On Finite Rank Deformations of Wigner Matrices

Introduction and Formulation of Main Results

Let $X_{N}=\frac{1}{\sqrt{N}}W_{N}$ be a random real symmetric (Hermitian) Wigner matrix with independent entries up from the diagonal. In the real symmetric case, we assume that the off-diagonal entries

are independent random variables such that

are independent random variables (that are also independent from the off-diagonal entries), such that

In a similar fashion, in the Hermitian case, we assume that the off-diagonal entries

are independent random variables such that

are independent centered random variables, independent from the off-diagonal entries, with uniformly bounded third moment of the absolute values.

For a real symmetric (Hermitian) matrix $M$ of order $N,$ its empirical distribution of the eigenvalues is defined as $\mu_{M}=\frac{1}{N}\sum_{i=1}^{N}\delta_{\lambda_{i}},$ where $\lambda_{1}\geq\ldots\geq\lambda_{N}$ are the (ordered) eigenvalues of $M.$ Wigner semicircle law (see e.g. , , ) states that almost surely the empirical distribution $\mu_{X_{N}}$ of a random real symmetric (Hermitian) Wigner matrix $X_{N}$ converges weakly to the nonrandom limiting distribution $\mu_{sc}.$ The limiting distribution $\mu_{sc}$ is known as the semicircle distribution. It is absolutely continuous with respect to the Lebesgue measure and has the compact support $[-2\sigma,2\sigma].$ The density of the Wigner semicircle distribution is given by

converges to $\int\varphi(x)\*d\mu_{sc}(dx)$ almost surely; here and throughout the paper, we use the notation $\text{tr}_{N}=\frac{1}{N}\text{Tr}$ to denote the normalized trace.

The Stieltjes transform of the semicircle law is

In this paper, we study the fluctuations of the outliers in the spectrum of finite-dimensional deformations of Wigner matrices. Starting with the pioneering work by Füredi and Komlós , there have been several results on finite rank perturbations of matrices with i.i.d. entries, in particular , , , , , , , , , . We also note several papers on the eigenvalues of sample covariance matrices of spiked population models (, , , ).

This manuscript can be viewed as a companion paper to our recent works and on the non-Gaussian fluctuation of the matrix entries of regular functions of Wigner matrices. However, no knowledge of the machinery used in and is required, and the paper can be read independently from these papers.

Here $W_{N}$ is a random real symmetric (Hermitian) Wigner matrix as defined in (1.1-1.4) ((1.5-1.7)), and $A_{N}$ is a deterministic Hermitian matrix of fixed finite rank $r.$ We assume that the eigenvalues of $A_{N}$ and their multiplicities are fixed. Let

be the eigenvalues of $A_{N}$ each with fixed multiplicity $k_{j}$. Clearly, the eigenvalue $\theta_{j_{0}}=0$ has multiplicity $N-r$ and $\sum_{j\neq j_{0}}k_{j}=r.$

The first theorem of this section, Theorem 1.1, concerns the convergence of the extreme eigenvalues of the deformed random matrix. Let us denote $\rho_{\theta}=\theta+\frac{\sigma^{2}}{\theta}.$ We shall use the shorthand notation $\rho_{j}$ for $\rho_{\theta_{j}}.$ Theorem 1.1 was originally proved by Capitaine, Donati-Martin, and Feral in in the case when the common marginal distribution of the matrix entries is symmetric and satisfies a Poincaré inequality.

Let $X_{N}=\frac{1}{\sqrt{N}}W_{N}$ be a random real symmetric (Hermitian) Wigner matrix satisfying (1.1-1.4) (respectively (1.5-1.7)). Let $J_{\sigma^{+}}$ be the number of $j$’s such that $\theta_{j}>\sigma$ and $J_{\sigma^{-}}$ be the number of $j$’s such that $\theta_{j}<-\sigma$.

For all $j=1,\ldots,J_{\sigma^{+}}$ and $i=1,\ldots,k_{j}$, $\lambda_{k_{1}+\ldots+k_{j-1}+i}\to\rho_{j},$

$\lambda_{k_{1}+\ldots+k_{J_{\sigma^{+}}}+1}\to 2\sigma,$

$\lambda_{k_{1}+\ldots+k_{J-J_{\sigma^{-}}}}\to-2\sigma,$

For all $j=J-J_{\sigma^{-}}+1,\ldots,J$ and $i=1,\ldots,k_{j}$, $\lambda_{k_{1}+\ldots+k_{j-1}+i}\to\rho_{j}.$

In other words, the first $k_{1}$ largest eigenvalues of $M_{N}$ converge to $\rho_{1},$ the next $k_{2}$ largest eigenvalues converge to $\rho_{2},\ldots,$ the $J_{\sigma^{+}}$th bunch of the largest eigenvalues converge to $\rho_{J_{\sigma^{+}}},$ the next largest eigenvalue converges to $2\*\sigma$ (since it corresponds to a nonnegative eigenvalue of $A_{N}$ which is not bigger than $\sigma$), etc.

If random variables $(W_{N})_{ij},\ 1\leq i\leq j\leq N,$ satisfy a Poincaré inequality (1.12) with constant $\upsilon_{i,j,N}$ uniformly bounded from zero, $\upsilon_{i,j,N}\geq\upsilon>0,$ the convergence holds with probability one.

