Local Circular Law for Random Matrices

Introduction

A considerable literature about random matrices focuses on Hermitian or symmetric matrices with independent entries. These models are paradigms for local eigenvalues statistics of many random Hamiltonians, as envisioned by Wigner. The study of non-Hermitian random matrices goes back to Ginibre, then in Princeton and motivated by Wigner. Ginibre’s viewpoint on the problem was described as follows [Gin1965]:

Apart from the intrinsic interest of the problem, one may hope that the methods and results will provide further insight in the cases of physical interest or suggest as yet lacking applications.

This phenomenon is the non-Hermitian counterpart of the semicircular law for Wigner random Hermitian matrices, and the quarter circular limit for Marchenko-Pastur random covariance matrices.

In the case of real Gaussian entries, the join distribution of the eigenvalues is more complicated but still integrable, allowing Edelman [Ede1997] to prove the limiting circular law as well; for more precise asymptotic properties of the real Ginibre ensemble, see [ForNag2007, Sin2007, BorSin2009]. We note also that the (right) eigenvalues of the quaternionic Ginibre ensemble were recently shown to converge to a (non-uniform) measure on the unit ball of the quaternions field [BenCha2011].

From this formula, it is clear that the small eigenvalues of the Hermitian matrix $(X^{*}-z^{*})(X-z)$ play a special role due to the logarithmic singularity at . The key question is to estimate the smallest eigenvalues of $(X^{*}-z^{*})(X-z)$, or in other words, the smallest singular values of $(X-z)$. This problem was not treated in [Gir1984], but the gap was remedied in a series of papers. First Bai [Bai1997] was able to treat the logarithmic singularity assuming bounded density and bounded high moments for the entries of the matrix (see also [BaiSil2006]). Lower bounds on the smallest singular values were given in Rudelson, Vershynin [Rud2008, RudVer2008], and subsequently Tao, Vu [TaoVu2008], Pan, Zhou [PanZho2010] and Götze, Tikhomirov [GotTik2010] weakened the moments and smoothness assumptions for the circular law, till the optimal $\mbox{L}^{2}$ assumption, under which the circular law was proved in [TaoVuKri2010].

The purpose of this paper is to prove a local version of the circular law, up to the optimal scale $N^{-1/2+{\varepsilon}}$ (see Section 2 for a precise statement). Below this scale, detailed local statistics will be important and that is beyond the scope of the current paper. The main tool of this paper is a detailed analysis of the self-consistent equations of the Green functions

Finally, we remark that the local circular law demonstrates that the eigenvalue distribution in the unit disk is extremely “uniform”. If the eigenvalues are distributed in the unit disk by a uniform statistics or any other statistics with summable decay of correlations, then there will be big holes or some clusterings of eigenvalues in the disk. While the usual circular law does not rule out these phenomena, the local law established in this paper does. This implies that the eigenvalue statistics cannot be any probability laws with summable decay of correlations

The local circular law

We first introduce some notations. Let $X$ be an $N\times N$ matrix with independent centered entries of variance $N^{-1}$. The matrix elements can be either real or complex, but for the sake of simplicity we will consider real entries in this paper. Denote the eigenvalues of $X$ by $\mu_{j}$, $j=1,\ldots,N$. We will use the following notion of stochastic domination which simplifies the presentation of the results and their proofs.

Let $W=(W_{N})_{N\geqslant 1}$ be family a random variables and $\Psi=(\Psi_{N})_{N\geqslant 1}$ be deterministic parameters. We say that $W$ is stochastically dominated by $\Psi$ if for any $\sigma>0$ and $D>0$ we have

for sufficiently large $N$. We denote this stochastic domination property by

In this paper, we will assume that the probability distributions for the matrix elements have the uniform subexponential decay property, i.e.,

for some constant $\vartheta>0$ independent of $N$. This condition can of course be weakened to an hypothesis of boundedness on sufficiently high moments, but the error estimates in the following Theorem would be weakened as well. We now state our local circular law, which holds up to the optimal scale $N^{-1/2+{\varepsilon}}$.

Let $X$ be an $N\times N$ matrix with independent centered entries of variance $N^{-1}$. Suppose that the probability distributions of the matrix elements satisfy the uniformly subexponentially decay condition (2.1). We assume that for some fixed $\tau>0$, for any $N$ we have $\tau\leqslant||z_{0}|-1|\leqslant\tau^{-1}$ ($z_{0}$ can depend on $N$). Let $f$ be a smooth non-negative function which may depend on $N$, such that $\|f\|_{\infty}\leqslant C$, $\|f^{\prime}\|_{\infty}\leqslant N^{C}$ and $f(z)=0$ for $|z|\geqslant C$, for some constant $C$ independent of $N$. Let $f_{z_{0}}(z)=N^{2a}f(N^{a}(z-z_{0}))$ be the approximate delta function obtained from rescaling $f$ to the size order $N^{-a}$ around $z_{0}$. We denote by $D$ the unit disk. Then for any $a\in(0,1/2]$,

Hermitization and local Green function estimate

In the following, we will use the notation

where $I$ is the identity operator. Let $\lambda_{j}(z)$ be the $j$-th eigenvalue (in the increasing ordering) of $Y^{*}_{z}Y_{z}$. We will generally omit the $z-$dependence in these notations. Thanks to the Hermitization technique of Girko [Gir1984], the first step in proving the local circular law is to understand the local statistics of eigenvalues of $Y^{*}_{z}Y_{z}$, for $z$ strictly inside the unit circle. In this section, we first recall some well-known facts about the Stieltjes transform of the empirical measure of eigenvalues of $Y^{*}_{z}Y_{z}$. We then present the key estimate concerning the Green function of $Y^{*}_{z}Y_{z}$ in almost optimal spectral windows. This result will be used later on to prove a local version of the circular law.

