getdocaa07. 159KB Jun 04 2011 12:04:50 AM

in PROBABILITY

ON THE SPECTRUM OF SUM AND PRODUCT OF NON-HERMITIAN

RANDOM MATRICES

CHARLES BORDENAVE

Institut de Mathématiques Toulouse, CNRS and Université de Toulouse III, 118 route de Narbonne, 31062 Toulouse, France

email:

❝❤❛r❧❡s✳❜♦r❞❡♥❛✈❡❅♠❛t❤✳✉♥✐✈✲t♦✉❧♦✉s❡✳❢r

SubmittedOctober 15,2010, accepted in final formJanuary 16,2011 AMS 2000 Subject classification: 60B20 ; 47A10; 15A18

Keywords: generalized eigenvalues, non-hermitian random matrices, spherical law.

Abstract

In this note, we revisit the work of T. Tao and V. Vu on large non-hermitian random matrices with independent and identically distributed (i.i.d.) entries with mean zero and unit variance. We prove under weaker assumptions that the limit spectral distribution of sum and product of non-hermitian random matrices is universal. As a byproduct, we show that the generalized eigenvalues distribution of two independent matrices converges almost surely to the uniform measure on the Riemann sphere

1

Introduction

We start with some usual definitions. We endow the space of probability measures onC_{with the}

topology of weak convergence: a sequence of probability measures(µ_n)_n≥1converges weaklytoµ

is for any bounded continuous function f :C_→R_,

Z

f dµ_n−

Z

f dµ

converges to 0 asngoes to infinity. In this note, we shall denote this convergence byµ_n   n→∞µ. Similarly, for two sequences of probability measures(µn)n≥1,(µ′n)n≥1, we will useµn−µ′n_n →∞0,

or say thatµ_n−µ′_ntends weakly to 0, if Z

f dµ_n−

Z

f dµ′_n

converges to 0 for any bounded continuous function f. We will say that a measurable function

f :C_→R_{isuniformly boundedfor(µ} n)n≥1if

lim sup

n→∞ Z

|f|dµ_n<_∞.

Finally, recall that a function f isuniformly integrablefor(µ_n)_n≥1if lim

t→+∞lim supn→∞ Z

|f|≥t

|f_|dµn=0.

The above definitions will also be used for probability measures on R

+ = [0,∞)and functions

f :R_+→R_.

Theeigenvaluesof ann_×ncomplex matrixM are the roots inC_{of its characteristic polynomial.}

We label themλ₁(M), . . . ,λ_n(M)so that_|λ₁(M)_{| ≥ · · · ≥ |}λ_n(M)_{| ≥}0. We also denote bys₁(M)_≥

· · · ≥s_n(M)_≥0 thesingular valuesof M, defined for every 1_≤k _≤nbys_k(M):=λ_k(pM M∗₎ where M∗ is the conjugate transpose of M. We define the empirical spectral measure and the empirical singular values measure as

µ_M= 1 n

n

X

k=1

δλk(M) and νM =

1

n n

X

k=1 δ_sk(M).

Note that µ_M is a probability measure on C _{while ν}

M is a probability measure on R+. The

generalized eigenvalues of(M,N), two n_×n complex matrices, are the zeros of the polynomial det(M₋zN). IfNis invertible, it is simply the eigenvalues ofN−1_M.

Let(X_{i j})_i,j≥1and(Y_{i j})_i,j≥1be independent i.i.d. complex random variables with mean 0 and vari-ance 1. Similarly, let(G_{i j})_i,j≥1and(H_{i j})_i,j≥1be independent complex centered gaussian variables with variance 1, independent of (X_{i j},Y_{i j}). We consider the random matrices X_n = (X_{i j})1≤i,j≤n, Y_n= (Y_{i j})1≤i,j≤n,Gn= (Gi j)1≤i,j≤nandHn= (Hi j)1≤i,j≤n. For ease of notation, we will sometimes

drop the subscriptn. It is known that almost surely (a.s.) fornlarge enough,X is invertible (see the forthcoming Theorem 11) and thenµX−1_Y is a well defined random probability measure onC. Now, letµ be the probability measure whose density with respect to the Lebesgue measure on

C_≃R2_is

1

π(1+_|z_|2₎2.

Through stereographic projection, µ is easily seen to be the uniform measure on the Riemann sphere. Haagerup and several authors afterwards have independently observed the following beautiful identity (see Krishnapur[17], Rogers[21]and Forrester and Mays[7]).

Lemma 1(Spherical ensemble). For each integer n_≥1, Eµ_G−1_H=µ.

By reorganizing the results of Tao and Vu[23,24], we will prove a universality result.

