**E**l e c t ro nic

**J**ourn a l
of

**P**r

ob a b il i t y

Vol. 16 (2011), Paper no. 64, pages 1750–1792.

Journal URL

http://www.math.washington.edu/~ejpecp/

**Free convolution with a semicircular distribution and** **eigenvalues of spiked deformations of Wigner matrices**

^{∗}

M. Capitaine^{†}, C. Donati-Martin^{‡}, D. Féral^{§} and M. Février^{¶}

**Abstract**

We investigate the asymptotic behavior of the eigenvalues of spiked perturbations of Wigner
matrices defined by*M** _{N}* =

^{p}

^{1}

_{N}*W*

*+*

_{N}*A*

*, where*

_{N}*W*

*is a*

_{N}*N*×

*N*Wigner Hermitian matrix whose entries have a distribution

*µ*which is symmetric and satisfies a Poincaré inequality and

*A*

*is a deterministic Hermitian matrix whose spectral measure converges to some probability measure*

_{N}*ν*with compact support. We assume that

*A*

*has a fixed number of fixed eigenvalues (spikes) outside the support of*

_{N}*ν*whereas the distance between the other eigenvalues and the support of

*ν*uniformly goes to zero as

*N*goes to infinity. We establish that only a particular subset of the spikes will generate some eigenvalues of

*M*

*which will converge to some limiting points outside the support of the limiting spectral measure. This phenomenon can be fully described in terms of free probability involving the subordination function related to the free additive convolution of*

_{N}*ν*by a semicircular distribution. Note that only finite rank perturbations had been considered up to now (even in the deformed GUE case).

**Key words:** Random matrices; Free probability; Deformed Wigner matrices; Asymptotic spec-
trum; Extreme eigenvalues; Stieltjes transform; Subordination property.

∗This work was partially supported by the*Agence Nationale de la Recherche grant ANR-08-BLAN-0311-03.*

†CNRS, Institut de Mathématiques de Toulouse, Equipe de Statistique et Probabilités, F-31062 Toulouse Cedex 09.

E-mail: mireille.capitaine@math.univ-toulouse.fr

‡Université de Versailles-St Quentin, Laboratoire de Mathématiques, 45 avenue de Etats Unis, F-78035 Versailles Cedex.

E-mail: catherine.donati-martin@uvsq.fr

§Institut de Mathématiques de Bordeaux, Université Bordeaux 1, 351 Cours de la Libération, F-33405 Talence Cedex.

E-mail: delphine.feral@math.u-bordeaux1.fr

¶Institut de Mathématiques de Toulouse, Equipe de Statistique et Probabilités, F-31062 Toulouse Cedex 09. E-mail:

fevrier@math.univ-toulouse.fr

**AMS 2010 Subject Classification:**Primary 15B52, 60B20, 46L54, 15A1.

Submitted to EJP on November 8, 2010, final version accepted August 9, 2011.

**1** **Introduction**

In the fifties, in order to describe the energy levels of a complex nuclei system by the eigenvalues of
large Hermitian matrices, E. Wigner introduced the so-called Wigner*N*×*N* matrix*W** _{N}*. According
to Wigner’s work[36],[37]and further results of different authors (see[3]for a review), provided
the common distribution

*µ*of the entries is centered with variance

*σ*

^{2}, the large

*N-limiting spectral*distribution of the rescaled complex Wigner matrix

*X*

*=*

_{N}^{p}

^{1}

_{N}*W*

*is the semicircle distribution*

_{N}*µ*

*whose density is given by*

_{σ}*dµ*_{σ}

*d x* (x) = 1
2πσ^{2}

p4σ^{2}−*x*^{2}11_{[−}_{2}_{σ}_{,2}* _{σ]}*(

*x*). (1.1) Moreover, if the fourth moment of the measure

*µ*is finite, the largest (resp. smallest) eigenvalue of

*X*

*converges almost surely towards the right (resp. left) endpoint 2σ (resp. −2σ) of the semicircular support (cf.[7]or Theorem 2.12 in[3]).*

_{N}Now, how does the spectrum behave under a deterministic Hermitian perturbation*A** _{N}*? The set of
possible spectra for

*M*

*=*

_{N}*X*

*+*

_{N}*A*

*depends in a complicated way on the spectra of*

_{N}*X*

*and*

_{N}*A*

*(see [21]). Nevertheless, when*

_{N}*N*becomes large, free probability provides us a good understanding of the global behavior of the spectrum of

*M*

*. Indeed, if the spectral measure of*

_{N}*A*

*weakly converges to some probability measure*

_{N}*ν*andk

*A*

*kis uniformly bounded in*

_{N}*N*, the spectral distribution of

*M*

*weakly converges to the free convolution*

_{N}*µ*

**

_{σ}*ν*almost surely and in expectation (cf[1],[27]and [33],[19]for pioneering works). We refer the reader to[35]for an introduction to free probability theory. Note that when

*A*

*is of finite rank, the spectral distribution of*

_{N}*M*

*still converges to the semicircular distribution (*

_{N}*ν*≡

*δ*0and

*µ*

**

_{σ}*ν*=

*µ*

*).*

_{σ}In [30], S. Péché investigated the deformed GUE model *M*_{N}* ^{G}* =

*W*

_{N}

^{G}*/*p

*N* +*A** _{N}*, where

*W*

_{N}*is a GUE matrix, that is a Wigner matrix associated to a centered Gaussian measure with variance*

^{G}*σ*

^{2}and

*A*

*is a deterministic perturbation of finite rank with fixed eigenvalues. This model is the additive analogue of the Wishart matrices with spiked covariance matrix previously considered by J.*

_{N}Baik, G. Ben Arous and S. Péché[8]who exhibited a striking phase transition phenomenon for the
fluctuations of the largest eigenvalue according to the values of the spikes. S. Péché pointed out an
analogous phase transition phenomenon for the fluctuations of the largest eigenvalue of *M*_{N}* ^{G}* with
respect to the largest eigenvalue

*θ*of

*A*

*[30]. These investigations imply that, if*

_{N}*θ*is far enough from zero (

*θ > σ*), then the largest eigenvalue of

*M*

_{N}*jumps above the support[−2*

^{G}*σ*, 2

*σ]*of the limiting spectral measure and converges (in probability) towards

*ρ*

*=*

_{θ}*θ*+

^{σ}

_{θ}^{2}. Note that Z. Füredi and J. Koml´os already exhibited such a phenomenon in[22]dealing with non-centered symmetric matrices.