. Define the Green function of $Y^{*}_{z}Y_{z}$ and its trace by

We will also need the following version of the Green function later on:

As we will see, with high probability $m(w,z)$ converges to $m_{\rm c}(w,z)$ pointwise, as $N\to\infty$ where $m_{\rm c}(w,z)$ is the unique solution of

with positive imaginary part (see Section 3 in [GotTik2010] for the existence and uniqueness of such a solution). The limit $m_{\rm c}(w,z)$ is the Stieltjes transform of a density $\rho_{\rm c}(x,z)$ and we have

whenever $\eta>0$. The function $\rho_{\rm c}(x,z)$ is the limiting eigenvalue density of the matrix $Y^{*}_{z}Y_{z}$ (cf. Lemmas 4.2 and 4.3 in [Bai1997]). Let

Note that $\lambda_{-}$ has the same sign as $|z|-1$. The following two propositions summarize the properties of $\rho_{\rm c}$ and $m_{\rm c}$ that we will need to understand the main results in this section. They will be proved in Appendix A. In the following, we use the notation $A\sim B$ when $cB\leqslant A\leqslant c^{-1}B$, where $c>0$ is independent of $N$.

The limiting density $\rho_{\rm c}$ is compactly supported and the following properties regarding $\rho_{\rm c}$ hold.

The support of $\rho_{\rm c}(x,z)$ is $[\max\{0,\lambda_{-}\},\lambda_{+}]$.

As $x\to\lambda_{+}$ from below, the behavior of $\rho_{\rm c}(x,z)$ is given by $\rho_{\rm c}(x,z)\sim\sqrt{\lambda_{+}-x}.$

For any ${\varepsilon}>0$, if $\max\{0,\lambda_{-}\}+{\varepsilon}\leqslant x\leqslant\lambda_{+}-{\varepsilon}$, then $\rho_{\rm c}(x,z)\sim 1$.

Near $\max\{0,\lambda_{-}\}$, the behavior of $\rho_{\rm c}(x,z)$ can be classified as follows.

If $|z|\geqslant 1+\tau$ for some fixed $\tau>0$, then $\lambda_{-}>{\varepsilon}(\tau)>0$ and $\rho_{\rm c}(x,z)\sim\mathds{1}_{x>\lambda_{-}}\sqrt{x-\lambda_{-}}$.

If $|z|\leqslant 1-\tau$ for some fixed $\tau>0$, then $\lambda_{-}<-{\varepsilon}(\tau)<0$ and $\rho_{\rm c}(x,z)\sim 1/\sqrt{x}$.

All of the estimates in this proposition are uniform in $|z|<1-\tau$, or $\tau^{-1}\geqslant|z|\geqslant 1+\tau$ for fixed $\tau>0$.

The preceding Proposition implies that, uniformly in $w$ in any compact set,

Moreover, the following estimates on $m_{\rm c}(w,z)$ hold.

If $|z|\geqslant 1+\tau$ for some fixed $\tau>0$, then $m_{\rm c}\sim 1$ for $w$ in any compact set.

If $|z|\leqslant 1-\tau$ for some fixed $\tau>0$, then $m_{\rm c}\sim|w|^{-1/2}$ for $w$ in any compact set.

2 Concentration estimate of the Green function up to the optimal scale.

We now state precisely the estimate regarding the convergence of $m$ to $m_{\rm c}$. Since the matrix $Y^{*}_{z}Y_{z}$ is symmetric, we will follow the approach of [ErdYauYin2010Adv]. We will use extensively the following definition of high probability events.

Let $\zeta>0$. We say that an $N$-dependent event $\Omega$ holds with $\zeta$-high probability if there is some constant $C$ such that

For $\alpha\geqslant 0$, define the $z$-dependent set

Suppose $\tau\leqslant||z|-1|\leqslant\tau^{-1}$ for some $\tau>0$ independent of $N$. Then for any $\zeta>0$, there exists $C_{\zeta}>0$ such that the following event holds with $\zeta$-high probability:

Moreover, the individual matrix elements of the Green function satisfy, with $\zeta$-high probability,

For $|z|<1$, we have $\lambda_{-}<0$ (see Proposition 3.1), so in this case we define $\kappa:=|E-\lambda_{+}|$.

There exists $\tau_{0}>0$ such that for any $\tau\leqslant\tau_{0}$ if $|z|\leqslant 1-\tau$ and $|w|\leqslant\tau^{-1}$ then the following properties concerning $m_{\rm c}$ hold. All constants in the following estimates depend on $\tau$.

$E\geqslant\lambda_{+}$ and $|w-\lambda_{+}|\geqslant\tau$. We have

$|w-\lambda_{+}|\leqslant\tau$ (Notice that there is no restriction on whether $E\leqslant\lambda_{+}$ or not ). We have

$|w|\geqslant\tau$, $|w-\lambda_{+}|\geqslant\tau$ and $E\leqslant\lambda_{+}$. We have

Here Case 1 covers the regime where $E\geqslant\lambda_{+}$ and $w$ is far away from $\lambda_{+}$. Case 2 concerns the regime that $w$ is near $\lambda_{+}$, while Case 3 is for $w$ is near the origin. Finally Case 4 is for $w$ not covered by the first three cases.

There exists $\tau_{0}>0$ such that for any $\tau\leqslant\tau_{0}$, if $|z|\geqslant 1+\tau$ and $|w|\leqslant\tau^{-1}$ then the following properties concerning $m_{\rm c}$ hold. All constants in the following estimates depend on $\tau$. Recall from (3.2) that $\lambda_{-}=\frac{(\alpha-3)^{3}}{8(\alpha-1)}>0$.

$E\geqslant\lambda_{+}$ and $|w-\lambda_{+}|\geqslant\tau$. We have

$E\leqslant\lambda_{-}$ and $|w-\lambda_{-}|\geqslant\tau$. We have

$|w|\geqslant\tau$, $|w-\lambda_{+}|\geqslant\tau$ and $\lambda_{-}\leqslant E\leqslant\lambda_{+}$. We have

Here Case 1 covers the regime $E\geqslant\lambda_{+}$ and $w$ is far away from $\lambda_{+}$. Case 2 concerns the regime $E\leqslant\lambda_{-}$ and $w$ is far away from $\lambda_{-}$. Case 3 is for $w$ near $\lambda_{\pm}$. Finally Case 4 is for $w$ not covered by the first three cases.

The following lemma concerns the two cases covered in Lemmas 4.1 and 4.2, i.e., $z$ is either strictly inside or outside of the unit disk.