Note that the Poincaré inequality tensorizes and the probability measures satisfying the Poincaré inequality have subexponential tails (, ) . In particular, if the marginal distributions of the matrix entries of $W_{N}$ satisfy the Poincaré inequality with constant $\upsilon>0,$ then the joint distribution of $(W_{N})_{ij},\ 1\leq i\leq j\leq N,$ also satisfies the Poincaré inequality with the same constant $\upsilon.$ By a standard scaling argument, we note that if the marginal distributions of the matrix entries of $W_{N}$ satisfy the Poincaré inequality with $\upsilon>0$ then the marginal distributions of the matrix entries of $X_{N}=\frac{1}{\sqrt{N}}\*W_{N}$ satisfy the Poincaré inequality with constant N\*\upsilon.\

Theorem 1.1 follow from Theorem 1.2 formulated below. Theorem 1.2 is concerned with the distribution of the outliers, i.e. the eigenvalues of $M_{N}$ corresponding to $\theta_{j}>\sigma.$ Namely, we are interested in the fluctuation of the outliers around $\rho_{j},\ 1\leq j\leq J_{\sigma^{+}}.$ Let us consider a fixed eigenvalue $\theta_{j}$ of $A_{N}$ such that $\theta_{j}>\sigma.$ In general, if one does not assume some additional information about the structure of the eigenvectors of $A_{N}$ corresponding to $\theta_{j},$ the sequence of random vectors

Let $X_{N}=\frac{1}{\sqrt{N}}W_{N}$ be a random real symmetric (Hermitian) Wigner matrix defined in (1.1-1.4) (respectively (1.5-1.7)). Let $1\leq j\leq J_{\sigma^{+}},$ so the eigenvalue $\theta_{j}$ of $A_{N}$ satisfies $\theta_{j}>\sigma$. Then the sequence of random vectors

is bounded in probability. In addition, if the marginal distributions of the matrix entries of $W_{N}$ satisfy the Poincaré inequality (1.12) with constant $\upsilon_{i,j,N}$ uniformly bounded from zero, the following holds with probability $1$

Theorem 1.2 clearly implies parts (a) and (d) of Theorem 1.1. To see that parts (b) and (c) of Theorem 1.1 also follow, we note that for any fixed positive integer $l\geq 1$ the $l$-th largest eigenvalue of $X_{N}$ converges in probability to $2\*\sigma.$ This is a simple consequence of the convergence of the largest eigenvalue of $X_{N}$ to $2\*\sigma$ and the semicircle law. Then the interlacing property and Theorem 1.2 imply the desired result.

The bound (1.15) means that there exists a sufficiently large deterministic constant $C=C(\sigma,\upsilon,\theta_{1},\ldots,\theta_{r})>0,$ such that with probability $1$

To study the fluctuations of the outliers in more detail, we consider two special cases following .

Case A (“The eigenvectors don’t spread out”)

Case B (“The eigenvectors are delocalized”)

The $l^{\infty}$ norm of every orthonormal eigenvector of $A_{N}$ corresponding to $\theta_{j}$ goes to zero as $N\to\infty.$

The next theorem is a consequence of Proposition 1.1 below and Theorems 1.1 and 1.5 in . We use a standard notation $\beta=1$ in the real symmetric case and $\beta=2$ in the Hermitian case.

Let $X_{N}=\frac{1}{\sqrt{N}}W_{N}$ be a random real symmetric (Hermitian) Wigner matrix defined in (1.1-1.4) (respectively (1.5-1.7)) such that the off-diagonal entries $(W_{N})_{ij},\ 1\leq i<j\leq N,$ are i.i.d. real (complex) random variables with probability distribution $\mu$ and the diagonal entries $(W_{N})_{ii},\ 1\leq i<N,$ are i.i.d. random variables with probability distribution $\mu_{1}.$ In Case A, the $k_{j}$-dimensional vector

converges in distribution to the distribution of the ordered eigenvalues of the $k_{j}\times k_{j}$ random matrix $V_{j}$ defined as

(i) $W_{j}$ is a Wigner random matrix of size $K_{j}$ with the same marginal distribution of the matrix entries as $W_{N},$

(ii) $H_{j}$ is a real symmetric (Hermitian) Gaussian matrix of size $K_{j},$ independent of $W_{j},$ with centered independent entries $H_{st},\ 1\leq s\leq t\leq K_{j},(\operatorname{\mathfrak{Re}}H_{st},\operatorname{\mathfrak{Im}}H_{st},\ 1\leq s<t\leq K_{j},\ H_{pp},1\leq p\leq K_{j}$ in the Hermitian case) with the variance of the entries given by

(iii) $U_{j}$ is a $K_{j}\times k_{j}$ such that the ($K_{j}$-dimensional) columns of $U_{j}$ are written from the first $K_{j}$ coordinates of the orthonormal eigenvectors corresponding to $\theta_{j}.$

In , Theorem 1.3 was proved for symmetric marginal distribution satisfying the Poincaré inequality (1.12) under an additional technical assumption that k=o(\sqrt{N}),\ where $k$ is defined in the paragraph above (1.16).