In[20], D. Féral and S. Péché proved that the results of[30]still hold for a non-necessarily Gaussian
Wigner Hermitian matrix*W** _{N}* with sub-Gaussian moments and in the particular case of a rank one
perturbation matrix

*A*

*whose entries are all*

_{N}

^{θ}*for some real number*

_{N}*θ*. In[18], we considered a deterministic Hermitian matrix

*A*

*of arbitrary fixed finite rank*

_{N}*r*and built from a family of

*J*fixed non-null real numbers

*θ*1

*>*· · ·

*> θ*

*J*independent of

*N*and such that each

*θ*

*j*is an eigenvalue of

*A*

*of fixed multiplicity*

_{N}*k*

*(withP*

_{j}*J*

*j*=1*k** _{j}*=

*r*). In the following, the

*θ*

*j*’s are referred as the spikes of

*A*

*. We dealt with general Wigner matrices associated to some symmetric measure satisfying a Poincaré inequality. We proved that eigenvalues of*

_{N}*A*

*with absolute value strictly greater than*

_{N}*σ*generate some eigenvalues of

*M*

*which converge to some limiting points outside the support of*

_{N}*µ*

*. To be more precise, we need to introduce further notations. Given an arbitrary Hermitian matrix*

_{σ}*B*of size

*N*, we denote by*λ*1(B)≥ · · · ≥*λ**N*(B) its*N* ordered eigenvalues. For each spike *θ**j*, we denote by
*n*_{j}_{−}_{1}+1, . . . ,*n*_{j}_{−}_{1}+*k** _{j}* the descending ranks of

*θ*

*j*among the eigenvalues of

*A*

*(multiplicities of eigenvalues are counted) with the convention that*

_{N}*k*

_{1}+· · ·+

*k*

_{j−}_{1}=0 for

*j*=1. One has that

*n*_{j−}_{1}=*k*_{1}+· · ·+*k*_{j−}_{1} if*θ*_{j}*>*0 and *n*_{j−}_{1}=*N*−*r*+*k*_{1}+· · ·+*k*_{j−}_{1} if*θ*_{j}*<*0.

Letting*J*_{+σ} (resp. *J*_{−σ}) be the number of *j’s such thatθ**j* *> σ*(resp. *θ**j* *<*−σ), we established in
[18]that, when*N* goes to infinity,

a) for all *j* such that 1≤ *j*≤*J*_{+σ} (resp. *j* ≥*J*−*J*_{−σ}+1), the *k** _{j}* eigenvalues(λ

*n*

*1+i(*

_{j−}*M*

*), 1≤*

_{N}*i*≤

*k*

*)converge almost surely to*

_{j}*ρ*

_{θ}*=*

_{j}*θ*

*j*+

^{σ}

_{θ}^{2}

*j*

which is*>*2*σ*(resp.*<*−2*σ*).

b) *λ**k*_{1}+···+*k*_{J}_{+σ}+1(M*N*)−→* ^{a.s.}* 2

*σ*and

*λ*

*N*−(

*k*

*+···+*

_{J}*k*

_{J}_{−}

_{J}_{−σ}

_{+1})(M

*N*)−→ −

*2*

^{a.s.}*σ*.

Actually, this phenomenon may be described in terms of free probability involving the subordination
function related to the free convolution of*ν* =*δ*0 by a semicircular distribution. Let us present it
briefly. For a probability measure*τ* on R, let us denote by *g** _{τ}* its Stieltjes transform, defined for

*z*∈C\R

^{by}

*g** _{τ}*(z) =
Z

R

*dτ(x*)
*z*−*x* .

Let*ν* and*τ*be two probability measures onR. It is proved in[13]Theorem 3.1 that there exists an
analytic map*F*:C^{+}→C^{+}, called subordination function, such that

∀*z*∈C^{+}^{,}^{g}* _{τν}*(z) =

*g*

*(F(z)),*

_{ν}where C^{+} denotes the set of complex numbers *z* such that ℑ*z* *>* 0. When *τ*= *µ** _{σ}*, let us denote
by

*F*

_{σ}_{,}

*the corresponding subordination function. When*

_{ν}*ν*=

*δ*0 and

*τ*=

*µ*

*, the subordination function is given by*

_{σ}*F*

_{σ,δ}_{0}= 1

*/g*

_{µ}*. According to Lemma 4.4 in [18], one may notice that the complement of the support of*

_{σ}*µ*

**

_{σ}*δ*0(=

*µ*

*)can be described as:*

_{σ}R\[−2*σ*, 2*σ] =*{*x*,∃*u*∈R^{∗}^{,}|*u*|*> σ* such that *x*=*H*_{σ,δ}_{0}(*u*)},

where*H*_{σ}_{,}_{δ}_{0}(*z*) =*z*+^{σ}_{z}^{2} is the inverse function of the subordination function*F*_{σ}_{,}_{δ}_{0}onR\[−2*σ*, 2*σ]*.
Now, the characterization of the spikes of*A** _{N}* that generate jumps of eigenvalues of

*M*

*i.e.|θ*

_{N}*j*|

*> σ*is obviously equivalent to the following

*θ**j*∈R\supp(δ0)(=R^{∗}) and *H*^{0}_{σ,δ}

0(θ*j*)*>*0.

Moreover the relationship between a spike*θ**j* of*A** _{N}* such that|θ

*j*|

*> σ*and the limiting point

*ρ*

_{θ}*of the corresponding eigenvalues of*

_{j}*M*

*(which is then outside[−2*

_{N}*σ*; 2

*σ]*) is actually described by the inverse function of the subordination function as:

*ρ*_{θ}* _{j}* =

*H*

_{σ}_{,}

_{δ}_{0}(θ

*j*).

Actually this very interpretation in terms of subordination function of the characterization of the
spikes of*A** _{N}* that generate jumps of eigenvalues of

*M*

*as well as the values of the jumps provides*

_{N}the intuition to imagine the generalization of the phenomenon dealing with non-finite rank pertur-
bations just by replacing*δ*0 by the limiting spectral distribution*ν* of*A** _{N}* in the previous lines. Up to
now, no result has been established for non-finite rank additive spiked perturbation. Moreover, this
paper shows up that free probability can also shed light on the asymptotic behavior of the eigenval-
ues of the deformed Wigner model and strengthens the fact that free probability theory and random
matrix theory are closely related.

