The single ring theorem

The problem

Horn asked the question of describing the eigenvalues of a square matrix with prescribed singular values. If $A$ is a $n\times n$ matrix with singular values $s_{1}\geq\ldots\geq s_{n}\geq 0$ and eigenvalues $\lambda_{1},\ldots,\lambda_{n}$ in decreasing order of absolute values, then the inequalities

were shown by Weyl to hold. Horn established that these were all the relationships between singular values and eigenvalues.

In this paper we study the natural probabilistic version of this problem and show that for “typical matrices”, the singular values almost determine the eigenvalues. To frame the problem precisely, fix $s_{1}\geq\ldots\geq s_{n}\geq 0$ and consider $n\times n$ matrices with these singular values. They are of the form $A=PTQ$, where $T$ is diagonal with entries $s_{j}$ on the diagonal, and $P,Q$ are arbitrary unitary matrices.

We make $A$ into a random matrix by choosing $P$ and $Q$ independently from Haar measure on $\mathcal{U}(n)$, the unitary group of $n\times n$ matrices, and independent from $T$. Let $\lambda_{1},\ldots,\lambda_{n}$ be the (random) eigenvalues of $A$. The following natural questions arise.

Are there deterministic or random sets $\{s_{j}\}$, for which one can find the exact distribution of $\{\lambda_{j}\}$?

For finite $n$, for fixed $S$, is $L_{\Lambda}$ concentrated in the space of probability measures on the plane?

In this paper, we concentrate on the second question and answer it in the affirmative, albeit with some restrictions. In this context, we note that Fyodorov and Wei [8, Theorem 2.1] gave a formula for the mean eigenvalues density of $A$, yet in terms of a large sum which does not offer an easy handle on asymptotic properties (see also for the case where $T$ is a projection). The authors of explicitely state the second question as an open problem.

Of course, questions 1–3. above are not new, and have been studied in various formulations. We now describe a partial and necessarily brief history of what is known concerning questions 1. and 2.; partial results concerning question 3. will be discussed elsewhere.

The most famous case of a positive answer to question 1. is the Ginibre ensemble, see , and its asymmetric variant, see . (There are some pitfalls in the standard derivation of Ginibre’s result. We refer to for a discussion.) Another situation is the truncation of random unitary matrices, described in .

Concerning question 2., the convergence of the empirical measure of eigenvalues in the Ginibre ensemble (and other ensembles related to question 1.) is easy to deduce from the explicit formula for the joint distribution of eigenvalues. Generalizations of this convergence in the absence of such explicit formula, for matrices with iid entries, is covered under Girko’s circular law, which is described in ; the circular law was proved under some conditions in and finally, in full generality, in and . Such matrices, however, do not possess the invariance properties discussed in connection of question 2. The single ring theorem of Feinberg and Zee is, to our knowledge, the first example where a partial answer to this question is offered. (Various issues of convergence are glossed over in and, as it turns out, require a significant effort to overcome.) As we will see in Section 3, the asymptotics of the spectral measure appearing in question 2. are described by the Brown measure of $R$-diagonal operators. (The Brown measure is a continuous analogue of the spectral distribution of non-normal operators, introduced in .) $R$-diagonal operators were introduced by Nica and Speicher in the context of free probability; they represent the weak*-limit (or more precisely, the limit in $*$-moments) of operators of the form $UT$ with $U$ unitary with size going to infinity and $T$ diagonal, and were intensively studied in the last decade within the theory of free probability, in particular in connection with the problem of classifying invariant subspaces .

Limiting spectral density of a non-normal matrix

$G_{\mu}$ is analytic off the support of $\mu$. We let ${\mathcal{H}}_{n}$ denote the Haar measure on the $n$-dimensional unitary group $\mathcal{U}(n)$. Let $\{P_{n},Q_{n}\}_{n\geq 1}$ denote a sequence of independent, ${\mathcal{H}}_{n}$-distributed matrices. Let $T_{n}$ denote a sequence of diagonal matrices, independent of $(P_{n},Q_{n})$, with real positive entries $S_{n}=\{s^{(n)}_{i}\}$ on the diagonal, and introduce the empirical measure of the symmetrized version of $T_{n}$ as

Let $A_{n}=P_{n}T_{n}Q_{n}$, let $\Lambda_{n}=\{\lambda_{i}^{(n)}\}$ denote the set of eigenvalues of $A_{n}$, and set

There exist constants $\kappa,\kappa_{1}>0$ such that

$L_{A_{n}}$ converges in probability to a limiting probability measure $\mu_{A}$.