Theorem 2(Universality of generalized eigenvalues). Almost surely, µ_X−1_Y−µ_G−1_H  

n→∞0. Applying once Lemma 1 and twice Theorem 2, we get

Corollary 3(Spherical law). Almost surely

µ_X−1_Y  

n→∞µ.

This statement was recently conjectured in[21,7]. More generally, our argument also leads to the following universality result for sums and products of random matrices.

Theorem 4(Universality of sum and product of random matrices). For every integer n, let Mn,Kn,Ln be n×n complex matrices such that, for someα >0,

(i) x7→ x−αis uniformly bounded for(ν_K

n)n≥1, and(νLn)n≥1and x7→ x

α_{is uniformly bounded} for(ν_Mn)_n≥1,

(ii) for almost all (a.a.) z∈C_,ν_K−1

n MnL−1n −Kn−1L−1n z converges weakly to a probability measureνz. Then, almost surely,

µ_{M+K X L/}p

n_n 

→∞µ,

whereµdepends only(ν_z)_z∈C.

For M =K = L= I, the identity matrix, this statement gives the famous circular law theorem, that was established through a long sequence of partial results[19,8,10,16,6,9,1,11,2,20,

12,23,24]. In this note, the steps of proof are elementary and they borrow all difficult technical statements from previously known results. Nevertheless, this theorem generalizes Theorem 1.18 in[24]in two directions. First, we have removed the assumption of uniformly bounded second moment forν_{M+K X L/}p_n,ν_K−1_{M L}−1 andν_K−1_L−1. Secondly, it proves the convergence of the spectral measure. The explicit form ofµin terms ofν_z is quite complicated. It is given by the forthcoming equations (2-3). This expression is not very easy to handle. However, following ideas developed in

[21]or using tools from free probability as in[22,14], it should be possible to find more elegant formulas. For nice examples of limit spectral distributions, see e.g.[21]. It is interesting to notice that we may deal with non-centered variables(X_{i j})by including the average matrix ofX/pninto

M, and recover[4]. Finally, as in [13], it is also possible by induction to apply Theorem 4 to product of independent copies ofX (with the use of the forthcoming Theorem 8).

2

Proof of Theorem 2

2.1

Replacement Principle

The following is an extension of Theorem 2.1 in[24]. The idea goes back essentially to Girko.

Lemma 5(Replacement principle). Let An,Bnbe n×n complex random matrices. Suppose that for a.a. z∈C_{, a.s.}

(i) ν_An−z−ν_Bn−z tends weakly to0,

(ii) ln(_·)is uniformly integrable for(ν_An−z)_n≥1and(ν_Bn−z)_n≥1.

Then a.s.µ_An−µ_Bntends weakly to0. Moreover the same holds if we replace (i) by (i’) Rln(x)dνAn−z−

R

ln(x)dνBn−z tends to0.

Proof. It is a straightforward adaptation of[3, Lemma A.2]. ƒ Corollary 6. Let An,Bn,Mnbe n×n complex random matrices. Suppose that a.s. Mnis invertible and for a.a. z∈C_{, a.s.}

(i) ν_A

n−z Mn−1−νBn−z M−1n tends weakly to0, (ii) ln(_·)is uniformly integrable for(ν_{An−z M}−1

Then a.s.µ_M

nAn−µMnBntends weakly to0. Proof. IfMnis invertible, note that

Z

ln(x)dνMnAn−z=

1

nln|det(MnAn−z)|=

Z

ln(x)dνAn−z Mn−1+

1

nln|detMn|.

We may thus apply Lemma 5(i’)-(ii). Indeed, in the expressionRln(x)dν_A n−z−

R

ln(x)dν_B n−z, the

term 1

nln|detMn|cancels. ƒ

2.2

Convergence of singular values

The following result is due to Dozier and Silverstein.

Theorem 7 (Convergence of singular values, [5]). Let(Mn)n≥1 be a sequence of n×n complex matrices such thatνMn converges weakly to a probability measureν. Then a.s. νXn/pn+Mn converges weakly to a probability measureρwhich depends only onν.

The measureρhas an explicit characterization in terms ofν. Its exact form is not relevant here.

2.3

Uniform integrability

In order to use the replacement principle, it is necessary to prove the uniform integrability of ln(_·)

for some empirical singular values measures. This is achieved by proving that, for someβ >0,

x_7→x−β+xβ is uniformly bounded.

Theorem 8 (Uniform integrability). Let (M_n)_n≥1 be a sequence of n×n complex matrices, and assume that x7→xαis uniformly bounded for(ν_M

n)n≥1for someα >0. Then there existsβ >0such that a.s. x_7→x−β+xβ is uniformly bounded for(νXn/pn+Mn)n≥1.

