Free probability and random matrices

Alice GuionnetMIT

19 novembre 2012

Abstract

In these lectures notes we will present and focus on free probabilityas a tool box to study the spectrum of polynomials in several (even-tually) random matrices, and provide some applications.

Introduction

Free probability was introduced by Voiculescu as a non-commutativeprobability theory equipped with a notion of freeness that is very simi-lar to independence in classical probability theory. Voiculescu [Voi91]showed at the early stage of free probability that freeness is describingthe asymptotic behavior of a large class of random matrices, that ismatrices whose eigenbasis are distributed very randomly, for instancewith law invariant under unitary conjugation. Since then, free probabi-lity appeared as the natural set up to study the asymptotics of randommatrices, at least when one is interested in polynomials of several (even-tually random) matrices.

The basic object of interest in free probability, at least as far as ran-dom matrices are concerned, is related with the asymptotics of quanti-ties

τXN (P ) :=1

NTr(P (XN

1 , . . . , XNd ))

1

2

where (XN1 , . . . , X

Nd ) is a sequence of N ×N (eventually) random ma-

trices and P a polynomial in d non-commutative variables. Of course,such quantities not only depend on the eigenvalues of the matrices butalso of the respective positions of the eigenvectors. Voiculescu consi-dered the case where the latter is as random as possible, meaning forinstance that the asymptotics of the random matrices under study aresimilar to those of XN

i = U iND

Ni (UN

i )∗ with UNi independent unitary

matrices following the Haar measure and DNi = diag(λj(X

Ni )) is a dia-

gonal matrix with entries given by the eigenvalues of XNi ), and the

spectral measure

LXNi

:=1

N

N∑j=1

δλj(XNi )

converges in moments (that is∫xkdLXN

i(x) converges as N goes to

infinity for each k. Then, Voiculescu showed that

limN→∞

τXN (P ) = τX(P )

converges almost surely as N goes to infinity and the limit can bedescribed by the so-called notion of freeness. This notion is related withthe notion of freeness on groups and can be used to effectively computethe limits. In particular, if P is a fixed self-adjoint polynomial, that isthat P (XN

1 , . . . , XNd ) = P (XN

1 , . . . , XNd )∗ for almost all realizations of

XN1 , . . . , X

Nd , then its spectral measure converges as∫xkdLP (XN

1 ,...,XNd )(x) =

1

NTr(P (XN

1 , . . . , XNd )k)

→ τX(P k) =:

∫xkdLP (x) .

In the cases under considerations the matrices are often asymptoti-cally bounded so that P (XN

1 , . . . , XNd ) is a asymptotically bounded and

hence the probability measure LP has bounded support and is thereforeuniquely determined by its moments. Therefore, free probability allowsto analyze the convergence of the spectral measure of polynomials inseveral matrices. If the matrices possess the kind of randomness for thedistribution of the eigenvectors that we described above, the limit will

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be described by freeness and free probability is the natural place wherethe analysis of this limit can be performed. For instance it allows tocharacterize the spectrum of the sum or the multiplication of randommatrices, or to state general property such as the connectivity of thesupport of a large family of polynomials in random matrices. In fact, weshall see that it allows to study not only the spectral measure of poly-nomials in random matrices but more complicated objects. It allows aswell to determine the asymptotics of extreme eigenvalues. The theoryalso developed to include the study of the fluctuations of the spec-tral measure [CD01, Gui02, MS06], large deviations [Voi00, BCG03]or Brownian motion. However we shall restrict ourselves to spectrumconvergence in these notes.

There are many instances when one would like to study the spec-trum of polynomial in several matrices, simply for instance when oneis interested in a matrix which is given as such a polynomial.

The first example which comes to mind is the so-called GaussianWishart matrix X = Y Y ∗ where Y = (Y1, . . . , YN) are iid copies ofa p dimensional Gaussian vector with covariance matrix Σ. Then if Zdenote a p×N matrix with independent centered Gaussian entries withcovariance one, then X has the same law as ZΣZ∗ which is a polynomialin the matrix Z and the deterministic matrix Σ. Note that the law ofZ is the same as the law of UZV where U, V are deterministic unitary(resp. orthogonall) matrices. Hence, such Wishart matrices naturallypertain to the domain of application of free probability when their sizegoes to infinity and indeed the spectrum of Y can be described byits tools. Eventhough the spectrum of Wishart matrices was studiedwithout using such ideas, they appear to give a general and unifiedanalysis [BG09].

