5.35) holds, for fixed \(t\in [0,6\log N]\). Using a lattice argument and the continuity of the Dyson matrix flow, we easily obtain (5.35) uniformly in \(t\in [0,6\log N]\). \(\square \)
We are now well-prepared for the proof of Theorem 2.9. It follows immediately from the next result.
Lemma 5.3
Let \(H_0\) satisfy Assumption (2.3) with \(\phi >0\). Then, uniformly in \(t\in [0,6\log N]\),
Proof of Lemma 5.3 and Theorem 2.9
Fix \(t\in [0,\log N]\). Consider the largest eigenvalue \(\lambda _1^{H_t}\). In Proposition 5.2, we already showed that \((L_t-\lambda _1^{H_t})_-\prec q_t^{-4}+N^{-2/3}\). It thus suffices to consider \((L_t-\lambda _1^{H_t})_+\). By Lemma 4.1 there is \(c>0\) such that \( c(L_t-\lambda _1^{H_t})_+^{3/2}\le n_{\widetilde{\rho }_t}(\lambda _1^{H_t},L_t)\), with \(n_{\widetilde{\rho }_t}\) as in (2.21). Hence, by Corollary 2.5 (and its obvious generalization to \(H_t\)), we have the estimate
so that \((L_t-\lambda _1^{H_t})_+\prec q_t^{-4}+N^{-2/3}\). Thus \(|\lambda _1^{H_t}-L_t|\prec q_t^{-4}+N^{-2/3}\). Similarly, one shows the estimate \(|\lambda _N^{H_t}+L_t|\prec q_t^{-4}+N^{-2/3}\) for the smallest eigenvalue \(\lambda _N^{H_t}\). This proves (5.43) for fixed \(t\in [0,6\log N]\). Uniformity follows easily from the continuity of the Dyson matrix flow. \(\square \)
6 Recursive moment estimate: Proof of Lemma 5.1
In this section, we prove Lemma 5.1. Recall the definitions of the Green functions \(G_t\) and \(m_t\) in (3.29). We fix \(t\in [0,6\log N]\) throughout this section, and we will omit t from the notation in the matrix \(H_t\), its matrix elements and its Green functions. Given a (small) \(\epsilon >0\), we introduce the z-dependent control parameter \(\Phi _\epsilon \equiv \Phi _\epsilon (z)\) by setting
Recall the domain \(\mathcal {D}\) defined in (3.2). Lemma 5.1 states that, for any (small) \(\epsilon >0\),
for N sufficiently large. We say that a random variable Z is negligible if \(|\mathbb {E}[Z]| \le C \Phi _\epsilon \) for some N-independent constant C.
To prove the recursive moment estimate of Lemma 5.1, we return to (3.25) which reads
where \(\partial _{ik} = \partial /(\partial H_{ik})\) and \(\kappa _t^{(k)}\) are the cumulants of \((H_t)_{ij}\), \(i\not =j\). The detailed form of the error \(\mathbb {E}\Omega _\ell (\cdot )\) is discussed in Sect. 6.1.
It is convenient to condense the notation a bit. Abbreviate
We rewrite the cumulant expansion (6.3) as
where we set
(By convention \(\partial _{ik}^0G_{ki}=G_{ki} \).) The weights \( w_{I_{r,s}}\) are combinatoric coefficient given by
Returning to (3.22), we have in this condensed form the expansion
6.1 Truncation of the cumulant expansion
In this subsection, we bound the error term \(\mathbb {E}\Omega _\ell (I)\) in (6.8) for large \(\ell \). We need some more notation. Let \(E^{[ik]}\) denote the \(N\times N\) matrix determined by
For each pair of indices (i, k), we define the matrix \(H^{(ik)}\) from H through the decomposition
With this notation we have the following estimate.
Lemma 6.1
Suppose that H satisfies Assumption 2.3 with \(\phi >0\). Let \(i,k\in \llbracket 1,N\rrbracket \), \(D\in \mathbb {N}\) and \(z\in {\mathcal D}\). Define the function \(F_{ki}\) by
where \(G\equiv G^H(z)\) and \(P\equiv P(m^H(z))\). Choose an arbitrary \(\ell \in \mathbb {N}\). Then, for any (small) \(\epsilon >0\),
uniformly \(z\in \mathcal {D}\), for N sufficiently large. Here \(\partial _{ik}^\ell \) denotes the partial derivative \(\frac{\partial ^\ell }{\partial H_{ik}^\ell }\).
Proof
Fix two pairs of indices (a, b) and (i, k). From the definition of the Green function and (6.10) we easily get
where we omit the z-dependence. Letting \(\Lambda _o^{H^{(ik)}}\mathrel {\mathop :}=\displaystyle {\max _{a,b}}| G^{H^{(ik)}}_{ab}|\) and \(\Lambda _o^{H}\mathrel {\mathop :}=\displaystyle {\max _{a,b}}| G^{H}_{ab}|\), we get
since \(|H_{ik}|\prec q_t^{-1}\) by (2.12). Moreover by (2.24) we have \(\Lambda _o^{H}\prec 1\), uniformly in \(z\in \mathcal {D}\). It thus follows that \(\Lambda _o^{H^{(ik)}}\prec \Lambda _o^{H}\prec 1\), uniformly in \(z\in \mathcal {D}\), where we used (2.24). Similarly, for \(x\in \mathbb {R}\), we have
and we get
uniformly in \(z\in \mathcal {D}\), where we used once more (2.24).
Recall that P is a polynomial of degree 4 in m. Then \(F_{ki}\) is a multivariate polynomial of degree \(4(2D-1)+1\) in the Green function entries and the normalized trace m whose number of member terms is bounded by \(4^{2D-1}\). Hence \(\partial _{{ik}}^{\ell }F_{ki}\) is a multivariate polynomial of degree \(4(2D-1)+1+\ell \) whose number of member terms is roughly bounded by \(4^{2D-1}\times (4(2D-1)+1+2l)^l\). Next, to control the individual monomials in \(\partial _{{ik}}^{\ell }F_{ki}\), we apply (6.13) to each factor of Green function entries (at most \(4(2D-1)+1+\ell \) times). Thus, altogether we obtain
for any small \(\epsilon '>0\) and sufficiently large N. Choosing \(\epsilon '=\epsilon /(2(8D+\ell ))\) with get (6.12). \(\square \)
Recall that we set \(I= {(1+zm)} P(m)^{D-1}\overline{P(m)^D}\) in (6.4). To control the error term \(\mathbb {E}\Omega _{\ell }(I)\) in (6.8), we use the following result.