In the remainder of the paper, the notation n ≫ 1 means large enough n. We start with an elementary lemma.

Lemma 9(Large singular values). Almost surely, for n≫1,

Z

x2dν_X/p_n≤2.

Proof. We have 1

n

Pn i=1s2i(X/

p

n) = 1

n2trX∗X =

1 n2

P

1≤i,j≤n|Xi j|2, and the latter converges a.s. to

1 by the law of large number. ƒ

Corollary 10. Let0< α_≤2and let(Mn)n≥1be a sequence of n×n complex matrices such that x7→ xαis uniformly bounded for(νMn)n≥1. Then, a.s. x7→x

α_{is uniformly bounded for(ν}

Xn/pn+Mn)n≥1. Proof. IfM,N aren_×ncomplex matrices, from[15, Theorem 3.3.16], for all 1_≤i,j_≤nwith 1≤i+j≤n+1,

si+j−1(M+N)≤si(M) +sj(N).

Hence,

We deduce that for any non-decreasing function, f :R_+→R_+andt>0, Z

f(x)dν_M+N≤2 Z

f(2x)dν_M+2 Z

f(2x)dν_N, where we have used the inequality

f(x+y)_≤ f(2x) +f(2y).

Now, in view of Lemma 9, we may apply the above inequality to f(x) = xα _{and deduce the}

statement. ƒ

The above corollary settles the problem of the uniform integrability of ln(_·)at+_∞ forν_X/p_n+M. The uniform integrability at 0+is a much more delicate matter. The next theorem is a deep result of Tao and Vu.

Theorem 11(Small singular values,[23,24]). Let(M_n)_n≥1be a sequence of n×n complex invertible matrices such that x7→xαis uniformly bounded for(ν_M

n)n≥1for someα >0. There exist c1,c0>0 such that a.s. for n≫1,

sn(Xn/

p

n+Mn)≥n−c1. Moreover for i_≥n1−γ_{withγ=0.01, a.s. for n}

≫1,

s_n−i(X_n/pn+M_n)_≥c₀i n.

Proof. The first statement is Theorem 2.1 in[23]and the second is contained in [24](see the proof of Theorem 1.20 and observe that the statement of Proposition 5.1 remains unchanged if

we consider a row of the matrixXn+pnMn). ƒ

Proof of Theorem 8.

By Corollary 10, it is sufficient to prove that x 7→ x−β is uniformly bounded for(ν_X/p

n+M)and

someβ >0. We have

lim sup n 1 n n X

i=1

s−_iβ(X/pn+M)<_∞, By Theorem 11, we may a.s. write forn_≫1,

1

n n

X

i=1

s−_iβ(X/pn+M) _≤ 1 n

⌊n1−γ_⌋ X

i=1

nβc1₊1

n n

X

i=⌊n1−γ_⌋+1

c2

_n

i

‹β

≤ nβc1−γ+1 n

n

X

i=1 c₂

_n

i

‹β

.

This last expression is uniformly bounded if 0< β <min(γ/c₁, 1). ƒ

2.4

End of proof of Theorem 2

Ifρis a probability measure onC_{\{0}, we define ˇ}ρas the pull-back measure ofρunderφ:z7→

1/z, for any BorelEinC_{\{0}, ˇ}ρ(E) =ρ(φ−1_{(E)). Obviously, if(ρ}

n)n≥1is a sequence of probability

Note that by Theorem 8, a.s. forn≫1,Xnis invertible and x7→ x−β+xβ is uniformly bounded

for(ν_X

n/pn)n≥1. Also, from the quarter circular law theorem,νXn/pnconverges a.s. to a probability

distribution with density

1

π

p 4₋x2₁

[0,2](x),

(see Marchenko-Pastur theorem[18,25,26]). From the independence of(Xi j),(Yi j),(Gi j),(Hi j),

we may apply Corollary 6, Theorem 7 and Theorem 8 conditioned on(X_{i j})toMn=zXn/pn. We

get a.s.

µ_X−1_Y−µ_X−1_H  

n→∞0.

By Theorem 11, a.s. forn_≫1,X−1HandG−1H are invertible, it follows that

µ_X−1_H−µ_G−1_H  

n→∞0 is equivalent to µˇX−1H−µˇG−1Hn →∞0. However sinceµ_{M N}=µ_{N M} and ˇµ_M =µ_M−1, we get

µX−1_H−µ_G−1_H  

n→∞0 is equivalent to µH−1X−µH−1Gn →∞0. The right hand side holds by applying again, Corollary 6, Theorem 7 and Theorem 8.