Another natural question is for instance, being given two sequencesof random matrices A and B, to understand the spectrum of its sumA + B. This of course also depends on their respective eigenbasis asif A and B have the same eigenvectors, the eigenvalues will simply bethe sum of the eigenvalues of A and B, whereas if the eigenbasis of Aand B are as much independent as possible in the sense that one is the

4

“random rotation” of the other, then the spectrum will be describedasymptotically by free convolution.

In some sense, in the case of non-normal matrices, it is even naturalto study the spectrum of a matrix X by considering it together withits adjoint. If X is normal, that is can be decomposed as X = UDU∗

with D a diagonal matrix whose entries are the eigenvalues of X, andU a unitary matrix, X and X∗ commute. To retrieve the spectrum(λ1, . . . , λN) of X, and for instance its empirical distribution, it may bewise to consider

Tr(Xp(X∗)k) =∑

λpi λki

to characterize the spectral measure by moments. In the case of non-normal matrices, that is matrices which do not commute with theiradjoint, it turns out that Green formula describes the spectral measurein terms of the spectral measure of (z − X)(z − X)∗ for z ∈ C. Eventhough free probability theory does not provide all the tools to proveconvergence of the spectrum of non-normal matrices it permits to guessthe asymptotics in terms of the so-called Brown measure. Indeed, freeprobability set up is mainly designed to handle smooth functions ofmatrices such as polynomials whereas it is well known that the spectrumof non-normal matrices can be very unstable, leading to the notion ofpseudo-spectrum ; changing only one entry by a very tiny number canchange drastically the spectrum. However, if one forget this instability,or somehow neglect a singularity, free probability allows to predict andstudy the limit of the spectrum of non-normal matrices.

As we have emphasized earlier, free probability deals with matriceswhose eigenbasis are very randomly chosen, at least one with respect tothe other. This applies to many different models of random matrices,but not to all. For instance, it does not apply to covariance matricesof a Erdos-Renyi graph, that is a symmetric matrix with independententries above the diagonal which vanish with probability 1− p/N andequal one otherwise, or more generally with moments which do notvanish sufficiently fast. Another question that can not be solved byfree probability is to analyze the spectrum of A+ PBP ∗ where P is auniformly chosen permutation matrix. Then, it turns out that moments

5

are not sufficient to describe the joint laws of such matrices but a notionof distribution of traffics [Mal11a] can be defined. Some of the apparatusof free probability can be generalized to this setting.

1 Random matrices and freeness

Free probability appears as the natural setting to study the asymp-totics of traces of words in several (possibly random) matrices.

Adopting the point of view that traces of words in several matricesare fundamental objects is fruitful because it leads to the study of somegeneral structure such as freeness ; freeness in turns simplifies the ana-lysis of convergence of moments. The drawback is that one needs toconsider more general objects than empirical measures of eigenvaluesconverging towards a probability measure, namely, traces of noncom-mutative polynomials in random matrices converging towards a linearfunctional on such polynomials, called a tracial state. Analysis of suchobjects is then achieved using free probability tools.

1.1 Non-commutative laws and freeness

During this course, we shall call non-commutative law of d self-adjoint non-commutative variables any linear form τX defined on theset C〈X1, . . . , Xd〉 of polynomial functions of d self-adjoint variablesand with value in C which can be constructed thanks to a sequence(XN

1 , . . . , XNd ) of N ×N self-adjoint random matrices by

τX(P ) = limN→∞

τXN (P )

where

τXN (P ) = E[1

NTr(P (XN

1 , . . . , XNd ))] .

We shall call τXN the empirical distribution of the matrices XN . Ifd = 1, as τXN (P ) = E[N−1

∑Ni=1 P (λi)] = E[

∫P (x)dLXN (x)] if λi

denote the eigenvalues ofXN , we conclude that τXN is the mean spectral

6

distribution. Hence, the space of non-commutative laws in one self-adjoint variable is simply the set of probability measures on R.

We see that any τ = τX which can be constructed as above arelinear forms on C〈X1, . . . , Xd〉 which satisfy

– A positivity property : we may endow C〈X1, . . . , Xd〉 with theinvolution(∑

zpXip1· · ·Xipnp

)∗=∑

zpXipnpXipnp−1

· · ·Xip1.

