The local circular law III: general case

The local circular law III: general case

Jun Yin

Department of Mathematics, University of Wisconsin-Madison Madison, WI 53706-1388, USA jyin@math.wisc.edu

Abstract

In the first part [9] of this article series, Bourgade, Yau and the author of this paper proved a local version of the circular law up to the finest scaleN^−1/2+εfor non-Hermitian random matrices at any point z∈Cwith||z| −1|> cfor any constantc >0independent of the size of the matrix. In the second part [10], they extended this result to include the edge case|z| −1 = o(1), under the main assumption that the third moments of the matrix elements vanish. (Without the vanishing third moment assumption, they proved that the circular law is valid near the spectral edge|z| −1 = o(1)up to scaleN^−1/4+ε.) In this paper, we will remove this assumption, i.e. we prove a local version of the circular law up to the finest scaleN^−1/2+εfor non-Hermitian random matrices at any pointz∈C.

AMS Subject Classification (2010): 15B52, 82B44 Keywords: local circular law, universality.

∗Partially supported by NSF grant DMS-1001655 and DMS- 1207961

1 Introduction and Main result

The circular law in random matrix theory describes the macroscopic limiting spectral measure of normalized non-Hermitian matrices with independent entries. Its origin goes beck to the work of Ginibre [18], who found the joint density of the eigenvalues of such Gaussian matrices. More precisely, for anN ×N matrix with independent entries^√1_Nzijsuch thatzijis identically distributed according to the measureµg= _π¹e^−|z|2dA(z) (dAdenotes the Lebesgue measure onC), its eigenvaluesµ1, . . . , µN have a probability density proportional

to Y

i<j

|µi−µj|²e^−NPk|µk|2 (1.1) with respect to the Lebesgue measure onC^N. These random spectral measures define a determinantal point process with the explicit kernel (see [18])

KN(z1, z2) =N

πe^{−N2(|z1|2+|z2|2)}

NX−1 ℓ=0

(N z1z2)^ℓ

ℓ! (1.2)

with respect to the Lebesgue measure onC. This integrability property allowed Ginibre to derive the circular law for the eigenvalues, i.e., _N¹ρ^(N)₁ converges to the uniform measure on the unit circle,

1

π1_|z|<1dA(z). (1.3)

This limiting law also holds for real Gaussian entries [14], for which a more detailed analysis was performed in [8, 17, 28].

For non-Gaussian entries, Girko [19] argued that the macroscopic limiting spectrum is still given by (1.3). His main insight is commonly known as theHermitization technique, which converts the convergence of complex empirical measures into the convergence of logarithmic transforms of a family of Hermitian matrices. If we denote the original non-Hermitian matrix byX and the eigenvalues ofX byµj, then for any C²function F we have the identity

1 N

XN j=1

F(µj) = 1 4πN

Z

∆F(z) Tr log(X^∗−z^∗)(X−z)dA(z). (1.4) Due to the logarithmic singularity at 0, it is clear that the small eigenvalues of the Hermitian matrix (X^∗−z^∗)(X−z)play a special role. A key question is to estimate the small eigenvalues of(X^∗−z^∗)(X−z), or in other words, the small singular values of(X−z). This problem was not treated in [19], but the gap was remedied in a series of papers. First Bai [3] was able to treat the logarithmic singularity assuming bounded density and bounded high moments for the entries of the matrix (see also [4]). Lower bounds on the smallest singular values were given in Rudelson, Vershynin [26, 27], and subsequently Tao, Vu [30], Pan, Zhou [23]

and Götze, Tikhomirov [20] weakened the moments and smoothness assumptions for the circular law, till the optimal L²assumption, under which the circular law was proved in [31]. On the other hand, Wood [33]

showed that the circular law also holds for sparse randomnbynmatrices where each entry is nonzero with probabilityn^α−1 where0< α61.

In the first part of this article [9], Bourgade, Yau and the author of this paper proved a local version of the circular law, up to the optimal scaleN^−1/2+ε, in the bulk of the spectrum. In the second part [10], they

extended this result to include the edge case, under the assumption that the third moments of the matrix elements vanish. (Without the vanishing third moment assumption, they also proved that the circular law is valid near the spectral edge|z| −1 = o(1)up to scaleN^−1/4+ε.) This vanishing third moment condition is also the main assumption in Tao and Vu’s work on local circular law [32]. In the current paper, we will remove this assumption, i.e. we prove a local version of the circular law up to the finest scaleN^−1/2+ε for non-Hermitian random matrices at any pointz∈C.

More precisely, we considered anN×N matrixX with independent real¹centered entries with variance N⁻¹. Letµj, j ∈ J1, NK denote the eigenvalues of X. To state the local circular law, we first define the notion ofstochastic domination.

Definition 1.1. Let W =W^(N) be a family of random variables andΨ = Ψ^(N) be a family of deterministic parameters. We say thatW is stochastically dominatedby Ψif for anyσ >0andD >0 we have

PhW> N^σΨi

6 N^−D (1.5)

for sufficiently largeN. We denote this stochastic domination property by W ≺ Ψ, or W = O_≺(Ψ).

Furthermore, LetU^(N) be a possibly N-dependent parameter set. We say W(u)is stochastically dominated byΨ(u)uniformly inu∈U^(N), if for anyσ >0 andD >0 we have

sup

u∈U^(N)

PhW(u)> N^σΨ(u)i

6 N^−D (1.6)

for uniformly sufficiently large N (may depends on σ andD).

