Local spectral statistics of Gaussian matrices with correlated entries

Introduction

Most rigorous works on random matrix ensembles concern either Wigner matrices with independent entries (up to the real symmetric or complex hermitian symmetry constraint), or invariant ensembles where the correlation structure of the matrix elements is very specific. Since the existing methods to study Wigner matrices heavily rely on independence, only very few results are available on ensembles with correlated entries, see for the Gaussian case. The global semicircle law in the non Gaussian case with (appropriately) weakly dependent entries has been established via moment method in and via resolvent method in . A similar result for sample covariance matrices was given in . All these works establish limiting spectral density on the macroscopic scale and in models where the dependence is sufficiently weak so that the limiting density of states coincides with that of the independent case. A more general correlation structure was explored in with a nontrivial limit density, but still only on the global scale, see also . We also mention the very recent proof of the local semicircle law and bulk universality for the adjacency matrix of the $d$-regular graphs which has a completely different specific correlation (due to the requirement that every row contains the same number of ones).

which constitutes a small perturbation of the Quadratic Vector Equation (QVE),

Gaussian random matrices with translation invariant covariance structure have been analyzed earlier and it has also been realized that the equation (1.2) via Fourier transform plays a key role in identifying the limiting density of eigenvalues, see Khorunzhy and Pastur , Girko , as well as Anderson and Zeitouni in . These works, however, were concerned only with the density on macroscopic scales. The off-diagonal decay of the resolvent and the bulk universality require much more detailed information. The current paper in combination with and presents such a precise analysis.

Set-up and main results

Consider a real symmetric or complex hermitian random matrix,

indexed by the large discrete torus of size $N$,

We assume that the matrix is centered, i.e.,

Power law decay: There is a positive integer $\kappa$, such that

Exponential decay: There is a constant $\nu>0$ such that

Non-resonance: There is a constant $\xi_{1}>0$, such that

Strong non-resonance: There is a constant $\xi_{2}>0$, such that

The restrictions on the correlation structure are quantified by the $N$-independent model parameters $\nu,\kappa,\xi_{1},\xi_{2}$ appearing above. We remark that the normalization of (2.4) and (2.5) is chosen for convenience, e.g., we could replace $1$ on the right hand side of (2.4) by some finite constant. The set of model parameters depends on our assumptions, e.g., if only (D1) and (R2) are assumed, then $\nu$ and $\xi_{2}$ are the model parameters. We allow constants appearing in the statements to depend on the model parameters.

For compact statements of our results we define the notion of stochastic domination, introduced in and . This notion is designed to compare sequences of random variables in the large $N$ limit up to small powers of $N$ on high probability sets.

In this case we write $\varphi\prec\psi$.

respectively. It was shown in that the Quadratic Vector Equation (QVE)

with the constants $C>0$ and $\nu^{\prime}>0$ depending only on the model parameters.

Similarly, as in the case of Wigner type matrices the local law implies the bulk universality for Gaussian matrices with correlated entries. However, the $q$-fullness condition (Definition 1.14 in ) is replaced by a different non-generacy condition.

In particular, the eigenvectors are completely delocalized, i.e., $\lVert\mspace{1.0mu}\mathbf{u}^{(i)}\rVert_{\infty}\prec N^{-1/2}$.

The following result shows a practical way to construct real symmetric random matrices with translation invariant correlation structure. A similar, but slightly more complicated convolution representation exists for complex hermitian random matrices.

This lemma is proven at the end of Subsection 5. We introduce the following conventions and notations used throughout this paper.

Symbols $c,c_{1},c_{2},\dots$ and $C,C_{1},C_{2},\dots$ denote generic positive and finite constants that depend only on the model parameters. They have a local meaning within a specific proof. For two arbitrary non-negative functions $\varphi$ and $\psi$ defined on some domain $U$, we write $\varphi\,\lesssim\,\psi$, or equivalently $\psi\,\gtrsim\,\varphi$, if $\varphi(u)\leq C\mspace{2.0mu}\psi(u)$, holds for all $u\in U$. The notation $\psi\sim\varphi$ is equivalent to both $\psi\lesssim\varphi$ and $\psi\gtrsim\varphi$ holding at the same time. In this case we say that $\psi$ and $\varphi$ are comparable. In general the relation $\gtrsim$ is called the comparison relation. We also write $\psi=\varphi+\mathcal{O}(\vartheta)$ if $\lvert\psi-\varphi\rvert\lesssim\vartheta$.