More precisely, in this paper, we consider the following general deformed Wigner models *M** _{N}* =

*X*

*+*

_{N}*A*

*such that:*

_{N}• *X** _{N}*=

^{p}

^{1}

_{N}*W*

*where*

_{N}*W*

*is a*

_{N}*N*×

*N*Wigner Hermitian matrix associated to a distribution

*µ*of variance

*σ*

^{2}and mean zero:

(*W** _{N}*)

*ii*,p

2ℜ((*W** _{N}*)

*i j*)

_{i<}*j*,p

2ℑ((*W** _{N}*)

*i j*)

_{i<}*j*are i.i.d., with distribution

*µ*which is symmetric and satisfies a Poincaré inequality (the definition of such an inequality is recalled in the Appendix).

• *A** _{N}* is a deterministic Hermitian matrix whose eigenvalues

*γ*

^{(N}

_{i}^{)}, denoted for simplicity by

*γ*

*, are such that the spectral measure*

_{i}*µ*

_{A}*:=*

_{N}

_{N}^{1}P

*N*

*i*=1*δ*_{γ}* _{i}* converges to some probability measure

*ν*with compact support. We assume that there exists a fixed integer

*r*≥0 (independent from

*N*) such that

*A*

*has*

_{N}*N*−

*r*eigenvalues

*β*

*j*(N)satisfying

1≤max*j*≤*N*−*r*dist(β*j*(*N*), supp(ν))−→

*N→∞*0,

where supp(ν)denotes the support of*ν*. We also assume that there are*J* fixed real numbers
*θ*1*>* . . .*> θ**J* independent of*N* which are outside the support of*ν* and such that each *θ**j* is
an eigenvalue of*A** _{N}* with a fixed multiplicity

*k*

*(withP*

_{j}

_{J}*j=1**k** _{j}*=

*r). Theθ*

*j*’s will be called the spikes or the spiked eigenvalues of

*A*

*.*

_{N}According to[1], the spectral distribution of *M** _{N}* weakly converges to the free convolution

*µ*

**

_{σ}*ν*almost surely (cf. Remark 4.1 below). It turns out that the spikes of

*A*

*that will generate jumps of eigenvalues of*

_{N}*M*

*will be the*

_{N}*θ*

*’s such that*

_{j}*H*

_{σ}^{0}

_{,}

*(θ*

_{ν}*)*

_{j}*>*0 where

*H*

_{σ}_{,}

*(z) =*

_{ν}*z*+

*σ*

^{2}

*g*

*(z)and the corresponding limiting points outside the support of*

_{ν}*µ*

**

_{σ}*ν*will be given by

*ρ*_{θ}* _{j}*=

*H*

_{σ}_{,}

*(θ*

_{ν}*j*).

It is worth noticing that the set{*u*∈R\supp(ν),*H*_{σ}^{0}_{,}* _{ν}*(u)

*>*0}is actually the complement of the closure of the open set

*U** _{σ,ν}*:=

¨
*u*∈R^{,}

Z

R

*dν*(*x*)
(*u*−*x*)^{2} *>* 1

*σ*^{2}

«

introduced by P. Biane in[12]to describe the support of the free additive convolution of a probability
measure *ν* on R by a semicircular distribution. Note that the deep study by P. Biane of the free
convolution by a semicircular distribution will be of fundamental use in our approach. In Theorem
8.1, which is the main result of the paper, we present a complete description of the convergence
of the eigenvalues of *M** _{N}* depending on the location of the

*θ*

*’s with respect to*

_{j}*U*

_{σ}_{,}

*and to the connected components of the support of*

_{ν}*ν*.

Our approach also allows us to study the “non-spiked" deformed Wigner models i.e. such that*r*=0.

Up to now, the results which can be found in the literature for such a situation concern the so-called

Gaussian matrix models with external source where the underlying Wigner matrix is from the GUE.

Many works on these models deal with the local behavior of the eigenvalues of*M** _{N}* (see for instance
[14], [2] and [15] for details). Moreover, the recent results of [26] (which investigate several
matrices in a free probability context) imply that the operator norm (i.e. the largest singular value)
of some non-spiked deformed GUE

*M*

_{N}*=*

^{G}*W*

_{N}

^{G}*/N*+

*A*

*converges almost surely to the*

_{N}*L*

^{∞}-norm of a(µ

**

_{σ}*ν*)-distributed random variable. Here, we readily deduce (cf. Proposition 8.1 below) from our results the almost sure convergence of the extremal eigenvalues of general non-spiked deformed Wigner models to the corresponding endpoints of the compact support of the free convolution

*µ*

*ν. The asymptotic behavior of the eigenvalues of the deformed Wigner model*

_{σ}*M*

*actually comes from two phenomena involving free convolution:*

_{N}1. the inclusion of the spectrum of*M** _{N}* in an

*ε*-neighborhood of the support of

*µ*

**

_{σ}*µ*

*A*

*, for all large*

_{N}*N*almost surely;

2. an exact separation phenomenon between the spectrum of *M** _{N}* and the spectrum of

*A*

*, in- volving the subordination function*

_{N}*F*

*of*

_{σ,ν}*µ*

**

_{σ}*ν*(i.e. to a gap in the spectrum of

*M*

*, it corresponds through*

_{N}*F*

_{σ}_{,}

*a gap in the spectrum of*

_{ν}*A*

*which splits the spectrum of*

_{N}*A*

*exactly as that of*

_{N}*M*

*).*

_{N}The key idea to prove the first point is to obtain a precise estimate of order ^{1}

*N* of the difference
between the respective Stieltjes transforms of the mean spectral measure of the deformed model
and of *µ** _{σ}*

*µ*

*A*

*. To get such an estimate, we prove an “approximative subordination equation"*

_{N}satisfied by the Stieltjes transform of the deformed model. Note that, even if the ideas and tools
are very close to those developed in[18], the proof in[18]does not use the above analysis from
free probability whereas this very analysis allows us to extend the results of[18]to non-finite rank
deformations. In particular, we didn’t consider in[18]*µ** _{σ}*

*µ*

*A*

*whose support actually makes the asymptotic values of the eigenvalues that will be outside the limiting support of the spectral measure of*

_{N}*M*

*appear.*

_{N}Note that phenomena 1. and 2. are actually the additive analogues of those described in[4],[5]in
the framework of spiked population models, even if the authors do not refer to free probability. In
[9], the authors use the results of[4],[5]to establish the almost sure convergence of the eigenval-
ues generated by the spikes in a spiked population model where all but finitely many eigenvalues
of the covariance matrix are equal to one. Thus, they generalize the pioneering result of [8] in
the Gaussian setting. Recently,[28], [6]extended this theory to a generalized spiked population
model where the base population covariance matrix is arbitrary. Our results are exactly the additive
analogues of theirs. It is worth noticing that one may check that these results on spiked population
models could also be fully described in terms of free probability involving the subordination function
related to the free multiplicative convolution of*ν* by a Marchenko-Pastur distribution.