Then

τ(PP ∗) ≥ 0 ∀P ∈ C〈X1, . . . , Xd〉 . (1)

– A mass condition

τ(1) = 1 . (2)

– A tracial property

τ(PQ) = τ(QP ) ∀P,Q ∈ C〈X1, . . . , Xd〉

It is still an open question whether any linear form τ satisfying theabove properties can be constructed as the limit of empirical distri-bution of matrices ; this is the general definition of the law of non-commutative variables.

The space of non-commutative laws thus appear as a kind of naturalgeneralization to the space of probability measures. It will be throu-ghout endowed with a notion of weak convergence : If τn is a sequenceof laws of d self-adjoint non-commutative variables then τn convergestowards τ off

limn→∞

τn(P ) = τ(P ) ∀P ∈ C〈X1, . . . , Xd〉 .

We say that matrices (XN1 , . . . , X

Nd ) converges in law towards τ if their

empirical distribution τXN converges weakly towards τ .

The notion of freeness gives an algebraic relation between moments.We say that random variables X = (X1, . . . , Xp) and Y = (Y1, . . . , Yd)with joint law τ are free iff for any polynomials P1, . . . , Pk ∈ C〈X1, . . . , Xp〉

7

and Q1, . . . , Qk ∈ C〈X1, . . . , Xd〉 so that τ(Pi(X)) and τ(Qi(Y )) vanish,we have

τ(P1(X)Q1(Y )P2(X) · · ·Pk(X)Qk(Y )) = 0 . (3)

It is easy to check by induction over the degree of polynomials thatthis relation defines uniquely joint moments if the marginals are known.Indeed, by definition for any polynomial P,Q

τ(P (X)Q(Y )) = τ(P (X))τ(Q(Y ))

is given by the marginals. More generally for any polynomial Pi, Qi

τ(P1(X)Q1(Y )P2(X) · · ·Pk(X)Qk(Y ))

−τ((P1(X)− τ(P1(X))I) · · · (Qk(Y )− τ(Qk(Y ))I))

is given by a sum of products of traces of polynomials which have(total) degree strictly smaller than P1(X)Q1(Y )P2(X) · · ·Pk(X)Qk(Y ).Hence, by induction we can compute all the moments since by freenessτ((P1(X)− τ(P1(X))I) · · · (Qk(Y )− τ(Qk(Y ))I)) vanishes.

Two sequences of N × N matrices (XNi , 1 ≤ i ≤ p) and (Y N

i , 1 ≤i ≤ d) is called asymptotically free if they converges in law as N goesto infinity to free random variables (Xi, 1 ≤ i ≤ p) and (Yi, 1 ≤ i ≤ d).In other words, for all P ∈ C〈X1, . . . , Xp, Y1, . . . , Yd〉

limN→∞

E[1

NTr(P (XN

1 , . . . , XNp , Y

N1 , . . . , Y N

d ))] = τ(P )

and X = (X1, . . . , Xp) and Y = (Y1, . . . , Yd) with joint law τ are free.

2 Wigner matrices are asymptotically free

Let us consider Wigner matrices, that is self-adjoint matrices withindependent entries above the diagonal. We assume that the law of theentries is independent of N and with all the moments finite. They canbe either real or complex valued, and are centered.

8

Theorem 2.1 Let XNi , 1 ≤ i ≤ d, be a family of Wigner matrices.

Assume that for all k ∈ N,

supN∈N

sup1≤i≤d

sup1≤m≤`≤N

E[|XNi (m, `)|k] ≤ ck <∞ , (4)

that (XNi (m, `), 1 ≤ m ≤ ` ≤ N, 1 ≤ i ≤ p) are independent, and that

E[XNi (m, `)] = 0 and E[|XN

i (m, `)|2] = 1.

Then, the empirical distribution τN := τ 1√NXN

i 1≤i≤dof 1√

NXNi 1≤i≤d

converges almost surely and in expectation to the law of d free semi-circular variables that is free variables with marginal distribution givenby the semi-circle law

σ(dx) =√

4− x2dx/π .

Proof.

We shall prove that if we represent a monomial q(X) = Xi1 · · ·Xik

by k colored ordered points (called the points of type q) so that thefirst has color i1, the second i2 etc till the last which has color ik, then

τ(q) = limN→∞

E[1

NTr(q(

XN1√N, . . . ,

XNd√N

))]

= # non-crossing pair partitions of the points of type q

with one color blocks

The right hand side is the number of non-crossing pair partitions withmono-colored blocks, that is the number of partitions of 1, . . . , k sothat if B1 = (p1, p2) and B2 = (q1, q2) are two blocks of this partition,p1 < p2 and q1 < q2, they are mono-colored (that is ip1 = ip2 as well asiq1 = iq2) and non-crossing

p2 < q1 or p1 < q1 < q2 < p2 .