In the following we analyze random matrices which have independent entries modulo two reflection symmetries.

The next result shows that the discrete Fourier transform maps Gaussian translation invariant random matrices into Wigner type random matrices with an extra dependence. This connection was first realized by Girko and Khorunzhy and Pastur . It has been later used in .

We remark that if $a_{xy}$ satisfies the decay estimate (2.4), then $\widehat{a}_{\phi\mspace{1.0mu}\theta},\lvert\mspace{1.0mu}\widehat{b}_{\phi\mspace{1.0mu}\theta}\rvert\lesssim N^{-1}\!$.

In this subsection we sketch how to prove a local law for the elements of the Fourier-transformed resolvent

then the conclusions of Theorem 1.6 from hold.

only in the proofs of Lemma 2.1 and Theorem 3.5 in .

2 Anisotropic local law for 444-fold correlation

In order to translate the statements of the local law in Fourier coordinates back to the original coordinates we need an anisotropic local law. Here we consider $\lvert z\rvert$ to be bounded to get simpler estimates. This condition can be easily dropped out if needed.

The starting point of the argument is to write the right hand side of (3.12) in the form:

The key idea of the proof is to apply recursively the general resolvent identities (cf. (2.9) in ) to express the product of resolvent entries in (3.13) as a sum over so-called trivial leaves (cf. Subsection 5.10 of ) and the sum over $C(p)$ terms (corresponding to the non-trivial leaves in ) of the form

Let us illustrate the modifications by considering the simplest leading order terms of the type (3.14) when $p=2$. Considering the contribution of such terms to the right hand side of (3.13) yields

Properties of QVE

If (R0) is assumed we will treat the associated parameters $\kappa,K,\xi_{0}$ as model parameters. By definition (R2) implies $\widetilde{a}(\phi,\theta)\geq\xi_{2}$ for every $\phi,\theta$, and thus (R0) holds with $\xi_{0}=\xi_{2}$ and $\mathcal{D}=\{{}\}$. Assumption (R1) does not imply (R0), but (R1) and (D2) together do ( cf. Lemma 4.2 below).

Instead of directly analyzing the discrete QVE (2.9) we will first establish the correct properties for the solution of the continuous version

of (2.9). Afterwards we deduce these properties for the discrete version (2.9) as well. For the transition from the discrete to the continuous version we need certain stability properties of the QVE that were established in .

We recall several notations and results from . We will consider QVEs defined on a probability space $(\mathfrak{X},\pi)$ with an operator $S$ in two different setups. When we discuss the discrete QVE (2.9), the setup is

Finally, the normalization A3. of holds if we replace $\widetilde{A}$ and $\widetilde{m}(z;\phi)$ by $\lambda\mspace{1.0mu}\widetilde{A}$ and $\lambda^{1/2}\widetilde{m}(\lambda^{1/2}z;\phi)$, respectively, with $\lambda:=\lVert\widetilde{A}\rVert_{1\to 1}^{-1}$. From (4.8) it follows that $\lambda\sim 1$.

Next we show that $m(z)$ is uniformly bounded for $z\neq 0$. Indeed, using (4.8) we get

Since this implies the condition B1. of (i) of Theorem 4.1 in is applicable in the setup (4.5b). The theorem shows that $\lVert m(z)\rVert_{\infty}\leq C(\delta)$ for any $\lvert z\rvert\geq\delta$ with $C(\delta)$ depending on $\delta>0$. The property (R0) is equivalent to property B2. in . Hence by (ii) of Theorem 4.1 in $\widetilde{m}(z)$ is uniformly bounded in some neighborhood of $z=0$. Combining this with the uniform bound away from $z=0$ we get the uniform bound for $\lvert\widetilde{m}(z;\phi)\rvert$ for all $z$ and $\phi$. In order to bound also the derivative $\partial_{\phi}\widetilde{m}(z;\phi)$ we differentiate the continuous QVE (4.4) and get

Using (4.8) and the uniform boundedness of $\widetilde{m}$ we finish the proof of (4.6).

and hence Theorem 1.9 from yields $\operatorname{supp}\widetilde{\rho}\,=[-\widetilde{\beta}\mspace{1.0mu},\mspace{2.0mu}\widetilde{\beta}\,]$, for some constant $\widetilde{\beta}\sim 1$.