Corollary 6.2
Let \(\mathbb {E}\Omega _\ell (I)\) be as in (6.8). With the assumptions and notation of Lemma 6.1, we have, for any (small) \(\epsilon >0\),
uniformly in \(z\in \mathcal {D}\), for N sufficiently large. In particular, the error \(\mathbb {E}\Omega _{\ell }(I)\) is negligible for \(\ell \ge 8D\).
Proof
First, fix a pair of indices (k, i), \(k\not =i\). Recall the definition of \(F_{ik}\) in (6.11). Denoting \(\mathbb {E}_{ik}\) the partial expectation with respect to \(H_{ik}\), we have from Lemma 3.2, with \(Q=q_t^{-1/2}\), that
with \(C_\ell \le (C\ell )^\ell /\ell !\), for some numeral constant C. To control the full expectation of the first term on the right side, we use the moment assumption (2.12) and Lemma 6.1 to conclude that, for any \(\epsilon >0\),
for N sufficiently large. To control the second term on the right side of (6.16), we use the operator norm of G(z) is deterministically bounded by \(\eta ^{-1}\), i.e. \(\Vert G(z)\Vert \le \eta ^{-1}\), to conclude that
cf. the paragraph above (6.14). On the other hand, we have from Hölder’s inequality and the moment assumptions in (2.12) that, for any \(D'\in \mathbb {N}\),
for N sufficiently large. Using that \(q_t\ge N^\phi \) by (2.3), we hence obtain, for any \(D'\in \mathbb {N}\),
uniformly on \(\mathbb {C}^+\), for N sufficiently large.
Next, summing over i, k and choosing \(D'\ge \ell \) sufficiently large in (6.17) we obtain, for any \(\epsilon >0\)
uniformly on \(\mathcal {D}\), for N sufficiently large. This proves (6.15). \(\square \)
Remark 6.3
We will also consider slight generalizations of the cumulant expansion in (6.3). Let \(i,j,k\in \llbracket 1,N\rrbracket \). Let \(n\in \mathbb {N}_0\) and choose indices \(a_1,\dots , a_{n},\) \( b_1,\ldots , b_n\in \llbracket 1,N\rrbracket \). Let \(D\in \mathbb {N}\) and choose \(s_1,s_2,s_3,s_4\in \llbracket 0,D\rrbracket \). Fix \(z\in \mathcal {D}\). Define the function \(F_{ki}\) by setting
It is then straightforward to check that we have the cumulant expansion
where the error \(\mathbb {E}\Omega _\ell (\cdot )\) satisfies the same bound as in (6.15). This follows easily by extending Lemma 6.1 and Corollary 6.2.
6.2 Truncated cumulant expansion
Armed with the estimates on \(\mathbb {E}\Omega _\ell (\cdot )\) of the previous subsection, we now turn to the main terms on the right side of (6.8). In the remainder of this section we derive the following result from which Lemma 5.1 follows directly. Recall the definition of \(\Phi _\epsilon \) in (6.1).
Lemma 6.4
Fix \(D\ge 2\) and \(\ell \ge 8D\). Let \(I_{r,s}\) be given by (6.6). Then we have, for any (small) \(\epsilon >0\),
uniformly in \(z\in \mathcal {D}\), for N sufficiently large. Moreover, we have, for any (small) \(\epsilon >0\),
uniformly in \(z\in \mathcal {D}\), for N sufficiently large.
Proof of Lemma 5.1
By the definition of \(\Phi _\epsilon \) in (6.1) (with a sufficiently large (small) \(\epsilon >0\)), it suffices to show that \(\mathbb {E}[ |P|^{2D}(z) ] \le \Phi _\epsilon (z)\), for all \(z\in \mathcal {D}\), for N sufficiently large. Choosing \(\ell \ge 8D\), Corollary 6.2 asserts that \(\mathbb {E}\Omega _\ell (I)\) in (6.8) is negligible. By Lemma 6.4 the only non-negligible terms in the expansion of the first term on the right side of (6.8) are \(w_{I_{1,0}}\mathbb {E}I_{1,0}\) and \(w_{I_{3,0}}\mathbb {E}I_{3,0}\), yet these two terms cancel with the middle term on the right side of (6.8), up to negligible terms. Thus the whole right side of (6.8) is negligible. This proves Lemma 5.1. \(\square \)
We now choose an initial (small) \(\epsilon >0\). Below we use the factor \(N^\epsilon \) to absorb numerical constants in the estimates by allowing \(\epsilon \) to increase by a tiny amount from line to line. We often drop z from the notation; it is always understood that \(z\in \mathcal {D}\) and all estimates are uniform on \(\mathcal {D}\). The proof of Lemma 6.4 is done in the remaining Sects. 6.3–6.7 where \((\mathbb {E}I_{r,s})\) are controlled.
6.3 Estimate on \(I_{1, s}\)
Starting from the definition of \(I_{1,0}\) in (6.5), a direct computation yields
The middle term on the last line is negligible since
where we used \(|G_{i_1i_1}|\prec 1\), and so is the third term since
where we used Lemma 6.5. We thus obtain from (6.22) that
for N sufficiently large. This proves the first estimate in (6.20).
Consider next \(I_{1,1}\). Similar to (3.11), we have
Here the fresh summation index \(i_3\) originated from \(\partial _{i_1i_2}P(m)=P'(m)\frac{1}{N}{\sum _{i_3=1}^N}\partial _{i_1i_2}G_{i_3i_3}\). Note that we can add the terms with \(i_1=i_2\) at the expense of a negligible error, so that
In the remainder of this section, we will freely include or exclude negligible terms with coinciding indices. Using (3.14) and (3.15) we obtain from (6.24) the estimate
for N sufficiently large. This proves (6.21) for \(r=s=1\).
6.4 Estimate on \(I_{2,0}\)
We start with a lemma that is used in the power counting arguments below.