Note: In the most cases of this paper, theU^(N) is chosen as the product of the index sets16i, j6N and some compact set inC².

In this paper, as in [9], [10] and [32], we assume that the probability distributions of the matrix elements satisfy the following uniform subexponential decay property:

sup

(i,j)∈J1,NK²

P

|√

N Xi,j|> λ

6ϑ⁻¹e^−λϑ (1.7)

for some constant ϑ > 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.

Note: most constants appearing in this work may depend onϑ, but we will not emphasize this dependence in the proof.

Letf :C→Rbe a fixed smooth compactly supported function, and fz0(µ) =N^2sf(N^s(µ−z0)), where z0 depends onN, andsis a fixed scaling parameter in[0,1/2]. LetD denote the unit disk. Theorem 2.2 of [9] and Theorem 1.2 of [10] assert that the following estimate holds: (Note: Herekfz0k¹=O(1))



N⁻¹X

j

fz0(µj)− 1 π

Z

D

fz0(z) dA(z)



≺ N^−1+2s, s∈(0,1/2], (1.8)

1For the sake of notational simplicity we do not consider complex entries in this paper, but the statements and proofs are similar.

[9] if ||z0| −1| > cfor some c >0 independent of N or [10] if the third moments of matrix entries vanish.

This implies that the circular law holds after zooming up to scaleN^−1/2+ε(ε >0) under these conditions.

In particular, there are neither clusters of eigenvalues nor holes in the spectrum at such scales. We note that in [9] and [10], the scaling parameter was denoted asa, but the letterawill be used as a fixed index in this work.

We aim at understanding the circular law for anyz0∈Cwithout the vanishing third moment assumption.

The following theorem is our main result.

Theorem 1.2. Local circular law: Let X be anN×N matrix with independent centered entries of variances 1/N. Suppose that the distributions of the matrix elements satisfy the subexponential decay property (1.7).

Let fz0 be defined as above (1.8)andD denote the unit disk. Then for any s∈(0,1/2]and any z0∈C, we

have 

N⁻¹X

j

fz0(µj)−1 π

Z

D

fz0(z)dA(z)



≺ N^−1+2s. (1.9)

Notice that the main new assertion of (1.9) is for the case: |z0| −1 = o(1) and the third moments not vanishing, since the other cases were proved in [9]and[10], stated in (1.8).

Remark: Shortly after the preprint [9] appeared, a version of local circular law (both in the bulk and near the edge) was proved by Tao and Vu [32] under the assumption that the first three moments of the matrix entries match a Gaussian distribution, i.e., the third moment vanish.

In the next section we will introduce our main strategy and improvements.

2 Proof of Theorem 1.2

Proof of Thm. 1.2: The bulk case of Thm. 1.2 was proved in Theorem 2.2 of [9]. Furthermore, it is easy to see that the results in Thm. 1.2 fors= 1/2 follow from the results in fors <1/2. Hence in this proof, we can assume that

||z0| −1|=o(1), s∈(0,1/2)

In the edge case, our Thm. 1.2 was proved in the Thm 1.2 of [10] with the vanishing third moment assumption.

Hence the goal of this paper is to improve the proof of Thm. 1.2 of [10]. One can easily check that in the proof of Thm. 1.2 of [10], the conditionEX_ij³ = 0was only used in the Lemma 2.13 of [10]. Therefore, we only need to prove a stronger version of Lemma 2.13 in [10], i.e., the one without vanishing third moment condition. More precisely, it only remains to prove the following lemma 2.2. (Here we use the same notations as in [10], except for the scaling parameter)

Before stating lemma 2.2, i.e., the stronger version of Theorem 1.2 of [10], we introduce some definitions and notations. First, we introduce the notation

Y :=Yz:=X−zI

where I is the identity operator. In the following, we use the notation A ∼ B when cB 6 |A| 6 c⁻¹B, wherec >0 is independent ofN. For any matrixM, we denoteM^T as the transpose ofM andM^∗ as the Hermitian conjugate. Usually we choosez−z0∼N^−s, hence we define the scaled parameterξ:

z=z0+N^−sξ, i.e., ξ:=N^s(z−z0)

Define the Green function ofY_z^∗Yz and its trace by, wherew∈CandImw >0, G(w) :=G(w, z) = (Y_z^∗Yz−w)⁻¹, m(w) :=m(w, z) = 1

N TrG(w, z) = 1 N

XN j=1

1

λj(z)−w. (2.1) Letmc:=mc(w, z)be the unique solution of

m⁻_c¹=−w(1 +mc) +|z|²(1 +mc)⁻¹ (2.2) with positive imaginary part. As proved in [9] and [10], for some regions of (w, z) with high probability, m(w, z)converges tomc(w, z)pointwise, asN → ∞. Letρc be the measure whose Stieltjes transform ismc. This measure is compactly supported andsuppρc = [max{0, λ₋}, λ+], where

λ_±:=λ_±(z) := (α±3)³

8(α±1), α:=p

1 + 8|z|². (2.3)

Note thatλ₋ has the same sign as |z| −1. It is well-known thatρc(x, z)can be obtained from its Stieltjes transformmc(x+ iη, z)via

ρc(x, z) = 1

πIm lim

η→0⁺mc(x+ iη, z) = 1

π1x∈[max{0,λ⁻},λ+]Im lim

η→0⁺mc(x+ iη, z).