3

Proof of Theorem 4

3.1

Bounds on singular values

Lemma 12 (Singular values of sum and product). If M,N are n_×n complex matrices, for any α >0,

Z

xαdν_M+N ≤ 21+α

‚Z

xαdν_M+

Z

xαdν_N

Π, Z

xαdν_{M N ≤} 2 ‚Z

x2αdν_M

Œ1/2‚Z

x2αdν_N

Œ1/2

.

Proof. The first statement was already treated in the proof of Corollary 10. Also, from [15, Theorem 3.3.16], for all 1_≤i,j_≤nwith 1_≤i+j_≤n+1,

s_i+j−1(M N)≤si(M)sj(N).

Hence,

s2i(M N)≤s2i−1(M N)≤si(M)si(N).

We deduce

Z

xαdν_{M N≤}2 n

n

X

i=1

sα_i(M)sα_i(N).

3.2

Logarithmic potential and Girko’s hermitization method

We denote by_D′(C_{)the set of Schwartz distributions endowed with its usual convergence with}

respect to all infinitely differentiable functions with bounded support. Let _P(C_{)be the set of}

probability measures onC_{which integrate ln|·|in a neighborhood of infinity. For everyµ∈ P(}C_),

thelogarithmic potential UµofµonCis the functionUµ:C→[−∞,+∞)defined for everyz∈C

by

Uµ(z) =

Z

C

ln_|z₋z′_|µ(dz′),

(in classical potential theory, the definition is opposite in sign). Since ln_|·| is Lebesgue locally integrable onC_{, one can check by using the Fubini theorem thatUµis Lebesgue locally integrable}

onC_{. In particular,Uµ}<_∞a.e. (Lebesgue almost everywhere) andUµ ∈ D′(C). Since ln|·|is

the fundamental solution of the Laplace equation inC_{, we have, inD}′(C),

∆Uµ=πµ, (1)

where∆is the Laplace differential operator onC_{is given by∆ =}1 4(∂

2 x +∂

2 y).

We now state an alternative statement of Lemma 5 which is closer to Girko’s original method, for a proof see[3, Lemma A.2].

Lemma 13 (Girko’s hermitization method). Let Anbe a n×n complex random matrix. Suppose that for a.a. z∈C_{, a.s.}

(i) ν_An−z tends weakly to a probability measureν_z onR

+,

(ii) ln(_·)is uniformly integrable for(ν_A n−z)n≥1.

Then there exists a probability measureµ_{∈ P}(C_{)such that a.s.} (j) µ_An converges weakly toµ

(jj) for a.a. z∈C_,

Uµ(z) =

Z

ln(x)dνz. Moreover the same holds if we replace (i) by

(i’) Rln(x)dν_An−ztends toRln(x)dν_z.

Corollary 14. Let An,Kn,Mnbe n×n complex random matrices. Suppose that a.s. Knis invertible andln(_·)is uniformly bounded for(νKn)n≥1, and for a.a. z∈C, a.s.

(i) ν_An+K−1

n (Mn−z)tends weakly to a probability measureνz, (ii) ln(_·)is uniformly integrable for(νAn+Kn−1(Mn−z))n≥1. Then there exists a probability measureµ_{∈ P}(C)such that a.s.

(j) µ_Mn+KnAnconverges weakly toµ, (jj) inD′₍C),

µ= 1 π∆

Z

Proof. IfKnis invertible, we write

Z

ln(x)dν_Mn+KnAn−z = 1

nln|det(An+K

−1

n (Mn−z))|+

1

nln|detKn| =

Z

ln(x)dν_An+K−1

n (Mn−z)+

1

nln|detKn|.

By assumption, 1

nln|detKn|=

R

ln(x)dν_K

nis a.s. bounded. We may thus consider any converging

subsequence and apply Lemma 5(i’)-(ii) together with (1). ƒ

3.3

End of proof of Theorem 4

We first notice that

µ_M+K X L/pn=µL M L−1_{+LK X/}p_n.

It is thus sufficient to prove that the right hand side converges. We setMe =L M L−1andKe=LK. SinceKe−1(Me −z) =K−1M L−1−K−1L−1z, we may apply Lemma 12 and deduce that x 7→ xα/4

is uniformly bounded for (ν_Kne(Mne−z))_n≥1. It only remains to invoke Theorem 8 and Theorem 7 applied toKe−1_{(Me−z), and use Corollary 14 forMe+K Xe /}p_n.

3.4

Explicit expression of the limit spectral measure

LetC₊=_{z∈C_:ℑ(z)>0}, for a probability measureρonR_{, itsCauchy-Stieltjes transformis}

defined as, for allz∈C_+,

mρ(z) =

Z 1

x₋zdρ(x).