Lemma 6.5
For any \(i,k\in \llbracket 1,N\rrbracket \),
Moreover, for fixed \(n\in \mathbb {N}\),
Proof
Let \(\lambda _1^{H_t} \ge \lambda _2^{H_t} \ge \dots \ge \lambda _N^{H_t}\) be the eigenvalues of \(H_t\), and let \({\varvec{u}}_1,\ldots ,{\varvec{u}}_N\), \({\varvec{u}}_{\alpha }\equiv {\varvec{u}}_{\alpha }^{H_t}\), denoted the associated normalized eigenvectors. Then, by spectral decomposition, we get
Since the eigenvectors are delocalized by Proposition 2.7, we find that
This proves the first inequality in (6.26) for \(i=k\). The inequality for \(i\not =k\), the second inequality in (6.26) and (6.27) then follow directly from the Schwarz inequality. \(\square \)
Recalling the definition of \(I_{r,s}\) in (6.6) we have
We then notice that \(I_{2,0}\) contains terms with one or three off-diagonal Green function entries \(G_{i_1i_2}\). We split accordingly
where \(I_{2,0}^{(1)}\) contains all terms with one off-diagonal Green function entries (and, necessarily, two diagonal Green function entries) and where \(I_{2,0}^{(3)}\) contains all terms with three off-diagonal Green function entries (and zero diagonal Green function entries), and \(w_{I_{2,0}^{(1)}}\), \(w_{I_{2,0}^{(3)}} \) denote the respective weights. Explicitly,
and \(w_{I_{2,0}}=1\), \(w_{I_{2,0}^{(1)}}=3\), \(w_{I_{2,0}^{(3)}}=1\).
We first note that \(I_{2,0}^{(3)}\) satisfies, for N sufficiently large,
Remark 6.6
(Power counting I) Consider the terms \(I_{r,0}\), \(r\ge 1\). For \(n\ge 1\), we then split
according to the parity of r. For example, for \(r=1\), \(I_{1,0}=I_{1,0}^{(0)}+I_{1,0}^{(2)}\), with
cf. (6.22). Now, using a simple power counting, we bound the summands in (6.31) as follows. First, we note that each term in \(I_{r,0}\) contains a factor of \(q_t^{(r-2)_+}\). Second, for \(\mathbb {E}I_{2n,0}^{(2l+1)}\) and \(\mathbb {E}I_{2n-1,0}^{(2l)}\), with \(n\ge 1\), \(l\ge 1\), we can by Lemma 6.5 extract one factor of \(\frac{{{\mathrm{\mathrm {Im}}}}m}{N\eta }\) (other Green function entries are bounded using \(|G_{ik}|\prec 1\)). Thus, for \(n\ge 1\), \(l\ge 1\),
for N sufficiently large, and we conclude that all these terms are negligible.
We next consider \(\mathbb {E}I_{2,0}^{(1)}\) which is not covered by (6.32). Using \(|G_{ii}|\prec 1\) and Lemma 6.5 we get
for N sufficiently large. Yet, this bound is not negligible. We need to gain an additional factor of \(q_t^{-1}\) with which it will become negligible. We have the following result.
Lemma 6.7
For any (small) \(\epsilon >0\), we have, for all \(z\in \mathcal {D}\),
for N sufficiently large. In particular, the term \(\mathbb {E}I_{2,0}\) is negligible.
Proof
Fix a (small) \(\epsilon >0\). Recalling (6.29), we have
The key feature here is that the Green function entries are \( G_{i_2i_1}G_{i_1i_1}G_{i_2i_2}\), where at least one index, say \(i_2\), appears an odd number of times. We say that the index \(i_2\) is unmatched. Using the resolvent formula (3.23) we expand in the unmatched index \(i_2\) to get
We now proceed in a similar way as in Remark 3.1 where we estimated \(|G^W_{i_1i_2}|\), \(i_1\not =i_2\), for the GOE. Applying the cumulant expansion to the right side of (6.36), we will show that the leading term is \(-\mathbb {E}[ mI_{2,0}^{(1)}]\). Then, upon substituting m(z) by the deterministic quantity \(\widetilde{m}(z)\) and showing that all other terms in the cumulant expansion of the right side of (6.36) are negligible, we will get that
for N sufficiently large. Since \(|z+\widetilde{m}(z) |\ge 1/6\) uniformly on \(\mathcal {E}\supset \mathcal {D}\), as shown in Remark 4.3, the lemma directly follows. The main efforts in the proof go into showing that the sub-leading terms in the cumulant expansion of the right side of (6.36) are indeed negligible.
For simplicity we abbreviate \(\widehat{I}\equiv I_{2,0}^{(1)}\). Then using Lemma 3.2 and Remark 6.3, we have, for arbitrary \(\ell '\in \mathbb {N}\), the cumulant expansion
with
and \(w_{\widehat{I}_{r',s'}}=\frac{1}{r'!}\left( {\begin{array}{c}r'\\ s'\end{array}}\right) \). Here, we used Corollary 6.2 to truncate the series in (6.38) at order \(\ell '\). Choosing \(\ell '\ge 8D\) the remainder is indeed negligible.
We first focus on \(\widehat{I}_{r',0}\). For \(r'=1\), we compute
where we organized the terms according to the off-diagonal Green functions entries. By Lemma 6.5,
Recall \(\widetilde{m} \equiv \widetilde{m}_t(z)\) defined in Proposition 3.3. We rewrite \(\widehat{I}_{1,0}^{(1)}\) with \(\widetilde{m}\) as
By the Schwarz inequality and the high probability bounds \(|G_{kk}|, |G_{ii}| \le N^{\epsilon /8}\), for N sufficiently large, the second term in (6.42) is bounded as
We thus get from (6.40), (6.41), (6.42) and (6.43) that
where we used (6.35). We remark that in the expansion of \(\mathbb {E}\widehat{I}=\mathbb {E}I_{2,0}^{(1)}\) the only term with one off-diagonal entry is \(\mathbb {E}\widehat{I}_{2,0}^{(1)}\). All the other terms contain at least two off-diagonal entries.
Remark 6.8
(Power counting II) Comparing (6.5) and (6.38), we have \(\widehat{I}_{r',s'}=(I_{2,0}^{(1)})_{r',s'}\). Consider now the terms with \(s'=0\). As in (6.31) we organize the terms according to the number of off-diagonal Green function entries. For \(r'\ge 2\),
A simple power counting as in Remark 6.6 then directly yields
for N sufficiently large. Here, we used that each term contains a factor \(\kappa _t^{(3)}\kappa _t^{(r'+1)}\le CN^{-2}q_t^{-r'}\). We conclude that all terms in (6.46) with \(r'\ge 2\) are negligible. Yet we remark that \(|\mathbb {E}\widehat{I}_{2,0}^{(1)}|\) is the leading error term in \(|\mathbb {E}I_{2,0}^{(1)}|\), which is explicitly listed on the right side of (6.34).