(Some basic properties ofmc andρc were discussed in section 2.2 of [10]) Definition 2.1. φ, χ,I andZ_X,c^(f)

Let h(x)be a smooth increasing function supported on [1,+∞]withh(x) = 1for x>2 andh(x) = 0for x61. For anyε >0, define φonR₊ by (note: λ+ depends onz)

φ(x) :=φε,z(x) :=h(N^2−2εx) (logx)

1−h x

2λ+

. (2.4)

Let χ be a smooth cutoff function supported in[−1,1]with bounded derivatives and χ(y) = 1 for |y|61/2.

Recall dA denotes the Lebesgue measure onC, for any fixed function g defined onC, we define:

Z_X,c^(g) :=Z_X,c^(g)(z0, ε, s) :=N Z

∆g(ξ) Z

I

χ(η)φ^′(E) Re(m(w)−mc(w))dEdηdA(ξ), w=E+iη, z=z0+N^−sξ and

I:=Iε:=n

w∈C:N^−1+ε√

E6η, E>N^−2+2ε, |w|6ε, w=E+iηo

. (2.5)

Note: the conditionE>N^−2+2εwas not in the definition of theIused in [10], but clearly this condition is implied byφ^′(E)6= 0, i.e., our newI does not change the value ofZ_X,c^(g). One can also easily check:

w∈Iε =⇒ |w|^1/262N^1−εη (2.6)

With these notations and definitions, we claim the following main lemma. It is a stronger version of Lemma 2.13 in [10], i.e., the one without vanishing third moment condition.

Lemma 2.2. Under the assumptions of Theorem 1.2, there exists a constant C >0 such that for any small enough ε >0(independent of N), if ||z0| −1|6ε,s∈(0,1/2), then

Z_X,c^(f) ≺N^Cεcf, wherecf is a constant depending only on the functionf.

As mentioned above, in the proof of Thm. 1.2 of [10], the vanishing third moment condition was only used in the Lemma 2.13 of [10]. Therefore with the improved Lemma (2.2), one can obtain our main result theorem 1.2 as in [10].

In the next step, the lemma 2.2 will be reduced to lemma 2.4.

We note that the bounds proved in [10] forGij’s are not strong enough for our purpose in this paper.

Unfortunately we noticed that it seems impossible to improve these bounds in general cases. On the other hand, we found that though the behaviorsG’s andG’s are unstable in the region |m| 6(N η)⁻¹, they are very stable in the region|m| ≫(N η)⁻¹ and many stronger bounds can be derived in this region. Therefore, in the following proof, we separate theZX,cinto two parts: the one comes for the region|m|6(N η)⁻¹ and the one comes for the region |m| ≫ (N η)⁻¹. The first part can be easily bounded, since the m is small, so as its contribution to ZX,c. For the second part, we will apply Green’s function comparison method (which was first introduced in [15] for generalized Wigner matrix) and our new stronger bounds in the region

|m| ≫(N η)⁻¹.

On the other hand, the old Green’s function comparison method was not enough for our purpose, which is also the reason that in [10], the authors needed the extra assumption on the third moment of the matrix entries. In this work, we will introduce an improved Green’s function comparison method, which provides an extraN^−1/2 factor than the previous method. This idea was motivated from the work in [6].

Definition 2.3. tX andA^(f)_X

ForN×N matrixX, we define

tX:=tX(ε, w, z) :=N^−εN ηRem i.e.,

tX:=N^−εηRe Tr ((X^∗−z^∗)(X−z)−w)⁻¹, η= Imw

Now we extend the function h defined in Def. 2.1 to the whole real lane, i.e., h(x) =h(−x), but still use the same notation h(x). With these notations, we define:

A^(f)_X :=A^(f)_X (z0, ε, s) =N Z

∆f(ξ) Z

I

χ(η)φ^′(E)

h(tX) Rem−Remc

dEdηdA(ξ), (2.7)

wherez=z0+N^−sξ,w=E+iη,φ=φε,z andtX=tX(ε, w, z).

Note the only difference betweenA^(f)_X andZ_X,c^(f) is theh(tX)in front ofRem. Then the difference ofA^(f)_X andZ_X,c^(f) only comes from the regionh(tX)6= 1, i.e,|Rem|62N^ε(N η)⁻¹. Therefore, by the definitions of φwe have

|A^(f)_X −Z_X,c^(f)|6 Z

|∆f(ξ)| Z

I

χ(η)|φ^′(E)| 2N^ε(N η)⁻¹

dEdηdA(ξ)6N^Cεcf (2.8) where we used|(1−h(tX)) Rem|62N^ε(N η)⁻¹.

Proof of Lemma 2.2: With (2.8), it only remains to prove the following lemma.

Lemma 2.4. Under the assumptions of Theorem 1.2, there exists a constant C >0 such that for any small enough ε >0(independent of N), if ||z0| −1|6εands∈(0,1/2), then

A^(f)_X ≺N^Cεcf

wherecf is a constant depending only on the functionf.

In the next subsection, we will introduce the basic idea of proving Lemma 2.4. The rigorous proof will start from section 3.

2.1 Basic strategy of proving Lemma 2.4: Before we give the complete proof of this lemma, we introduce the basic idea and main improvement in the remainder of this section. Lemma 2.2 was proved in [10] under the vanishing third moment condition. With (2.8), that result implies that if Xij’s are Gaussian variables, for all16i, j6N, then for any fixed p∈2N,

E|A^(f)_X,c|^p≺N^Cεp, Xij ∼ N(0,1/N) (2.9) As one can see thatA^(f)_X,cis basically a linear functional ofm(w, z). Hence as in [10], we will apply the Green function comparison method to show that for sufficiently largeN,

E|A^(f)_X |^p6CE|A^(f)_X^′|^p+N^Cεp, (2.10) for any two different ensemblesX andX^′ whose matrix elements satisfy the condition of Theorem 1.2. To complete the proof for lemma 2.4, we will chooseX^′ to be the Ginibre ensemble, whose matrix elements are Gaussian variables. The X will be the general ensembles in lemma 2.4. Combining (2.9) and (2.10), with Markov inequality, one immediately obtains Lemma 2.4.