Moreover, the results of F. Benaych-Georges and R. R. Nadakuditi in[11]about the convergence of
the extremal eigenvalues of a matrix*X** _{N}*+

*A*

*,*

_{N}*A*

*being a finite rank perturbation whereas*

_{N}*X*

*is a unitarily invariant matrix with some compactly supported limiting spectral distribution*

_{N}*µ, could*be rewritten in terms of the subordination function related to the free additive convolution of

*δ*0by

*µ*. Hence, we think that subordination property in free probability definitely sheds light on spiked deformed models.

Finally, one can expect that our results hold true in a more general setting than the one considered
here, namely only requires the existence of a finite fourth moment on the measure*µ*of the Wigner

entries. Nevertheless, the assumption that *µ* satisfies a Poincaré inequality is fundamental in our
approach since we need several variance estimates.

The paper is organized as follows. In Section 2, we first recall some results on free additive convo-
lution and subordination property as well as the description by P. Biane of the support of the free
convolution of some probability measure*ν* by a semicircular distribution. We then deduce a char-
acterization of this support via the subordination function when*ν* is compactly supported and we
exhibit relationships between the steps of the distribution functions of*ν* and*µ** _{σ}*ν. In Section 3, we
establish an approximative subordination equation for the Stieltjes transform

*g*

*of the mean spec- tral distribution of the deformed model*

_{N}*M*

*and explain in Section 4 how to deduce an estimation up to the order*

_{N}^{1}

*N*^{2} of the difference between*g** _{N}* and the Stieltjes transform of

*µ*

**

_{σ}*µ*

*A*

*when*

_{N}*N*goes to infinity. In Section 5, we show how to deduce the almost sure inclusion of the spectrum of

*M*

*in a neighborhood of the support of*

_{N}*µ*

**

_{σ}*µ*

*A*

*for all large*

_{N}*N; we use the ideas (based on inverse Stieltjes*tranform) of[23]and[31]in the non-deformed Gaussian complex, real or symplectic Wigner set- ting; nevertheless, since

*µ*

*µ*

_{σ}

_{A}*depends on*

_{N}*N, we need here to apply the inverse Stieltjes tranform*to functions depending on

*N*and we therefore give the details of the proof to convince the reader that the approach developped by[23]and[31]still holds. In Section 6, we show how the support of

*µ*

**

_{σ}*µ*

_{A}*makes the asymptotic values of the eigenvalues that will be outside the support of the limiting spectral measure appear since we prove that, for any*

_{N}*ε >*0, supp(µ

**

_{σ}*µ*

*A*

*)is included in an*

_{N}*ε*-neighborhood of supp(µ

**

_{σ}*ν*)Sn

*ρ*_{θ}* _{j}*,

*θ*

*j*such that

*H*

_{σ}^{0}

_{,}

*(θ*

_{ν}*j*)

*>*0o

, when *N* is large enough.

Section 7 is devoted to the proof of the exact separation phenomenon between the spectrum of*M** _{N}*
and the spectrum of

*A*

*, involving the subordination function*

_{N}*F*

*. In the last section, we show how to deduce our main result (Theorem 8.1) about the convergence of the eigenvalues of the deformed model*

_{σ,ν}*M*

*. Finally we present in an Appendix the proofs of some technical estimates on variances used throughout the paper.*

_{N}Throughout this paper, we will use the following notations.

- For a probability measure*τ*onR, we denote by*g** _{τ}*its Stieltjes transform defined for

*z*∈C\R by

*g** _{τ}*(z) =
Z

R

*dτ(x*)
*z*−*x* .

- *G** _{N}* denotes the resolvent of

*M*

*and*

_{N}*g*

*the mean of the Stieltjes transform of the spectral measure of*

_{N}*M*

*, that is*

_{N}*g** _{N}*(z) =E(tr

_{N}*G*

*(z)),*

_{N}*z*∈C\R

^{,}where tr

*is the normalized trace: tr*

_{N}*=*

_{N}

_{N}^{1}Tr.

We recall some useful properties of the resolvent (see[25],[17]).

**Lemma 1.1.** *For a N*×*N Hermitian or symmetric matrix M , for any z* ∈ C\Spect(M), we
*denote by G(z)*:= (z I* _{N}*−

*M*)

^{−}

^{1}

*the resolvent of M .*

*Let z*∈C\R^{,}

*(i)* k*G*(*z*)k ≤ |ℑ*z*|^{−1} *where*k.k*denotes the operator norm.*

*(ii)* |*G(z)**i j*| ≤ |ℑ*z*|^{−}^{1} *for all i,j*=1, . . . ,*N .*
*(iii) For p*≥2,

1
*N*

*N*

X

*i,j=1*

|*G*(*z*)*i j*|* ^{p}*≤(|ℑ

*z*|

^{−}

^{1})

*. (1.2)*

^{p}*(iv) The derivative with respect to M of the resolvent G(z)satisfies:*

*G*^{0}* _{M}*(z).B=

*G*(z)BG(z)

*for any matrix B.*

*(v) Let z*∈C* ^{such that}*|

*z*|

*>*k

*M*k

*; we have*

k*G*(*z*)k ≤ 1

|*z*| − k*M*k.

- ˜*g** _{N}* denotes the Stieltjes transform of the probability measure

*µ*

**

_{σ}*µ*

*A*

*.*

_{N}- When we state that some quantity∆*N*(z),*z*∈C\R^{, is}*O(*_{N}^{1}*p*), this means precisely that:

|∆*N*(z)| ≤ *P*(|ℑ*z*|^{−}^{1})
*N** ^{p}* ,

for some polynomial*P*with nonnegative coefficients which is independent of*N.*

- For any set*S*inR, we denote the set{*x* ∈R^{, dist}(*x,S*)≤*ε}*(resp.{*x* ∈R^{, dist}(*x*,*S*)*< ε}*) by
*S*+ [−ε,+ε](resp.*S*+ (−ε,+ε)).

**2** **Free convolution**

**2.1** **Definition and subordination property**

Let *τ* be a probability measure onR. Its Stieltjes transform *g** _{τ}* is analytic on the complex upper
half-planeC

^{+}. There exists a domain

*D*_{α}_{,}* _{β}* ={

*u*+

*i v*∈C

^{,}|

*u*|

*< αv,v> β}*

on which*g** _{τ}* is univalent. Let

*K*

*be its inverse function, defined on*

_{τ}*g*

*(D*

_{τ}*,*

_{α}*β*), and

*R*

*(z) =*

_{τ}*K*

*(z)−1*

_{τ}*z*.