Using Theorems 4.1 and 4.2 from , one can extend the results of Theorem 1.3 to the case when the entries of $W_{N}$ are not identically distributed provided the distribution of the entries $(W_{N})_{il},\ 1\leq i,l\leq K_{j}$ does not depend on $N.$

Let $X_{N}=\frac{1}{\sqrt{N}}W_{N}$ be a random real symmetric (Hermitian) Wigner matrix defined in (1.1-1.4) (respectively (1.5-1.7)) such that the distribution of the entries $(W_{N})_{il},\ 1\leq i,l\leq K_{j}$ does not depend on $N.$ Let us assume that the limits

Then in case A, the results of Theorem 1.3 hold with $\kappa_{4}(\mu)$ in (1.18) replaced by

Let $X_{N}=\frac{1}{\sqrt{N}}W_{N}$ be a random real symmetric (Hermitian) Wigner matrix defined in (1.1-1.4) (respectively (1.5-1.7)) such that the off-diagonal entries $(W_{N})_{ij},\ 1\leq i<j\leq N,$ are i.i.d. random variables with probability distribution $\mu$ and the diagonal entries $(W_{N})_{ii},\ 1\leq i<N,$ are i.i.d. random variables with probability distribution $\mu_{1}.$ In Case B, the $k_{j}$-dimensional vector

converges in distribution to the distribution of the (ordered) eigenvalues of a $k_{j}\times k_{j}$ GOE (GUE) matrix with the variance of the matrix entries given by $\frac{\theta_{j}^{2}\*\sigma^{2}}{\theta_{j}^{2}-\sigma^{2}}$ provided $k=o(\sqrt{N}).$

We recall that $k$ has been defined above as the minimal number of canonical basis vectors $e_{1},\ldots r_{N}$ required to span the eigenvectors corresponding to the eigenvalues $\theta_{1},\ldots\theta_{J_{\sigma^{+}}}.$

Theorem 1.5 is an immediate extension of the result of Capitaine, Donati-Martin, and Féral from to our setting since their arguments apply essentially unchanged as soon as Theorem 1.1 is established.

It should be noted that Benaych-Georges, Guionnet, and Maida consider in perturbations of a random Wigner matrix by a finite rank random matrix with eigenvectors that are either independent copies of a random vector $v$ with i.i.d. centered components satisfying the log-Sobolev inequality or are obtained by Gram-Schmidt orthonormalization of such independent copies. The distribution of the outliers is given in Proposition 5.3. of . Let us denote the distribution of the first component of $v$ by $\nu.$ If the fourth cumulant $\kappa_{4}(\nu)$ of $\nu$ vanishes, the limiting distribution of the outliers is similar to the result of Theorem 1.5, and given by the distribution of the ordered eigenvalues of a GOE (GUE) matrix. If the fourth cumulant does not vanish, one has to add a diagonal matrix with i.i.d. real Gaussian entries to a GOE (GUE) matrix.

One of the most important results of , concerns the distribution of the “sticking” eigenvalues (i.e. the eigenvalues that correspond to $|\theta_{j}|<\sigma).$ In Theorem 5.3 of , Benaych-Georges, Guionnet, and Maida prove that their limiting distribution is given by the Tracy-Widom law.

Let us briefly describe a key ingredient of the proofs of Theorems 1.2-1.4. We use the notation

Let us consider a fixed eigenvalue $\theta_{j}$ of $A_{N}$ such that $\theta_{j}>\sigma$ and denote by $v^{(1)},\ldots,v^{(k_{j})}$ the orthonormal eigenvectors of $A_{N}$ that correspond to the eigenvalue $\theta_{j}.$ Denote by $\Xi^{(j)}_{N}$ the $k_{j}\times k_{j}$ matrix with the entries

where we recall that \rho_{j}=\theta_{j}+\frac{\sigma^{2}}{\theta_{j}}\. The following proposition plays an important part in our proofs.

Let $y_{1}\geq\ldots\geq y_{k_{j}}$ be the ordered eigenvalues of the matrix $\Xi^{(j)}_{N}.$ Then

It should be mentioned that the key part of the proof of Proposition 1.1 is a lemma from which is stated as Lemma 4.2 in Section 4. Proposition 1.1 indicates that the question of the limiting distribution of the outliers of the spectrum of the deformed Wigner matrix $M_{N}$ can be reduced to the question about the limiting distribution of the entries of (1.25).

Without additional assumptions on $u^{(N)}$ and $v^{(N)},$ the sequence

does not necessarily converge in distribution. However, one can show that it is tight.

Let $X_{N}=\frac{1}{\sqrt{N}}W_{N}$ be a random real symmetric (Hermitian) Wigner matrix defined in (1.1-1.4) (respectively (1.5-1.7)). Then the following statements hold:

where $Const(\sigma^{2},m_{5},c_{3})$ depends on $\sigma^{2},m_{5},$ and $c_{3}.$

(iii) If the marginal distributions of the entries of $W_{N}$ satisfy the Poincaré inequality (1.12) with a uniform constant $\upsilon>0$, and $f$ is a Lipschitz continuous function on $[-2\*\sigma-\delta,2\*\sigma+\delta]$ that satisfies a subexponential growth condition

for some positive constants $a$ and $b,$ then

where $|f|_{\mathcal{L},\delta}$ is defined in (1.32),

and $\upsilon$ is the constant in the Poincaré inequality (1.12).

We finish this section by formulating our last theorem, Theorem 1.7, which allows us to extend Theorem 1.3 (see Remark 5.1 in Section 5). Assume that that the off-diagonal entries $(W_{N})_{ij},\ 1\leq i<j\leq N,$ are i.i.d. random variables with probability distribution $\mu$ and the diagonal entries $(W_{N})_{ii},\ 1\leq i<N,$ are i.i.d. random variables with probability distribution $\mu_{1}.$

converges in distribution as $N\to\infty.$ Without loss of generality, we will consider the real symmetric case; the Hermitian case is essentially identical. Let $m$ be an arbitrary fixed positive integer. Denote by $R^{(m)}(z)$ the $m\times m$ upper-left corner of the matrix $R_{N}(z).$ Theorem 1.1 in states that a matrix-valued random field

with values in the space of complex symmetric $m\times m$ matrices, converges in finite-dimensional distributions to a random field

where $W^{(m)}$ is the $m\times m$ upper-left corner submatrix of a Wigner matrix $W_{N},\ g_{\sigma}(z)$ is the Stieltjes transform (1.9) of the Wigner semicircle law, and

is a Gaussian random field with the covariance matrix given by the formulas (1.18)-(1.23) in the real-symmetric case and (1.50)-(1.55) in the Hermitian case in . It is important to note that $Y_{ij}(z),\ 1\leq i\leq j\leq m,$ are independent random processes for different indices $(ij).$