In applying the Green function comparison method, we estimate the expectation value of the functionals of Y, G= (Y^∗Y −w)⁻¹ and G = (Y Y^∗−w)⁻¹, i.e., EF(Y, G,G). In [10] and most previous applications of Green function comparison method, one can only bound the expectation value of these functionals with their stochastically dominations. For example, in [10], fori6=j and|w|^1/2≪(N η), one has

|(Y G)ij| ≺1

With this stochastically domain, the authors in [10] obtained that |E(Y G)ij| 6N^σ for anyσ > 0. In the present paper, under the condition|Rem| ≫(ηN)⁻¹, i.e.,h(tX)>0, we will first show an improved bound:

fori6=j and|w|^1/2≪(N η)

|h(tX)(Y G)ij| ≺ s

|w|^1/2 N η

Then using a new idea on Green’s function comparison method, we will show that the expectation value of this term will obtain an extra factorN^−1/2, i.e.,

|Eh(tX)(Y G)ij|6CN^−1/2+σ s

|w|^1/2

N η (2.11)

This extra factorN^−1/2plays a key role in our new proof. A similar method was used in the [6].

Now we explain the basic idea of proving (2.11)-type bounds, i.e.,where the extra N^−1/2 factor comes from. For simplicity we assume Xij ∈ R. Let Yz^(i,i) be the matrix obtained by removing i−th row and column ofYz, and define

G^(i,i):= ((Y_z^(i,i))^∗Y_z^(i,i)−w)⁻¹, G^(i,i):= (Y_z^(i,i)(Y_z^(i,i))^∗−w)⁻¹

We writeh(tX)(YzG)ij as the polynomials of thei−th row/column ofX: Xik, Xki (16k6N),G^(i,i) and G^(i,i), i.e.,

h(tX)(YzG)ij=P({Xik}^Nk=1, {Xki}^Nk=1, G^(i,i),G^(i,i)) + negligible error

where P is a polynomial. By definition, Xik, Xki are independent of G^(i,i) and G^(i,i). In this polynomial, we will show that the degrees of every monomials w.r.t. Xik andXki’s are always odd numbers. Therefore, in taking the expectation value, with assumption EXij = 0and |EX_ij^k|6O(N^−k/2), one will see an extra combination factor N^−1/2. The following simple example will show why the odd powers give an extra factorN^−1/2. Suppose we estimate EP

kstXikG^(i,i)_kl XisG^(i,i)_st Xit. SinceXik,Xki are independent ofG^(i,i) and G^(i,i), EXij = 0and |EX_ij^k| 6O(N^−k/2), the nonzero contributions only come from the terms where k=s=t, therefore

|EX

kst

XikG^(i,i)_kl XisG^(i,i)_st Xti|=|EX

k

XikG^(i,i)_kl XikG^(i,i)_st Xki|6CN^−1/2E(max

ab |G^(i,i)_ab |)²

On the other hand, withoutE, this term can only be bounded without thisN^−1/2factor (with large deviation theory).

|X

kst

XikG^(i,i)_kl XisG^(i,i)_st Xti|=|X

k

XikG^(i,i)_kl ||X

st

XisG^(i,i)_st Xti|6(logN)^C(max

ab |G^(i,i)_ab |)²

Note: one will not see thisN^−1/2factor if the degree is even number, e.g., EPXisG^(i,i)_st Xit. Based on this new idea, the main task of proving lemma 2.4 and (2.11)-type bounds is writing the functionals ofYz’s,G’s andG’s as the polynomials ofXik,Xki (16k6N) ,G^(i,i) andG^(i,i) for some16i6N, (up to negligible error) and counting the degree of each monomial.

3 Proof of Lemma 2.4

In this section, we apply the Green’s function comparison method to prove the Lemma 2.4. We will see the key input of proving Lemma 2.4 is the lemma 3.2. This new lemma is similar to (3.62)-(3.63) of [10], but without the third moments vanishing assumption. More precisely, the (3.62)-(3.63) of [10] is similar to the (3.4) of this work, and lemma 3.2 is the key step of proving (3.4). The proof of lemma 3.2 will start from section 4. In [10], the (3.62)-(3.63) can be easily proved by bounding the expectation value of these terms with their stochastically dominations. In this paper, as introduced in subsection 2.1, we will introduce a new comparison method to show that, for the contribution comes from Xij’s third moment, their expectation values have an extra factorN^−1/2, i.e., lemma 3.2.

First of all, we state the following lemma. It will be used to estimate the expectation value of some random variables which are stochastically dominated, but notL_∞ bounded.

Lemma 3.1. Let v = v^(N) be a family of centered random variables with variance 1/N, satisfying the sub exponential decay (1.7). LetAe=Ae^(N) andA=A^(N) be families of random variables. Suppose A≺1, and

A=PC

n=0Anvⁿ, where |An|6N^C for some fixed constantC >0. We also assume thatAeis independent of v and|Ae|6N^C for someC >0. Then for any fixedp∈N and fixed (small)δ >0,

|EAA ve ^p|6(E|Ae|)N^−p/2+δ+N^−1/δ for large enough N.