In terms of these quantities (2.9) can be written as

We will now consider $g$ as the solution of the perturbed continuous QVE

Using (4.13) we see that the perturbation $d$ is indeed small:

Comparing (4.15) and (4.4) we show that (4.17) implies that the corresponding solutions $g$ and $\widetilde{m}$ are close in the sense of (4.7). For this purpose we use the rough stability statement from Theorem 1.10 of to get

where $c_{0}\sim 1$ and $\lambda_{1}\sim 1$ are sufficiently small constants left unspecified until the end of the proof.

This means that we get stability as long as we stay away from the points $\pm\mspace{1.0mu}\widetilde{\beta}$. The necessary initial bound inside the indicator function is satisfied for large enough values of $|z|$, since

It remains to show (4.7) close to the edges by using that the instability at these two points is quadratic. The argument is a simplified version of the one used in a random setting in Section 4 of . For the convenience of the reader we show a few details. We restrict to the case $|z-\widetilde{\beta}|\leq c_{0}$, close to the right edge. The left edge is treated in the same way. For the following analysis we use the stability result, Theorem 4.2 of , in the continuous setup (cf. Proposition 8.1 in ). The theorem yields

where the quantity $\Theta=\Theta(z)\geq 0$ is continuous in $z$ and satisfies the cubic inequality

Here the constant $\lambda_{2}\sim 1$ is independent of $c_{0}$.

Note that (4.21) corresponds to (4.10) in and (8.5) in , respectively. Combining (4.11), (4.14b) and (4.5d) in , the coefficients $\pi_{k}=\pi_{k}(z)$ of the cubic equation (4.21) satisfy

provided $c_{0}\sim 1$ is sufficiently small. Since $\pi_{1}(z)\to 0$ as $z\to\widetilde{\beta}$, by decreasing the size $c_{0}$ of the neighborhood we are working on, the value of $|\pi_{1}|$ can be made arbitrarily small. This, in turn, implies that the solution $\Theta$ of the cubic inequality (4.21) is small,

Using this we can make the right hand side of (4.20) smaller than $\lambda_{2}/2$, say, by decreasing the value of $c_{0}$. Thus, there is a gap in the possible values of the continuous function $z\mapsto\lVert g(z)-\widetilde{m}(z)\rVert_{\infty}$, in the sense that $\lVert g-\widetilde{m}\rVert_{\infty}\notin(\mspace{1.0mu}\lambda_{2}/2,\mspace{2.0mu}\lambda_{2})$. Since on the boundary, $|z-\widetilde{\beta}\mspace{1.0mu}|=c_{0}$, the initial bound, $\lVert g-\widetilde{m}\rVert_{\infty}\leq\lambda_{2}$, holds by (4.19), it propagates to all $z$ with $\lvert z-\widetilde{\beta}\rvert\leq c_{0}$. Thus, (4.20) and (4.21) remain true without the indicator functions.

It still remains to bound $\Theta$ in (4.20). Since $|\pi_{2}|\sim 1$, we may absorb the cubic term in $\Theta$ in (4.21). We find that $\Theta$ satisfies

where $\varpi:=\pi_{1}/(\pi_{2}+\Theta\mspace{1.0mu})$. From this it is easy to see that the bound $\Theta\leq N^{-1/2}$ can be propagated from the boundary $|z-\widetilde{\beta}|=c_{0}$ inside the neighborhood $|z-\widetilde{\beta}|\leq c_{0}$ of the right edge to give $\Theta\lesssim N^{-1/2}$ everywhere. Using this in (4.20) without the indicator function proves the bound (4.7) at the right edge. ∎

The part of the proof considering the exponentially decaying correlation matrices relies on the following technical result that is proven in the appendix.