Let us extend the definition of $\Upsilon(z)$ to that of an infinite-dimensional matrix \Upsilon(z)_{pq},\ 1\leq p,q<\infty,\ using the formulas (1.18)-(1.23) (respectively (1.50)-(1.55)) from . Thus, the r.h.s. in (1.43) defines now the $m\times m$ upper-left corner of the infinite matrix $\Upsilon(z).$ Then Theorem 1.1 of implies that

Let $X_{N}=\frac{1}{\sqrt{N}}W_{N}$ be a random real symmetric (Hermitian) Wigner matrix defined in (1.1-1.4) (respectively (1.5-1.7)) such that that the off-diagonal entries $(W_{N})_{ij},\ 1\leq i<j\leq N,$ are i.i.d. random variables with probability distribution $\mu$ and the diagonal entries $(W_{N})_{ii},\ 1\leq i<N,$ are i.i.d. random variables with probability distribution $\mu_{1}.$

converges weakly to the joint distribution of $\ \langle u_{p},\Upsilon(z)u_{q}\rangle,\ \ 1\leq p,q\leq l.$

We would like to thank A. Guionnet for bringing our attention to the preprints and .

Mathematical Expectation and Variance of Resolvent Sesquilinear Form

This section is devoted to the proof of the main building block Theorem 1.6, namely Proposition 2.1.

When it does not lead to ambiguity we will use the shorthand notation, $R_{ij}$, for the $ij$-th entry $(R_{N}(z))_{ij},$ of the resolvent matrix $R_{N}(z).$

In the case when $u^{(N)}$ and $v^{(N)}$ are standard basis vectors, $u=e_{i},\ v=e_{j},$ the mathematical expectation and the variance of $\langle u^{(N)},R_{N}(z)v^{(N)}\rangle$ have been studied in . In particular, it has been shown there in Proposition 2.1 and (3.27) that

In , Erdös, Yau, and Yin studied generalized Wigner matrices (defined at the beginning of Section 2 of ), and obtained the following estimates provided the marginal distributions have subexponential tails

where $0<\phi<1,\ C\geq 1,c>0$ are some constants, $4/\phi\leq l\leq C\log N/\log\log N,$ $N^{-1}\*(\log N)^{10\*l}<\operatorname{\mathfrak{Im}}z\leq 10,\ |\operatorname{\mathfrak{Re}}z|\leq 5\sigma,$ and $N$ is sufficiently large.

It follows from our proofs that the error term on the r.h.s. of (2.2) can be replaced by O\left(\frac{\min(\|u\|_{1},\|v\|_{1})}{|\operatorname{\mathfrak{Im}}z|^{7}\*N}\right),\ where $\|u\|_{1}=\sum_{i=1}^{N}|u_{i}|.$

The rest of the section is devoted to the proof of Proposition 2.1.

Without loss of generality, we can restrict our attention to the real symmetric case. The proof in the Hermitian case is very similar. We start by proving (2.2). Using $(z\*I_{N}-X_{N})\*R_{N}(z)=I_{N},$ we write

where $\eta_{N}$ is defined in (2.1), and $r_{N}$ contains the third and the fourth cumulant terms corresponding to $p=2$ and $p=3$ in the decoupling formula (6.1) for $i\not=k,$ and the error terms due to the truncation of the decoupling formula (6.1) for $i\not=k$ at $p=3$ and for $i=k$ at $p=1.$

where by $\kappa_{3}(i,k)$ we denote the third cumulant of $(W_{N})_{ik}.$ We note that

uniformly in $i\not=k$ and $N.$ To estimate the absolute value of the first term in (2.15), we first sum with respect to $j$ and then use the Cauchy-Schwarz inequality and (6.7) to obtain

To estimate the absolute value of the second term in (2.15), we write

Finally, we bound the last of the third cumulant terms in (2.15) as

Combining the bounds (2.16-2.18), we see that the contribution of the third cumulant terms to $r_{N}$ in (2.12-2.13) is bounded from above by $O\left(\frac{1}{|\operatorname{\mathfrak{Im}}z|^{3}\*\sqrt{N}}\right).$ The fourth cumulant terms give

To estimate the absolute value of the first term in (2.19), we note that

(6.7), and the fact that the fourth cumulants of $(W_{N})_{ik}$ are uniformly bounded in absolute value by some constant $Const(m_{5}).$

To estimate the second term in (2.19), we write

The other two terms in (2.19) are estimated in a similar fashion. Each of them is $O\left(\frac{N\*\|u\|\*\|v\|}{|\operatorname{\mathfrak{Im}}z|^{2}}\right).$ Therefore, the fourth cumulant terms give the contribution $O\left(\frac{1}{N\*|\operatorname{\mathfrak{Im}}z|^{4}}\right)$ to $r_{N}$ in (2.12-2.13).