There exists $\tau_{0}>0$ such that for any $\tau\leqslant\tau_{0}$ if either the conditions $|z|\leqslant 1-\tau$ and $|w|\leqslant\tau^{-1}$ hold or the conditions $|z|\geqslant 1+\tau$, $|w|\leqslant\tau^{-1}$, $\operatorname{Re}\omega\geqslant\lambda_{-}/5$ hold, then we have the following three bounds concerning $m_{\rm c}$ (all constants in the following estimates depend on $\tau$):

Proof of Theorem 2.2, local circular law in the bulk

Our main tool in this section will be Theorem 3.4, which critically uses the hypothesis $||z|-1|\geqslant\tau$: when $z$ is on the unit circle the self-consistent equation (which is a fixed point equation for the function $g(m)=(1+wm(1+m)^{2})/(|z|^{2}-1)$ see (6.21) later in this paper) becomes unstable

We follow Girko’s idea [Gir1984] of Hermitization, which can be reformulated as the following identity (see e.g. [GuiKriZei2009]): for any smooth $F$

We will use the notation $z=z(\xi)=z_{0}+N^{-a}\xi$. Choosing $F=f_{z_{0}}$ defined in Theorem 2.2 and changing the variable to $\xi$, we can rewrite the identity (5.1) as

Recall that $\lambda_{j}(z)$’s are the ordered eigenvalues of $Y_{z}^{*}Y_{z}$, and define $\gamma_{j}(z)$ as the classical location of $\lambda_{j}(z)$, i.e.

Thanks to Proposition 3.1, one can check that uniformly in $|z|<1-\tau$, and also in the domain $1+\tau\leqslant|z|\leqslant\tau^{-1}$ ($\tau>0$), for any $\delta>0$ we have

It is known, by Lemma 4.4 of [Bai1997], that

Combining (5.4) and (5.5), we have proved (2.2) provided that we can prove (5.3). To prove (5.3), we need the following rigidity estimate which is a consequence of Theorem 3.4.

Suppose $\tau\leqslant||z|-1|\leqslant\tau^{-1}$ for some $\tau>0$ independent of $N$. Then for any $\zeta>0$, there exists $C_{\zeta}>0$ such that the following event holds with $\zeta$-high probability: for any $\varphi^{C_{\zeta}}<j<N-\varphi^{C_{\zeta}}$ we have

First, with (3.5) and the definition (3.4), for any $\zeta$ there exists $C_{\zeta}>0$ such that

holds with with $\zeta$-high probability. It also implies that for $\eta=\varphi^{C_{\zeta}}N^{-1}|m_{\rm c}|^{-1}$,

and define $\eta_{j}\geqslant 0$ as the smallest positive solution of

To prove (5.6), we choose $q$ to be supported in $[E_{1},E_{2}]$ such that $q(x)=1$ if $x\in[E_{1}+\eta_{1},E_{2}-\eta_{2}]$ and $|q^{\prime}|\leqslant C(\eta_{i})^{-1}$, $|q^{\prime\prime}|\leqslant C(\eta_{i})^{-2}$ if $|x-E_{i}|\leqslant\eta_{i}$. We now claim that

Combining (5.12) and (5.11), we have for any $1\leqslant j\leqslant N$,

which implies (5.6) with $C_{\zeta}$ in (5.6) replaced by $2C_{\zeta}$.

It remains to prove (5.12). Since $q$ and $\chi$ are real, with $\Delta m=m-m_{\rm c}$

using (3.5) and that on the support of $\chi^{\prime}$ is in $1\geqslant|\eta|\geqslant 1/2$.

For the second term in the r.h.s. of (5.13), with $|q^{\prime\prime}|\leqslant C\eta_{i}^{-2}$, (5.9) and (5.10), we obtain

We now integrate the third term in (5.13) by parts first in $E$, then in $\eta$ (and use the Cauchy-Riemann equation $\frac{\partial}{\partial E}\operatorname{Im}(\Delta m)=-\frac{\partial}{\partial\eta}\operatorname{Re}(\Delta m)$) so that

We therefore can bound the third term in (5.13) with absolute value by

where the last term can be bounded as the first term in r.h.s. of (5.13). By using (5.9) we have

where we used $\eta_{i}\geqslant N^{-C}$. Together with (5.14) and (5.15), we obtain (5.12) and complete the proof of (5.6).

Now we prove (5.7). Using (5.2) and Proposition 3.1, we have

Combining (5.18) with (5.6), we obtain (5.7).

For (5.8), the proof is similar to the above reasoning, but simpler: in this case $\gamma_{j}\sim 1$ for $j\leqslant N/2$. For $j\geqslant N/2$, $\gamma_{j}$ is bounded as (5.17), and one can check if $1+\tau\leqslant|z|\leqslant\tau^{-1}$, Proposition 3.1, we have

We return to the proof of the local circular law, Theorem 2.2. We now only need to prove (5.3) from Lemma 5.1. From (5.7) and (5.8), we have

Notice that, for large enough $C$, there is a constant $c>0$ such that for any $j$ we have

with probability larger than $1-\exp({{-N^{c}}})$ (for this elementary fact, one can for example see that the entries of $X$ are smaller that $1$ with probability greater than $1-\vartheta^{-1}e^{-N^{\vartheta}}$ by the subexponential decay assumption (2.1) and then use $\sum\lambda_{j}=\operatorname{Tr}Y^{*}Y$), so together with the above bounds on $\left|\log\lambda_{j}(z)-\log\gamma_{j}(z)\right|$ this proves that for any $\zeta>0$, there exists $C_{\zeta}>0$ such that

with $\zeta$-high probability. Furthermore, one can see that or estimates hold uniformly for $z$’s in this region.

On the other hand, the following important Lemma 5.2 holds, concerning the smallest eigenvalue. It implies that

holds uniformly for $z$ in any fixed compact set. It is easy to check that for any $\delta>0$, for large enough $N$,

Hence we can extend the summation in (5.19) to all $j\geqslant 1$, which gives (5.3) and completes the proof of Theorem 2.2.