The support of $\mu_{A}$ is a single ring: there exist constants $0\leq a<b<\infty$ so that

Further, $a=0$ if and only if $\int x^{-2}d\Theta(x)=\infty$.

See Remark 7 for an explicit characterization of the free convolution appearing in Theorem 1, and [1, Ch. 5] for general background. A different characterization of $\rho_{A}$, borrowed from and instrumental in the proof of part (c) of Theorem 1, is provided in Remark 8 in Section 3.1.

As a corollary of Theorem 1, we prove the Feinberg-Zee “single ring theorem”.

Let $V$ denote a polynomial with positive leading coefficient. Let the $n$-by-$n$ complex matrix $X_{n}$ be distributed according to the law

where $Z_{n}$ is a normalization constant and $dX$ the Lebesgue measure on $n$-by-$n$ complex matrices. Let $L_{X_{n}}$ be the ESD of $X_{n}$. Then $\{L_{X_{n}}\}_{n}$ satisfies the conclusions of Theorem 1 with $\Theta$ the unique minimizer of the functional

Theorem 3 will follow by checking that the assumptions of Theorem 1 are satisfied for the spectral decomposition $X_{n}=U_{n}T_{n}V_{n}$, see Section 6.

The second hypothesis in Theorem 1 may seem difficult to verify in general; we show in the next proposition that adding a small Gaussian matrix guarantees it.

Let $(T_{n})_{n\geq 0}$ be a sequence of matrices satisfying the assumptions of Theorem 1 except for (3) and assume that $\|T_{n}^{-1}\|$ is uniformly bounded. Let $N_{n}$ be a $n\times n$ matrix with independent (complex) Gaussian entries of zero mean and covariance equal identity. Let $U_{n},V_{n}$ follow the Haar measure on unitary $n\times n$ matrices, independently of $T_{n},N_{n}$. Then, the empirical measure of the eigenvalues of $Y_{n}:=U_{n}T_{n}V_{n}+n^{-\gamma}N_{n}$ converges weakly in probability to $\mu_{A}$ as in Theorem 1 for any $\gamma\in(\frac{1}{2},\infty)$.

satisfies the hypotheses of Proposition 4.

A rather straightforward generalization of Theorem 1 concerns the limiting spectral measure of $P_{n}+B_{n}$, where $P_{n}$ is ${\mathcal{H}}_{n}$ distributed and the sequence of $n\times n$ matrices $B_{n}$ converges in $*$-moments to an operator $b$ in a non-commutative probability space $({\mathcal{A}},\tau)$. (The latter means that for all polynomial $P$ in two non-commutative variables,

An example of matrices $B_{n}$ which satisfy the hypotheses of Proposition 6 is given by the diagonal matrices $B_{n}=\mbox{diag}(s_{1}^{n},\ldots,s_{n}^{n})$ with entries $s_{i}^{n}$ satisfying the hypotheses of Example 5. This is easily verified from the fact that the eigenvalues of $D_{n}(w)$ are given by $(|w-s_{1}^{n}|,\ldots,|w-s_{n}^{n}|)$.

The main difficulty in studying the ESD $L_{A_{n}}$ is that $A_{n}$ is not a normal matrix, that is $A_{n}A^{*}_{n}\not=A^{*}_{n}A_{n}$, almost surely. For normal matrices, the limit of ESDs can be found by the method of moments or by the method of Stieltjes’ transforms. For non-normal matrices, the only known method of proof is more indirect and follows an idea of Girko that we describe now (the details are a little different from what is presented in Girko or Bai ).

From Green’s formula, for any polynomial $P(z)=\prod_{j=1}^{n}(z-\lambda_{j})$, we have

It will be convenient for us to introduce the $2n\times 2n$ matrix

It may be checked easily that eigenvalues of $H_{n}^{z}$ are the positive and negative of the singular values of $zI-A_{n}$. Therefore, if we let $\nu_{n}^{z}$ denote the ESD of $H_{n}^{z}$,

This is Girko’s formula in a different form and its utility lies in the following attack on finding the limit of $L_{A_{n}}$.