Note: HereAorAi’s may depend onv.

Proof of Lemma 3.1: By definition 1.1, the assumptionA≺1, and the fact thatv has sub exponential decay (1.7), for any fixedδ >0andD >0 there is a probability subsetΩsuch thatP(Ω)>1−N^−Dand

|1ΩAv^p|6N^−p/2+δ Then

|EAAve ^p|6(E|Ae|)N^−p/2+δ+|E_ΩcAAve ^p|6(E|Ae|)N^−p/2+δ+O(N^−D/2+2C)

for the second inequality, we used Cauchy Schwarz inequality. Choosing large enoughD, we complete the proof of lemma 3.1.

Because of this lemma, for any centered random variablesvwith variance1/N, satisfying the sub expo- nential decay (1.7), we define

M^C(v) :=

(

A:A= XC n=0

Anvⁿ, |An|6N^C )

(3.1) Now we return to prove Lemma 2.4.

Proof of Lemma 2.4: For simplicity, we assume that the matrix entries are real numbers. Let X and X^′be two ensembles which satisfy the assumption of Theorem 1.2. To prove Lemma 2.4, as we explained in the beginning of subsection 2.1 (near (2.10)), one only needs to show that for any fixed small enoughε >0, s∈(0,1/2), andp∈2N, if||z0| −1|6εthen

E|A^(f)_X |^p6CE|A^(f)_X^′|^p+N^Cεp, (3.2) for large enoughN. For integer k, 0 6k 6N², define the following matrix Xk interpolating between X^′ andX:

Xk(i, j) =

X(i, j) if k>N(i−1) +j X^′(i, j) if k < N(i−1) +j .

Note that X^′ =X0 and X = XN². As one can see that the difference between Xk and Xk−1 is just one matrix entry. We denote the index of this entry as (a, b) := (ak, bk) (ak, bk ∈ Z, 1 6 ak, bk 6N), here k= (ak−1)N+bk

Furthermore, we definetXk−1,tXk,A^(f)_Xk−1,A^(f)_Xk withXk−1 andXk, as in Def. 2.3. We are going to show that if this special matrix entry is in the diagonal line, i.e.,a=bthen

E

A^(f)_Xkp

−E

A^(f)_Xk−1p6N^−3/2

N^ε+ 2E

A^(f)_Xk−1p

(3.3) otherwise, i.e.,a6=b,

E

A^(f)_Xkp

−E

A^(f)_Xk−1p6N⁻²

N^ε+ 2E

A^(f)_Xk−1p

(3.4)

for sufficiently largeN (independent ofk). Clearly, (3.3) and (3.4) imply (3.2).

We are going to compare the these functionals corresponding to Xk and X_k−1 with a third one, corre- sponding to the matrixQehereafter with deterministic (a, b)entry. We define the followingN×N matrices (hereafter,Yℓ=Xℓ−zI,ℓ=kor k−1):

v=vabeab=X^′(a, b)eab, (3.5)

u=uabeab=X(a, b)eab, (3.6)

Qe=Xk−1−v=Xk−u, (3.7)

Q=Y_k−1−v=Yk−u, (3.8)

R= (Q^∗Q−wI)⁻¹ (3.9)

R= (QQ^∗−wI)⁻¹ (3.10)

S= (Y_k^∗₋₁Yk−1−wI)⁻¹ (3.11)

T = (Y_k^∗Yk−wI)⁻¹ (3.12)

Furthermore, we define t_Qe, A^(f)_e

Q with Q, as in Def. 2.3. To prove (3.3) and (3.4), we will estimatee A^(f)_Xk−1−A^(f)_X

Qe andA^(f)_Xk−A^(f)_X

Qe . First we introduce the notations

mS = 1

N TrS, mR= 1

N TrR, mT = 1 N TrT We note: with Cauchy’s interlace theorem, for someC >0,

|mS−mR|6C(N η)⁻¹, η= Imw (3.13) holds for anywandz. It implies

|A^(f)_Xk−1−A^(f)_e

Q |6C (3.14)

To estimateA^(f)_Xk−1−A^(f)_Qe , from (2.7), we have A^(f)_Xk−1−A^(f)_Qe =N

Z

∆f(ξ) Z

I

χ(η)φ^′(E)

h(tXk−1) RemS−h(t_Qe) RemR

dEdηdA(ξ), (3.15)

wherez=z0+N^−sξ,w=E+iη,φ=φε,z. RecalltXk−1 andt_Qeare defined withmS andmR respectively.

Applying Taylor’s expansion on the termh(tXk−1) RemS−h(t_Qe) RemRin (3.15) and lettingh^(k)be thekth derivative ofh, we have

h(tXk−1) RemS−h(t_Qe) RemR= X3 n=1

Bn(Q) (Ree mS−RemR)ⁿ+B4(Xk−1,Q) (Ree mS−RemR)⁴ (3.16) whereBn(Q)e (16n63) andB4(X_k−1,Q)e are defined as

Bn(Q) :=e 1

n!(N^1−εη)⁽ⁿ⁻¹⁾

nh⁽ⁿ⁻¹⁾(t_Qe) +h⁽ⁿ⁾(t_Qe)t_Qe

(3.17) B4(Xk−1,Q) :=e 1

24(N^1−εη)³

4h⁽³⁾(ζ) +h⁽⁴⁾(ζ)ζ

whereζis betweentXk−1 andt_Qe, and only depends ontXk−1,t_Qeandh. As one can see thatB1,B2andB3

are independent ofvab. For the definition ofB’s, we note that if n>1, then h⁽ⁿ⁾(x)6= 0 =⇒ x∼1