By Corollary 14 and Theorem 7, in_D′₍C_), µ= 1

2π∆

Z

ln(x)dρz(x), (2)

where for z _∈ C_{, ρ}

z is a probability distribution on R+. From[5], for a.a. z ∈ C, ρz has a

Cauchy-Stieltjes transform that satisfies the integral equation: for allw_∈C_+, mρz(w) =

Z _2x(1+m

ρz(w)) x2_−(1+m

ρz(w))2w

dνz(x), (3)

whereν_z is as in Theorem 4.

Acknowledgment

The author is indebted to Tim Rogers for pointing reference[21] which has initiated this work, and thanks Djalil Chafaï and Manjunath Krishnapur for sharing their enthusiasm on non-hermitian random matrices.

References

[1] Z. D. Bai,Circular law, Ann. Probab.25(1997), no. 1, 494–529.MR1428519

[2] Z. D. Bai and J. W. Silverstein,Spectral Analysis of Large Dimensional Random Matrices, Math-ematics Monograph Series 2, Science Press, Beijing, 2006.

[3] Ch. Bordenave, P. Caputo, and D. Chafaï,Circular Law Theorem for Random Markov Matrices, preprint

❛r❳✐✈✿✵✽✵✽✳✶✺✵✷✈✷

, 2010.MR2644041

[4] D. Chafaï,Circular law for noncentral random matrices, J. Theoret. Probab.23(2010), no. 4, 945–950.

[5] B. Dozier and J. Silverstein, On the empirical distribution of eigenvalues of large dimen-sional information-plus-noise-type matrices, J. Multivariate Anal.98(2007), no. 4, 678–694. MR2322123

[6] A. Edelman,The probability that a random real Gaussian matrix has k real eigenvalues, re-lated distributions, and the circular law, J. Multivariate Anal.60 (1997), no. 2, 203–232. MR1437734

[7] P. Forrester and A. Mays, Pfaffian point process for the gaussian real generalised eigenvalue problem, preprint

❛r❳✐✈✿✵✾✶✵✳✷✺✸✶

, 2009.

[8] V. L. Girko, The circular law, Teor. Veroyatnost. i Primenen. 29 (1984), no. 4, 669–679. MR0773436

[9] ,Strong circular law, Random Oper. Stochastic Equations5(1997), no. 2, 173–196. MR1454343

[10] , The circular law. Twenty years later. III, Random Oper. Stochastic Equations 13

(2005), no. 1, 53–109.MR2130247

[11] I. Y. Goldsheid and B. A. Khoruzhenko,The Thouless formula for random non-Hermitian Jacobi matrices, Israel J. Math.148(2005), 331–346, Probability in mathematics.MR2191234

[12] F. Götze and A. Tikhomirov,The Circular Law for Random Matrices, preprint to appear in the Annals of Probability

❛r❳✐✈✿♠❛t❤✴✵✼✵✾✸✾✾✺

, 2010.

[13] , On the asymptotic spectrum of products of independent random matrices, preprint

❛r❳✐✈✿♠❛t❤✴✶✵✶✷✳✷✼✶✵

, 2010.

[14] A. Guionnet, M. Krishnapur, and O. Zeitouni, The single ring theorem, preprint

❛r❳✐✈✿✵✾✵✾✳✷✷✶✹

, 2009.

[15] R. Horn and C. Johnson,Topics in matrix analysis, Cambridge University Press, Cambridge, 1991.MR1091716

[16] C.-R. Hwang,A brief survey on the spectral radius and the spectral distribution of large ran-dom matrices with i.i.d. entries, Random matrices and their applications (Brunswick, Maine, 1984), Contemp. Math., vol. 50, Amer. Math. Soc., Providence, RI, 1986, pp. 145–152. MR0841088

[17] M. Krishnapur,From random matrices to random analytic functions, Ann. Probab.37(2009), no. 1, 314–346.MR2489167

[18] V.A. Marchenko and L.A. Pastur,The distribution of eigenvalues in sertain sets of random ma-trices, Mat. Sb.72(1967), 507–536.MR0208649

[19] M. L. Mehta,Random matrices and the statistical theory of energy levels, Academic Press, New York, 1967.MR0220494

[20] G.M. Pan and W. Zhou,Circular law, extreme singular values and potential theory, J. Multivar. Anal.101(2010), no. 3, 645–656.MR2575411

[21] T. Rogers,Universal sum and product rules for random matrices, J. Math. Phys.51(2010), no. 093304.MR2742824

[22] P. ´Sniady,Random regularization of Brown spectral measure, J. Funct. Anal.193(2002), no. 2, 291–313.MR1929504

[23] T. Tao and V. Vu,Random matrices: the circular law, Commun. Contemp. Math.10(2008), no. 2, 261–307.MR2409368

[24] ,Random matrices: Universality of ESDs and the circular law, preprint to appear in the Annals of Probability

❛r❳✐✈✿✵✽✵✼✳✹✽✾✽ ❬♠❛t❤✳P❘❪

, 2010.