Justify that $\int\log|x|d\nu_{n}^{z}(x)\rightarrow\int\log|x|d\nu^{z}(x)$ for (almost every) $z$. But for the fact that “$\log$” is not a bounded function, this would have followed from the weak convergence of $\nu_{n}^{z}$ to $\nu^{z}$. As it stands, this is the hardest technical part of the proof.

A standard weak convergence argument is then used in order to convert the convergence for (almost every) $z$ of $\nu_{n}^{z}$ to a convergence of integrals over $z$. Indeed, setting $h(z):=\int\log|x|d\nu^{z}(x)$, we will get from (6) that

Show that $h$ is smooth enough so that one can integrate the previous equation by parts to get

which identifies $\Delta h(z)$ as the density (with respect to Lebesgue measure) of the limit of $L_{A_{n}}$.

Identify the function $h$ sufficiently precisely to be able to deduce properties of $\Delta h(z)$. In particular, show the single ring phenomenon, which states that the support of the limiting spectral measure is a single annulus (the surprising part being that it cannot consist of several disjoint annuli).

Girko’s equation (6) and these five steps give a general recipe for finding limiting spectral measures of non-normal random matrices. Whether one can overcome the technical difficulties depends on the model of random matrix one chooses. For the model of random matrices with i.i.d. entries having zero mean and finite variance, this has been achieved in stages by Bai , Götze and Tikhomirov , Pan and Zhou and Tao and Vu . While we heavily borrow from that sequence, a major difficulty in the problem considered here is that there is no independence between entries of the matrix $A_{n}$. Instead, we will rely on properties of the Haar measure, and in particular on considerations borrowed from free probability and the so called Schwinger–Dyson (or master-loop) equations. Such equations were already the key to obtaining fine estimates on the Stieltjes transform of Gaussian generalized band matrices in . In , they were used to study the asymptotics of matrix models on the unitary group. Our approach combines ideas of to estimate Stieltjes transforms and the necessary adaptations to unitary matrices as developped in . The main observation is that one can reduce attention to the study of the ESD of matrices of the form $(T+U)(T+U)^{*}$ where $T$ is real diagonal and $U$ is Haar distributed. In the limit (i.e., when $T$ and $U$ are replaced by operators in a $C^{*}$-algebra that are freely independent, with $T$ bounded and self adjoint and $U$ unitary), the limit ESD has been identified by Haagerup and Larsen . The Schwinger–Dyson equations give both a characterization of the limit and, more important to us, a discrete approximation that can be used to estimate the discrepancy between the pre-limit ESD and its limit. These estimates play a crucial role in integrating the singularity of the log in Step two above, but only once an a-priori (polynomial) estimate on the minimal singular value has been obtained. The latter is deduced from assumption 3. In the context of the Feinberg–Zee single ring theorem, the latter assumption holds due to an adaptation of the analysis of .

Notation

We describe our convention concerning constants. Throughout, by the word constant we mean quantities that are independent of $n$ (or of the complex variables $z$, $z_{1}$). Generic constants denoted by the letters $C$,$c$ or $R$, have values that may change from line to line, and they may depend on other parameters. Constants denoted by $C_{i}$, $K$, $\kappa$ and $\kappa^{\prime}$ are fixed and do not change from line to line.

where $W_{n}^{z}=\overline{z}Q_{n}P_{n}/|z|$ is unitary and $\mathcal{H}_{n}$ distributed. Throughout, we will write $\rho=|z|$. We also will assume in this section that the sequence $T_{n}$ is deterministic. We are thus led to the study of the ESD for a sequence of matrices of the form

with $B_{n}=\rho U_{n}+T_{n}$, $T_{n}$ being a real, diagonal matrix of uniformly bounded norm, and $U_{n}$ a ${\mathcal{H}}_{n}$ unitary matrix. Because $\|T_{n}\|$ is uniformly bounded, it will be enough to consider throughout $\rho$ uniformly bounded.

where we used the notation $A\otimes B\sharp C=ACB$.

By the invariance of $\mu$ under unitary conjugation, see [27, Proposition 5.17] or [1, (5.4.31)], we have the Schwinger–Dyson equation

We continue to use the notation ${\bf Y}$, ${\bf U},{\bf U^{*}}$ and ${\bf T}$ in a way similar to (17) and (18). So, we let ${\bf Y}=\rho({\bf U}+{\bf U}^{*})+{\bf T}$ with

We extend $\mu$ to the algebra generated by ${\bf U},{\bf U}^{*}$ and ${\bf T}$ by putting for any $A,B,C,D\in{\cal A}$,

Observe that this extension is still tracial.