Therefore, with|h|61, we obtain the following uniform bounds forB’s:

|Bn|6(N^1−εη)⁽ⁿ⁻¹⁾, 16n64 (3.18) To estimate themS−mRin (3.16), we study the difference between mS and mRin the parameter set:

(k, z, w)∈Z×C² : 06k6N², ||z| −1|62ε, w∈Iε (3.19) Recall in (3.59) of [2] and the discussion below (3.61) of [2], it was proved that with the notations:

P1(Q) :=e 1

N Re −2(QR²)ab

(3.20) P2(Q) :=e 1

N Re wR^aa(R²)bb+ 2(QR²)ab(RQ^∗)ba+ (QR²Q^∗)aaRbb P3(Q) :=e

1

N Re −2(RQ^∗)²_ba(QR²)ab−2(RQ^∗)ba(QR²Q^∗)aaRbb−2(RQ^∗)bawR^aa(R²)bb−2wR^aaRbb(QR²)ab) the difference between RemS and RemR, i.e., (_N¹ Re TrS− N¹ Re TrR) can be written as (recall vab = X^′(a, b))

RemS−RemR= X3 n=1

Pn(Q)e ·(vab)³+P4(Xk−1,Q)e ·(vab)⁴, (3.21) whereP4(X_k−1,Q)e depends onX_k−1 andQ, and thee P’s can be bounded as

P1(Q), Pe 2(Q), Pe 3(Q), Pe 4(Xk−1,Q)e ≺(N η)⁻¹, (3.22) uniformly for(k, z, w) in (3.19). In [2], the uniformness was not emphasized, but it can be easily checked.

From (3.7)-(3.10) and the definition ofP1,2,3(Q), we can see thate P1,2,3(Q)e only depend onQe and they are independent ofvab.

Now we collect some simple bounds on Pi’s. For L_∞ norm, by definition, it is easy to prove that the following inequalities always hold:

kSk, kRk, kRk, kR²k, kQRk, kQR²k, kQR²Q^∗k6N^C

for any(k, z, w)in (3.19) and some fixed constantC >0. Then with the definition in (3.20), we also have that for any(k, z, w)in (3.19) and some constantC >0

P1, P2, P3=O(N^C). (3.23)

ExpandingS aroundR with the fact: S= (R⁻¹+ (Y_k^∗Yk−Q^∗Q))⁻¹, we obtain that for any fixedm∈N S−R=

Xm n=1

(−R(Y_k^∗Yk−Q^∗Q))ⁿR+ (−R(Y_k^∗Yk−Q^∗Q))^m+1S (3.24)

Let m= 5 in (3.24). Now we take _N¹ Re Tron the both sides of (3.24) and compare it with (3.21). Since Y_k^∗Yk−Q^∗Q=vab(ebaQ) +vab(Q^∗eab) +v²_abeaa, we can see that for 16l63, thePl(Q)is the coefficient of the(vab)^l term in the r.h.s. of _N¹ Re Tr(3.24) and

P4(Xk−1,Q)e ∈ M^C(vab) (3.25)

Similarly, using this expansion (m= 5), and the fact:

∂wR=R²=O(N^C), ∂zR=R(Q+Q^∗)R=O(N^C), and∂wS,∂zS=O(N^C), we can improve (3.22) to the following one:

(k,z,w)max∈(3.19)N η

|P1(Q)e |+|P2(Q)e |+|P3(Q)e |+|P4(Xk−1,Q)e |

≺1 (3.26)

We note: this statement shows that (3.22) can hold for different(k, z, w)∈(3.19) with the same probability subset.

Inserting (3.16) and (3.21) into (3.15), we writeA^(f)_Xk−1−A^(f)_Q as a polynomial ofvabas follows.

A^(f)_Xk−1−A^(f)_e

Q =P₁(Q)e ·vab+P₂(Q)e ·(vab)²+P₃(Q)e ·(vab)³+P₄(Xk−1,Q)e ·(vab)⁴. (3.27) where

P₁(Q) :=Ne Z

∆f(ξ) Z

I

B1P1

χ(η)φ^′(E)dEdηdA(ξ) (3.28)

P₂(Q) :=Ne Z

∆f(ξ) Z

I

B1P2+B2P₁²

χ(η)φ^′(E)dEdηdA(ξ) P₃(Q) :=Ne

Z

∆f(ξ) Z

I

B1P3+ 2B2P1P2+B3P₁³

χ(η)φ^′(E)dEdηdA(ξ)

P₄(X_k−1,Q) :=Ne Z

∆f(ξ) Z

I

X

n

Bn

X

P

jij>4

(vab)^(Pjij)−4 Yn j=1

Pij

χ(η)φ^′(E)dEdηdA(ξ)

whereBn=Bn(Q),e Pn =Pn(Q)e (16n63),B4=B4(Xk−1,Q)e andP4=P4(Xk−1,Q). We note:e P₁(Q),e P₂(Q)e andP₃(Q)e are independent ofvab.