This is the law of free semicircular variables. Indeed, first note thatthe moments of only one variable are given by the numbers of non-crossing pair partitions, which are given by the Catalan numbers. Weleave the reader to check that they are also equal to the moments of

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the semi-circle law. Secondly, it is enough by linearity to prove thefreeness conditions for monomials. But, if we decompose q as q(X) =q1(Xi1)q2(Xi2) · · · qr(Xir) with ij 6= ij+1 then

τ((q1(Xi1)− σ(q1))(q2(Xi2)− σ(q2)) · · · (qr(Xir)− σ(qr)))

is the number of non crossing pair partitions so that the sets of pointsSi+1 =

∑ij=1 degqj + 1, . . . ,

∑i+1j=1 degqj, 1 ≤ i ≤ r have pairings with

each other. But we claim that there is no such non-crossing pair parti-tions. Indeed, let us consider the pairing of the first point. Either theother point of the pairing is inside the first set of points S1 and then wereplace the first point by its right neighbor. We can continue till eitherS1 has no matching with the other points, in which case we are done,or there is one. In which case we can focus on the points s, . . . , t inthe interior of this block. In the interior of s, . . . , t there are at leastpoints of two different colors. We can continue until there is exactly twodifferent colors in the points considered in which case the conclusionfollows.

To prove the convergence, we expand the trace of q = Xi1 · · ·Xikas

E[1

Nk2

+1Tr(q(

XN1√N, . . . ,

XNd√N

))] =1

Nk2

+1

N∑j1,...,jk=1

E[Xi1(j1j2) · · ·Xik(jk, j1)]

To any set of indices j1, . . . , jk we associate the connected graph G =(V,E) so that E = jq, jq+1, 1 ≤ q ≤ k with the convention jk+1 = j1.The term E[Xj1j2 · · ·Xjkj1 ] vanishes unless the edges of G appear atleast with multiplicity 2 as non-oriented edges by independence andcentering. But since G is connected, if |E| is the number of differentnon-oriented edges in G and |V | the numbers of distinct vertices, wehave

|V | ≤ |E|+ 1 ≤ k

2+ 1 .

Hence, there are at most k2

+ 1 different vertices, that is at most Nk2

+1

choices of the indices. As we divide the sum by Nk2

+1, we see that graphswith |V | ≤ k

2will not contribute asymptotically. Hence, only graphs G

so that |V | = |E| + 1 contribute, that is trees. Moreover, each edge

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has multiplicity two. Furthermore, there is a connected path ip→ip+1

around this tree. Unfolding this path gives a bijection between the treeand a non-crossing pair partition. Finally, the graph contributes onlyif the edges which are paired correspond to the same matrix, that is ifthe non-crossing pair partition pairs only points of the same color.

Theorem 2.1 generalizes to the case of polynomials including somedeterministic matrices.

Theorem 2.2 Let DN = DNi 1≤i≤d be a sequence of Hermitian deter-

ministic matrices and let XN = XNi 1≤i≤d, X

Ni : Ω→ H(β)

N , 1 ≤ i ≤ d,be Wigner matrices satisfying the hypotheses of Theorem 2.1. Assumethat

D := supk∈N

max1≤i≤d

supN

1

NTr((DN

i )k)1k <∞ , (5)

and that the law τDN of DN converges to a noncommutative law µ.Then,the noncommutative variables 1√

NXN and DN are asymptotically

free. In particular, the empirical distribution τ 1√NXN ,DN converges al-

most surely and in expectation to the law of X,D, X and D beingfree, D with law µ and X being d free semi-circular variables.

Theorem 2.2 can be proved as Theorem 2.1 if the DNi are diagonal

matrices. However, this strategy is not straightforward to follow other-wise. The proof is give in [AGZ10, Section 5] by showing an approximateintegration by parts formula. In some sense, Theorem 2.2 is a universa-lity result and shows that Wigner matrices whose entries have enoughfinite moments have the same asymptotics as Gaussian. Gaussian ma-trices have the particularity that their law is invariant by multiplicationXN→UXNU∗. The next result shows that this is the real essence offreeness.

We now consider conjugation by unitary matrices following the Haarmeasure dU on the set of N ×N unitary matrices.

