Convergence of the spectra measure of non norma matrices Aice Guionnet Phiip Wood Ofer Zeitouni October, 2 Abstract We discuss reguarization by noise of the spectrum of arge random non- Norma matrices. Under suitabe conditions, we show that the reguarization of a sequence of matrices that converges in -moments to a reguar eement a, by the addition of a poynomiay vanishing Gaussian Ginibre matrix, forces the empirica measure of eigenvaues to converge to the Brown measure of a. Introduction Consider a sequence A N of N N matrices, of uniformy bounded operator norm, and assume that A N converges in -moments toward an eement a in a W probabiity space (A,,,ϕ), that is, for any non-commutative poynomia P, N trp(a N,A N ) N ϕ(p(a,a )). We assume throughout that the tracia state ϕ is faithfu; this does not represent a oss of generaity. If A N is a sequence of Hermitian matrices, this UMPA, CNRS UMR 5669, ENS Lyon, 46 aée d Itaie, 697 Lyon, France. aguionne@umpa.ens-yon.fr. This work was partiay supported by the ANR project ANR-8- BLAN-3-. Schoo of Mathematics, University of Minnesota and Facuty of Mathematics, Weizmann Institute, POB 26, Rehovot 76, Israe. zeitouni@math.umn.edu. The work of this author was partiay supported by NSF grant DMS-8433 and by a grant from the Israe Science Foundation.
is enough in order to concude that the empirica measure of eigenvaues of A N, that is the measure L A N := N N δ λi (A N ), where λ i (A N ),i =...N are the eigenvaues of A N, converges weaky to a imiting measure µ a, the spectra measure of a, supported on a compact subset of R. (See [, Coroary 5.2.6, Lemma 5.2.9] for this standard resut and further background.) Significanty, in the Hermitian case, this convergence is stabe under sma bounded perturbations: with B N = A N + E N and E N < ε, any subsequentia imit of L B N wi beong to B L(µ a,δ(ε)), with δ(ε) ε and B L (ν a,r) is the ba (in say, the Lévy metric) centered at ν a and of radius r. Both these statements fai when A n is not sef adjoint. For a standard exampe (described in [6]), consider the nipotent matrix T N =..................................... Obviousy, L T N = δ, whie a simpe computation reveas that T N converges in -moments to a Unitary Haar eement ofa, that is { N tr(t α N (T N )β...t α N k (TN, if k )β k ) αi = k β i, N, otherwise. Further, adding to T N the matrix whose entries are a except for the bottom eft, which is taken as ε, changes the empirica measure of eigenvaues drasticay - as we wi see beow, as N increases, the empirica measure converges to the uniform measure on the unit circe in the compex pane. Our goa in this note is to expore this phenomenun in the context of sma random perturbations of matrices. We reca some notions. For a A, the Brown measure ν a on C is the measure satisfying ogdet(z a) = og z z dν a (z ), z C, where det is the Fugede-Kadison determinant; we refer to [2, 4] for definitions. We have in particuar that ogdet(z a) = og xdν z a(x) z C, () 2
where ν z a denotes the spectra measure of the operator z a. In the sense of distributions, we have ν a = ogdet(z a). 2π That is, for smooth compacty supported function ψ on C, ψ(z)dν a (z) = dz ψ(z) og z z dν a (z ) 2π = dz ψ(z) og xdν z 2π a(x). A crucia assumption in our anaysis is the foowing. Definition (Reguar eements). An eement a A is reguar if ε im dz ψ(z) ogxdν z a(x) =, (2) ε C for a smooth functions ψ on C with compact support. Note that reguarity is a property of a, not merey of its Brown measure ν a. We next introduce the cass of Gaussian perturbations we consider. Definition 2 (Poynomiay vanishing Gaussian matrices). A sequence of N-by-N random Gaussian matrices is caed poynomiay vanishing if its entries (G N (i, j)) are independent centered compex Gaussian variabes, and there exist κ >, κ +κ so that N κ E G i j 2 N κ. Remark 3. As wi be cear beow, see the beginning of the proof of Lemma, the Gaussian assumption ony intervenes in obtaining a uniform ower bound on singuar vaues of certain random matrices. As pointed out to us by R. Vershynin, this uniform estimate extends to other situations, most notaby to the poynomia rescae of matrices whose entries are i.i.d. and possess a bounded density. We do not discuss such extensions here. Our first resut is a stabiity, with respect to poynomiay vanishing Gaussian perturbations, of the convergence of spectra measures for nonnorma matrices. Throughout, we denote by M op the operator norm of a matrix M. 3