The non-commutative derivative $\partial$ in (20) extends naturally to the algebra generated by the matrix-valued ${\bf U},{\bf U}^{*},{\bf T}$, using the Leibniz rule (19) together with the relations

where $A(B\otimes C)D=(AB)\otimes(CD)$. Further, (21) extends also in this context.

We apply the derivative $\partial$ to the analytic function $P=(z_{1}-{\bf Y})^{-1}(z_{2}-{\bf T})^{-1}{\bf U}\,,$ while noticing that, by (19) and (23),

Applying (21), with $\mu(P)=G_{U}(z_{1},z_{2})$ and $\mu(p)=1/2$, we find

Note that $Pp=P$ and thus $\mu(pP)=\mu(P)$. Further, for any smooth function $Q$, $\mu({\bf U}^{*}Q{\bf U})$ equals $\mu((1-p)Q)$ due to the traciality of $\mu$ and ${\bf U}{\bf U}^{*}=1-p$. By symmetry (note that $(1-p)(z_{1}-{\bf Y})^{-1}(z_{2}-{\bf T})^{-1}$ and $p(z_{1}-{\bf Y})^{-1}(z_{2}-{\bf T})^{-1}$ are given by the same formula up to replacing $({\bf U},{\bf U}^{*})$ by $({\bf U}^{*},{\bf U})$, which has the same law) we get that $\mu({\bf U}^{*}P)$ equals

The first equality holds without the last factor $(z_{2}-{\bf T})^{-1}$, thus implying that $\mu((z_{1}-{\bf Y})^{-1}p)=\mu((z_{1}-{\bf Y})^{-1})/2=G(z_{1})/2$ and so we get from (26) that

Noticing that $G_{U}(z_{1})$ is the limit of $z_{2}G_{U}(z_{1},z_{2})$ as $z_{2}\to\infty$, we find by (28) that

and therefore, as $G_{U}(z_{1})$ goes to zero as $z_{1}\to\infty$,

(Again, here and in the rest of this subsection, the proper branch of the square root is determined by analyticity.) Let $R_{\rho}$ denote the $R$-transform of the Bernoulli law $\lambda_{\rho}:=(\delta_{-\rho}+\delta_{+\rho})/2$, that is,

see [1, Definition 5.3.22 and Exercise 5.3.27], so that we have

Repeating the computation with $G_{U^{*}}$, we have $G_{U^{*}}=G_{U}$. Algebraic manipulations yield

Therefore, we get by substituting (31) and (32) into (33) that

and has an analytic continuation to a neighborhood of ${\cal D}$, and $F^{\prime}>0$ on ${\cal D}$. Further, with $\mu_{A}$ as above, $\rho_{A}(re^{i\theta})=\rho_{A}(r)$ and it holds that

Finally, $\rho_{A}$ has an analytic continuation to a neighborhood of $(a,b]$, and $\mu_{A}$ is a probability measure, see [13, Pg 333].

In the next section, we will need the following estimate.

If $|\Im G_{T}(\cdot)|\leq\kappa_{1}$ on $\{z:\Im(z)\geq\epsilon\}$ then $|\Im G(\cdot)|\leq\kappa_{1}$ on $\{z:\Im(z)\geq\epsilon\}$.

2 Finite n𝑛n equations and convergence

We next turn to the evaluation of the law of ${\bf Y}_{n}$. We assume throughout that the sequence $T_{n}$ is uniformly bounded by some constant $M$, that $L_{T_{n}}\to\mu_{T}$ weakly in probability, and further that (4) is satisfied. All constants in this section are independent of $\rho$, but depend implicitly on $M$, the uniform bound on $\|T_{n}\|$ and on $\rho$.