[25] K. W. Wachter,The strong limits of random matrix spectra for sample matrices of independent elements, Ann. Probability6(1978), no. 1, 1–18.MR0467894

[26] Y. Q. Yin,Limiting spectral distribution for a class of random matrices, J. Multivariate Anal.20

We deduce that for any non-decreasing function, f :R_+→R_+andt>0, Z

f(x)dν_M+N≤2 Z

f(2x)dν_M+2 Z

f(2x)dν_N, where we have used the inequality

f(x+y)_≤ f(2x) +f(2y).

Now, in view of Lemma 9, we may apply the above inequality to f(x) = xα _{and deduce the}

statement. ƒ

The above corollary settles the problem of the uniform integrability of ln(_·)at+_∞ forν_X/p_n+M.

The uniform integrability at 0+is a much more delicate matter. The next theorem is a deep result of Tao and Vu.

Theorem 11(Small singular values,[23,24]). Let(M_n)_n≥1be a sequence of n×n complex invertible matrices such that x7→xαis uniformly bounded for(ν_M

n)n≥1for someα >0. There exist c1,c0>0

such that a.s. for n≫1,

sn(Xn/ p

n+Mn)≥n−c1. Moreover for i_≥n1−γ_{withγ=0.01, a.s. for n}

≫1,

s_n−i(X_n/pn+M_n)_≥c₀i n.

Proof. The first statement is Theorem 2.1 in[23]and the second is contained in [24](see the proof of Theorem 1.20 and observe that the statement of Proposition 5.1 remains unchanged if

we consider a row of the matrixXn+pnMn). ƒ

Proof of Theorem 8.

By Corollary 10, it is sufficient to prove that x 7→ x−β is uniformly bounded for(ν_X/p

n+M)and someβ >0. We have

lim sup n 1 n n X

i=1

s−_iβ(X/pn+M)<_∞, By Theorem 11, we may a.s. write forn_≫1,

1 n

n X i=1

s−_iβ(X/pn+M) _≤ 1 n

⌊n1−γ

⌋

X i=1

nβc1₊1 n

n X i=⌊n1−γ_⌋+1

c2

_n i

‹β

≤ nβc1−γ₊1 n

n X

i=1

c₂ _n

i ‹β

.

This last expression is uniformly bounded if 0< β <min(γ/c₁, 1). ƒ

2.4

End of proof of Theorem 2

Ifρis a probability measure onC_{\{0}, we define ˇ}ρas the pull-back measure ofρunderφ:z7→

1/z, for any BorelEinC_{\{0}, ˇ}ρ(E) =ρ(φ−1_{(E)). Obviously, if(ρ}

n)n≥1is a sequence of probability

Note that by Theorem 8, a.s. forn≫1,Xnis invertible and x7→ x−β+xβ is uniformly bounded for(ν_X

n/pn)n≥1. Also, from the quarter circular law theorem,νXn/pnconverges a.s. to a probability

distribution with density

1 π

p 4₋x2₁

[0,2](x),

(see Marchenko-Pastur theorem[18,25,26]). From the independence of(Xi j),(Yi j),(Gi j),(Hi j), we may apply Corollary 6, Theorem 7 and Theorem 8 conditioned on(X_{i j})toMn=zXn/pn. We get a.s.

µ_X−1_Y−µ_X−1_H   n→∞0.

By Theorem 11, a.s. forn_≫1,X−1HandG−1H are invertible, it follows that µ_X−1_H−µ_G−1_H  

n→∞0 is equivalent to µˇX−1H−µˇG−1Hn →∞0.

However sinceµ_{M N}=µ_{N M} and ˇµ_M =µ_M−1, we get µX−1_H−µ_G−1_H  

n→∞0 is equivalent to µH−1X−µH−1Gn →∞0.

The right hand side holds by applying again, Corollary 6, Theorem 7 and Theorem 8.

3

Proof of Theorem 4

3.1

Bounds on singular values

Lemma 12 (Singular values of sum and product). If M,N are n_×n complex matrices, for any α >0,

Z

xαdν_M+N ≤ 21+α ‚Z

xαdν_M+ Z

xαdν_N Œ

, Z

xαdν_{M N ≤} 2 ‚Z

x2αdν_M

Œ1/2‚Z

x2αdν_N Œ1/2

.