Theorem 4. Assume that the uniformy bounded (in the operator norm) sequence of N-by-N matrices A N converges in -moments to a reguar eement a. Assume further that L A N converges weaky to the Brown measure ν a. Let G N be a sequence of poynomiay vanishing Gaussian matrices, and set B N = A N + G N. Then, L B N ν a weaky, in probabiity. Theorem 4 puts rather stringent assumptions on the sequence A N. In particuar, its assumptions are not satisfied by the sequence of nipotent matrices T N in (). Our second resut corrects this defficiency, by showing that sma Gaussian perturbations reguarize matrices that are cose to matrices satisfying the assumptions of Theorem 4. Theorem 5. Let A N, E N be a sequence of bounded (for the operator norm) N-by-N matrices, so that A N converges in -moments to a reguar eement a. Assume that E N op converges to zero poynomiay fast in N, and that LN A+E ν a weaky. Let G N be a sequence of poynomiay vanishing Gaussian matrices, and set B N = A N +G N. Then, L B N ν a weaky, in probabiity. Theorem 5 shoud be compared to earier resuts of Sniady [6], who used stochastic cacuus to show that a perturbation by an asymptoticay vanishing Ginibre Gaussian matrix reguarizes arbitrary matrices. Compared with his resuts, we aow for more genera Gaussian perturbations (both structuray and in terms of the variance) and aso show that the Gaussian reguarization can decay as fast as wished in the poynomia scae. On the other hand, we do impose a reguarity property on the imit a as we as on the sequence of matrices for which we assume that adding a poynomiay sma matrix is enough to obtain convergence to the Brown measure. A coroary of our genera resuts is the foowing. Coroary 6. Let G N be a sequence of poynomiay vanishing Gaussian matrices and et T N be as in (). Then L T+G N converges weaky, in probabiity, toward the uniform measure on the unit circe in C. In Figure, we give a simuation of the setup in Coroary 6 for various N. We wi now define cass of matrices T for which, if b is chosen correcty, adding a poynomiay vanishing Gaussian matrix G N is not sufficient to reguarize T + G N. Let b be a positive integer, and define T to be an N by N bock diagona matrix which each b+ by b+ bock on the diagona equa T b+ (as defined in (). If b+ does not divide N eveny, a bock of zeros is inserted at bottom of the diagona. Thus, every entry of T is zero except for entries on the superdiagona (the superdiagona is the ist of 4
(a) N = 5 (b) N = (c) N = 5 (d) N = 5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5 Figure : The eigenvaues of T N +N 3 /2 G N, where G N is iid compex Gaussian with mean, variance entries. entries with coordinates (i,i+) for i N ), and the superdiagona of T is equa to (,,...,,,,,...,,,...,,,...,,,,...,). }{{}}{{}}{{}}{{} b b b b Reca that the spectra radius of a matrix is the maximum absoute vaue of the eigenvaues. Aso, we wi use A = tr(a A) /2 to denote the Hibert- Schmidt norm. Proposition 7. Let b = b(n) be a sequence of positive integers such that b(n) ogn for a N, and et T be as defined above. Let R N be an N by N matrix satisfying R N g(n), where for a N we assume that g(n) < 3b N. Then ρ(t + R N ) (Ng(N)) /b + o(), where ρ(m) denotes the spectra radius of a matrix M, and o() denotes a sma quantity tending to zero as N. Note that T converges in -moments to a Unitary Haar eement of A (by a computation simiar to ()) if b(n)/n goes to zero, which is a reguar eement. The Brown measure of the Unitary Haar eement is uniform measure on the unit circe; thus, in the case where (Ng(N)) /b <, Proposition 7 shows that T + R N does not converge to the Brown measure for T. Coroary 8. Let R N be an iid Gaussian matrix where each entry has mean zero and variance one. Set b = b(n) og N be a sequence of integers, and et γ > 5/2 be a constant. Then, with probabiity tending to as N, we have ( ρ(t + exp( γb)r N ) exp γ+ 2og N ) + o(), b 5