Remark 6.9
(Power counting III) Consider the terms \(\widehat{I}_{r',s'}\), with \(1\le s'\le r'\). For \(s'=1\), note that \(\partial _{i_2i_3} \big (P^{D-1} \overline{P^D}\big )\) contains two off-diagonal Green function entries. Explicitly,
where the fresh summation index \(i_4\) is generated from \(\partial _{i_2i_3} P\). Using Lemma 6.5 we get, for \(r'\ge 1\),
for N sufficiently large, where we used that \(\partial _{i_2i_3}^{r'-1}(G_{i_3i_1}G_{i_2i_2}G_{i_1i_1})\), \(r'\ge 1\), contains at least one off-diagonal Green function entry.
For \(2\le s'\le r'\), we first note that, for N sufficiently large,
Next, since \(s'\ge 2\), the partial derivative in \(\partial _{i_2i_3}^{s'} \big ( P^{D-1} \overline{P^D}\big )\) acts on P and \(\overline{P}\) (and on their derivatives) more than once. For example, for \(s'=2\),
where \(\partial _{i_1i_2}\) acted twice on P, respectively \(P'\), to produce the first two terms. More generally, for \(s'\ge 2\), consider a resulting term containing
with \(1\le s_1' \le D\), \(0\le s_2' \le D\) and \(\sum _{n=1}^8s_n'\le s'\). Since \(P^{(4)}\) is constant we did not list it. We see that such a term above was generated from \(P^{D-1}\overline{P^D}\) by letting the partial derivative \(\partial _{i_2i_3}\) act \((s_1'-1)\)-times on P and \(s_2'\)-times on \(\overline{P}\), which implies that \( s_1'-1\ge s_3'\) and \(s_2'\ge s_4'\). If \( s_1'-1>s_3'\), then \(\partial _{i_2i_3}\) acted on the derivatives of \(P, \overline{P}\) directly \((s_1'-1-s_3')\)-times, and a similar argument holds for \(\overline{P'}\). Whenever \(\partial _{i_2i_3}\) acted on P, \(\overline{P}\) or their derivatives, it generated a term \( 2N^{-1} \sum _{i_l} G_{i_2i_l}G_{i_li_3}\), with \(i_l\), \(l\ge 3\), a fresh summation index. For each fresh summation index we apply Lemma 6.5 to gain a factor \(\frac{{{\mathrm{\mathrm {Im}}}}m}{N\eta }\). The total number of fresh summation indices in a term corresponding to (6.49) is
with \(\tilde{s}_0\mathrel {\mathop :}=s_1'+s_2'\) and \(\tilde{s}\mathrel {\mathop :}=s_3+s_4\), and we note this number does not decrease when \(\partial _{i_2i_3}\) acts on off-diagonal Green functions entries later. Thus, from (6.48) we conclude, upon using \(|G_{i_1i_2}|,|P''(m)|\), \(|P'''(m)|, |P^{(4)}(m)| \prec 1\) that, for \(2\le s'\le r'\),
for N sufficiently large. Here the last term on the right corresponds to \(\widetilde{s}=\widetilde{s}_0-1\). Thus, we conclude from (6.50) and the definition of \(\Phi _\epsilon \) in (6.1) that \(\mathbb {E}[\widehat{I}_{r',s'}]\), \(2\le s'\le r'\), is negligible.
To sum up and conclude this remark, we established that all \(\mathbb {E}[\widetilde{I}_{r',s'}]\) with \(1\le s'\le r'\) are negligible.
From (6.35), (6.38), (6.44), (6.46), (6.47) and (6.50) we find that
for N sufficiently large. Since \((z+ \widetilde{m})\) is deterministic and \(|z+\widetilde{m}| > 1/6\), as we showed in Remark 4.3, we obtain that \(|\mathbb {E}I_{2,0}^{(1)}|\le \Phi _\epsilon \). This concludes the proof of (6.34). \(\square \)
Combining (6.28), (6.30) and (6.34) we showed that
for N sufficiently large, i.e. all terms in \(\mathbb {E}I_{2,0}\) are negligible and the second estimate in (6.20) is proved.
6.5 Estimate on \(I_{r,0}\), \(r \ge 4\)
For \(r \ge 5\) we use the bounds \(|G_{i_1i_1}|,|G_{i_1i_2}|\prec 1\) to get
for N sufficiently large. For \(r=4\), \(\partial _{i_1i_2}^r G_{i_2i_1}\) contains at least one off-diagonal term \(G_{i_1i_2}\). Thus
for N sufficiently large, where we used Lemma 6.5 to get the last line. We conclude that all terms \(\mathbb {E}I_{r,0}\) with \(r\ge 4\) are negligible. This proves the fourth estimate in (6.20).
6.6 Estimate on \(I_{r,s}\), \(r\ge 2\), \(s\ge 1\)
For \(r\ge 2\) and \(s=1\), we have
Note that each term in \(\mathbb {E}I_{r,1}\), \(r\ge 2\), contains at least two off-diagonal Green function entries. For the terms with at least three off-diagonal Green function entries, we use the bound \(| G_{i_1i_2}|, |G_{i_1i_1}| \prec 1\) and
for N sufficiently large, where we used Lemma 6.5. Note that the right side is negligible since \({{\mathrm{\mathrm {Im}}}}m \prec 1\).
Denoting the terms with two off-diagonal Green function entries in \(\mathbb {E}I_{r,1}\) by \(\mathbb {E}I_{r,1}^{(2)}\), we have
where \(i_3\) is a fresh summation index and where we noted that r is necessarily even in this case. Lemma 6.5 then give us the upper bound
which is negligible for \(r> 2\). However, for \(r=2\), we need to gain an additional factor \(q_t^{-1}\). This can be done as in the proof of Lemma 6.7 by considering the off-diagonal entries \(G_{i_2i_3}G_{i_3i_1} \), generated from \(\partial _{i_1i_2} P(m)\), since the index \(i_2\) appears an odd number of times.
Lemma 6.10
For any (small) \(\epsilon >0\), we have
uniformly on \({\mathcal D}\), for N sufficiently large. In particular, the term \(\mathbb {E}I_{2,1}\) is negligible.
Proof
We start with the first term on the right side of (6.55). Using (3.23), we write
As in the proof of Lemma 6.7, we now apply the cumulant expansion to the right side. The leading terms of the expansion is
and, thanks to the additional factor of \(q_t^{-1}\) from the cumulant \(\kappa _t^{(3)}\), all other terms in the cumulant expansion are negligible, as can be checked by power counting as in the proof of Lemma 6.7. Replacing in (6.57) m by \(\widetilde{m}\), we then get
for N sufficiently large; cf. (6.37). Since \(|z+\widetilde{m}(z)|\ge 1/6\), \(z\in \mathcal {D}\) by Remark 4.3, we conclude that the first term on the right side of (6.55) is negligible. In the same way one shows that the second term is negligible, too. We leave the details to the reader. \(\square \)
We hence conclude from (6.54) and (6.56) that \(\mathbb {E}I_{r,1}\) is negligible for all \(r\ge 2\).