Suppose $f$ is an analytic function on the complex strip,

then for every $\varepsilon>0$ there exists $\delta>0$ depending only on $\varepsilon,\nu,C_{1}$ such that

Using (4.28) we now show that $\widetilde{A}$ is a block fully indecomposable operator, i.e., (R0) holds. From (4.8) we see that

Denoting $D_{I}:=\cup_{i\mspace{1.0mu}\in\mspace{1.0mu}I}D_{i}$, we have $\widetilde{a}(\phi,\theta)\leq\delta$ for $(\phi,\theta)\in D_{I}\times D_{J}$. Thus (4.28) implies

Recall from Lemma 4.1 that $\widetilde{m}(z)$ is the bounded solution of the continuous QVE (4.4). We will first prove that

where $e_{x}$ is the Fourier-basis function. Then we show that $q_{x}(z)$ and $\widetilde{q}_{x}(z)$ are so close that (2.13) holds.

Thus, we need to show that $\lVert\mspace{1.0mu}\partial^{\mspace{1.0mu}\kappa}\widetilde{m}(z)\rVert_{\infty}\lesssim 1$ uniformly in $z$. The proof is by induction on the number of derivatives of $\widetilde{m}$. It is based on

which follows from (4.9), and the following consequence of (2.4):

As the second step of the proof we show that

This proves (4.41) for $\lvert x\rvert\leq N^{1/2}$.

For $\lvert x\rvert\geq N^{1/2}$ we bound $q_{x}=q_{x}(z)$ directly by using the summation of parts

where we have dropped two boundary terms of size $\mathcal{O}(N^{-1})$. Here, $\lvert\mspace{1.0mu}m_{(j+1)/N}-m_{j/N}\rvert\leq C/N$, while the geometric sum is $\mathcal{O}(N/x)=\mathcal{O}(N^{1/2})$ for $N^{1/2}\leq\lvert x\rvert\leq N/2+1$. Thus, estimating each term in the sum over $j$ separately shows that $\lvert q_{x}(z)\rvert\lesssim N^{-1/2}$ also in this case. ∎

Next we show that the probability density $\rho$ corresponding to the discrete QVE, via (2.14), is also regular and supported on a single interval.

Proofs for local law and bulk universality

The following is the strongest version of the local law we prove here.

If (D1) and (R2) are assumed, then (R0) holds with $\xi_{0}=\xi_{2}$ and $\mathcal{D}=\{{}\}$, and Proposition 5.1 yields the proof. If on the other hand, (D2) and (R1) are assumed, then (R0) holds by Lemma 4.2. The proof is hence again reduced to Proposition 5.1. ∎

satisfy (3.4). This immediately yields (2.3) for the matrix (5.1).

From the proof of Corollary 2.4 we read off the convolution representation for symmetric translation invariant random matrices.

The assumption (2.17) implies $\widehat{a}_{\phi\mspace{1.0mu}\theta}\geq 0$. This guarantees that $\widehat{H}$ defined through (5.2) is self-adjoint. Expressing (5.2) in the original coordinates yields the representation (2.18). ∎

Appendix A Appendix

where $C_{2}<\infty$ does not depend on $f$. From the first inclusion of (A.1) it follows that

Let $\log_{+}$ and $\log_{-}$ be the positive and negative parts of the logarithm, respectively, so that $\log\tau=\log_{+}\tau-\log_{-}\tau$, for $\tau>0$. Using Chebyshev’s inequality we get

Using (A.5) to bound the derivative and writing $\widetilde{f}:=f\circ\Phi_{\zeta_{0}}$ we get

We will now bound the last integral using the Jensen-Poisson formula,

where we have used (A.6) to get the second inequality. For the last bound we have used $\lvert\mspace{1.0mu}\widetilde{f}(\omega)\rvert=\lvert f(\Phi_{\zeta_{0}}(\omega))\rvert\leq C_{1}$. Plugging this into (A.7) and recalling (A.3) we get

This finishes the proof as $C_{3}(\zeta_{0})$ and $C_{1}$ are independent of $\delta$. ∎

References