We get by taking $P=(z_{1}-{\bf Y}_{n})^{-1}(z_{2}-{\bf T}_{n})^{-1}{\bf U}_{n}$ that

with $\|P\|_{L}$ the Lipschitz constant of $P$ given by

if $D$ is the cyclic derivative given by $D=m\circ\partial$ with $m(A\otimes B)=BA$ and $\|DP\|_{\infty}$ denotes the operator norm. (The appearance of the cyclic derivative in the evaluation of the Lipshitz constant can be seen by approximating $P$ by polynomials.) Applying (41) to each term of $\partial P$ (recall formula (25)), we get that for $\Im(z_{1}),\Im(z_{2})>0$, and with $a\wedge b=\min(a,b)$,

(The inequality uses that for any Hermitian matrix, $\|(z-H)^{-1}\|_{\infty}\leq 1/|\Im(z)|$.) Multiplying by $z_{2}$ and taking the limit as $z_{2}\to\infty$ we deduce from (40) that

with again the choice of the square root determined by analyticity and behavior at infinity.

and therefore, as $(\Re(z)-x)^{2}+\Im(z)^{2}\leq 4R^{2}$ for all $z,x\in B(0,R)$

Moreover, since $|G_{U}^{n}(z)|\leq 1/|\Im(z)|$, we deduce from (42) that for some constant $c$ independent of $n$ and all $n$ large,

Combining this estimate and (48), we get that

as soon as $\Im(z)>C_{1}n^{-1/3}$ for an appropriate $C_{1}$, and $|z|<R$. The conclusion follows.

There exists a constant $C_{3}$ such that if $\Im(z)>C_{3}n^{-1/4}$, then $\Im(\psi_{n}(z))\geq\Im(z)/2.$

Then, for any $z\in\Delta_{n}$, and whatever choice of branch of the square root made in (43), if ${\bf e}_{n}^{-1/2}O_{1}(n,z)$ is small enough (smaller than ${\bf e}_{n}/2$ is fine), then that choice can be extended to include a neighborhood of the point $w=G^{n}(z)$ such that with this choice, the function $r_{\rho}(w)=\frac{1}{4\rho}(-1+\sqrt{1+4\rho^{2}w^{2}})$ is Lipschitz in the sense that

Combining the last display with the relation $R_{\rho}(\theta)=2r_{\rho}(\theta)/\theta$, (50) and (48), one obtains that for $z\in\Delta_{n}$,

Since the above right hand side is smaller than $\Im(z)/2$ for $\Im(z)>n^{-1/4}$, we conclude that for $z\in\Delta_{n}\cap\{\Im(z)>n^{-1/4}\}$

as, regardless of the branch taken in the definition of $R_{\rho}(\cdot)$, $\Im R_{\rho}(G^{n}(z))\leq 0$.

We thus conclude from the last display and (51) the existence of a constant $C_{3}$ such that if $\Im(z)>C_{3}n^{-1/4}$ then

We have made all preparatory steps in order to state the main result of this subsection.

There exist positive finite constants $C_{6},C_{7},C_{8}$ such that, for $n>C_{6}$ and all $z\in{\cal E}_{n}:=\{z:\Im(z)>n^{-C_{7}}\}$,

Proof This is immediate from Lemma 11, Lemma 12, the definition of $\psi_{n}$, the assumption (4) on $G_{T_{n}}$, and the equality (46). ∎

in probability. (ii) Fix $R>0$. For any smooth compactly supported deterministic function $\varphi$ on $B_{R}$,

Before bringing the proof of Proposition 14, we recall the following elementary lemma.

We can now provide the Proof of Proposition 14

(i) Assume $z\in B_{R}$ for some $R>0$. By (3), we can replace the lower limit of integration in (53) with $n^{-\delta}$. Let $G_{n}^{z}$ denote the Stieltjes transform of $E[\nu_{n}^{z}]$. By Lemma 13 and Lemma 9, there exist positive constants $c_{1}=c_{1}(R),c_{2}=c_{2}(R)$ such that whenever $\Im(u)>n^{-c_{1}}$, it holds that $|\Im G_{n}^{z}(u)|<c_{2}$. We may and will assume that $c_{1}<\delta$.