Under the same assumptions of Theorem 2.2,

holds uniformly for $z$ in any fixed compact set.

where $u_{k}(z)$ is a unit vector independent of $X(z)e_{k}$. By conditioning on $u_{k}(z)$, the result of this lemma is straightforward since the matrix entries have a density. ∎

Weak local Green function estimate

In this section, we make a first step towards Theorem 3.4, with a weaker version of it, stated hereafter.

This theorem will be proved in the subsequent subsections.

By definition, $m^{(\emptyset,\emptyset)}=m$. Since the eigenvalues of $Y^{*}Y$ and $YY^{*}$ are the same except the zero eigenvalue, it is easy to check that

For $i,j\neq k$ ( $i=j$ is allowed) we have

By the row-column reflection symmetry, we only need to prove those formulas involving $G$. We first prove (6.4). In [ErdYauYin2010PTRF]-[ErdYauYin2010Adv], was proved a lemma concerning Green functions of matrices and their minors. This lemma is stated as Lemma B.2 in Appendix B. Let

We now prove (6.5). Recall the rank one perturbation formula

where ${\bf{v}}$ is a row vector and ${\bf{v}}^{*}$ is its Hermitian conjugate. Together with

By (6.3), (6.10) holds for $m_{\mathcal{G}}^{(i,\emptyset)}$ as well. Similar arguments can be used to prove (6.6) for $m_{\mathcal{G}}^{(i,j)}$, $m_{G}^{(i,j)}$ and the general cases. This completes the proof of Lemma 6.3. ∎

The next step is to derive equations between the matrix and its minors. The main results are stated as the following Lemma 6.5. We first need the following definition.

By the row-column reflection symmetry, we only need to prove the $G$ part of this lemma. Furthermore, for simplicity, we prove the case $T=\emptyset$, the general case can be proved in the same way.

We first prove (6.11). Let $H=Y^{*}Y$. Similarly to (6.7) and (6.8), we define $G^{[i]}$ and $H^{[i]}$. Then using (B.2) and (6.9), we have

From the definition of $H$, we have $h_{ik}={\bf{y}}_{i}^{*}{\bf{y}}_{k}$. Then

We now prove (6.12). As above, using now (B.3), we have

Then using (6.16) again, we obtain (6.12). ∎

2 The self-consistent equation and its stability.

We now derive the self-consistent equation for $m(w)$ and its stability estimates. Following [ErdYauYin2010Adv], we introduce the following control parameter:

The quantity $|m_{\rm c}|^{-1}\Psi$ will be our controlling small parameter in this paper.

Suppose $|z|\leqslant 1-\tau$ for some $\tau>0$. Then there exists a small constant $\alpha>0$ independent of $N$ such that if the estimate

holds for some $|w|\leqslant C$ on a set $A$ in the probability space of matrix elements for $X$, then in the set $A$ we have with $\zeta$-high probability

By (4.15), (4.16) and (6.20), for $|z|\leqslant 1-t$ the following inequalities hold on the set $A$:

Furthermore, using (6.22), (4.15), (4.16), (6.20) and (3.1), we have in the set $A$

Moreover, we have from (C.1) that with $\zeta$-high probability in $A$

where we have used (6.22) and $|m_{c}|\sim|w|^{-1/2}$. Together with (6.23), we thus have with $\zeta$-high probability

Using this estimate, (6.6) and (6.29), we can estimate $\mathcal{Z}_{i}:=\mathcal{Z}_{i}^{(\emptyset)}$ by

We can now use (6.32), (6.29) and (6.6) to estimate the right hand side of (6.11) such that

where $\mathcal{E}_{1}$ and $\mathcal{Z}_{i}$ are bounded in (6.30) and (6.32) and $\mathcal{E}_{2}$ is bounded by

In the last inequality, we have used (6.24) to bound $1+m-\frac{|z|^{2}}{w(1+m)}$ and (4.15) for $m_{\rm c}$.

Summing over the index $i$ in (6.34), we have

Together with the assumption (6.20) on $\Lambda$ and (4.15) on the order of $m_{\rm c}$, this proves (6.21). ∎

Comparing (6.38) with (6.34), we have proved (6.37).

We will omit $A$ in the following argument.

We define for any sequence $A_{i}$ ($1\leqslant i\leqslant N$) the quantity

In application, we often use $A=Z$ or $A=\mathcal{Z}$. Define

The following lemma is our stability estimate for the equation $\mathcal{D}(m)=0$. Notice that it is a deterministic result. It assumes that $|\mathcal{D}(m)|$ has a crude upper bound and then derives a more precise estimate on $\Lambda=|m-m_{c}|$.

Suppose that, for a fixed $E$ with $0\leqslant E\leqslant C$ for some constant $C$ independent of $N$, (6.20) and the estimate

The three upper bounds (i.e., the first inequalities in (6.42)-(6.44)) can be summarized as

By definition of $\Upsilon$ (6.21), we have

By definition of $P_{w,z}$, we can express $A$ and $B$ by

Case 1: In this case, we claim that the following estimates concerning $A$ and $B$ hold:

Since $A$ and $B$ are explicit functions of $m_{\rm c}$, equation (6.46) is just properties of the solution $m_{\rm c}$ of the third order polynomial $P_{w,z}(m)$. We now give a sketch of the proof. Consider first the case $|w|\ll 1$. Then (6.46) follows from (4.15), (4.16), (4.9) and the definitions of $A$ and $B$.

We now assume that $w\sim 1$ . Clearly, $|B|\leqslant\operatorname{O}(1)\sim|w^{1/2}|$, which gives (6.46) for $B$. To prove $|A|\geqslant C/M$, by definition of $m_{\rm c}$ (3.1), we have $w=\frac{-1-m_{\rm c}+m_{\rm c}|z|^{2}}{m_{\rm c}(1+m_{\rm c})^{2}}$. Thus we can rewrite $A$ as

By (4.15) and (4.17) (where $\alpha=\sqrt{1+8|z|^{2}}$), we obtain (6.46).

We now prove (6.42) by contradiction. If (6.42) is violated then with $u=m-m_{c}$ we have

where $M$ is a large constant in the last inequality. By (6.41) and (4.15), $|\Upsilon|\leqslant C\delta/|w|$. Thus we have

which is a contradiction provided that $M$ is large enough.