(a) N = 5 (b) N = (c) N = 5 (d) N = 5.5.5.5.5.5.5.5.5.5.5..5.5..5.5..5.5 Figure 2: The eigenvaues of T ogn,n + N 3 /2 G N, where G N is iid compex Gaussian with mean, variance entries. The spectra radius is roughy.7, and the bound from Coroary 8 is exp( ).37. where ρ denotes the spectra radius and where o() denotes a sma quantity tending to zero as N. Note in particuar that the bound on the spectra radius is stricty ess than exp( /2) < in the imit as N, due to the assumptions on γ and b. Coroary 8 foows from Proposition 7 by noting that, with probabiity tending to, a entries in R N are at most C ogn in absoute vaue for some constant C, and then checking that the hypotheses of Proposition 7 are satisfied for g(n) = exp( γb)cn(og N) /4. There are two instances of Coroary 8 that are particuary interesting: when b = N, we see that a exponentiay decaying Gaussian perturbation does not reguarize T N = T N,N, and when b = og(n), we see that poynomiay decaying Gaussian perturbation does not reguarize T ogn,n (see Figure 2). We wi prove Proposition 7 in Section 5. The proof of our main resuts (Theorems 4 and 5) borrows from the methods of [3]. We introduce notation. For any N-by-N matrix C N, et C N = ( CN C N We denote by G C the Cauchy-Stietjes transform of the spectra measure of the matrix C N, that is ). G C (z) = 2N tr(z C N ), z C +. The foowing estimate is immediate from the definition and the resovent identity: G C (z) G D (z) C D op Iz 2. (3) 6
2 Proof of Theorem 4 We keep throughout the notation and assumptions of the theorem. The foowing is a crucia simpe observation. Proposition 9. For a compex number ξ, and a z so that Iz N δ with δ < κ/4, E IG BN +ξ(z) E IG AN +ξ(z) + Proof. Noting that E B N A N k op = E G N k op C k N κk/2, (4) the concusion foows from (3) and Höder s inequaity. We continue with the proof of Theorem 4. Let ν z A N denote the empirica measure of the eigenvaues of the matrix à N z. We have that, for smooth test functions ψ, dz ψ(z) og x dν z A N (x) = ψ(z)dl A 2π N(z). In particuar, the convergence of L A N toward ν a impies that E dz ψ(z) og x dν z A N (x) ψ(z)dν a (z)= dz ψ(z) ogxdν z a (x). On the other hand, since x ogx is bounded continuous on compact subsets of (, ), it aso hods that for any continuous bounded function ζ : R + R compacty supported in (, ), E dz ψ(z) ζ(x)og xdν z A N (x) dz ψ(z) ζ(x)og xdν z a(x). Together with the fact that a is reguar and that A N is uniformy bounded, one concudes therefore that ε im im E og x dν z ε N A N (x)dz =. Our next goa is to show that the same appies to B N. In the foowing, we et ν z B N denote the empirica measure of the eigenvaues of B N z. Lemma. im im ε N E[ ε og x dν z B N (x)]dz = 7
Because E B N A N k op for any k >, we have for any fixed smooth w compacty supported in (, ) that E dz ψ(z) w(x)og xdν z A N (x) dz ψ(z) w(x)og xdν z B N (x) N, Theorem 4 foows at once from Lemma. Proof of emma : Note first that by [5, Theorem 3.3] (or its generaization in [3, Proposition 6] to the compex case), there exists a constant C so that for any z, the smaest singuar vaue σ z N of B N + zi satisfies P(σ z N x) C (N 2 +κ x with β = or 2 according whether we are in the rea or the compex case. Therefore, for any ζ >, uniformy in z E[ N ζ og x dν z B N (x)] ) β E[og(σ z N ) σ z N N ζ] ( ) = C N β N ζ 2 +κ ζ og(n ζ ( )+ x C N β 2 x) +κ dx goes to zero as N goes to infinity as soon as ζ > 2 + κ. We fix hereafter such a ζ and we may and sha restrict the integration from N ζ to ε. To compare the integra for the spectra measure of A N and B N, observe that for a probabiity measure P, with P γ the Cauchy aw with parameter γ P([a,b]) P P γ ([a η,b+η])+p γ ([ η,η] c ) P P γ ([a η,b+η])+ γ η (5) whereas for b a > η Reca that P([a,b]) P P γ ([a+η,b η]) γ η. (6) P P γ ([a,b]) = b a IG(x + iγ) dx. (7) Set γ = N κ/5, κ = κ/2 and η = N κ /5. We have, whenever b a 4η, Eν z B N ([a,b]) b+η a η E IG Bn +z(x+iγ) dx+n (κ κ )/5 (b a+2n κ /5 )+ν z A N P N κ/5([a N κ/,b+n κ/ ])+N κ/ (b a+2n κ/ )+ν z A N ([(a 2N κ/ ) +,(b+2n κ/ )])+2N κ/, 8