Since $G_{n}^{z}$ is the Stieltjes transform of $E[\nu_{n}^{z}]$, by Lemma 15, we have for any $y>0$ that

Thus, we get that for any $z\in B_{R}$ and with $\alpha\in$,

where $2^{J-1}n^{-c_{1}}<\epsilon\leq 2^{J}n^{-c_{1}}$. Note that by Lemma 15 and the estimate on $G_{n}^{z}$, for $j\geq 0$,

where the constant $C=C(R)$. To obtain the estimate (53), we will consider $\alpha=1$ and argue as follows. Due to (3), for $\alpha<2$ we have

by Hölder’s inequality. The first factor goes to zero because

By (3), the second factor is bounded by $(\delta^{\prime})^{\alpha/2}$. We thus get (53) from (57). By Chebycheff’s inequality, the convergence in expectation implies the convergence in probability and therefore for any $\delta,\delta^{\prime}>0$ there exists $\epsilon>0$ small enough so that

On the other hand, $\int_{\epsilon}^{\infty}\log|x|d\nu_{n}^{z}(x)$ converges to $\int_{\epsilon}^{\infty}\log|x|d\nu^{z}(x)$ by the weak convergence of $\nu_{n}^{z}$ to $\nu^{z}$ in probability for any $\epsilon>0$, and $\int_{0}^{\epsilon}\log|x|d\nu^{z}(x)$ converges to as $\epsilon\to 0$ since $\nu^{z}$ has a bounded density by Lemma 9. Hence, we get (54).

and set $f_{n}(z)=f_{n}^{1}(z)+f_{n}^{2}(z)$. Because $\nu_{n}^{z}$ is supported in $B_{R+M}$ on $\|T_{n}\|\leq M$ for all $z\in B_{R}$, $f_{n}$ is bounded above by $\log(R+M)$. By (57), $E[|f_{n}^{2}(\cdot)|^{2}$ is bounded, uniformly in $z\in B_{R}$. On the other hand, by (3), again uniformly in $z\in B_{R}$, $E(f_{n}^{1}(z)^{2})<\delta^{\prime}$, and therefore

is bounded in probability. This uniform integrability and the weak convergence (54) are enough to conclude, using dominated convergence (see [25, Lemma 3.1] for a similar argument). ∎

Proof of Theorem 1

in probability. Since the sequence $L_{A_{n}}$ is tight, it thus follows that it converges, in the sense of distribution, to the measure

From Remark 8 (based on [13, Corollary 4.5]), we have that $\mu_{A}$ is a probability measure that possesses a radially symmetric density $\rho_{A}$ satisfying the properties stated in parts b and c of the theorem. ∎

Proof of Theorem 3

which goes to zero for $\kappa\in(0,(1-2\kappa^{\prime})/4)$. Together with [22, Equation (2.32)], this proves point 3 of the assumptions. Thus, it remains only to check point 2 of the assumptions. Toward this end, define ${\cal G}_{n}=\{\sigma_{1}^{n}<M+1\}$ and note that we may and will restrict attention to $|z|<M+2$ when checking (3). We begin with the following proposition, due to .

Let $\overline{A}$ be an arbitrary $n$-by-$n$ matrix, and let $A=\overline{A}+\sigma N$ where $N$ is a matrix with independent (complex) Gaussian entries of zero mean and unit variances. Let $\sigma_{n}(A)$ denote the minimal singular value of $A$. Then, there exists a constant $C_{12}$ independent of $\overline{A}$, $\sigma$ or $n$ such that

The proof of Proposition 16 is identical to [23, Theorem 3.3], with the required adaptation in moving from real to complex entries. (Specifically, in the right side of the display in [23, Lemma A.2], $\epsilon\sqrt{2/\pi}/\sigma$ is replaced by its square.) We omit further details.

On the event ${\cal G}_{n}$, all entries of the matrix $X_{n}$ are bounded by a constant multiple of $\sqrt{n}$. Let $N_{n}$ be a Gaussian matrix as in Proposition 16. With $\alpha>2$ a constant to be determined below, set

Turning to the construction, observe first that from (59),

Let $X_{n}^{(\alpha)}=X_{n}+n^{-\alpha}N_{n}{\bf 1}_{{\cal G}_{n}^{\prime}}$. Let $\{\theta_{i}\}$ and $\{\mu_{i}\}$ denote the eigenvalues of $W_{n}=X_{n}X_{n}^{*}$ and of $W_{n}^{(\alpha)}=(X_{n}^{(\alpha)})(X_{n}^{(\alpha)})^{*}$, respectively, arranged in decreasing order. Note that the density of $X_{n}$ is of the form

where the variable ${\bf x}=\{x_{i,j}\}_{1\leq i,j\leq n}$ is matrix valued and $d{\bf x}=\prod_{1\leq i,j\leq n}dx_{i,j}$, while that of $X_{n}^{(\alpha)}$ is of the form