Finally, we estimate the error terms due to the truncation of the decoupling formula at $p=3$ for $i\not=k$ and at $p=1$ for $i=k.$ Here, we treat the error term due to the truncation of the decoupling formula at $p=3$ for $i\not=k.$ The second error term can be treated in a similar way. To estimate the error term, we have to consider expressions of the following form

where a,b,c,d,e,f,p,q,s\in\{i,k\},\ the supremum in (2.22) is considered over the resolvents $R^{(l)}=(z-X_{N}^{(l)})^{-1},\ l=1,\ldots 5$ of rank two perturbations $X_{N}^{(l)}=X_{N}+x\*E_{ik}$ of $X_{N}$ with $(E_{ik})_{jh}=\delta_{ij}\*\delta_{kh}+\delta_{ih}\*\delta_{kj}.$ Estimating each entry of $R^{(l)}$ by $\frac{1}{|\operatorname{\mathfrak{Im}}z|},$ taking into account that

and using the fact that the fifth cumulants of the off-diagonal entries of $W_{N}$ are uniformly bounded, we bound (2.22) from above by O\left(\frac{1}{N\*|\operatorname{\mathfrak{Im}}z|^{5}}\right).\

Combining the estimates of the third and the fourth cumulant terms and the truncation error term, we can rewrite the Master equation (2.12) as

where we recall that by $P_{l}$ we denote a polynomial of degree $l$ with positive coefficients that do not depend on $N.$

which is exactly the estimate (2.2) of Proposition 2.1.

To prove (2.3), we note that (2.25-2.26), (2.28) and (2.6) imply

Now, we turn our attention to the proof of (2.4). The key part of the proof is the following lemma.

where $r_{N}$ contains the third and the fourth cumulant terms corresponding to $p=2$ and $p=3$ in (6.1) for $k=i$, and the error due to the truncation of the decoupling formula (6.1) at $p=3$ for $k\not=i$ and at $p=1$ for $k=i.$ Clearly,

Using (2.42) and (2.47), one can write the last term in (2.39) as

The third cumulant terms in $r_{N}$ in (2.40) can be written as

We are going to estimate the terms (2.53-2.55) separately. We start with the last two. We claim that both (2.54) and (2.55) are $O\left(\frac{1}{|\operatorname{\mathfrak{Im}}z|^{4}\*N}\right)$. Indeed, consider first (2.54). It follows from (6.4-6.5), (2.42), and (2.47), that it is equal to

Combining (2.59) and (2.58), we estimate (2.57) as $O\left(\frac{1}{|\operatorname{\mathfrak{Im}}z|^{4}\*N}\right).$ The other terms in (2.56) can be estimated in a similar way, which implies that (2.54) is $O\left(\frac{1}{|\operatorname{\mathfrak{Im}}z|^{4}\*N}\right)$.

Now, we turn our attention to (2.55). Using (2.43-2.45) and (2.48), one can rewrite (2.55) as

We estimate (2.60). The subsums (2.61-2.63) can be estimated in a similar way. The summation with respect to $j$ in (2.60) gives

Combining the last two bounds, we obtain that (2.60) is $O\left(\frac{1}{|\operatorname{\mathfrak{Im}}z|^{4}\*N}\right).$

Finally, let us estimate (2.53). It can be written as

The subsums (2.64) and (2.66) are bounded from above by $O\left(\frac{1}{|\operatorname{\mathfrak{Im}}z|^{4}\*N}\right).$ The calculations are very similar to the ones used above and are left to the reader. The subsum (2.65) can be written as

It follows from the estimates in (2.17) that one has a deterministic upper bound

Combining the estimates (2.53-2.70), we obtain that the third cumulant term (2.52) contributing to $r_{N}$ in (2.38) can be written as

Somewhat long but straightforward calculations using (6.4-6.5) and (2.42-2.51) show that the fourth cumulant term in $r_{N}$ in (2.38) can be estimated from above by O\left(\frac{1}{|\operatorname{\mathfrak{Im}}z|^{5}\*N}\right).\ Since the calculations are very similar to those in (2.19- 2.21), we leave the details to the reader. In a similar fashion, the error terms in $r_{N},$ due to the truncation of the decoupling formula at $p=3$ for $i\not=k$ and at $p=1$ for $i=k$ are bounded from above by O\left(\frac{1}{|\operatorname{\mathfrak{Im}}z|^{6}\*N}\right).\ The considerations are similar to those given in the analysis of (2.22).

Combining (2.41), (2.50-2.51), (2.71-2.72), and the bounds on the fourth cumulant term and the error terms discussed in the above paragraph, one rewrites the Master equation (2.38-2.39) as

Subtracting the r.h.s. in (2.35) from the r.h.s. in (2.76), we obtain (2.32). Lemma 2.1 is proven. ∎

Now, we are ready to finish the proof of Proposition 2.1. To obtain the estimate (2.4) from (2.32), we use the same arguments as in Section 3 of and Section 2 of . We note (see e.g. (3.9) in ) that

where the constant $L$ is chosen sufficiently large so that the $O\left(\frac{P_{4}(|\operatorname{\mathfrak{Im}}z|^{-1})}{N}\right)$ term on the r.h.s. of (2.77) is at most $1/2$ in absolute value. Multiplying both sides of (2.32) by $g_{N}(z),$ and using (6.8), we obtain that

for $z\in\mathcal{O}_{N}.$ It follows from (2.78) that

On the other hand, if $|\operatorname{\mathfrak{Im}}z|\leq L\*N^{-1/4},$ then $\frac{L^{4}}{N\*|\operatorname{\mathfrak{Im}}z|^{4}}\geq 1.$ Since $|\langle u^{(N)},R_{N}(z)v^{(N)}\rangle|\leq\frac{1}{|\operatorname{\mathfrak{Im}}z|},$ we have

for $z$ such that $|\operatorname{\mathfrak{Im}}z|\leq L\*N^{-1/4}.$ Combining (2.79) and (2.80), we obtain (2.4). This finishes the proof of Proposition 2.1. ∎

Proof of Theorem 1.6

Our exposition follows closely the ones in Section 3 of and Section 4 of . In order to extend the estimates of Proposition 2.1 to a more general class of test functions, we use the Helffer Sjöstrand functional calculus (see , ).