Theorem 2.3 Let DN = DNi 1≤i≤p be a sequence of Hermitian (even-

tually random) N ×N matrices. Assume that their empirical distribu-tion converges to a noncommutative law µ. Assume also that there exists

11

a deterministic D <∞ such that for all k ∈ N, all N ∈ N,

1

NTr((DN

i )2k) ≤ D2k, a.s.

Let UN = UNi 1≤i≤p be independent unitary matrices with Haar law

ρU(N), independent from DNi 1≤i≤p. Then, the matrices UN

i , (UNi )∗1≤i≤p,

and the matrices DNi 1≤i≤p,are asymptotically free. For all i ∈ 1, . . . , p,

the limit law of UNi , (U

Ni )∗ is given by

τ((UU∗ − 1)2) = 0, τ(Un) = τ((U∗)n) = 1n=0 .

This theorem can easily be deduced from the previous one by noticingthat unitary matrices following the Haar measure are approximatelysmooth functions of GUE matrices. Indeed, both U following the Haarmeasure and X following the GUE law are normal matrices and thuscan be decomposed as U = V DUV

∗ and X = V DX V∗ where V, V

follow the Haar measure on the unitary group. DU and DX are diagonalmatrices whose entries are the eigenvalues (eiθj)1≤i≤j of U and (λi)1≤i≤Nof X so that j→θj and j→λj are increasing. We can couple these twomatrices so that V = V and moreover consider the map fN which sendthe ordered eigenvalues of X onto the eigenvalues eiθj of U chosen withincreasing θj’s. As shown in [CM12], if we put

FX(t) = N−1

N∑j=1

1λj≤t FU(t) = N−1

N∑j=1

1θj≤t

then we can take fN(t) = exp(iF−1U FX(t)) which converges almost

surely towards a deterministic smooth function. Hence, we can alwaysapproximate polynomials in the variables UN

i by polynomials in GNi

and therefore deduce the last theorem from Theorem 2.1.

We have the following corollary, which shows that freeness can beseen as a consequence of independence of the respective eigenbasis.

Corollary 2.4 Let DNi 1≤i≤p and DN

j 1≤j≤d be sequences of uni-formly bounded self-adjoint matrices which converge in law towards τD

12

and τD respectively. Let UN be an independent unitary matrix followingthe Haar measure. Then, the noncommutative variables UNDN

i (UN)∗1≤i≤dand DN

j 1≤j≤p are asymptotically free, the law of the marginals beinggiven by τD and τD.

Assuming the convergence of the joint law of DN , DN, the Corollaryis just due to τ(U) = τ(U∗) = 0 since

P (UNDNi (UN)∗, 1 ≤ i ≤ p) = UNP (DN

i , 1 ≤ i ≤ p)(UN)∗

so that monomials in UNDNi (UN)∗1≤i≤p and DN

j 1≤j≤p can be writ-ten as products of monomials in the D’s interlaced by one unitary UN orits adjoint (UN)∗, the asymptotic freeness of UNDN

i (UN)∗1≤i≤p andDN

j 1≤j≤p is a consequence of the asymptotic freeness of UNi , (U

Ni )∗1≤i≤p,

and the matrices DNi 1≤i≤p, DN

j 1≤j≤d. A closer look shows that wedo not need to have convergence of the joint law but only of the mar-ginals.

2.1 Strong asymptotic freeness

Until this point, free probability was introduced as a tool to studythe asymptotics of polynomials in several matrices, which gives informa-tion on “macroscopic quantities” such as the empirical measure of theeigenvalues. However, it is well known that for many models of randommatrices, the extreme eigenvalues stick to the bulk, that is converge tothe boundary of the support of the limiting spectral measure. It turnsout that this phenomenon can be extended to all polynomials in ran-dom matrices. Strong asymptotic freeness precisely means that not onlythe empirical distribution of the matrices converge to the law of freevariables, but that also the operator norm of these matrices convergeto their free limit. Namely, if DN = DN

i 1≤i≤p be a sequence of Hermi-tian (eventually random) N ×N matrices. Assume that their empiricaldistribution converges to a noncommutative law τ . Assume also thatthere exists a deterministic D <∞ such that for all k ∈ N, all N ∈ N,

1

NTr((DN

i )2k) ≤ D2k, a.s.