where $E_{N}$ denotes expectation with respect to the law of $N_{n}$, and $Z_{n}$ is the same in both expressions. Note that $\sigma_{1}(X_{n}^{(\alpha)})\in[\sigma_{1}(X_{n})-1,\sigma_{1}(X_{n})+1]$. Because $V(\cdot)$ is locally Lipschitz, we have that if either $\sigma_{1}(X_{n})\leq M+1$ or $\sigma_{1}(X_{n}^{(\alpha)})\leq M+1$, then there exists a constant $C_{13}$ independent of $\alpha$ so that

where the Cauchy–Schwarz inequality was used in the third inequality and the Hoffman–Wielandt inequality in the next (see e.g. [1, Lemma 2.1.19]). On the event ${\cal G}_{n}$, all entries of $W_{n}-W_{n}^{\alpha}$ are bounded by $n^{(3-\alpha)/2}$. Therefore,

where the constant $C_{14}$ does not depend on $\alpha$. In particular, if $\alpha>(C_{14}+1)\vee 2$ we obtain that on ${\cal G}_{n}$, the ratio of the functions $f_{n}=e^{-n{\mbox{\rm tr}}(V(W_{n}))}$ and $g_{n}=e^{-n{\mbox{\rm tr}}(V(W_{n}^{(\alpha)}))}$ is bounded e.g. by $1+n^{(C_{14}+1-\alpha)/2}$; in particular, it holds that

Therefore, the variational distance between the law of $X_{n}$ conditioned on $\sigma_{1}(X_{n})<M$ and that of $X_{n}^{(\alpha)}$ conditioned on $\sigma_{1}(X_{n}^{(\alpha)})<M$, is bounded by

It follows that one can construct a matrix $Y_{n}$ of law identical to the law of $X_{n}^{(\alpha)}$ conditioned on $\sigma_{1}(X_{n}^{\alpha})<M$, together with $X_{n}$, on the same probability space so that

where $\alpha$ was chosen as function of $x$. This yields immediately point 2 of the assumptions of Theorem 1, if $\delta>5C_{17}/2$.

We have checked now that in the setup of Theorem 3, all the assumptions of Theorem 1 hold. Applying now the latter theorem completes the proof of Theorem 3. ∎

The proof of Theorem 3 carries over to more general situations; indeed, $V$ does not need to be a polynomial, it is enough that its growth at infinity is polynomial and that it is locally Lipschitz, so that the results of still apply. We omit further details.

Proof of Proposition 4

with $C(\|T_{n}^{-1}\|,\|T_{n}\|)$ a finite constant depending only on $\|T_{n}^{-1}\|,\|T_{n}\|$ which we assumed bounded. (In deriving the last estimate, we used that $\|(I+B)^{1/2}-I\|\leq\|B\|$ when $\|B\|<1/2$.) As a consequence, the third condition is satisfied since

with $\gamma^{\prime}=\min\{\kappa,\frac{1}{2}(\gamma-\frac{1}{2})\}$ and $\Im(z)\geq n^{-\max\{\frac{1}{2}(\gamma-\frac{1}{2}),\kappa^{\prime}\}}$. Hence, the results of Lemma 13 hold and we need only check, as in Proposition 14, that with $\nu_{n}^{z}$ the empirical measure of the singular values of $zI-Y_{n}$,

where we finally used that $F^{-1}$ is Hölder continuous with index $\alpha$. ∎

Extension to orthogonal conjugation

In this section, we generalize Theorem 1 to the case where we conjugate $T_{n}$ by orthogonal matrices instead of unitary matrices.

Proof of Proposition 6

whereas our hypotheses allow us to bound uniformly the Stieltjes transform of $\nu^{n}_{z}$ on $\{z_{1}:\Im(z_{1})\geq n^{-C_{7}}\}$ as in Lemma 13, hence providing a control of the integral on the interval $[n^{-C_{7}},\epsilon]$. The control of the integral for $x<n^{-C_{7}}$ uses a regularization by the Gaussian matrix $n^{-\gamma}N_{n}$ as in Proposition 4 .∎

Acknowledgments: We thank Greg Anderson for many fruitful and encouraging discussions. We thank Yan Fyodorov for pointing out the paper and Philippe Biane for suggesting that our technique could be applied to the examples in . We thank the referee for a careful reading of the manuscript.

References