Consider next the terms
with \(2\le s\le r\). We proceed in a similar way as in Remark 6.8. We note that each term in \(\partial _{i_1i_2}^{r-s}G_{i_2i_1}\) contains at least one off-diagonal Green function when \(r-s\) is even, yet when \(r-s\) is odd there is a term with no off-diagonal Green function entries. Since \(s\ge 2\), the partial derivative \(\partial _{i_1i_2}^s\) acts on P or \(\overline{P}\) (or their derivatives) more than once in total; cf. Remark 6.8. Consider such a term with
for \( 1\le s_1 \le D\) and \(0\le s_2\le D\). Since \(P''(m), P'''(m), P^{(4)}(m) \prec 1\) and \(P^{(5)}=0\), we do not include derivatives of order two and higher here. We see that such a term was generated from \(P^{D-1}\overline{P^D}\) by letting the partial derivative \(\partial _{i_1i_2}\) act \((s_1 -1)\)-times on P and \(s_2\)-times on \(\overline{P}\), which implies that \(s_3 \le s_1 -1\) and \(s_4 \le s_2\). If \(s_3 < s_1 -1\), then \(\partial _{i_1i_2}\) acted on \(P'\) as well \([(s_1 -1)-s_3]\)-times, and a similar argument holds for \(\overline{P'}\). Whenever \(\partial _{i_1i_2}\) acts on P or \(\overline{P}\) (or their derivatives), it generates a fresh summation index \(i_l\), \(l\ge 3\), with a term \( 2N^{-1} \sum _{i_l} G_{i_2i_l}G_{i_li_1}\). The total number of fresh summation indices in this case is
Assume first that \(r=s\) so that \( \partial _{i_1i_2}^{r-s}G_{i_2i_1}=G_{i_2i_1}\). Then applying Lemma 6.5 \((2s_1 + 2s_2 - s_3 - s_4 -2)\)-times and letting \(s_0\mathrel {\mathop :}=s_1+s_2\) and \(s'\mathrel {\mathop :}=s_3+s_4\), we obtain the upper bound, \(r=s\ge 2\),
for N sufficiently large, i.e. \(\mathbb {E}I_{r,r}\), \(r\ge 2\), is negligible.
Second, assume that \(2\le s<r\). Then applying Lemma 6.5 \((2s_1 + 2s_2 - s_3 - s_4 -2)\)-times, we get
for N sufficiently large with \(2\le s<r\). In particular, \(|\mathbb {E}I_{r,s}|\le \Phi _\epsilon \), \(2\le s<r\). In (6.59) the second term bounds the terms corresponding to \(s_0-1=s'\) obtained by acting on \(\partial _{i_1i_2}\) exactly \((s_1 -1)\)-times on P and \(s_2\)-times on \(\overline{P}\) but never on their derivatives.
To sum up, we showed that \(\mathbb {E}I_{r,s}\) is negligible, for \(1\le s<r\). This proves (6.21) for \(1\le s<r\).
6.7 Estimate on \(I_{3,0}\)
We first notice that \(I_{3,0}\) contains terms with zero, two or four off-diagonal Green function entries and we split accordingly
When there are two off-diagonal entries, we can use Lemma 6.5 to get the bound
for N sufficiently large. A similar estimate holds for \(|\mathbb {E}I_{3,0}^{(2)}|\). The only non-negligible term is \(I_{3,0}^{(0)}\).
For \(n\in \mathbb {N}\), set
By definition \(S_1=m\). We remark that \(|S_n| \prec 1\) on \(\mathcal {D}\), for any fixed n, by Proposition 2.6.
Lemma 6.11
We have
Proof
Recalling the definition of \(I_{r,s}\) in (6.5), we have
We then easily see that the terms with no off-diagonal entries in \(\partial _{i_1i_2}^3 G_{i_2i_1}\) are of the form
We only need to determine the weight \(w_{I_{3,0}^{(0)}}\). With regard to the indices, taking the third derivative corresponds to putting the indices \(i_2i_1\) or \(i_1i_2\) three times. In that sense, the very first \(i_2\) and the very last \(i_1\) are from the original \(G_{i_2i_1}\). The choice of \(i_2i_1\) or \(i_1i_2\) must be exact in the sense that the connected indices in the following diagram must have been put at the same time:
Thus, the only combinatorial factor we have to count is the order of putting the indices. In this case, we have three connected indices, so the number of terms must be \(3!=6\). Thus, \(w_{I_{3,0}^{(0)}}=1\) and (6.62) indeed holds. \(\square \)
Lemma 6.12
For any (small) \(\epsilon >0\), we have, for all \(z\in \mathcal {D}\),
Proof
Fix \(\epsilon >0\). We first claim that
The idea is to expand the term \(\mathbb {E}[zm S_2^2 P^{D-1} \overline{P^D}]\) in two different ways and compare the results. Using the resolvent identity (3.23) and Lemma 3.2, we get
for arbitrary \(\ell '\in \mathbb {N}\). Using the resolvent identity (3.23) once more, we write
Thus, using Lemma 3.2, we also have
for arbitrary \(\ell '\in \mathbb {N}\). By Corollary 6.2 and Remark 6.3, the two error terms \(\mathbb {E}[\Omega _{\ell '}(\cdot )]\) in (6.65) and (6.66) are negligible for \(\ell '\ge 8D\).
With the extra factor \(N\kappa _t^{(4)}\), we then write
with
and 
.
For \(r=1\), \(s=0\), we find that
and similarly
where we used (6.61). We hence conclude that 
equals 
up to negligible error.
Following the ideas in Sect. 6.4, we can bound
and similarly 
, for N sufficiently large. In fact, for \(r\ge 2\), \(s\ge 0\) we can use, with small notational modifications the power counting outlined in Remark 6.8 and Remark 6.9 to conclude that
Therefore the only non-negligible terms on the right hand side of (6.67) are \(N\kappa _t^{(4)}\mathbb {E}[S_2^2 P^{D-1} \overline{P^D}] \), \(N\kappa _t^{(4)}\mathbb {E}[m^2 S_2 P^{D-1} \overline{P^D}]\) as well as 
and 
. Since, by (6.69) and (6.70), the latter agree up to negligible error terms, we conclude that the former two must be equal up do negligible error terms. Thus (6.64) holds.