Given two probability measures*τ*and*ν*, there exists a unique probability measure*λ*such that
*R** _{λ}*=

*R*

*+*

_{τ}*R*

_{ν}on a domain where these functions are defined. The probability measure*λ*is called the free convo-
lution of*τ*and*ν* and denoted by*τ**ν*.

The free convolution of probability measures has an important property, called subordination, which
can be stated as follows: let*τ*and*ν* be two probability measures onR; there exists an analytic map
*F*:C^{+}→C^{+}^{such that}

∀*z*∈C^{+}^{,} ^{g}* _{τν}*(z) =

*g*

*(F(z)).*

_{ν}This phenomenon was first observed by D. Voiculescu under a genericity assumption in[34], and then proved in generality in[13]Theorem 3.1. Later, a new proof of this result was given in[10], using a fixed point theorem for analytic self-maps of the upper half-plane.

**2.2** **Free convolution by a semicircular distribution**

In[12], P. Biane provides a deep study of the free convolution by a semicircular distribution. We first recall here some of his results that will be useful in our approach.

Let*ν* be a probability measure onR^{. P. Biane}[12]introduces the set
Ω* _{σ}*,

*ν*:={

*u*+

*i v*∈C

^{+}

^{,}

^{v}>v

_{σ}_{,}

*(u)}, where the function*

_{ν}*v*

_{σ}_{,}

*:R→R*

_{ν}^{+}is defined by

*v*_{σ}_{,}* _{ν}*(u) =inf

¨
*v*≥0,

Z

R

*dν*(x)

(u−*x*)^{2}+*v*^{2} ≤ 1
*σ*^{2}

«

and proves the following
**Proposition 2.1.** *[12]The map*

*H*_{σ}_{,}* _{ν}* :

*z*7−→

*z*+

*σ*

^{2}

*g*

*(*

_{ν}*z*)

*is a homeomorphism from*Ω* _{σ}*,

*ν*

*to*C

^{+}∪R

*which is conformal from*Ω

*,*

_{σ}*ν*

*onto*C

^{+}

^{. Let F}*,*

_{σ}*ν*:C

^{+}∪R→ Ω

*,*

_{σ}*ν*

*be the inverse function of H*

_{σ}_{,}

_{ν}*. One has,*

∀*z*∈C^{+}^{,} ^{g}_{µ}_{σ}_{ν}(*z*) =*g** _{ν}*(

*F*

*(*

_{σ,ν}*z*))

*and then*

*F*_{σ}_{,}* _{ν}*(

*z*) =

*z*−

*σ*

^{2}

*g*

_{µ}

_{σ}_{ν}(

*z*). (2.1) Note that in particular the Stieltjes transform ˜

*g*

*of*

_{N}*µ*

**

_{σ}*µ*

*A*

*satisfies*

_{N}∀*z*∈C^{+}^{,} ^{g}^{˜}*N*(z) = *g*_{µ}

*AN*(z−*σ*^{2}˜*g** _{N}*(z)). (2.2)
Considering

*H*

_{σ}_{,}

*as an analytic map defined in the whole upper half-planeC*

_{ν}^{+}, it is clear that

Ω* _{σ,ν}*=

*H*

_{σ}^{−}

_{,}

^{1}

*(C*

_{ν}^{+}). (2.3)

Let us give a quick proof of (2.3). Let*v>*0. Since
ℑ*H*_{σ}_{,}* _{ν}*(u+

*i v) =v(1*−

*σ*

^{2}

Z

R

*dν(x*)
(u−*x*)^{2}+*v*^{2}),
we have

ℑ*H*_{σ}_{,}* _{ν}*(u+

*i v)>*0⇐⇒

Z

R

*dν*(x)

(u−*x*)^{2}+*v*^{2} *<* 1

*σ*^{2}. (2.4)

Consequently one can easily see thatΩ* _{σ}*,

*ν*is included in

*H*

_{σ}^{−}

_{,}

^{1}

*(C*

_{ν}^{+}). Moreover if

*u*+

*i v*∈

*H*

_{σ}^{−}

_{,}

^{1}

*(C*

_{ν}^{+}) then (2.4) implies that

*v*≥

*v*

_{σ}_{,}

*(u). If we assume that*

_{ν}*v*=

*v*

_{σ}_{,}

*(u), then*

_{ν}*v*

_{σ}_{,}

*(u)*

_{ν}*>*0 and finally

Z

R

*dν*(x)

(*u*−*x*)^{2}+*v*^{2} = 1
*σ*^{2}

by Lemma 2 in[12]. This is a contradiction : necessarily*v>v** _{σ,ν}*(u)or, in other words,

*u+i v*∈Ω

*and we are done.*

_{σ,ν}The previous results of P. Biane allow him to conclude that *µ** _{σ}*

*ν*is absolutely continuous with respect to the Lebesgue measure and to obtain the following description of the support.

**Theorem 2.1.** *[12]Define*Ψ* _{σ,ν}* :R→R

*by:*

Ψ* _{σ}*,

*ν*(u) =

*H*

_{σ}_{,}

*(u+*

_{ν}*i v*

_{σ}_{,}

*(u)) =*

_{ν}*u*+

*σ*

^{2}Z

R

(u−*x*)d*ν(x*)
(u−*x*)^{2}+*v** _{σ}*(u)

^{2}.

Ψ* _{σ}*,

*ν*

*is a homeomorphism and, at the point*Ψ

*,*

_{σ}*ν*(u), the measure

*µ*

**

_{σ}*ν*

*has a density given by*

*p*

_{σ}_{,}

*(Ψ*

_{ν}*,*

_{σ}*ν*(u)) =

*v*

*(*

_{σ,ν}*u*)

*πσ*^{2} .
*Define the set*

*U** _{σ,ν}* :=

¨
*u*∈R^{,}

Z

R

*dν*(x)
(*u*−*x*)^{2} *>* 1

*σ*^{2}

«

=¦

*u*∈R^{,}^{v}* _{σ,ν}*(

*u*)

*>*0© .