where the first inequaity is due to (5) and (7), the second is due to Proposition 9, and the ast uses (6) and (7). Therefore, if b a = CN κ/ for some fixed C arger than 4, we deduce that there exists a finite constant C which ony depends on C so that Eν z B N ([a,b]) C (b a)+ν z A N ([(a 2N κ/ ) +,(b+2n κ/ )]). As a consequence, as we may assume without oss of generaity that κ > κ/, ε E[ og x dν z B N (x)] N ζ [N κ/ ε] k= og(n ζ + 2CkN κ/ ) E[ν z B N ]([N ζ + 2CkN κ/,n ζ + 2C(k+)N κ/ ]). We need to pay specia attention to the first term that we bound by noticing that og(n ζ ) E[ν z B N ([N ζ,n ζ + 2CN κ/ ])] ζ κ og(n κ/ ) E[ν z B N ([,2(C + )N κ/ ])] ζ κ og(n κ/ ) (2C N κ/ + ν z A N ([,(C + 2)N κ/ ])) 2C ζ κ 2(C+2)N κ/ og(n κ/ ) N κ/ +C og x dν z A N (x) For the other terms, we have [N κ/ ε] k= og(n ζ + 2CkN κ/ ) E[ν z B N ]([N ζ + 2CkN κ/,n ζ + 2C(k+)N κ/ ]) [Nκ/ ε] 2C k= [N κ/ ε] + k= og(ckn κ/ ) CN κ/ og(ckn κ/ ) ν z A N ([2C(k )N κ/,2c(k+2)n κ/ ]). Finay, we can sum up a these inequaities to find that there exists a finite constant C so that E[ ε N ζ og x dν z B N (x)] C ε ε og x dν z A N (x)+c og x dx 9
and therefore goes to zero when n and then ε goes to zero. This proves the caim. 3 Proof of Theorem 5. From the assumptions, it is cear that (A N + E N ) converges in -moments to the reguar eement a. By Theorem 4, it foows that L A+E+G N converges (weaky, in probabiity) towards ν a. We can now remove E N. Indeed, by (3) and (4), we have for any χ < κ /2 and a ξ C and therefore for Iz N χ/2, G N A+G+ξ (z) GN N χ A+G+E+ξ (z) Iz 2 IG N A+G+ξ (z) IGN A+G+E+ξ (z) +. Again by [5, Theorem 3.3] (or its generaization in [3, Proposition 6]) to the compex case), for any z, the smaest singuar vaue σ z N of A N + G N + z satisfies ) P(σ z N (N x) C β 2 +κ x with β = or 2 according whether we are in the rea or the compex case. We can now rerun the proof of Theorem 4, repacing A N by A N = A N +E N +G N and B N by A N E N. 4 Proof of Coroary 6 We appy Theorem 5 with A N = T N, E N the N-by-N matrix with E N (i, j) = { δ N = N (/2+κ ), i =, j = N, otherwise, where κ > κ. We check the assumptions of Theorem 5. We take a to be a Unitary Haar eement in A, and reca that its Brown measure ν a is the uniform measure on {z C : z = }. We now check that a is reguar. Indeed, x k dν z a (x) = if k is odd by symmetry whie for k even, x k dν z a (x) = ϕ([(z a)(z ( a) ] k/2 ) = ( z 2 + ) k j k 2 j k/2 j= )( 2 j j ),
and one therefore verifies that for k even, x k dν z a (x) = ( z 2 + +2 z cosθ) k/2 dθ. 2π It foows that ε ogxdν z a (x)= 4π 2π og( z 2 ++2 z cosθ) { z 2 ++2 z cos θ<ε}dθ ε, proving the required reguarity. Further, we caim that LN A+E converges to ν a. Indeed the eigenvaues λ of A N + E N are such that there exists a non-vanishing vector u so that that is u N δ N = λu,u i = λu i, λ N = δ N. In particuar, a the N-roots of δ N are (distinct) eigenvaues, that is the eigenvaues λ N j of A N are λ N j = δ N /N e 2iπ j/n, j N. Therefore, for any bounded continuous g function on C, as caimed. im N N N g(λ N j ) = 2π g(θ)dθ, 5 Proof of Proposition 7 In this section we wi prove the foowing proposition: Proposition. Let b = b(n) be a sequence of positive integers, and et T be as in Proposition 7. Let R N be an N by N matrix satisfying R N g(n), where for a N we assume that g(n) < 3b N. Then ( ( Nb ρ(t + R N ) O (2N )) /4 g /2) b /(b+) + ( b 2 Ng ) /(b+).