Next, expanding the term \(\mathbb {E}[zm^3 S_2 P^{D-1} \overline{P^D}]\) in two different ways similar to above, we further get
Together with (6.64) this shows (6.63) and concludes the proof of the lemma. \(\square \)
Finally, from Lemmas 6.11 and 6.12, we conclude that
This proves the third estimate in (6.20).
Proof of Lemma 6.4
The estimates in (6.20) were obtained in (6.23), (6.51), (6.52), (6.53) and (6.72). Estimate (6.21) follows from (6.25), (6.54), (6.58) and (6.59). \(\square \)
7 Tracy–Widom limit: Proof of Theorem 2.10
In this section, we prove the Tracy–Widom limit of the extremal eigenvalues, Theorem 2.10, by following the strategy outlined in Sect. 3.4. In the first step, Proposition 7.1, we introduce a smooth cutoff to express the distribution of the largest eigenvalue of H as a functional of the imaginary part of the normalized trace of the Green function m of H. Then in Proposition 7.2, we compare the expectations of such functionals with respect to H and \(\mathrm {GOE}\), respectively.
To make the first step more precise in Proposition 7.1 below, we need some more notation. For \(\eta > 0\), we set
Using the functional calculus and the definition of the Green function, we then get
We have the following proposition, which corresponds to Corollary 6.2 in [23], respectively to Lemma 6.5 in [16]. Recall the definition of L in (2.28).
Proposition 7.1
Suppose that H satisfies Assumption 2.3 with \(\phi > 1/6\). Denote by \(\lambda _1^H\) the largest eigenvalue of H. Fix \(\epsilon > 0\). Let \(E \in \mathbb {R}\) be such that \(|E-L| \le N^{-2/3 + \epsilon }\). Let \(E_+ \mathrel {\mathop :}=L + {2}N^{-2/3 + \epsilon }\) and define \(\chi _E \mathrel {\mathop :}=\mathbbm {1}_{[E, E_+]}\). Set \(\eta _1 \mathrel {\mathop :}=N^{-2/3 - 3\epsilon }\) and \(\eta _2 \mathrel {\mathop :}=N^{-2/3 - 9\epsilon }\). Let moreover \(K: \mathbb {R}\rightarrow [0, \infty )\) be a smooth function satisfying
which is monotone decreasing on \([0, \infty )\). Then, for any \(D > 0\),
and
for N sufficiently large, with \(\theta _{\eta _2}\) as in (7.1).
We postpone the proof of Proposition 7.1 to Sect. 7.1 and move on to the Green function comparison theorem. Let \(W^{\mathrm {GOE}}\) be a GOE matrix independent of H with vanishing diagonal entries as introduced in Sect. 3.4 and denote by \(m^{\mathrm {GOE}}\equiv m^{W^{\mathrm {GOE}}}\) the normalized trace of its Green function.
Proposition 7.2
(Green function comparison) Let \(\epsilon >0\) and set \(\eta _0 \mathrel {\mathop :}=N^{-2/3 - \epsilon }\). Let \(E_1, E_2\in \mathbb {R}\) satisfy \(|E_1|, |E_2| \le N^{-2/3 + \epsilon }\). Let \(F : \mathbb {R}\rightarrow \mathbb {R}\) be a smooth function satisfying
Then, for any sufficiently small \(\epsilon > 0\), there exists \(\delta >0\) such that
for sufficiently large N, where L is given in (2.28).
Proposition 7.2 is proved in Sect. 7.2. We are ready to prove Theorem 2.10.
Proof of Theorem 2.10
Fix \(\epsilon > 0\) and set \(\eta _1 \mathrel {\mathop :}=N^{-2/3 - 3\epsilon }\) and \(\eta _2 \mathrel {\mathop :}=N^{-2/3 - 9\epsilon }\). Consider \(E = L + sN^{-2/3}\) with \(s \in (-N^{-2/3+\epsilon }, N^{-2/3+\epsilon })\). For any \(D > 0\), we find from Proposition 7.1 that
for N sufficiently large. Applying Proposition 7.2 with \(9\epsilon \) instead of \(\epsilon \) and setting \(E_1 \mathrel {\mathop :}=E-L +\eta _1\), \(E_2 \mathrel {\mathop :}=E_+ -L\), we find that there is \(\delta >0\) such that
for N sufficiently large. Hence, applying Proposition 7.1 to the matrix \(W^{\mathrm {GOE}}\), we get
Similarly, we can also check that
Since the right sides of Eqs. (7.8) and (7.9) converge both to \(F_1(s)\), the \(\mathrm {GOE}\) Tracy–Widom distribution, as N tends to infinity we conclude that
This proves Theorem 2.10.
In the rest of this section, we prove Propositions 7.1 and 7.2
7.1 Proof of Proposition 7.1
In principle, we could adopt the strategy in the proof of Corollary 6.2 of [23] after proving the optimal rigidity estimate at the edge with the assumption \(q \ggg N^{1/6}\) and checking that such an optimal bound is required only for the eigenvalues at the edge. However, we introduce a slightly different approach that directly compares \({{\mathrm{Tr}}}(\chi _E *\theta _{\eta _2})(H)\) and \({{\mathrm{Tr}}}\chi _{E-\eta _1}(H)\) by using the local law.
Proof of Proposition 7.1
For an interval \(I \subset \mathbb {R}\), let \({\mathcal N}_I\) be the number of the eigenvalues in I, i.e.
We compare \({{\mathrm{Tr}}}(\chi _E *\theta _{\eta _2})(H)\) and \({{\mathrm{Tr}}}\chi _{E-\eta _1}(H)\) by considering the following four cases:
Case 1 If \(x \in [E + \eta _1, E_+ - \eta _1)\), then \(\chi _{E-\eta _1}(x) = 1\) and
For any \(E' \in [E + \eta _1, E_+ - \eta _1)\), with the local law, Proposition 3.3, we can easily see that
with high probability, where we used \({{\mathrm{\mathrm {Im}}}}\widetilde{m}(E'+\mathrm {i}\eta _1)\le C\sqrt{\varkappa (E')+\eta _1}\lll N^{\epsilon /2}/(N\eta _1)\). Thus, considering at most \([E_+ - (L-N^{-2/3+\epsilon })]/\eta _1 = 3 N^{4\epsilon }\) intervals, we find that
and
with high probability.