*The support of the measureµ** _{σ}*ν

*is the image of the closure of the open set U*

_{σ,ν}*by the homeomorphism*Ψ

*,*

_{σ}*ν*

*.*Ψ

*,*

_{σ}*ν*

*is strictly increasing on U*

_{σ}_{,}

_{ν}*.*

Hence,

R\supp(µ* _{σ}*

*ν) = Ψ*

*,*

_{σ}*ν*(R\

*U*

_{σ}_{,}

*).*

_{ν}One hasΨ* _{σ}*,

*ν*=

*H*

_{σ}_{,}

*onR\*

_{ν}*U*

_{σ}_{,}

*andΨ*

_{ν}^{−}

_{σ,ν}^{1}=

*F*

_{σ}_{,}

*onR\supp(µ*

_{ν}**

_{σ}*ν*). In particular, we have the following description of the complement of the support:

R\supp(µ* _{σ}*

*ν*) =

*H*

_{σ}_{,}

*(R\*

_{ν}*U*

_{σ}_{,}

*). (2.5) Let*

_{ν}*ν*be a compactly supported probability measure. We are going to establish a characterization of the complement of the support of

*µ*

**

_{σ}*ν*involving the support of

*ν*and

*H*

*. We will need the following preliminary lemma.*

_{σ,ν}**Lemma 2.1.** *The support ofν* *is included in U*_{σ}_{,}_{ν}*.*

**Proof of Lemma 2.1:**Let *x*_{0} be inR\*U** _{σ,ν}*. Then, there is some

*ε >*0 such that[

*x*

_{0}−

*ε*,

*x*

_{0}+

*ε]*⊂ R\

*U*

*. For any integer*

_{σ,ν}*n*≥1, we define

*α*

*k*=

*x*

_{0}−

*ε*+2kε/nfor all 0≤

*k*≤

*n. Then, as the sets*[α

*k*,

*α*

*k*+1]are trivially contained inR\

*U*

_{σ}_{,}

*, one has that:*

_{ν}∀*u*∈[α* _{k}*,

*α*

*1], 1*

_{k+}*σ*

^{2}≥

Z _{α}_{k+}_{1}

*α**k*

*dν*(x)

(u−*x*)^{2} ≥ *ν*([α*k*,*α** _{k+1}*])
(α

*1−*

_{k+}*α*

*)*

_{k}^{2}. This readily implies that

*ν([x*_{0}−*ε*,*x*_{0}+*ε])*≤

*n−*1

X

*k=0*

*ν*([α*k*,*α**k*+1])≤(2*ε)*^{2}
*σ*^{2}*n* .

Letting*n*→ ∞, we get that*ν([x*_{0}−*ε*,*x*_{0}+*ε]) =*0, which implies that*x*_{0}∈R\supp(ν).
From the continuity and strict convexity of the function*u*−→R

R
*dν(x)*

(*u*−*x*)^{2} on R\supp(ν), it follows
that

*U*_{σ}_{,}* _{ν}* =supp(ν)∪ {

*u*∈R\supp(ν), Z

R

*dν*(*x*)
(u−*x*)^{2} ≥ 1

*σ*^{2}} (2.6)

and

R\*U*_{σ}_{,}* _{ν}*={

*u*∈R\supp(ν), Z

R

*dν*(*x*)
(*u*−*x*)^{2} *<* 1

*σ*^{2}}.

Now, as*H*_{σ}_{,}* _{ν}* is analytic onR\supp(ν), the following characterization readily follows:

R\*U*_{σ}_{,}* _{ν}* ={

*u*∈ R\supp(ν),

*H*

_{σ}^{0}

_{,}

*(u)*

_{ν}*>*0}. and thus, according to (2.5), we get

**Proposition 2.2.**

*x*∈R\supp(µ* _{σ}*

*ν*)⇔ ∃

*u*∈R\supp(ν)such that

*x*=

*H*

_{σ}_{,}

*(u),*

_{ν}*H*

_{σ}^{0}

_{,}

*(u)*

_{ν}*>*0.

**Remark 2.1.** *Note that H*_{σ}_{,}_{ν}*is strictly increasing on*R\*U*_{σ}_{,}_{ν}*since, if a<* *b are in*R\supp(ν), one
*has, by Cauchy-Schwarz inequality, that*

*H** _{σ,ν}*(

*b*)−

*H*

*(*

_{σ,ν}*a*) = (

*b*−

*a*)

1−*σ*^{2}
Z

R

*dν*(x)
(a−*x*)(b−*x*)

≥ (*b*−*a)*

1−*σ*^{2}p

(−*g*_{ν}^{0}(a))(−*g*_{ν}^{0}(b))

.
*which is nonnegative if a and b belong to*R\*U*_{σ}_{,}_{ν}*.*

**Remark 2.2.** *Each connected component of U*_{σ}_{,}_{ν}*contains at least one connected component of*supp(ν).

Indeed, let[s*l*,*t** _{l}*]be a connected component of

*U*

_{σ}_{,}

*. If*

_{ν}*s*

*or*

_{l}*t*

*is in supp(ν), [s*

_{l}*l*,

*t*

*]contains at least a connected component of supp(ν)since supp(ν)is included in*

_{l}*U*

_{σ}_{,}

*. Now, if neither*

_{ν}*s*

*nor*

_{l}*t*

*is in supp(ν), according to (2.6), we have*

_{l}Z

R

*dν*(x)
(*s** _{l}*−

*x*)

^{2}=

Z

R

*dν*(x)
(*t** _{l}*−

*x*)

^{2}= 1

*σ*^{2}.

Assume that [*s** _{l}*,

*t*

*] ⊂ R\supp(ν), then, by strict convexity of the function*

_{l}*u*7−→ R

R
*d**ν(**x*)
(*u*−*x*)^{2} on
R\supp(ν), one obtains that, for any*u*∈]*s** _{l}*,

*t*

*[,*

_{l}Z

R

*dν(x*)
(*u*−*x*)^{2} *<* 1

*σ*^{2},
which leads to a contradiction.

**Remark 2.3.** *One can readily see that*

*U*_{σ}_{,}* _{ν}* ⊂ {

*u, dist*(

*u, supp*(ν))≤

*σ}*

*and deduce, since*supp(ν)*is compact, that U*_{σ,ν}*is a relatively compact open set. Hence, U*_{σ,ν}*has a*
*finite number of connected components and may be written as the following finite disjoint union*

*U** _{σ,ν}*=

1

[

*l*=*m*

*s** _{l}*,

*t*

_{l}*with s*_{m}*<t*_{m}*<*. . .*<s*_{1}*<t*_{1}. (2.7)

We close this section with a proposition pointing out a relationship between the distribution func-
tions of*ν* and*µ** _{σ}*

*ν*.