13

Let UN = UNi 1≤i≤p be independent unitary matrices with Haar law

ρ, independent from DNi 1≤i≤p. Let XN = XN

i , 1 ≤ i ≤ d be inde-pendent GUE matrices. Let P be a polynomial in (DN ,UN ,XN) anddenote by ‖P (DN ,UN ,XN)‖ the operator norm of P (DN ,UN ,XN)given by

‖P (DN ,UN ,XN)‖ = limp→∞

(Tr(P (DN ,UN ,XN)P (DN ,UN ,XN)∗)p

)1/2p

On the other hand, let (D,U,X) be free variables with marginal dis-tribution τ , free unitaries and free semi-circle. We let

‖P (D,U,X)‖ = limp→∞

τ((P (D,U,X)P (D,U,X)∗)p)1/2p

Then we have the following result, proved for Gaussian matrices in thebreakthrough paper [HT05], for Wigner matrices in [CDM07, And11],for joint deterministic and GUE [Mal11b] and under this form in [CM12] :

Theorem 2.5 Assume that for each polynomial P

limN→∞

‖P (DN)‖ = ‖P (D)‖

Then, for any polynomial P

limN→∞

‖P (DN ,UN ,XN)‖ = ‖P (D,U,X)‖ a.s.

The proof of such a result requires finer estimate than just finite mo-ments : the idea developed in [HT05] (and then in [CDM07, And11]) isto realize that the spectrum of a polynomial P is related with that ofa larger matrix which is linear in the random matrices (the so-called li-near trick) and then study the Stieljes transform of this matrix by usingthe so-called loop equations. The extension to unitary matrices is per-formed in [CM12] by approximating unitary matrices by polynomialsin GUE matrices.

Interestingly enough, free probability also can be used to study ei-genvalues which do not stick to the bulk. Indeed, in [CDMFF11], the

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extreme eigenvalues of M = A+X were studied with X a Wigner ma-trix with entries satisfying Poincare inequality and A a deterministicmatrix with converging spectral distribution but also with some out-liers, that is eigenvalues which do not stick to the bulk. It was provedthat the positions of the outliers of M , when it has some, can be des-cribed by the subordination function which describes free convolution,see section ??.

2.2 Convergence of the spectral measure of poly-nomials in random matrices

We next specialize the previous theorem to study the spectrum ofa polynomial in uniformly bounded deterministic matrices DN , andrandom matrices UN following the Haar measure and XN being inde-pendent Wigner matrices. By the previous point their empirical distri-bution converges towards the law of free variables D,U,X which arebounded. Hence, if P is a self-adjoint polynomial function, P (D,U,X)is bounded uniformly and in particular the law µP of P (D,U,X) isdetermined by its moments :∫

xkdµP (x) := τ((P (D,U,X))k) .

From the previous section we infer the following result :

Corollary 2.6 Let DN = DNi 1≤i≤p be a sequence of Hermitian (even-

tually random) N ×N matrices. Assume that their empirical distribu-tion converges to a noncommutative law τ . Assume also that there existsa deterministic D <∞ such that for all k ∈ N, all N ∈ N,

1

NTr((DN

i )2k) ≤ D2k, a.s.

Let UN = UNi 1≤i≤p be independent unitary matrices with Haar law

ρ, independent from DNi 1≤i≤p. Let XN = XN

i , 1 ≤ i ≤ d be in-dependent Wigner matrices with entries with finite moments. Let P bea self-adjoint polynomial in (DN ,UN ,XN). Then the spectral measure

15

of P (DN ,UN ,XN) converges towards the law µP of P (D,U,X) whereD,U,X are free with marginal distribution given respectively by τ , freeunitaries and free semi-circle distribution, whereas the extreme eigenva-lues of P (DN ,UN ,XN) converges towards the extremes of the supportof µP .

One of the nice corollary of the representation of the limit law interms of free variables is that the eigenvalues are asymptotically connec-ted in the sense that :

Corollary 2.7 For any polynomial P in U,X, free unitaries and semi-circle respectively, the law µP has a connected support.

Indeed, the C∗ algebra generated by free semi-circle is projection lessaccording to [PV82, GS09], which will not be the case if the support wasdisconnected. Indeed, otherwise the projection onto one of the connec-ted component S of the support could be written A =

∫Γ(z − P )−1dz

where Γ is a contour around S whose interior does not intersect therest of the support. Since A belongs to the C∗ algebra of P (as a conti-nuous function of P ) and therefore of a finite number of free semi-circlelaw (as (X,U) are in the C∗-algebra of free semi-circle variables), theconclusion follows.

2.3 R-transform and S-transform

We specialize the previous result to the case of the sum and themultiplication of free matrices.