Case 2 For \(x < E-\eta _1\), choose \(k\ge 0\) such that \(3^k \eta _1 \le E-x < 3^{k+1} \eta _1\). Then, \(\chi _{E-\eta _1}(x)=0\) and
Abbreviate \({\mathcal N}_k = {\mathcal N}_{(E- 3^{k+1} \eta _1, E - 3^k \eta _1]}\). Consider
With the local law, Proposition 3.3, and the estimate \({{\mathrm{\mathrm {Im}}}}\widetilde{m}(x+ \mathrm {i}y) \asymp \sqrt{|x-L| + y}\), we find that
and hence
with high probability. Thus, with high probability,
Case 3 By Proposition 2.9 there are with high probability no eigenvalues in \([E_+ - \eta _1, \infty )\).
Case 4 For \(x \in [E-\eta _1, E+\eta _1)\), we use the trivial estimate
Combining the four cases above, we find that
with high probability. From the definition of the cutoff K and the fact that \({{\mathrm{Tr}}}\chi _{E-\eta _1}(H)\) is an integer,
Thus, since K is monotone decreasing on \([0, \infty )\), (7.22) implies that
with high probability. Taking the expectation, we get for any fixed \(D>0\) that
for N sufficiently large. This proves the first part of Proposition 7.1. The second part can also be proved in a similar manner by showing that
with high probability, applying the cutoff K and taking the expectation. In this argument (7.21) gets replaced by
for \(x \in [E-\eta _1, E+\eta _1)\). This proves Proposition 7.1. \(\square \)
7.2 Green function comparison: Proof of Proposition 7.2
Recall the definition of the matrix \(H_t\) in (3.26). We first state a lemma that has the analogue role of Lemma 3.2 in calculations involving \(H \equiv H_t\) and \(\dot{H} \equiv \mathrm {d}H_t / \mathrm {d}t\).
Lemma 7.3
Fix \(\ell \in \mathbb {N}\) and let \(F\in C^{\ell +1}(\mathbb {R};\mathbb {C}^+)\). Let \(Y \equiv Y_0\) be a random variable with finite moments to order \(\ell +2\) and let W be a Gaussian random variable independent of Y. Assume that \(\mathbb {E}[Y] = \mathbb {E}[W] = 0\) and \(\mathbb {E}[Y^2] = \mathbb {E}[W^2]\). Define
and abbreviate \(\dot{Y}_t\equiv \frac{\mathrm {d}Y_t}{\mathrm {d}t}\). Then,
where \(\mathbb {E}\) denotes the expectation with respect to Y and W, \(\kappa ^{(r+1)}(Y)\) denotes the \((r+1)\)-th cumulant of Y and \(F^{(r)}\) denotes the r-th derivative of the function F. The error term \(\Omega _\ell \) in (7.26) satisfies
where \(Q\ge 0\) is an arbitrary fixed cutoff and \(C_\ell \) satisfies \(C_\ell \le \frac{(C\ell )^\ell }{\ell !}\) for some numerical constant C.
Proof
We follow the proof of Corollary 3.1 in [41]. First, note that
Thus,
Applying Lemma 3.2 and (3.9), we get (7.26) since the first two moments of W and Y agree. \(\square \)
Proof of Proposition 7.2
Fix a (small) \(\epsilon >0\). Consider \(x \in [E_1, E_2]\). Recall the definition of \(H_t\) in (3.26). For simplicity, let
with \(\eta _0=N^{-2/3-\epsilon }\), and define
Note that \(X \prec N^{\epsilon }\) and \(|F^{(l)}(X)| \prec N^{C\epsilon }\), for \(l \in \llbracket 1,11\rrbracket \). Recall from (3.32) that
where \(q_t=\mathrm {e}^{t/2}q_0\). Let \(z \mathrel {\mathop :}=x + L+ \mathrm {i}\eta _0\) and \(G \equiv G(z)\). Differentiating F(X) with respect to t, we get
where by definition
Thus, from Lemma 7.3, we find that
for \(\ell = 10\), where we abbreviate \(\partial _{i_2i_3} \equiv \partial /(\partial H_{i_2i_3})\). Note that the \(O(N^{1/3 + C\epsilon })\) error term in (7.34) originates from \(\Omega _\ell \) in (7.26), which is \(O(N^{C\epsilon } N^2 q_t^{-10})\) for the choice \(Y = H_{i_2i_3}\).
We claim the following lemma.
Lemma 7.4
Let \(z \mathrel {\mathop :}=x + L+ \mathrm {i}\eta _0\) and \(G \equiv G(z)\). For any integer \(r \ge 2\), set
Then,
and, for \(r=2\) and for \(r \ge 4\),
Assuming that Lemma 7.4 holds, we obtain that there exists \(\epsilon ' > 2\epsilon \) such that, for all \(t\in [0,6\log N]\),
which implies that the right side of (7.32) is \(O(N^{-\epsilon '/2})\). Integrating from \(t=0\) to \(t=6\log N\), we get
Comparing \({{\mathrm{\mathrm {Im}}}}m|_{t=6\log N}\) and \({{\mathrm{\mathrm {Im}}}}m^{\mathrm {GOE}}\) is trivial; if we let \(\lambda _k(6\log N)\) be the k-th largest eigenvalue of \(H_{6\log N}\) and \(\lambda _k^{\mathrm {GOE}}\) the k-th largest eigenvalue of \(W^{\mathrm {GOE}}\), then \(|\lambda _k(6\log N) - \lambda _k^{\mathrm {GOE}}| \prec N^{-3}\), hence
This proves Proposition 7.2. \(\square \)
It remains to prove Lemma 7.4. The proof uses quite similar ideas as used in Sect. 6. We thus sometimes omit some details and refer to the corresponding paragraph in Sect. 6.
Proof of Lemma 7.4
First, we note that in the definition of \(J_r\) we may freely include or exclude terms with \(i_1=i_2\) or \(i_1=i_3\) in the summation \(\sum _{i_1,i_2,i_3}\), since they contain at least one off-diagonal Green function entry, \(G_{i_1i_3}\) or \(G_{i_1i_2}\), and the sizes of these terms are, in expectation, at most of order
for any sufficiently small \(\epsilon ' > 0\). (When \(i_2=i_3\) the summand is zero by construction.)
For \(r \ge 5\), it is easy to see that \(\mathbb {E}J_r = O(N^{2/3 -\epsilon '})\), since it contains at least two off-diagonal entries in \(\partial _{i_2i_3}^r \left( F'(X) G_{i_1i_2} G_{i_3i_1} \right) \) and \(|\mathbb {E}J_r|\) is bounded by
which can be checked using Lemma 6.5 and a simple power counting.