Proposition 7 foows from Proposition by adding the assumption that b(n) og(n) and then simpifying the upper bound on the spectra radius. Proof of Proposition : To bound the spectra radius, we wi use the fact that ρ(t + R N ) (T + R N ) k /k for a integers k. Our genera pan wi be to bound (T + R N ) k and then take a k-th root of the bound. We wi take k = b+, which aows us to take advantage of the fact that T is (b+)-step nipotent. In particuar, we make use of the fact that for any positive integer a, { T a = (b a+) /2 N /2 b+ if a b (8) if b+ a. We may write (T + R N ) b+ b+ T λ i λ {,} b+ b+ b+ = = λ {,} b+ λ has ones R λ i N T λ i R λ i N When is arge, we wi make use of the foowing emma. Lemma 2. If λ {,} k has ones and (k+)/2, then k T λ i R λ i N T k + k + R N k. We wi prove Lemma 2 in Section 5.. Using Lemma 2 with k = b+ aong with the fact that AB A B, 2
we have (T + R N ) b+ b+2 2 ( ) b+ T R n b + = b+ ( ) b+ + T b +2 b+2 2 = + = b+2 2 ( b+ b+ = b+2 2 b +2 R N b +. ) T g b + (9) ( ) b+ T b +2 b +2 g b +, () where the second inequaity comes from the assumption R N g = g(n). We wi bound (9) and () separatey. To bound (9) note that b+2 2 = ( b+ ) T g b + b+2 2 = b+4 2 ( b+ )( ) N /2 (b+) g b + b+ ( b+ (b+)/2 ) N (b+2)/4 g b/2 = O( Nb(2N /4 g /2 ) b). () Next, we turn to bounding (). We wi use the foowing emma to show that the argest term in the sum () comes from the = b term. Note that when = b+, the summand in () is equa to zero by (8). Lemma 3. ; If T b + + > and b and then ( ) b+ T b +2 b +2 g 2 e 3/2 N /2 b, g b + We wi prove Lemma 3 in Section 5.. ( ) b+ T + + b + b + g b. 3
Using Lemma 3 we have b+ = b+2 2 ( ) b+ T b +2 b +2 g b + b 2 (b+) T b 2 2 g b N (b+)(b b/2 +) 2 b+ g b 2 Ng. (2) Combining () and (2) with (9) and (), we may use the fact that (x+y) /(b+) x /(b+) + y /(b+) for positive x,y to compete the proof of Proposition. It remains to prove Lemma 2 and Lemma 3, which we do in Section 5. beow. 5. Proofs of Lemma 2 and Lemma 3 Proof of Lemma 2: Using (8), it is easy to show that T a T c T < a T c+ for integers 3 c+2 a b. (3) It is aso cear from (8) that T a T a for a positive integers a. (4) Let λ {,} k have ones. Then, using the assumption that k +, we may write k T λ i R λ i N = T a Rb N T a 2 Rb 2 N Ta k where a i for a i and b i for a i. Thus k T λ i R λ i N R N k Rb k N k + T a i. T a k +, Appying (3) repeatedy, we may assume that two of the a i differ by more than, a without changing the fact that k + a i =. Thus, some of the a i are equa to k + and some are equa to k +. Finay, appying (4), we have that k + T a i T k + k +. 4
Proof of Lemma 3: Using (8) and rearranging, it is sufficient to show that ( + b b + b +2 ) /2 N + b+ ( /2 b g b + + b b +2 ) b + 2 + Using a variety of manipuations, it is possibe to show that ( ) b + b b + + 2 ( ) b (b +2)(b +) exp + (b+2)(b +2) b+2 (b+2)(b +2) b +2 Thus, it is sufficient to have exp( 3/2). b 2 N/2 g exp( 3/2), which is true by assumption. References [] Anderson, G. W., Guionnet, A. and Zeitouni, O., An introduction to random matrices, Cambridge University Press, Cambridge (2). Brown s spectra measure in [2] Brown, L. G., Lidskii s theorem in the type II case, in Proceedings U.S. Japan, Kyoto/Japan 983, Pitman Res. Notes. Math Ser. 23, 35, (983). [3] Guionnet, A., Krishnapur, M. and Zeitouni, O., The singe ring theorem, arxiv:99.224v (29). [4] Haagerup, U. and Larsen, F., Brown s spectra distribution measure for R-diagona eements in finite von Neumann agebras, J. Funct. Ana. 2, 33 367, (2). [5] Sankar, A., Spieman, D. A. and Teng, S.-H., Smoothed anaysis of the conditioning number and growth factor of matrices, SIAM J. Matrix Ana. 28, 446 476, (26). [6] Sniady, P., Random reguarization of Brown spectra measure, J. Funct. Ana. 93 (22), pp. 29 33. [7] Voicuescu, D., Limit aws for random matrices and free products Inventiones Mathematicae 4, 2 22, (99). 5