Case 2: ${\varepsilon}^{2}:=\kappa+\eta\leqslant 1/M^{2}$. Note in this case $w\sim 1$. Then by (4.3) we have

where the last equation can be checked by direct computation and we used $|z|^{2}<1-t<1$. There is a more intrinsic reason why the last equation for $A$ holds. Notice that $\lambda_{+}$ is a point that the polynomial $P_{w,z}(m)|_{w=\lambda_{+}}$ has a double root. Therefore, we have $0=P^{\prime}_{w,z}(m_{\rm c}(\lambda_{+},z))=A(\lambda_{+},z)$.

Notice that in the case $\kappa+\eta$ is small enough, we can approximate $A(w,z)$ by linearizing w.r.t. $w=\lambda_{+}$. Thus by the defining equation $P^{\prime}_{w,z}(m_{\rm c}(\lambda_{+},z))=A(\lambda_{+},z)$, we have

where we have used that $P_{w,z}^{\prime\prime}(m_{\rm c}(\lambda_{+},z))=B(\lambda_{+},z)\sim 1$, $\frac{\partial P_{w,z}}{\partial w}(m_{\rm c}(\lambda_{+},z))\sim 1$ and, by (4.3), that $(m_{\rm c}(w,z)-m_{\rm c}(\lambda_{+},z))\sim\sqrt{\kappa+\eta}$. While we can also check the conclusion of (6.48) by direction computation, the current derivation provides a more intrinsic reason why it is correct.

Case 2a: Suppose (6.43) is violated. We first choose $M$ large enough so that $|m_{\rm c}(1+m_{\rm c})|\leqslant M^{1/4}$ in this regime. Then by (6.47) and (6.48), with $w\sim 1$, we have

which is a contradiction provided that $M$ is large enough. Here we have used that, by the restriction of ${\varepsilon}$ and $\delta$ in (6.43) that ${\varepsilon}\geqslant M^{3/4}\sqrt{\delta}$, $M$ is large enough constant and $\delta\ll 1$.

Case 2b: Suppose (6.44) is violated. Similarly we have

which is a contradiction. Here we have used, by the restriction of ${\varepsilon}$ and $\delta$ in (6.44) and $M$ is large enough constant, that $C_{2}{\varepsilon}\leqslant C_{2}M^{3/4}\sqrt{\delta}\leqslant M\sqrt{\delta}/20$. ∎

With a slighter strong condition on $\delta$ and an initial estimate $\Lambda\ll 1$ when $\eta\sim 1$, the first inequalities in (6.42)-(6.44), i.e., (6.45), always hold. We state this as the following Corollary, which is a deterministic statement.

Suppose that the assumptions of Lemma 6.9 hold. If we have

Suppose that as $\eta$ decreases, we get to Case 2a. Notice that when we decrease $\eta$, by the conditions on ${\varepsilon}$ we will not go back to Case 1 from either Case 2a or Case 2b. For any ${\varepsilon}\leqslant 1/M$ with $M$ large, we have

Hence at the transition point from Case 1 to Case 2a, the inequality $\Lambda(E+i\eta)\leqslant\frac{M\delta}{{{\varepsilon}}}$ holds. Thus by continuity of $\Lambda$, the bound $\Lambda(E+i\eta)\leqslant\frac{M\delta}{{{\varepsilon}}}$ in (6.44) holds until we leave Case 2a.

It is possible that we cross from Case 2a to Case 2b. At the transition point, we have $\delta=\frac{{\varepsilon}^{2}}{M^{3/2}}$ and thus

for $M$ large. Hence the first inequality of Case 2b, i.e., $\Lambda\leqslant M\sqrt{\delta}$ holds. By continuity, this bound continues to hold unless we leave Case 2b. Since $\delta$ is decreasing in $\eta$ when ${\varepsilon}$ is small, once we get to Case 2b, we will not go back to Case 2a (or Case 1 as explained before).

It is possible that the Case 2a is omitted and we get to Case 2b directly from Case 1. Notice that ${\varepsilon}=1/M$ at such a transition point and we have $|w|\sim 1$. Furthermore, by (6.40), we get $\delta\leqslant 1/\log N$ at the transition point. Putting these together, we have for $M$ large,

Hence the bound $\Lambda(E+i\eta)\leqslant M\sqrt{\delta}$ in (6.44) holds. ∎

3 The large η𝜂\eta case.

Our method to estimate the Green functions and the Stieltjes transform is to fix the energy $E$ and apply a continuity argument in $\eta$ by first showing that the crude bound in Lemma 6.9 holds for large $\eta$. In order to start this scheme, we need to establish estimates on the Green functions when $\eta=\operatorname{O}(1)$. This is the main focus of this subsection. We start with the following lemma which provide a crude bound on the Green functions.

For any $w\in{\rm S}(0)$ and $\eta>c>0$ for fixed $c$, we have the bound

for some $C>0$. Notice that this bound is deterministic and is independent of the randomness.

The main result of this subsection is the following bound on $\Lambda$.

For any $\zeta>0$ and ${\varepsilon}>0$, we have

From (6.25)-(6.27), for $\eta=\operatorname{O}(1)$ we have

From (6.49), we have $|G_{ij}|+|\mathcal{G}_{ij}|\leqslant\eta^{-1}\leqslant\operatorname{O}(1)$ and $|m_{G}^{(i,i)}|\leqslant\operatorname{O}(1)$. Hence the large deviation estimate (6.27) becomes, with $\zeta$-high probability,

By an argument similar to the one used in (6.51), we can estimate $\mathcal{Z}_{i}$ by

for any ${\varepsilon}>0$ with $\zeta$-high probability. This implies that, with $\zeta$-high probability,

For any $\eta$ fixed, we claim that the following inequality between the real and imaginary parts of $m$ holds:

Assume that $\operatorname{Im}m\leqslant c(\log N)^{-1}$. From (6.53), we have $|m|\leqslant c(\log N)^{-1/2}$. Together with $\operatorname{Im}w=\eta\sim 1$,

for some constant $C$. This contradicts $|m|\leqslant c(\log N)^{-1/2}$ and we can thus assume that $\operatorname{Im}m\geqslant c(\log N)^{-1}$ when $\eta\sim 1$ and $w=\operatorname{O}(1)$. In this case, we also have