Therefore, we only need to consider the cases \(r=2, 3, 4\). In the following subsections, we check each case and complete the proof of Lemma 7.4. See Eqs. (7.48), (7.52) and (7.53) below.
7.2.1 Proof of Lemma 7.4 for \(r=2\)
We proceed as in Lemma 6.7 of Sect. 6.4 and apply the idea of an unmatched index. Observe that
We first consider the expansion of \(F'(X)\,\partial _{i_2i_3}^2 (G_{i_1i_2} G_{i_3i_1})\). We can easily estimate the terms with four off-diagonal Green function entries, since, for example,
where we used Lemma 6.5. Thus, for sufficiently small \(\epsilon \) and \(\epsilon '\),
For the terms with three off-diagonal Green function entries, the bound we get from Lemma 6.5 is
which is not sufficiently small. To gain an additional factor of \(q_t^{-1}\), making the above bound \(q_t^{-2} N^{1+C\epsilon } \lll N^{2/3 -\epsilon '}\), we use Lemma 3.2 to expand in an unmatched index. For example, such a term is of the form
and we focus on the unmatched index \(i_3\) in \(G_{i_3i_2}\). Then, multiplying by z and expanding, we get
for \(\ell = 10\).
For \(r'=1\), we need to consider \(\partial _{i_3i_4} (F'(X) G_{i_1i_2} G_{i_4i_2} G_{i_3i_3} G_{i_2i_1})\). When \(\partial _{i_3i_4}\) acts on \(F'(X)\) it creates a fresh summation index \(i_5\) and we get a term
where we abbreviate \(\widetilde{G} \equiv G(y + L + \mathrm {i}\eta _0)\). Applying Lemma 6.5 to the index a and \(\widetilde{G}\), we get,
for any \(i_3\) and \(i_4\), which also shows that
Applying Lemma 6.5 to the remaining off-diagonal Green function entries, we obtain from (7.42) that
When \(\partial _{i_3i_4}\) acts on \(G_{i_1i_2} G_{i_4i_2} G_{i_3i_3} G_{i_2i_1}\), then we always get four or more off-diagonal Green function entries with the only exception being
To the terms with four or more off-diagonal Green function entries, we apply Lemma 6.5 and obtain a bound similar to (7.44) by power counting. For the term of the exception, we rewrite it as
Here, the last term is again bounded by \(q_t^{-1} N^{2/3 + C\epsilon }\) as we can easily check with Proposition 3.3 and Lemma 6.5. We thus arrive at
On the right side, the summation starts from \(r'=2\), hence we have gained a factor \((Nq_t)^{-1}\) from \(\kappa ^{(r'+1)}\) and added the fresh summation index \(i_4\), so the net gain is \(q_t^{-1}\). Since \(|z + \widetilde{m}| \asymp 1\), this shows that
Together with (7.41), this takes care of the first term on the right side of (7.40).
For the second term on the right side of (7.40), we focus on
and apply the same argument to the unmatched index \(i_3\) in \(\widetilde{G}_{i_3i_4}\).
For the third term, we focus on the factor \(G_{i_1i_2} G_{i_3i_1}\) and apply the same argument once more with the unmatched index \(i_3\) in \(G_{i_3i_1}\). We omit the detail.
Estimating all terms in this way, we eventually get the desired estimate
7.2.2 Proof of Lemma 7.4 for \(r=3\)
We proceed as in Sect. 6.7. Note that there will be no unmatched indices in the Green function entries for this case. If \(\partial _{i_2i_3}\) acts on \(F'(X)\) at least once, then that term is bounded by
where we used (7.43) and the fact that \(G_{i_1i_2} G_{i_3i_1}\) or \(\partial _{i_2i_3} (G_{i_1i_2} G_{i_3i_1})\) contains at least two off-diagonal entries. Moreover, in the expansion of \(\partial _{i_2i_3}^2 (G_{i_1i_2} G_{i_3i_1})\), the terms with three or more off-diagonal Green function entries entries can be bounded by
Thus,
where the combinatorial factor 4! is computed as in Lemma 6.11 and \(S_2\) is as in (6.61). As in Lemma 6.12 of Sect. 6.7, the first term on right side of (7.49) is computed by expanding
in two different ways. We then obtain that
where we used that \(m \equiv m(z) = -1 + O(N^{-1/3 + \epsilon })\) with high probability. Indeed, since \(\widetilde{m}(L)\), which was denoted by \(\tau \) in the proof of Lemma 4.1, satisfies \(\widetilde{m}(L) = -1 + O(\mathrm {e}^{-t}q_t^{-2})\) by (4.6), and since \(|\widetilde{m}(z) -\widetilde{m}(L)| \asymp \sqrt{\varkappa +\eta _0} \le N^{-1/3 + \epsilon }\) by (4.8), we have from Proposition 3.3 that \(m(z) = -1 + O(N^{-1/3 + \epsilon })\) with high probability.
Consider next the first term on the very right of (7.50). Since \(|z-2|\le CN^{-1/3}\), we get
where we used (7.43). Expanding on left hand side the factor \(G_{i_2i_2}\) using (3.23), we obtain
Applying Lemma 3.2 to the second term on the right side, we find that most of the terms are \(O(N^{2/3 -\epsilon '})\) either due to three (or more) off-diagonal entries, the partial derivative \(\partial _{i_2i_3}\) acting on \(F'(X)\), or higher cumulants. The only term that does not fall into one these categories is
which is generated when \(\partial _{i_2i_3}\) acts on \(G_{i_3i_2}\). From this argument, we find that
Hence, combining it with (7.51) and the fact that \(m = -1 + O(N^{-1/3 + \epsilon })\) with high probability, we get
In combination with (7.49) and (7.50), we conclude that
7.2.3 Proof of Lemma 7.4 for \(r=4\)
In this case, we estimate the term as in the case \(r=2\) and get
We leave the details to the interested reader.
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Acknowledgements
We thank László Erdős for useful comments and suggestions. Ji Oon Lee is grateful to the department of mathematics, University of Michigan, Ann Arbor, for their kind hospitality during the academic year 2014–2015.
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J. O. Lee: Supported by Samsung Science and Technology Foundation project number SSTF-BA1402-04.
K. Schnelli: Supported by ERC Advanced Grant RANMAT No. 338804 and the Göran Gustafsson Foundation.
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Lee, J.O., Schnelli, K. Local law and Tracy–Widom limit for sparse random matrices. Probab. Theory Relat. Fields 171, 543–616 (2018). https://doi.org/10.1007/s00440-017-0787-8
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DOI: https://doi.org/10.1007/s00440-017-0787-8