To prove (1.34), we let $l=7$ in (3.2) and assume that $f$ has compact support. It follows from (2.2) that

uniformly on $\{z:\operatorname{\mathfrak{Re}}z\in supp(f),\ |\operatorname{\mathfrak{Im}}z|\leq 1\},$ and $C_{2}$ is a constant depending on $supp(f).$ We conclude that the second term on the r.h.s. of (3.8) can be estimated as follows

where $\chi_{f}$ and $\chi_{\sigma}$ are the characteristic functions of the support of $f$ and of $\sigma$ respectively, and $L$ is such that $supp(f)\subset[-L,L].$ This proves (1.34).

where $z=x+iy,\ w=s+it.$ Taking into account (2.4), we get

Plugging (3.5) with $l=4$ in (3.15), we prove (1.33). Thus, we have proved the parts (i) and (ii) of Theorem 1.6.

Now, let us assume that the marginal distributions of the entries of $W_{N}$ satisfy the Poincaré inequality (1.12) with a uniform constant $\upsilon$ and prove the parts (iii)-(v), i.e. the estimates (1.37), (1.39), and (1.40). Since the proof of (1.37-1.40) is very similar to the proof of Proposition 3.3 in , we discuss here only the main ingredients.

where the Hilbert-Schmidt norm is defined as

In particular, if $u$ and $v$ are unit vectors, then

is a complex-valued Lipschitz continuous function on the space of $N\times N$ real symmetric (Hermitian) matrices with the Lipschitz constant

The second observation is that joint distribution of the matrix entries

of $X_{N}$ satisfies the Poincaré inequality with the constant $\frac{1}{2}\*N\*\upsilon$ since the Poincaré inequality tensorizes (, ). Therefore, for any complex-valued Lipschitz continuous function of the matrix entries with the Lipschitz constant $|G|_{\mathcal{L}},$ the distribution of $G(X_{N})$ has exponential tails (see e.g. Lemma 4.4.3 and Exercise 4.4.5 in ), i.e.

Applying (3.19) to the spectral norm $\|X\|$ of the matrix $X_{N}$ and using the universality results for the largest eigenvalues (see and references therein), we obtain

Outliers in the Spectrum of Finite Rank Perturbations of Wigner Matrices

This section is devoted to the proof of Theorem 1.2

is decreasing and $\ g_{\sigma}(2\*\sigma+0)=1/\sigma.$ Let us choose $\delta>0$ in such a way that

i.e. for all $\theta_{j}$ that correspond to the outliers (so $\theta_{j}>\sigma$). Let

where $\zeta_{N}(x)$ is defined in (4.6).

where x_{i+1}-x_{i}=N^{-1/3},\ 0\leq i\leq l(N)-1,\ and $x_{l(N)-1}\leq L<x_{l}(N).$ Clearly, the number of elements in the sequence is $O(N^{1/3}).$ We have

uniformly in $0\leq i\leq l(N)$ and N\geq 1.\ Indeed,

Now, we are ready to start the proof of Theorem 1.2. Let us denote by $u^{(1)},\ldots,u^{(r)},$ the orthonormal eigenvectors of $A_{N}$ corresponding to the non-zero eigenvalues. We recall that we used the notation $\theta_{1}>\ldots>\theta_{j_{0}}=0>\ldots>\theta_{J}$ for the (fixed) eigenvalues of $A_{N},$ and denoted the (fixed) multiplicity of $\theta_{j}$ by $k_{j}$. The zero eigenvalue $\theta_{j_{0}}=0$ has multiplicity $N-r.$ Clearly, $\sum_{j\neq j_{0}}k_{j}=r.$ Let us denote by $\Theta$ the $r\times r$ diagonal matrix built from the non-zero eigenvalues of $A_{N},$

Let us also denote by $U_{N}$ the $N\times r$ matrix whose columns are given by the orthonormal eigenvectors $u^{(1)},\ldots,u^{(r)}$ of $A_{N}.$ Clearly,

For any $x\in[2\*\sigma+2\delta,L],$ we define the $r\times r$ matrix $\Xi_{N}(x)$ as follows. Let

The first step in the proof of Theorem 1.2 is the following lemma from .

Suppose that $x$ is not an eigenvalue of $X_{N}.$ Then $x$ is an eigenvalue of $X_{N}+A_{N}$ with multiplicity $n\geq 1$ if and only if $g_{\sigma}(x)$ is an eigenvalue of the $r\times r$ matrix

For the convenience of the reader, we sketch the proof of Lemma 4.2 below.

Let $x\not\in Sp(X_{N}).$ Therefore $R_{N}(x)=(x\*I_{N}-X_{N})^{-1}$ is well defined, and

We obtain that for $x\not\in Sp(X_{N})$ that $\ x\in Sp(X_{N}+A_{N})$ if and only if

where one uses the identity \det(I-B\*C)=\det(I-C\*B).\ Rewriting

Proposition 1.1 plays an important role in the proof of Theorem 1.2. Before we prove Proposition 1.1, we need to introduce some notations and prove Lemma 4.3.