**Proposition 2.3.** *Let*[s* _{l}*,

*t*

*]*

_{l}*be a connected component of U*

_{σ}_{,}

_{ν}*, then*(µ

**

_{σ}*ν)([Ψ*

*,*

_{σ}*ν*(s

*l*),Ψ

*,*

_{σ}*ν*(t

*l*)]) =

*ν([s*

*l*,

*t*

*]).*

_{l}**Proof of Proposition 2.3:** Let ]a,*b[* be a connected component of *U*_{σ}_{,}* _{ν}*. Since

*a*and

*b*are not atoms of

*ν*and

*µ*

**

_{σ}*ν*is absolutely continuous, it is enough to show

(µ* _{σ}*

*ν*)([Ψ

*,*

_{σ}*ν*(a),Ψ

*,*

_{σ}*ν*(b)]) =

*ν([a,b])*.

From Cauchy’s inversion formula,*µ** _{σ}*

*ν*has a density given by

*p*

*(*

_{σ}*x*) =−

_{π}^{1}ℑ(

*g*

*(*

_{ν}*F*

_{ν}_{,}

*(*

_{σ}*x*))and (µ

**

_{σ}*ν*)([Ψ

*(*

_{σ,ν}*a*),Ψ

*(*

_{σ,ν}*b*)]) =−1

*π*ℑ

Z Ψ* _{σ,ν}*(b)
Ψ

*σ,ν*(a)

*g** _{ν}*(

*F*

*(*

_{ν,σ}*x*))

*d x*

! .

We set*z*=*F*_{σ}_{,}* _{ν}*(

*x*), then

*x*=

*H*

_{σ}_{,}

*(z)and*

_{ν}*z*=

*u*+

*i v*

_{σ}_{,}

*(u). Note that*

_{ν}*v*

_{σ}_{,}

*(u)*

_{ν}*>*0 for

*u*∈]a,

*b[*and

*v*

_{σ}_{,}

*(a) =*

_{ν}*v*

_{σ}_{,}

*(b) =0 (see[12]). Then,*

_{ν}(µ* _{σ}*

*ν*)([Ψ

*,*

_{σ}*ν*(a),Ψ

*,*

_{σ}*ν*(

*b)])*

= −1
*π*ℑ

Z *b*
*a*

*g** _{ν}*(u+

*i v*

_{σ}_{,}

*(u))H*

_{ν}

_{σ}^{0}

_{,}

*(u+*

_{ν}*i v*

_{σ}_{,}

*(u))(1+*

_{ν}*i v*

_{σ}^{0}

_{,}

*(u))du*

_{ν}!

= −1
*π*ℑ

Z *b*
*a*

*g** _{ν}*(u+

*i v*

_{σ}_{,}

*(u))(1+*

_{ν}*σ*

^{2}

*g*

_{ν}^{0}(u+

*i v*

_{σ}_{,}

*(u)))(1+*

_{ν}*i v*

_{σ}^{0}

_{,}

*(u))du*

_{ν}!

= −1

*π* ℑ

Z *b*
*a*

*g** _{ν}*(

*u*+

*i v*

_{σ}_{,}

*(*

_{ν}*u*))(1+

*i v*

_{σ}^{0}

_{,}

*(*

_{ν}*u*))

*du*+

*σ*

^{2}

2 ℑ[*g*_{ν}^{2}(*u*+*i v*_{σ}_{,}* _{ν}*(

*u*))]

^{b}

_{a}!

= −1
*π*ℑ

Z *b*
*a*

*g** _{ν}*(

*u*+

*i v*

*(*

_{σ,ν}*u*))(1+

*i v*

_{σ}^{0}

_{,}

*(*

_{ν}*u*))

*du*=−1

*π*ℑ

Z

*γ*

*g** _{ν}*(

*z*)

*dz,*where

*γ*={*z*=*u*+*i v*_{σ}_{,}* _{ν}*(u),

*u*∈[a,

*b]}*.

Now, we recall that, since*a*and*b*are points of continuity of the distribution function of*ν*,
*ν*([*a,b*]) =lim

*ε→0*−1
*π*ℑ

Z *b*
*a*

*g** _{ν}*(

*u*+

*iε)du*

!

=lim

*ε→0*−1
*π*ℑ

Z

*γ*_{ε}

*g** _{ν}*(

*z*)

*dz*

!
,
where*γ** _{ε}*={

*z*=

*u*+

*iε*,

*u*∈[a,

*b]}*. Thus, it remains to prove that:

*ε→*lim0 ℑ
Z

*γ*

*g** _{ν}*(z)dz

!

− ℑ Z

*γ**ε*

*g** _{ν}*(z)dz

!!

=0. (2.8)

Let*ε >*0 such that*ε <*sup_{[}_{a,b}_{]}*v** _{σ,ν}*(

*u*). We introduce the contour ˆ

*γ** _{ε}*={

*z*=

*u*+

*i(v*

_{σ}_{,}

*(u)∧*

_{ν}*ε),u*∈[a,

*b]}*.

From the analyticity of*g** _{ν}* onC

^{+}

^{, we have}Z

*γ*

*g** _{ν}*(z)dz=
Z

ˆ
*γ**ε*

*g** _{ν}*(z)dz.

Let*I** _{ε}*={

*u*∈[a,

*b]*,

*v*

_{σ}_{,}

*(u)*

_{ν}*< ε}*=∪

*C*

*(ε), where*

_{i}*C*

*(ε)are the connected components of*

_{i}*I*

*. Then,*

_{ε}*I*

*↓*

_{ε}*ε→0*{

*a,b*}. For

*u*∈

*I*

*,*

_{ε}|ℑ*g** _{ν}*(u+

*iε)|*=

*ε*

Z *dν(x*)

(u−*x*)^{2}+*ε*^{2} ≤*ε*

Z *dν*(*x*)

(u−*x*)^{2}+*v*_{σ,ν}^{2} (u)≤ *ε*
*σ*^{2}

and Z

*I*_{ε}

|ℑ*g** _{ν}*(u+

*iε)|du*≤

*ε*

*σ*^{2}(*b*−*a)*.
On the other hand, for*u*∈*I** _{ε}*,

|ℑ*g** _{ν}*(

*u*+

*i v*

*(*

_{σ,ν}*u*))|=

*v*

*(*

_{σ,ν}*u*)

Z *dν*(x)

(*u*−*x*)^{2}+*v** _{σ,ν}*(

*u*)