Proof. The first statement was already treated in the proof of Corollary 10. Also, from [15, Theorem 3.3.16], for all 1_≤i,j_≤nwith 1_≤i+j_≤n+1,

s_i+j−1(M N)≤si(M)sj(N). Hence,

s2i(M N)≤s2i−1(M N)≤si(M)si(N). We deduce

Z

xαdν_{M N≤}2 n

n X

i=1

sα_i(M)sα_i(N).

3.2

Logarithmic potential and Girko’s hermitization method

We denote by_D′(C_{)the set of Schwartz distributions endowed with its usual convergence with} respect to all infinitely differentiable functions with bounded support. Let _P(C_{)be the set of} probability measures onC_{which integrate ln|·|in a neighborhood of infinity. For everyµ∈ P(}C_), thelogarithmic potential UµofµonCis the functionUµ:C→[−∞,+∞)defined for everyz∈C by

Uµ(z) = Z

C

ln_|z₋z′_|µ(dz′),

(in classical potential theory, the definition is opposite in sign). Since ln_|·| is Lebesgue locally integrable onC_{, one can check by using the Fubini theorem thatUµis Lebesgue locally integrable} onC_{. In particular,Uµ}<_∞a.e. (Lebesgue almost everywhere) andUµ ∈ D′(C). Since ln|·|is the fundamental solution of the Laplace equation inC_{, we have, inD}′(C),

∆Uµ=πµ, (1)

where∆is the Laplace differential operator onC_{is given by∆ =}1

4(∂ 2

x +∂

2

y).

We now state an alternative statement of Lemma 5 which is closer to Girko’s original method, for a proof see[3, Lemma A.2].

Lemma 13 (Girko’s hermitization method). Let Anbe a n×n complex random matrix. Suppose that for a.a. z∈C_{, a.s.}

(i) ν_An−z tends weakly to a probability measureν_z onR

+,

(ii) ln(_·)is uniformly integrable for(ν_An−z)_n≥1.

Then there exists a probability measureµ_{∈ P}(C_{)such that a.s.} (j) µ_An converges weakly toµ

(jj) for a.a. z∈C_,

Uµ(z) = Z

ln(x)dνz. Moreover the same holds if we replace (i) by

(i’) Rln(x)dν_An−ztends toRln(x)dν_z.

Corollary 14. Let An,Kn,Mnbe n×n complex random matrices. Suppose that a.s. Knis invertible andln(_·)is uniformly bounded for(νKn)n≥1, and for a.a. z∈C, a.s.

(i) ν_An+K−1

n (Mn−z)tends weakly to a probability measureνz,

(ii) ln(_·)is uniformly integrable for(νAn+Kn−1(Mn−z))n≥1.

Then there exists a probability measureµ_{∈ P}(C)such that a.s. (j) µ_Mn+KnAnconverges weakly toµ,

(jj) inD′₍C),

µ= 1 π∆

Z

Proof. IfKnis invertible, we write Z

ln(x)dν_Mn+KnAn−z = 1

nln|det(An+K

−1

n (Mn−z))|+ 1

nln|detKn| =

Z

ln(x)dν_An+K−1

n (Mn−z)+

1

nln|detKn|. By assumption, 1

nln|detKn|= R

ln(x)dν_K

nis a.s. bounded. We may thus consider any converging

subsequence and apply Lemma 5(i’)-(ii) together with (1). ƒ

3.3

End of proof of Theorem 4

We first notice that

µ_M+K X L/pn=µL M L−1_{+LK X/}p_n.

It is thus sufficient to prove that the right hand side converges. We setMe =L M L−1andKe=LK. SinceKe−1(Me −z) =K−1M L−1−K−1L−1z, we may apply Lemma 12 and deduce that x 7→ xα/4 is uniformly bounded for (ν_Ken(Men−z))_n≥1. It only remains to invoke Theorem 8 and Theorem 7 applied toKe−1_{(Me−z), and use Corollary 14 forMe+K Xe /}p_n.

3.4

Explicit expression of the limit spectral measure

LetC₊=_{z∈C_:ℑ(z)>0}, for a probability measureρonR_{, itsCauchy-Stieltjes transformis} defined as, for allz∈C_+,

mρ(z) = Z

1

x₋zdρ(x). By Corollary 14 and Theorem 7, in_D′₍C_),

µ= 1 2π∆

Z

ln(x)dρz(x), (2)

where for z _∈ C_{, ρ}

z is a probability distribution on R+. From[5], for a.a. z ∈ C, ρz has a Cauchy-Stieltjes transform that satisfies the integral equation: for allw_∈C_+,

mρz(w) =

Z _2x(1+m ρz(w))

x2_−(1+m

ρz(w))

2_wdνz(x), (3)

whereν_z is as in Theorem 4.