Corollary 2.8 Let DN and DN be sequences of uniformly boundedself-adjoint matrices so that the empirical measure of their eigenvaluesconverge to µ and µ respectively. Let UN be an independent unitarymatrix following the Haar measure. Then

1. The spectral distribution of DN + UNDNU∗N converges weakly al-

most surely to µ µ as N goes to infinity.

16

2. Assume that DN is nonnegative. Then, the spectral distributionof (DN)

12UND

NU∗N(DN)12 converges weakly almost surely to µ µ

as N goes to infinity.

It turns out that even though freeness looks much more complicatedthan independence, it still allows to make some computations, such asfor instance computing µ µ or µ µ. This is done by characterizing agenerating function of the moments of this law which is suitable for theproblem under consideration. We shall in the following only considerthese generating functions as formal series to simplify.

Let for a probability µ, Gµ(z) =∫

(z − x)−1dµ(x) and let zµ(g) beits inverse (which exists by Lagrange inversion theorem) whereas wedenote sµ the inverse of

∑n≥1 z

nµ(xn). We then put

Rµ(g) = zµ(g)− g−1 Sµ(s) =1 + s

ssµ(s) .

Rµ and Sµ both characterize the measure µ as it characterize its mo-ments. Then, we have

Theorem 2.9Rµµ = Rµ +Rµ

Sµµ′ = SµSµ′

This result is often proved by using cumulants and the combinatoricsof non-crossing partitions, see [NS06]. We here follow the point of viewof T. Tao [Tao12] which gives a formal and clear argument for theadditive property of the R-transform, but apply it to the S-transformfor a change. We shall be sketchy below. We write SX and sX for theS-transform of the law µ (resp. sµ) if X has law µ. The point is to writesX as the solution of ∫

xsX(z)

1− sX(z)xdµ(x) = z

which is equivalent to the existence of a centered random variable EXso that

Xsµ(z)

1−Xsµ(z)= z + EX a.s .

17

Note that in fact EX is a nice function of X, and can be approximatedby polynomials at list for =z large enough. Some algebra reveals thatthis is equivalent to

SX(z)X =1 + 1

zEX

1 + 1z+1

EX.

Doing the same for Y with law µ we deduce

SY (z)SX(z)XY =

(1 + 1

zEX

1 + 1z+1

EX

)(1 + 1

zEY

1 + 1z+1

EY

).

the statement will follow if we see that there exists EXY centered sothat

A :=

(1 + 1

zEX

1 + 1z+1

EX

)(1 + 1

zEY

1 + 1z+1

EY

)=

1 + 1zEXY

1 + 1z+1

EXY

or in other words if we can check that

EXY := z(z + 1)A− 1

z + 1− zAis centered. This is equivalent to

EXY =1

z(z + 1)− EXEY(EX + EY +

2z + 1

z(z + 1)EXEY )

But it is not hard to verify that freeness implies that for all n ∈ N

τ [EX(EXEY )n] = 0 τ [(EXEY )nEY ] = 0 τ [(XY )n] = 0

which shows that at least if we expand formally the denominator, EXYis centered.

3 The spectrum of non-normal matrices

and the Brown measure

Let us remind that non-normal matrices do not commute with theiradjoint. This can also be written through the decomposition of the

18

matrix in its eigenbasis. X = QDQ−1 where the entries of D are theeigenvalues of X and Q is the matrix of the linearly independent eigen-vectors ; X is non-normal when we can not take Q unitary.

It turns out that the spectrum of non-normal matrices is not asmooth function of the entries. Indeed, the spectrum of non-normalmatrices can be quite unstable, the addition of a matrix with verysmall entries being able to change it drastically, hence leading to thestudy of the pseudo-spectrum, see e.g. [TE05]. The typical example isgiven by the matrix

Ξ(N) =

0 0 · · · 0 01 0 · · · 0 0

0 1. . .

......

.... . . . . . 0 0

0 · · · 0 1 0

whose spectrum is reduced to the null eigenvalue. However, if one addsa very tiny matrix

0 0 · · · 0 N−106

1 0 · · · 0 0

0 1. . .

......

.... . . . . . 0 0

0 · · · 0 1 0

the spectrum will be drastically modified as it is uniformly spread onthe unit disc when N goes to infinity.