ReplacingXk−1 withXk, with the same method, we obtain (Herevabis replaced withuab) A^(f)_Xk−A^(f)_e

Q =P₁(Q)ue ab+P₂(Q)ue ²_ab+P₃(Q)ue ³_ab+P₄(Xk,Q)ue ⁴_ab (3.29) From (3.18) and (3.26), it is easy to check thatP₁,P₂andP₃≺1uniformly hold for16k6N². For L_∞ bound, with (3.23), they are bounded byN^C for someC. Similarly, we can obtain that P₄≺1. With (3.25), we haveP₄(Xk−1,Q)e ∈ M^C(vab)andP₄(Xk,Q)e ∈ M^C(uab). So far, we proved

P_1,2,3,4≺1, P_1,2,3(Q)e 6N^C, P₄(Xk−1,Q)e ∈ M^C(vab), P₄(Xk,Q)e ∈ M^C(uab) (3.30) uniformly hold for16k6N².

Now we return to prove (3.3) and (3.4). First we write (A^(f)_Xk−1)^p−(A^(f)_Xk)^p=

p−1

X

j=0

p j

(A^(f)_Q˜ )^j

(A^(f)_Xk−1−A^(f)_e

Q )^p−j−(A^(f)_Xk−A^(f)_e

Q )^p−j .

We insert the (3.27) and (3.29) into the r.h.s. and write it in the following form (A^(f)_Xk−1)^p−(A^(f)_Xk)^p=

X4p m=1

(A^mv_ab^m− B^mu^m_ab) (3.31) whereA^m only containsA^(f)_Q˜ ,P_1,2,3(Q),e P₄(Xk−1,Q), ande B^monly containsA^(f)_Q˜ , P_1,2,3(Q),e P₄(Xk,Q)e For example,

A³=B³=Cp,3(A^(f)_e

Q )^p−3P³

1(Q) +e Cp,2(A^(f)_e

Q )^p−2P₁(Q)e P₂(Q) +e Cp,1(A^(f)_e

Q )^p−1P₃(Q)e (3.32) whereCp,n(16n63) are constants only depends onp. Since the first two moments ofvabanduabcoincide, uab, vabare independent ofQ, ande A¹=B¹,A²=B²only containA^(f)_Q˜ ,P_1,2,3(Q), we havee

E(A^(f)_Xk−1)^p−E(A^(f)_Xk)^p= X4p m=3

(A^mv_ab^m− B^mu^m_ab)

Recall the definition ofA^m andB^m from (3.31), for the termsm>4, using (3.30) and Lemma 3.1, we get

|E X4p m=4

(A^mv_ab^m− B^mu^m_ab)|6

p−1X

j=0

E(A^(f)_Qe )^jO_≺(N⁻²) +N⁻²6N⁻²

O_≺(1) +E|A^(f)_Qe |^p .

Therefore, withA³=B³,

E

A^(f)_Xk−1)p

−E

A^(f)_Xk)p6N⁻²

O_≺(1) +E|A^(f)_e

Q |^p

+|EA³| |Ev_ab³ |+|Eu³_ab|

, (3.33) Similarly, using (3.30), (3.32),A^(f)_Qe =O(N^C)and Lemma 3.1, we have

|EA³| |Ev³_ab|+|Eu³_ab|

6N^−3/2

O_≺(1) +E|A^(f)_e

Q |^p

(3.34) As in [2]-(3.64), using Hölder’s inequality and the bound (3.14), we have

E|A^(f)_e

Q |^p6E

(A^(f)_Xk−1)^p +

Xp j=1

j p

E

A^(f)_Xk−1

p−j

|A^(f)_Xk−1−A^(f)_e

Q |^j

(3.35)

6E

(A^(f)_Xk−1)^p +

Xp j=1

j p

EA^(f)_Xk−1

p^p−j_p

EA^(f)_Xk−1−A^(f)_e

Q

p^j_p ,

6

O_≺(1) + 2E|A^(f)_Xk−1|^p ,

Then combining (3.33)-(3.35), we obtain (3.3). (Note: p∈2Z.)

To prove (3.4), we claim the following lemma, which provides the stronger bound on the expectation value of the r.h.s. of (3.32).

Lemma 3.2. Assume1 6a 6=b 6N. Let X be defined as in Theorem 1.2, except that Xab = 0. For any fixed small enoughε >0, if ||z0| −1|6εands∈(0,1/2), defineA^(f)_X ,Pi(X),Bi(X),P_i(X),i= 1,2,3as in (2.7), (3.20),(3.17) and (3.28). (More precisely Q,e Q, R in (3.20) and (3.17) will be replaced with X, Y =X−zI and(Y^∗Y −wI)respectively.) Then

E(A^(f)_X )^p−3P³

1(X)+E(A^(f)_X )^p−2P₁(X)P₂(X)+E(A^(f)_X )^p−1P₃(X)≺N^−1/2

O_≺(1) +E|A^(f)_X |^p (3.36) uniformly for(a, b).

We return to prove (3.33) and prove lemma (3.2) in the next section. Inserting this lemma and (3.32) into (3.33), as in (3.34), we obtain that ifa6=b, then

|EA|6N⁻²

O_≺(1) +E|A^(f)_e

Q |^p

(3.37) Together with (3.33) and (3.35), we obtain (3.4). Clearly, (3.3) and (3.4) imply (3.2), and we complete the proof of lemma 2.4 and lemma 2.2.