Then (6.52) implies for any ${\varepsilon}>0$ that with $\zeta$-high probability

Summing up all $i$, we have the following equation for $m$ with $\zeta$-high probability:

We can rewrite this equation into the following form:

It can be checked (with computer calculation or rather complicated but elementary algebraic calculation) that for $0\leqslant E\leqslant 5\lambda_{+}$ and $\eta=O(1)$, the third order polynomial $P_{w,z}(m)$ has no double root and there is only one root with positive real part. We denote this root by $m_{1}$ and the other two roots by $m_{2}$ and $m_{3}$. For $0\leqslant E\leqslant 5\lambda_{+}$ and $t\leqslant\eta\leqslant t^{-1}$ for any $t$ fixed, the three roots are separate by order one due to compactness. Since there is no double root, we have $|P^{\prime}_{w,z}(m_{1})|\geqslant c>0$ whenever $0\leqslant E\leqslant 5\lambda_{+}$ and $t\leqslant\eta\leqslant t^{-1}$. Thus the stability of (6.54) is trivial and we have proved that in this range of parameters

for any ${\varepsilon}>0$ with $\zeta$-high probability. ∎

4 Proof of the weak local Green function estimates.

In this subsection, we finish the proof of Theorem 6.1. We fix an energy $E$ and we will decrease the imaginary part $\eta$ of $w=E+i\eta$. Recall all stability results are based on assumption (6.20), i.e., $\Lambda\leqslant\alpha|m_{c}|\sim\alpha|w|^{-1/2}$ for some small constant $\alpha$, which so far was established only for large $\eta$ in (6.50). We would like to know that this condition continue to hold for smaller $\eta$. More precisely, suppose that (6.20) holds in a set $A$ for all $w=E+\eta i$ with $\eta\in[\widetilde{\eta},10]$ where $\widetilde{\eta}$ satisfies

for all $w=E+i\eta_{j}$ for all $1\leqslant j\leqslant n$. Since $\Lambda(E+i\eta)$ is continuous in $\eta$ at a scale, say, $N^{-10}$, (6.56) holds for all $\eta\in[\widetilde{\eta},10]$ with $\zeta$-high probability in $A$. Hence for $\widetilde{\eta}$ satisfying (6.55) the estimate (6.41) holds with

With this choice, we can check that the assumption on $\delta$, (6.40), holds as well. Furthermore $\delta$ is decreasing in $\eta$ when ${\varepsilon}=\sqrt{\kappa+\eta}$ is small enough. By Corollary 6.10, (6.45) holds all $\eta\in[\widetilde{\eta},10]$.

For $|z|<1-t$ for some $t>0$, if $\kappa\ll 1$ then $|w|\sim 1$ and (6.45) implies

If $\kappa\geqslant c>0$ for some $c>0$ then

By definition of $\Psi$, (6.58) and $m_{c}\sim|w^{-1/2}|$, we have

Using the restriction on $\eta$ so that $N\eta\geqslant|w|^{1/2}\varphi^{5Q_{\zeta}}$, we have

With (6.57) and (6.59), we have thus proved that

To conclude Theorem 6.1, it remains to prove the estimate on the off-diagonal elements. Recall the identity (6.12) for $G_{ij}$ and the equations (C.3) and (C.4). We can estimate the off-diagonal Green function by

Here we have used $|G_{ii}G^{(i,\emptyset)}_{jj}|=O(|w|^{-1})$, which follows from (6.36), $\Lambda\ll m_{c}$ and $|m_{c}|\sim|w^{-1/2}|$

where we have used (C.4) and that, by definition, $\operatorname{Im}G^{(ij,ij)}_{ii}=0=\operatorname{Im}G^{(ij,ij)}_{jj}$. Therefore, we have with $\zeta$-high probability,

where we also used $|\mathcal{G}_{ii}^{(ij,\emptyset)}\mathcal{G}^{(ij,i)}_{jj}|\leqslant C|m_{c}|^{2}\leqslant C|w|^{-1}$. Together with (6.61) and (6.36), we have proved that with $\zeta$-high probability

Proof of the strong local Green function estimates

Lemma 6.7 provides an error estimate to the self-consistent equation of $m$ linearly in $\Psi$. The following Lemma improves this estimate to quadratic in $\Psi$. This is the key improvement leading to a proof of the strong local Green function estimates, i.e., Theorem 3.4.

For any $\zeta>1$, there exists $R_{\zeta}>0$ such that the following statement holds. Suppose for some deterministic number $\widetilde{\Lambda}(w,z)$ (which can depend on $\zeta$) we have

Notice that the probability deteriorates in the exponent by a $(\log N)^{-2}$ factor.

We remark that, by Lemma 4.1, $\operatorname{Im}m_{\rm c}\ll|m_{\rm c}|$ when $\eta+\kappa\ll 1$. Hence we have to track the dependence of $\operatorname{Im}m_{\rm c}$ carefully in the previous Lemma. This is one major difference between the weak and strong local Green function estimates. Similar phenomena occur for the Stieltjes transforms of the eigenvalue distributions of Wigner matrices. Lemma 7.1 will be proved later in this section; we now use it to prove Theorem 3.4. We first give a heuristic argument.

Suppose that we have the estimate (7.2) with $\widetilde{\Psi}$ replaced by $\Psi$. We assume $\Lambda\geqslant(N\eta)^{-1}$ for convenience so that $\Psi^{2}\sim(\operatorname{Im}m_{\rm c}+\Lambda)/(N\eta)$ (If this assumption is violated then then (3.5) holds automatically and we have nothing to prove). Then we can apply Corollary 6.10 by choosing

which implies (6.45). Consider first the case $\kappa+\eta\sim\operatorname{O}(1)$. Using (6.45) with the choice of $\delta$ in (7.3) and $\kappa+\eta+\delta\geqslant\operatorname{O}(1)$, we have

When $\eta$ satisfies the condition (6.55), the coefficient of $\Lambda$ on the right side of the last equation is smaller than $1/2$. Hence, using $\operatorname{Im}m_{\rm c}\leqslant|m_{\rm c}|\leqslant C|w|^{-1/2}$ (see Proposition 3.2), we have

We now consider the case $\kappa+\eta\ll 1$ and thus $|w|\sim\operatorname{O}(1)$. From the first inequality of (6.45), we have

Also, in the regime $\kappa+\eta\ll 1$, (4.7) asserts that

Using the choice of $\delta$ in (7.3), we have

where we have used (7.4) to absorb the last term involving $\Lambda$ in the last inequality with a change of constant $C$. This completes the heuristic proof of Theorem 3.4. We now give a formal proof of this theorem assuming Lemma 7.1.