19

Such a phenomena can not happen for normal matrices, as can beseen thanks to the Hoffman-Wielandt inequality which shows that fortwo normal matrices A,B with complex-valued entries with eigenvalues(λi(A))1≤i≤N and (λi(B))1≤i≤N respectively, we have

minπ permutation

N∑i=1

|λi(A)− λπ(i)(B)|2 ≤ Tr(A−B)(A−B)∗ .

This inequality shows that modifying the entries by a factor N−p cannot modify the spectrum of a normal matrix by an error greater thanN−p+2.

The point here is that even if the eigenvalues are smooth functionsof the entries when they are distinct, as the roots of the characteristicpolynomial, they are very close to each other so that in fact the Lip-schitz constant blows up with N faster than polynomially if they arespaced like N−1. Hence, adding a polynomially decaying matrix canaffect the spectrum drastically.

This can be analytically understood by the Green formula. :If (λNi )are the eigenvalues of XN , for any smooth function ψ, we have

N∑i=1

ψ(λNi ) =1

2π

∫C

∆ψ(z) log |N∏i=1

(z − λNi )|dz

=1

4π

∫C

∆ψ(z)

(N∑i=1

log |z − λNi |2)dz

=N

4π

∫C

∆ψ(z)

∫log |x|dL(z−XN )(z−XN )∗(x)dz

where L(z−XN )(z−XN )∗ denotes the spectral measure of the Hermitianmatrix (z−XN)(z−XN)∗. As we have seen before, the convergence ofthe spectral measure (z − XN)(z − XN)∗ can be studied by momentsmethods and behaves smoothly with the entries of XN . However, thelogarithm is singular at the origin and can be seen to be exactly thesource of the instability of the spectrum. The Brown measure is exactly

20

defined as the measure we would obtain if we would “forget” this sin-gularity. This is the idea originally proposed by Girko.

Namely, let us assume that the empirical distribution of XN , (XN)∗

converges to the non-commutative law τ and let LzX be the law of(z − X)(z − X∗) under τ , that is the probability measure on the realline so that for each k∫

xkdLzX(x) = τ(((z −X)(z −X)∗)k

)Then we define the Brown measure µX of X as the measure on C sothat for all smooth functions ψ∫

ψ(z)dµX(z) :=1

4π

∫C

∆ψ(z)

∫log |x|dLzX(x)dz

Even though the Brown measure may not characterize the asympto-tics of the spectrum of the matrices XN , it will describe those of a smallrandomization of them. Indeed, Sniady [Sni02] proved the following

Theorem 3.1 Let XN be a sequence of N × N matrices so that thejoint empirical distribution of (XN , (XN)∗) converges. Let GN be thenon-normal matrix with i.i.d centered Gaussian entries with varianceN−1. Then, there exists εN going to zero as N goes to infinity so thatthe spectral measure LXN+εNGN

converges towards the Brown measureof X.

A similar result was obtained by Haagerup by smoothing via Cauchyrandom matrices.

The size of εN depends on the matrices XN ; in fact it can be takento go to zero polynomially in the example XN = ΞN , or for any matricesfor which adding a polynomially small matrix (but eventually determi-nistic) is enough to obtain the convergence towards the Brown measure[GZW12]. However, some matrices require stronger regularization.

In the classical models of random non-normal matrices such as theGinibre matrices which correspond to matrices with independent equi-distributed entries, it turns out that the spectral measure converges

21

towards the Brown measure, that is the uniform law on the disc assoon as the entries have (slightly more than) a finite second moment[Bai97, TV08].

In some cases, the Brown measure can be explicitly computed byusing free probability tools. This is the case for instance for XN =UNDN with UN following the Haar measure on the unitary group andDN a diagonal matrix with real entries where this computation wasdone by Haagerup and Larsen [HL00]. Bitting the singularity of thelogarithm (in particular thanks to recent results on the smallest singularvalue of XN +UN obtained by M. Rudelson and R. Vershynin [RV12]),one can obtain [GKZ11]

Theorem 3.2 Assume that DN is uniformly bounded, with convergingspectral measure and with Cauchy-Stieljes transform uniformly boundedon C+. Then, the spectral measure of UNDN converges almost surelytowards the Brown measure µUD. Moreover, the Brown measure µUD isrotation invariant and

µUD(B(0, f(t))) = t

where f(t) = 1/√SD2(t− 1) with SD2 the S-transform of D2. Further-

more, the support of µUD is an annulus

supp(µUD) = reiθ, r ∈ [(µD(x−2))−12 , (µD(x2))

12 ], θ ∈ [0, 2π[

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