Acknowledgment

The author is indebted to Tim Rogers for pointing reference[21] which has initiated this work, and thanks Djalil Chafaï and Manjunath Krishnapur for sharing their enthusiasm on non-hermitian random matrices.

References

[1] Z. D. Bai,Circular law, Ann. Probab.25(1997), no. 1, 494–529.MR1428519

[2] Z. D. Bai and J. W. Silverstein,Spectral Analysis of Large Dimensional Random Matrices, Math-ematics Monograph Series 2, Science Press, Beijing, 2006.

[3] Ch. Bordenave, P. Caputo, and D. Chafaï,Circular Law Theorem for Random Markov Matrices, preprint

❛r❳✐✈✿✵✽✵✽✳✶✺✵✷✈✷

, 2010.MR2644041

[4] D. Chafaï,Circular law for noncentral random matrices, J. Theoret. Probab.23(2010), no. 4, 945–950.

[5] B. Dozier and J. Silverstein, On the empirical distribution of eigenvalues of large dimen-sional information-plus-noise-type matrices, J. Multivariate Anal.98(2007), no. 4, 678–694.

MR2322123

[6] A. Edelman,The probability that a random real Gaussian matrix has k real eigenvalues, re-lated distributions, and the circular law, J. Multivariate Anal.60 (1997), no. 2, 203–232.

MR1437734

[7] P. Forrester and A. Mays, Pfaffian point process for the gaussian real generalised eigenvalue problem, preprint

❛r❳✐✈✿✵✾✶✵✳✷✺✸✶

, 2009.

[8] V. L. Girko, The circular law, Teor. Veroyatnost. i Primenen. 29 (1984), no. 4, 669–679.

MR0773436

[9] ,Strong circular law, Random Oper. Stochastic Equations5(1997), no. 2, 173–196.

MR1454343

[10] , The circular law. Twenty years later. III, Random Oper. Stochastic Equations 13

(2005), no. 1, 53–109.MR2130247

[11] I. Y. Goldsheid and B. A. Khoruzhenko,The Thouless formula for random non-Hermitian Jacobi matrices, Israel J. Math.148(2005), 331–346, Probability in mathematics.MR2191234

[12] F. Götze and A. Tikhomirov,The Circular Law for Random Matrices, preprint to appear in the Annals of Probability

❛r❳✐✈✿♠❛t❤✴✵✼✵✾✸✾✾✺

, 2010.

[13] , On the asymptotic spectrum of products of independent random matrices, preprint

❛r❳✐✈✿♠❛t❤✴✶✵✶✷✳✷✼✶✵

, 2010.

[14] A. Guionnet, M. Krishnapur, and O. Zeitouni, The single ring theorem, preprint

❛r❳✐✈✿✵✾✵✾✳✷✷✶✹

, 2009.

[15] R. Horn and C. Johnson,Topics in matrix analysis, Cambridge University Press, Cambridge, 1991.MR1091716

[16] C.-R. Hwang,A brief survey on the spectral radius and the spectral distribution of large ran-dom matrices with i.i.d. entries, Ranran-dom matrices and their applications (Brunswick, Maine, 1984), Contemp. Math., vol. 50, Amer. Math. Soc., Providence, RI, 1986, pp. 145–152.

[17] M. Krishnapur,From random matrices to random analytic functions, Ann. Probab.37(2009), no. 1, 314–346.MR2489167

[18] V.A. Marchenko and L.A. Pastur,The distribution of eigenvalues in sertain sets of random ma-trices, Mat. Sb.72(1967), 507–536.MR0208649

[19] M. L. Mehta,Random matrices and the statistical theory of energy levels, Academic Press, New York, 1967.MR0220494

[20] G.M. Pan and W. Zhou,Circular law, extreme singular values and potential theory, J. Multivar. Anal.101(2010), no. 3, 645–656.MR2575411

[21] T. Rogers,Universal sum and product rules for random matrices, J. Math. Phys.51(2010), no. 093304.MR2742824

[22] P. ´Sniady,Random regularization of Brown spectral measure, J. Funct. Anal.193(2002), no. 2, 291–313.MR1929504

[23] T. Tao and V. Vu,Random matrices: the circular law, Commun. Contemp. Math.10(2008), no. 2, 261–307.MR2409368

[24] ,Random matrices: Universality of ESDs and the circular law, preprint to appear in the Annals of Probability

❛r❳✐✈✿✵✽✵✼✳✹✽✾✽ ❬♠❛t❤✳P❘❪

, 2010.

[25] K. W. Wachter,The strong limits of random matrix spectra for sample matrices of independent elements, Ann. Probability6(1978), no. 1, 1–18.MR0467894

[26] Y. Q. Yin,Limiting spectral distribution for a class of random matrices, J. Multivariate Anal.20