We first prove (3.6) assuming (3.5). By (6.63) and the definition of $\Psi$, we have for $i\neq j$,

where we have used (3.5) in the last step. This proves (3.6).

holds with the probability larger than $1-\exp(-\varphi^{\zeta+5}(\log N)^{-2})$. Notice that the application of Lemma 7.1 causes the probability in the exponent to deteriorate by a $(\log N)^{-2}$ factor.

Using (7.6), we can apply Corollary 6.10 with

holds with the probability larger than $1-\exp(-\varphi^{\zeta+5}(\log N)^{-2})$. We have thus proved (3.5) provided that $\kappa+\eta\geqslant(\log N)^{-1}$.

holds with probability larger than $1-\exp(-\varphi^{\zeta+5}(\log N)^{-2})$. Here $C_{1}$ depends only on $C_{0}$. From the definition of $\delta_{1}$ and $\Psi_{1}$, we have

holds with the probability larger than $1-\exp(-\varphi^{\zeta+5}(\log N)^{-2})$ for some $C_{3}$. Notice that we have used $N\eta\geqslant\varphi^{5R_{\zeta}}$ in the last step in (7.8).

holds with the probability larger than $1-\exp(-\varphi^{\zeta+5}(\log N)^{-4})$. Notice that the last constant $C_{3}$ is the same as the one appears in (7.8) and it does not change in the iteration procedure. We now iterate this process $K$ times to have

holds with the probability larger than $1-\exp(-\varphi^{\zeta+5}(\log N)^{-2K})$. We need $K$ so large that

On the other hand, we need $K$ small enough so that

We note that it also guarantees (7.1), since $\varphi^{\zeta+5}\geqslant p_{1}\geqslant p_{2}\geqslant\cdots\geqslant p_{K}\geqslant\varphi$. We choose $K=\log\log N/\log 2$ and we have thus proved that

with the probability larger than $1-\exp(-\varphi^{\zeta})$ which implies (3.5) when $\kappa+\eta\leqslant(\log N)^{-1}$. This completes the proof of Theorem 3.4. ∎

The first step in proving Lemma 7.1 is to derive a second order self-consistent equation which identifies the first order dependence of the correction in the self-consistent equation derived in Lemma 6.7. The second error terms will be bounded by $\Psi^{2}$; the first order terms are of the forms of averages of $Z^{(i)}_{i}$ and $\mathcal{Z}_{i}$. In Lemma 7.3, the averages of $Z^{(i)}_{i}$ and $\mathcal{Z}_{i}$ will be estimated by $\Psi^{2}$. This improvement from the naive order $\Psi$ to $\Psi^{2}$ is the key ingredient to obtain the strong local law. We remark that $\operatorname{Im}m_{\rm c}\ll|m_{\rm c}|$ when $\eta+\kappa\ll 1$. Hence the dependence of $\operatorname{Im}m_{\rm c}$ verses $m_{c}$ has to be tracked carefully. We now state the second order self-consistent equation: as the following lemma.

We first take the inverse of both sides of (6.33) and sum up $i$ to get, with $\zeta$-high probability,

where we have used (6.30) and the bound (6.22). Recall the estimates of $\mathcal{Z}_{i}$ and $Z^{(i)}_{i}$ by $\Psi$ in (6.27) and (6.32). Hence we have

where $b\geqslant 5Q_{\zeta}$ and $Q_{\zeta}$ is defined in Lemma C.1. We now perform the expansion $G_{ii}^{-1}=[(G_{ii}-m)+m]^{-1}$ to have

Using this approximation in (7.13), we have

The diagonal element $G_{ii}$ can be estimated by (7.14) so that

Notice that only the imaginary part of $m_{\rm c}$ appears through $\Psi$ instead of $m_{\rm c}$ which can be much bigger near the spectral edge.

We now estimate the last term in (7.16). Notice that $\mathcal{G}^{(i,\emptyset)}$ is the Green function of the matrix $A^{+}A$ where $A=(Y^{(i,\emptyset)})^{*}$. Then $m^{(i,i)}$ is the Green function of $A^{(i,),+}A^{(i,)}$ where we have used $A^{(i,)}=Y^{(i,i)}$. Thus we can apply (7.17) (which holds for matrices of the form $A^{+}A$ with A not necessarily a square matrix) to get

Inserting (7.18) and (7.19) into (7.15), we obtain

To conclude Lemma 7.2, we choose $C_{\zeta}=2Q_{\zeta}$ and it remains to prove $|\frac{1}{m_{\rm c}^{3}}\Psi^{2}|\geqslant\operatorname{O}(N^{-1})$. By definition of $\Psi$ and the fact that $|m_{\rm c}|\sim|w|^{-1/2}$ (4.15), this inequality follows from the following property of $\operatorname{Im}m_{c}$:

This estimate on $\operatorname{Im}m_{c}$ is a direct consequence of (4.2), (4.7), (4.9) and (4.10). This completes the proof of Lemma 7.2 ( with $C_{\zeta}$ increasing by 1).

We now estimate the averages $[\mathcal{Z}]$ and $[Z_{\ast}^{\ast}]$. Our goal is to catch cancellation effects due to the average over the indices $i$. This is the content of the next lemma, to be proved in next subsection. Clearly this lemma completes the proof of Lemma 7.1.