Consider a family of $r\times r$ matrices $Z_{N}(x)$ defined in (4.23) for $x\in[2\*\sigma+2\delta,L].$ Fix an eigenvalue $\theta_{j}$ of $A_{N}$ such that $\theta_{j}>\sigma$ and use the notation $v^{(1)},\ldots,v^{(k_{j})}$ for the eigenvectors of $A_{N}$ that correspond to the eigenvalue \theta_{j}.\ Without loss of generality we can assume that $j=1.$ We do it just to simplify notations. The case $1<j\leq J_{\sigma^{+}}$ is identical. We recall that $\Xi^{(j)}_{N}$ is defined in (1.25) as the $k_{j}\times k_{j}$ submatrix of $\Xi_{N}(\rho_{j})$ restricted to the rows and columns corresponding to $v^{(i)},\ 1\leq i\leq k_{j}.$ The central role in the proof of Proposition 1.1 is played by the following lemma.

Let $Z_{N}(x),\ x\in[2\*\sigma+2\delta,L],$ be as in (4.23), with $\Xi_{N}(x)$ defined in (4.22), and $\Theta$ defined in (4.20). Let

be the ordered eigenvalues of $Z_{N}(x).$ Then, for sufficiently large constant $C>0,$

in probability, i.e. $\sqrt{N}\*(z_{i}(\rho_{1})-\frac{1}{\theta_{1}})$ is bounded in probability, $\ 1\leq i\leq k_{1}.$

We claim that (4.27) follows from Lemma 4.1. Indeed, (4.11) and (4.6) imply that

as $N\to\infty.$ Since $|z_{i}(x)-z_{i}(y)|\leq\|Z_{N}(x)-Z_{N}(y)\|=\frac{1}{\sqrt{N}}\*\|\Xi_{N}(y)-\Xi_{N}(y)\|,\ 1\leq i\leq r,$ we conclude that (4.29) implies (4.27).

in probability. Indeed, the entries of the $r\times r$ matrix $\Xi_{N}(x)$ are bounded in probability since the expectation and variance of

almost surely. Thus, $\|\Xi_{N}(x)\|$ is also bounded in probability. Since the first $k_{1}$ eigenvalues of $\Theta^{-1}$ are equal to $\frac{1}{\theta_{1}},$ we obtain (4.28). Lemma 4.3 is proven. ∎

Now, we are ready to prove Proposition 1.1.

By Lemma 4.2, the outliers of $X_{N}+A_{N}$ are given by those values of $x\in[2\*\sigma+\delta,M]$ such that

We recall that $g_{\sigma}(x)$ is a monotonically decreasing function on $[2\*\sigma+\delta,M]$ and

Since for $1\leq i\leq k_{1},$ (4.28) gives us that $z_{i}(\rho_{1})-g_{\sigma}(\rho_{1})=O(\frac{1}{\sqrt{N}})$ in probability, it follows from (4.31) and (4.27) that with probability going to $1,$ there exist $M>x_{1}\geq x_{2}\geq\ldots\geq x_{k_{1}}>2\*\sigma+\delta$ such that g_{\sigma}(x_{i})=z_{i}(x_{i}),\ 1\leq i\leq k_{1},\ and

in probability. Applying (4.27) one more time, we get that

in probability. By a standard perturbation theory argument (see e.g. section XII.1 in ), one proves that the first $k_{1}$ smallest eigenvalues of the matrix $Z_{N}(\rho_{1})$ differ from the (increasingly ordered) eigenvalues of the $k_{1}\times k_{1}$ matrix $\frac{1}{\theta_{1}}\*Id-\frac{1}{\sqrt{N}}\*\Xi^{(m)}_{N}$ by at most O\left(\frac{1}{N}\right),\ in probability, where the matrix $\Xi^{(m)}_{N}$ has been defined in (1.25). To see this, we use the following standard lemma from the perturbation theory

Let $B$ be an $n\times n$ real symmetric (Hermitian) matrix that can be written in the block form as $B=(B_{ij})_{i,j=1,2},$ where $B_{ij}$ is an $n_{i}\times n_{j}$ matrix. Suppose that all eigenvalues of $B_{11}$ are smaller than all eigenvalues of $B_{22}$ and the gap between the spectra of $B_{11}$ and $B_{22}$ is at least $Const>0.$ In addition, suppose that the operator norm of the offdiagonal block $B_{12}$ is bounded from above by $\epsilon,$ so that $\|B_{12}\|=\|B_{21}\|\leq\epsilon.$

Then there exists $const(Const,n)$ such that the first $n_{1}$ smallest eigenvalues of $B$ differ from the (increasingly ordered) eigenvalues of $B_{11}$ by at most $const\*\epsilon^{2}.$

and $\lambda_{n_{1}+1}-\lambda_{n_{1}}>Const.$ Then it is easy to see that

is an approximate eigenvector of $B$ with the approximate eigenvalue $\lambda_{1}$ such that

Since $\|(B-\lambda_{j})\*e_{j}\|\leq\epsilon,\ 1\leq j\leq n,$ and $\lambda_{j}-\lambda_{1}\geq Const,\ n_{1}<j\leq n,$ we obtain that