^{2}≤

*ε*

*σ*

^{2}. Moreover,

ℜ*g** _{ν}*(u+

*i v*

_{σ}_{,}

*(u))v*

_{ν}

_{σ}^{0}

_{,}

*(u) =Ψ*

_{ν}*,*

_{σ}*ν*(u)−

*u*

*σ*

^{2}

*v*

_{σ}^{0}

_{,}

*(u) and*

_{ν}Z

*I*_{ε}

ℜ*g** _{ν}*(u+

*i v*

*(u))v*

_{σ,ν}

_{σ,ν}^{0}(u)du = Z

*I*_{ε}

Ψ* _{σ,ν}*(

*u*)−

*u*

*σ*^{2} *v*_{σ,ν}^{0} (u)du

= 1

*σ*^{2}
X

*i*

[(Ψ* _{σ}*,

*ν*(u)−

*u)v*

*,*

_{σ}*ν*(u)]

*C*

*(ε)*

_{i}− 1
*σ*^{2}

Z

*I*_{ε}

(Ψ^{0}_{σ}_{,}* _{ν}*(

*u*)−1)

*v*

_{σ}_{,}

*(*

_{ν}*u*)

*du,*by integration by parts. Now (see[12]or Theorem 2.1),

Z

*I*_{ε}

Ψ^{0}* _{σ,ν}*(u)v

*,*

_{σ}*ν*(u)du=

*πσ*

^{2}(µ

**

_{σ}*ν*)(Ψ

*,*

_{σ}*ν*(I

*))−→*

_{ε}*ε→*00.

Z

*I*_{ε}

*v*_{σ}_{,}* _{ν}*(u)du≤

*ε(b*−

*a)*. SinceΨ

*,*

_{σ}*ν*is increasing on[a,

*b],*

X

*i*

[Ψ* _{σ,ν}*(

*u*)

*v*

*(*

_{σ,ν}*u*)]

*C*

*(ε)≤*

_{i}*ε(Ψ*

*(*

_{σ,ν}*b*)−Ψ

*(*

_{σ,ν}*a*)) and

X

*i*

[*uv*_{σ}_{,}* _{ν}*(

*u*)]

*C*

*(ε)≤*

_{i}*ε(b*−

*a*). The above inequalities imply (2.8).

**3** **Approximate subordination equation for** *g*

_{N}### ( *z* )

We look for an approximative equation for *g** _{N}*(z) of the form (2.2). To estimate

*g*

*(z), we first handle the simplest case where*

_{N}*W*

*is a GUE matrix and then see how the equation is modified in the general Wigner case. We shall rely on an integration by parts formula. The first integration by parts formula concerns the Gaussian case; the distribution*

_{N}*µ*associated to

*W*

*is a centered Gaussian distribution with variance*

_{N}*σ*

^{2}and the resulting distribution of

*X*

*=*

_{N}*W*

_{N}*/*p

*N* is denoted
by GUE(N,*σ*^{2}*/N). Then, the integration by parts formula can be expressed in a matricial form.*

**Lemma 3.1.** *Let*Φ*be a complex-valued*C^{1} *function on*(M*N*(C)*sa*)*and X** _{N}* ∼

*GUE(N*,

^{σ}^{2}

*N* *). Then,*
E[Φ^{0}(X*N*).H] = *N*

*σ*^{2}E[Φ(X*N*)Tr(X*N**H)]*, (3.1)
*for any Hermitian matrix H, or by linearity for H* = *E*_{jk}*,* 1 ≤ *j,k* ≤ *N , where* (E*jk*)1≤*j,k*≤*N* *is the*
*canonical basis of the complex space of N*×*N matrices.*

For a general distribution*µ*, we shall use an “approximative" integration by parts formula, applied
to the variable*ξ*=p

2ℜ((X* _{N}*)

*)orp*

_{kl}2ℑ((X* _{N}*)

*),*

_{kl}*k<l*, or(X

*)*

_{N}*. Note that for*

_{kk}*k<l*the derivative ofΦ(

*X*

*)with respect top*

_{N}2ℜ((*X** _{N}*)

*kl*)(resp.p

2ℑ((*X** _{N}*)

*kl*)) isΦ

^{0}(

*X*

*).e*

_{N}*(resp.Φ*

_{kl}^{0}(

*X*

*).*

_{N}*f*

*), where*

_{kl}*e*

*=*

_{kl}^{p}

^{1}

_{2}(E

*kl*+

*E*

*)(resp.*

_{l k}*f*

*=*

_{kl}^{p}

^{i}_{2}(E

*kl*−

*E*

*)) and for any*

_{l k}*k, the derivative of*Φ(X

*N*)with respect to(X

*N*)

*kk*isΦ

^{0}(X

*N*).E

*.*

_{kk}**Lemma 3.2.** *Letξbe a real-valued random variable such that*E(|ξ|* ^{p+2}*)

*<*∞

*. Letφ*

*be a function*

*from*R

*C*

^{to}*such that the first p*+1

*derivatives are continuous and bounded. Then,*

E(ξφ(ξ)) =

*p*

X

*a=0*

*κ** _{a+}*1

*a!* E(φ^{(a)}(ξ)) +*ε*, (3.2)

*whereκ**a**are the cumulants ofξ,*|ε| ≤*C*sup* _{t}*|φ

^{(}

^{p}^{+}

^{1}

^{)}(t)|E(|ξ|

^{p}^{+}

^{2}), C only depends on p.

Let*U*(=*U*(N))be a unitary matrix such that

*A** _{N}*=

*U*

^{∗}diag(γ1, . . . ,

*γ*

*N*)U

and let*G*stand for*G** _{N}*(z). Consider ˜

*G*=

*U GU*

^{∗}. We describe the approach in the Gaussian case and present the corresponding results in the general Wigner case but detail some technical proofs in the Appendix.

**a) Gaussian case:** We apply (3.1) toΦ(X* _{N}*) =

*G*

*,*

_{jl}*H*=

*E*

*, 1≤*

_{il}*i,j,l*≤

*N, and then take*

^{1}

*N*

P

*l* to
obtain, using the resolvent equation*GX** _{N}* =−

*I*+

*zG*−

*GA*

*(see[18]),*

_{N}*Z** _{ji}*:=

*σ*

^{2}E[G

*tr*

_{ji}*(G)] +*

_{N}*δ*

*−*

_{i j}*z*E(G

*) +E[(GA*

_{ji}*)*

_{N}*] =0.*

_{ji}Now, let 1≤*k,p*≤*N* and consider the sumP

*i,j**U*_{ik}^{∗}*U*_{p j}*Z** _{ji}*. We obtain from the previous equation

*σ*

^{2}E[

*G*˜

*tr*

_{pk}*(G)] +*

_{N}*δ*

*pk*−

*z*E(

*G*˜

*) +*

_{pk}*γ*

*k*E[

*G*˜

*] =0. (3.3)*

_{pk}