For any $\zeta>1$, there exists $R_{\zeta}>0$ such that the following statement holds. Suppose for some deterministic number $\widetilde{\Lambda}(w,z)$ (which can depend on $\zeta$) we have

where $\widetilde{\Psi}$ is defined in (7.2).

2 Strong bounds on [Z]delimited-[]𝑍[Z].

In this subsection, we prove Lemma 7.3. The main tool is the abstract cancellation Lemma D.1.

We first perform a cutoff for all random variables $X_{ij}$ in $X$ so that $|X_{ij}|\leqslant N^{10}$. Due to the subexponential decay assumption, the probability of the complement of this event is $e^{-N^{c}}$, which is negligible.

Define $P_{i}$ and $\mathcal{P}_{i}$ as the operator for the expectation value w.r.t. the $i$-th row and $i$-th column. Let

With this convention and Lemma 6.5, we can rewrite $\mathcal{Z}_{i}$ and $Z_{i}^{(i)}$, from Definition 6.4, as

Choosing $C_{\zeta}=2D_{\zeta}+20\zeta$, one can see that (7.21) follows from (7.20), (7.30) and the Markov inequality.

It remains to prove (7.28) and (7.29). We prove (7.28) first. For simplicity, we assume that ${A=\{1,\ldots,\lvert A\rvert\}}$. Denote the first $|A|$ column of $Y_{z}$ by ${\bf a}$ so that $\bf a$ is a $N\times|A|$ matrix. Similarly, denote by $B$ the matrix obtained after removing the first $K$-columns of $Y$. Then we have the identity

Recall the identity (6.16): for any matrix $M$,

We will prove $\|R\|\ll 1$ with high probability. Using (3.1), $\Lambda\ll m_{\rm c}$ (7.24) and (6.6), we have

With (7.31) and the definition of $R$, we have $-w\alpha G_{ij}=[(I+R)^{-1}]_{ij}$ for $i,j\in A$. Therefore,

Then, together with (7.32), (7.24) and $m_{c}\sim|w^{-1/2}|\sim\alpha$, we have thus proved that, in $\Xi$,

where we used $|A|\leqslant p\leqslant\varphi^{2\zeta}$ and $U_{A}$ is a linear combination of the following products of $(R^{j})_{ii}$’s

Define $\Omega_{A}$ as the probability space for the columns $\{{\bf{y}}_{k}:k\in A\}$ and $\Omega_{A^{c}}$ the one for the columns $\{{\bf{y}}_{k}:k\in A^{c}\}$. Then the full probability space $\Omega$ equals to $\Omega=\Omega_{A}\times\Omega_{A^{c}}$. Define $\pi_{A^{c}}$ to be the projection onto $\Omega_{A^{c}}$ and $\Xi^{*}=\left(\pi^{-1}_{A^{c}}\cdot\pi_{A^{c}}\cdot\Xi\right)$. Then ${\bf 1}(\Xi^{*})$ is independent of $\{{\bf{y}}_{k}:k\in A\}$. Hence we can extend (7.35) to

By (7.33), $|\mathcal{Z}_{i,A}|\leqslant\operatorname{O}(|w|^{-1/2}(|A|\varphi^{D_{\zeta}+2\zeta}|w|^{1/2}\widetilde{\Psi})^{{|A|}})$ in $\Xi$. We now prove that

By (7.22), we have $\left(wG_{ii}\right)^{-1}=\operatorname{O}(N^{C})$. Notice that $\alpha$ is independent of $\{{\bf{y}}_{k}:k\in A\}$. Since $\alpha\sim|w^{-1/2}|$ in $\Xi$, the same asymptotic holds in $\Xi^{*}\setminus\Xi$. By definitions of $U_{A}$ (7.33) and $R$, and the assumption $X_{ij}=O(N^{C})$, we obtain (7.36) and this completes the proof of (7.28). Similarly, we can prove (7.29) and this completes the proof of Lemma 7.3.

In this appendix we are going to prove the lemma 4.1, 4.2 and 4.3. We can solve $m_{\rm c}$ explicitly by the following formula.

By definition, $m_{\rm c}$ is an analytic function, so we only need to prove (A.1). By definition, $m_{\rm c}$ is one of the three solutions of (3.1), and needs to have positive imaginary part. Solving explicitly this degree three polynomial equation proves that there is just one such solution, with the limit A.1 close to the critical axis. ∎

With Lemma A.1 and (A.2), one can easily prove Proposition 3.1.

Moreover, recall that $\alpha=\sqrt{1+8|z|^{2}}$, so (still in the first case)

We also have easily $|m_{\rm c}|\sim 1$ easily from (A.3), we therefore obtained the l.h.s. of (4.2). Similarly, one can prove $\operatorname{Im}m_{\rm c}\sim\eta$ thanks to

and complete the proof for the first case.

For the second case, it is easy to prove (4.3) when $w=\lambda_{+}$, as we did from an explicit calculation. Then one obtains (4.3) by expanding $m_{\rm c}$ around $m_{\rm c}(\lambda_{+},z)$, using (3.1). The estimate (4.7) directly follows from (4.3).

Similarly, for the third case, first $m_{\rm c}=\infty$, i.e., $m_{\rm c}^{-1}=0$ when $w=0$, then one can easily obtain (4.8) in case 3 by solving (3.1) with expanding $m_{\rm c}^{-1}$ around $(m_{\rm c}(0,z))^{-1}$. The estimate (4.9) directly follows from (4.8). The fourth case follows from

and the properties of $\rho$ stated in proposition 3.1. ∎

This is similar to the proof of Lemma 4.1. ∎

We are going to prove this lemma in the case $|z|\leqslant 1-\tau$, the other cases can be proved similarly. Note first that (4.15) is a consequence of all possible cases in Lemma 4.1.

We now prove (4.16) in the four different cases, which have been classified in Lemma 4.1. In the first case, if additionally $\eta\sim 1$, as $0>\operatorname{Re}(m_{\rm c})>-1/2$, the l.h.s. in (4.16) is bounded by $\operatorname{O}(1)$, which implies (4.16). For the first case if $\eta$ is small enough, since $|\operatorname{Re}w|\sim(1+m_{\rm c})\sim 1$ and $|\operatorname{Im}(m_{\rm c})|\sim\eta$, so