The result of the lemma can be immediately extended by induction to the case of $m\times m$ block matrices $B=(B_{ij})_{1\leq i,j\leq m}.$ To apply it in our setting, we note that the $k_{1}\times k_{1}$ matrix $\frac{1}{\theta_{1}}\*Id-\frac{1}{\sqrt{N}}\*\Xi^{(m)}_{N}$ is the upper-left block of $Z_{N}(\rho_{1}).$ The other diagonal blocks of $Z_{N}(\rho_{1})$ are given by $\Xi^{(i)}_{N},\ 1\leq i\leq m-1$ defined in (1.25). Since the operator norms of the off-diagonal blocks of $Z_{N}(\rho_{1})$ are $O(N^{-1/2})(see(\ref{andor})),$ the desired statement follows.

where $y_{1}\geq\ldots\geq y_{k_{1}}$ are the eigenvalues of the matrix $\Xi^{(m)}_{N}.$ The result of Proposition 1.1 now follows from (4.36) and (4.33). ∎

Since the eigenvalues of the matrix $\Xi^{(m)}_{N}(\rho_{1})$ are bounded in probability, the first part of Theorem 1.2, i.e. (1.14), follows from (1.26) in Proposition 1.1.

almost surely, where $Const_{1}>0,\ Const_{2}>0$ are sufficiently large, improving (4.11). Reasoning as before, (4.37) implies that

almost surely for sufficiently large constant $Const_{3}>0.$ Thus, we have

almost surely, which implies (1.15) since $g_{\sigma}(\rho_{1})=\frac{1}{\theta_{1}}.$ Theorem 1.2 is proven. ∎

Proof of Theorems 1.3, 1.4, and 1.7

In this section, we prove Theorems 1.3, 1.4, and 1.7. We start with Theorem 1.3.

Let $\theta_{j}>\sigma$ be an eigenvalue of $A_{N}$ with the multiplicity $k_{j}.$ Let us assume that Case A takes place. Thus, without loss of generality, we can assume that the eigenvectors of $A_{N}$ corresponding to the eigenvalue $\theta_{j}$ belong to $Span(e_{1},\ldots,e_{K_{j}}),$ where $K_{j}$ is a fixed positive integer. As always, we consider the real symmetric case. The treatment of the Hermitian case is very similar. Consider a $K_{j}\times k_{j}$ matrix $U_{j}$ such that the ($K_{j}$-dimensional) columns of $U_{j}$ are filled by the first $K_{j}$ coordinates of the $k_{j}$ orthonormal vectors of $A_{N}$ corresponding to the eigenvalue $\theta_{j}.$ We recall that the remaining $N-K_{j}$ coordinates of these orthonormal vectors are zero. Let us denote by $R_{N}^{(K_{j})}(z)$ the upper-left $K_{j}\times K_{j}$ submatrix of the resolvent matrix $R_{N}(z)=(z\*I_{N}-X_{N})^{-1}.$ Finally, we define the random matrix-valued field

We recall that Theorem 1.1 in states that $\Upsilon_{N}(z)$ converges weakly in finite-dimensional distributions to a random field

Now, Theorem 1.3 follows from Proposition 1.1 in this paper, and Theorem 1.1 in , since

Theorem 1.3 is proven. The proof of Theorem 1.4 is very similar to the given proof of Theorem 1.3. One has to use Theorems 4.1 and 4.2 and Remark 4.1 in that generalize Theorems 1.1 and 1.5 in to the non-i.i.d. case, and replace $\kappa_{4}(\mu)$ in (5.4) with $\kappa_{4}(i),\ 1\leq i\leq K_{j}.$ ∎

Now, we turn to the proof of Theorem 1.7.

converges weakly to the joint distribution of $\ \langle u^{(n)}_{p},\Upsilon(z)u^{(n)}_{q}\rangle,\ \ 1\leq p,q\leq l.$ Choosing $n$ sufficiently large, we can make

arbitrary small uniformly in $N\geq n.$ Indeed, the variance in (5.7) is bounded by O(\|u_{p}-u^{(n)}_{p}\|^{2}+\|u_{q}-u^{(n)}_{q}\|^{2})\ since the entries of $\Upsilon(z)$ are i.i.d random variables with bounded variance on the diagonal and i.i.d. random variables with bounded variance off the diagonal. In addition,

and we can use the bounds (1.33) and (1.34) in Theorem 1.6 rewritten as

are arbitrary small (uniformly in $N$) provided one chooses $n$ sufficiently large. This finishes the proof. ∎

Theorem 1.7 allows the following extension of Theorem 1.3:

where $u^{(p)}_{N}$ denotes the projection of $u^{(p)}$ onto the subspace spanned by the first $N$ standard basis vectors $e_{1},\ldots,e_{N}.$ Let $U_{N}$ be the $N\times r$ matrix whose columns are given by the vectors u^{(1)}_{N},\ldots,u^{(r)}_{N}.\ Also denote by $\Theta$ the $r\times r$ diagonal matrix

The result of Theorem 1.3 can be extended for such $A_{N}$, with the matrix $V_{j}$ given by

Appendix

The appendix contains several basic formulas used throughout the paper.

and can be immediately verified by integration by parts.

Next, we write a basic resolvent identity. For any two Hermitian matrices $X_{1}$ and $X_{2}$ and non-real $z$ we have:

As a corollary of (6.3), one has the following formulas. If $X$ is a real symmetric matrix with resolvent $R$ then

In a similar way, if $X$ is a Hermitian matrix then

Finally, we will use the following properties of the resolvent:

where by $Sp(X)$ we denote the spectrum of a real symmetric (Hermitian) matrix $X.$ The bound (6.6) implies

Therefore, all entries of the resolvent matrix are bounded by $|\operatorname{\mathfrak{Im}}(z)|^{-1}$. In a similar fashion, we have the following bound for the Stieltjes transform, $g(z)$, of any probability measure:

References