4 Proof of Lemma 3.2

Lemma 3.2 bounds the expectation values of some polynomials ofA^(f)_X and P_1,2,3(X). Roughly speaking Lemma 3.2 shows that the expectation value of these polynomials are much less than their stochastic dom- ination by a factor N^−1/2. (Note: a and b appear in the definitions of P1,2,3 and B1,2,3. The P_1,2,3 are defined withP1,2,3 andB1,2,3.) As introduced in the second part of subsection 2.1 (below (2.11)), the main strategy of showing this extra factor is

• writing them as the polynomials (up to negligible error) ofXak’s,Xka’s (16k6N),G^(a,a)andG^(a,a), which are defined as

G^(a,a):= ((Y_z^(a,a))^∗Y_z^(a,a)−w)⁻¹, G^(a,a):= (Y_z^(a,a)(Y_z^(a,a))^∗−w)⁻¹ andY^(a,a):=Yz^(a,a) is the matrix obtained by removing thei−th row and column ofYz.

• showing the degrees of the monomials ofXak’s andXka’s in above polynomials are always odd (except forXaa).

First of all, in lemma 4.5 and 4.7, we introduce some polynomials having the properties we need for Lemma 3.2, i.e., their expectation values have an extra factorN^−1/2 comparing with their stochastic domination.

In the next subsection, we introduce someF sets, whose elements are the "basic" polynomials in our proof, i.e., the bricks of the polynomials in lemma 4.5 and 4.7.

4.1 Basic polynomials and their properties. We first introduce some notations.

Definition 4.1. X^(T,U),Y^(T,U),G^(T,U) andG^(T,U)

Let T,Ube some subsets of {1,2,· · ·, N}. Then we define Y^(T,U) as the (N − |U|)×(N − |T|) matrix obtained by removing all columns of Y indexed byi∈Tand all rows of Y indexed byi∈U. Notice that we keep the labels of indices ofY when defining Y^(T,U). With the same method, we define X^(T,U)withX.

Letyi be thei-th column ofY andy^(S)_i be the vector obtained by removingyi(j)for allj∈S. Similarly we define yi be thei-th row ofY. Define

G^(T,U)=h

(Y^(T,U))^∗Y^(T,U)−wi−1

, m^(T,U)_G = 1

N TrG^(T,U), G^(T,U)=h

Y^(T,U)(Y^(T,U))^∗−wi−1

, m⁽_G^T,U)= 1

N TrG^(T,U).

By definition, m^(∅,∅)=m. Since the eigenvalues ofY^∗Y andY Y^∗ are the same except the zero eigenvalue, it is easy to check that

m⁽_G^T,U)(w) =m⁽_G^T,U)+|U| − |T|

N w (4.1)

For|U|=|T|, we define

m^(T,U):=m^(T,U)_G =m^(T,U)_G (4.2)

There is a crude bound for(m^(T,U)_G −m)proved in (6.6) of [9]:

m⁽_G^T,U)−m+m⁽_G^T,U)−m6C|T|+|U|

N η (4.3)

Definition 4.2. Notations for general sets.

As usual, ifx∈RorC, andS is a set of random variables then xS denotes the following set as xS:={xs:s∈ S}

For two setsS¹ andS² of random variables, we define the following set as S¹· S²:=

s1·s2

s1∈ S¹, s2∈ S²

For simplicity, we calls∈ⁿS if and only ifs can be written as the sum ofO(1)elements in S, i.e.,

s∈ⁿS ⇐⇒ s∈ ( _n

X

i=1

si

si∈ S, n∈N, n=O(1) )

Definition 4.3. Definition ofF⁰,F¹, F^1/2andF.

For fixed indecesa, band ensembleXin lemma 3.2, we defineF⁰as the set of random variables (depending onX) which are stochastically dominated by 1 and independent of any Xak andXka (16k6N), i.e.,

F⁰={V : V ≺1, V is independent of thea−th row and column of X} Note: F⁰ depends ona, notb. One example element inF⁰ isTrX−Xaa.

For simplicity, we define

X(a) i

:=X

i6=a

,

X(a) ij

:= X

ij6=a

Next we defineF¹ as the union of the set (N^1/2XaaF⁰)and the sets of some quadratic forms as follows

F¹:=

N^1/2XaaF⁰ [



 X(a)

kl

XkaVklXla or X(a)

kl

XakVklXal

max

kl |Vkl| ≺1, Vkl ∈ F⁰



 [



 X(a) k6=l

XakVklXla+N^1/2 X(a)

k

XakVkkXka

max

kl |Vkl| ≺1, Vkl∈ F⁰





(Note it is XkaVklXla or XakVklXal in the first line and XakVklXla in the second line, and the diagonal terms in the second case is allowed to be larger than the others by a factorN^1/2.)

Furthermore, we define F as the set of following random variables

F :=



V V ∈ⁿ F⁰[

 [

n=O(1)

(F¹)ⁿ









where(F¹)ⁿ represents the set of the products ofnelements in F¹. For simplicity, sometimes we writeF F =F∅

i.e., with the subscription empty set ∅. Similarly, we define

F^1/2=



 X(a)

k

XakVk or X(a)

k

VkXka

max

k |Vk| ≺1, Vk ∈ F⁰





Note: For fixed k6=a, the total number ofXak andXka (16k6N), in each monomial of the element inF is always even. On the other hand, this number inF1/2· F is always odd. By the definition, it is easy to see that

F⁰, F¹∈ F

F⁰· F^α=F^α, α= 0, 1/2, 1, ∅ and

F1/2· F1/2⊂ F¹, F · F ⊂ F (4.4)

Examples: by definition,G^(a,a)_kl 6η⁻¹for any k, l6=a. Hence we have X(a)

kl

XkaG^(a,a)_kl Xla∈η⁻¹F¹,



 X(a)

kl

XkaG^(a,a)_kl Xla







 X(a)

kl

XakG^(a,a)_kl Xal



∈η⁻²F,